Filter
Content
EE414 Handout #14: Spring 2001
Filters
1.0 Introduction
The subject of filter design is so vast that we have to abandon any hope of doing justice to
it as a subset of a textbook. Indeed, even though we have chosen to present this material
over two chapters, the limited aim here is to focus on important qualitative ideas and practical information about filters, instead of attempt a comprehensive review of all possible
filter types and supply complete mathematical details of their underlying theory. For those
interested in the rigor that we will tragically neglect, we will be sure to provide pointers to
the relevant literature. And for those who would rather ignore the small amount of rigor
that we do provide, the reader is invited to skip directly to the appendices, which summarize filter design information in “cookbook” form.
Although our planar focus would normally imply a discussion limited to microstrip implementations, many such filters derive directly from lower frequency lumped prototypes.
Because so many key concepts may be understood by studying those prototypes, we will
follow a roughly historical path and begin with a discussion of lumped filter design. It is
definitely the case that certain fundamental insights are universal, and it is these that will
be emphasized in this chapter, despite differences in implementation details between
lumped and distributed realizations.
Only passive filters will be considered here, partly to limit the length of the chapter to
something manageable. Another reason is that, compared to passive filters, active filters
generally suffer from higher noise and nonlinearity, limited operational frequency range,
higher power consumption, and relatively high sensitivity to parameter variations, particularly at the GHz frequencies with which we are concerned in this textbook.
2.0 Background
2.1 A quick history
The use of frequency selective circuits certainly dates back at least to the earliest research
on electromagnetic waves. In his classic experiments of 1887-1888 Hertz himself used
dipole and loop antennas (ring resonators) to clean up the spectrum generated by his spark
gap apparatus and thereby impart a small measure of selectivity to his primitive receivers.
Wireless pioneer Sir Oliver Lodge of the U.K. coined the term “syntony” to describe the
action of tuned circuits, showing a conscious appreciation of the value of such tuning,
despite the hopelessly broadband nature of spark signals.1 At nearly the same time, Nikola
Tesla and Guglielmo Marconi developed tuned circuits (Marconi’s patent #7777 was so
1. See H. Aitken’s excellent book, Syntony and Spark, Princeton, 1987, for a technically detailed and fascinating account of early work in wireless.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 1 of 38
EE414 Handout #14: Spring 2001
valuable that it became the subject of bitter and protracted litigation)2 for the specific purpose of rejecting unwanted signals, anticipating the advent of sinusoidal carrier based
communications.
Despite that foundation, however, modern filter theory does not trace directly back to
those early efforts in wireless. Rather the roots go back even further in time: it is research
into the properties of transmission lines for telegraphy and telephony that primarily inform
early filter theory. In 1854 William Thomson (who would later become Lord Kelvin), carried out the first analysis of a transmission line, considering only the line’s distributed
resistance and capacitance. His work, inspired by what was to be the 3000-kilometer
Atlantic Cable Project, established a relationship between practical transmission rates and
line parameters. A bit over 20 years later, Oliver Heaviside and others augmented Kelvin’s
analysis by including distributed inductance, thereby extending greatly the frequency
range over which transmission line behavior could be described accurately.3 Following up
on one particular implication of Heaviside’s work, both George Ashley Campbell of the
American Bell Company and Michael Idvorsky Pupin of Columbia University suggested
around 1900 the insertion of lumped inductances at regularly spaced intervals along telephone transmission lines to reduce dispersion (the smearing out of pulses).4 This suggestion is relevant to the filter story because Heaviside recognized that a lumped line differs
from a continuous one in possessing a definite cutoff frequency. Campbell and Pupin provided design guidelines for guaranteeing a certain minimum bandwidth.5
In true engineering fashion, the apparent liability of a lumped line’s limited bandwidth
was quickly turned into an asset, and thus was established the main evolutionary branch of
filter design. The first published formalism is Campbell’s, whose classic 1922 paper
describes in fuller detail ideas he had developed and patented during WWI.6 Karl Willy
Wagner also developed these ideas at about the same time, but German military authorities
delayed publication, giving Campbell priority.7 It is now acknowledged that credit should
be shared by these two pioneers, who independently and nearly simultaneously hit upon
the same great idea.
2. The U.S. Supreme Court eventually ruled it invalid (in 1943) because of prior work by Lodge, Tesla and
others.
3. For additional background on this story, see Paul J. Nahin’s excellent book, Oliver Heaviside: Sage in
Solitude, IEEE Press, 1987.
4. As with many key ideas of great commercial import, a legal battle erupted over this one. It is a matter of
record that the Bell System was already experimenting with loading coils developed by Campbell well
before publication of Pupin’s 1900 paper. Pupin’s self-promotional abilities were superior, though, and he
was able to obtain a patent nonetheless. He eventually earned royalties of over $400,000 from Campbell’s
employer (at a time when there was no U.S. income tax) for his “invention.” To add to the insult, Pupin’s
Pulitzer-prize winning autobiography of 1924 shamefully fails to acknowledge Campbell and Heaviside.
5. A. T. Starr, Electric Circuits and Wave Filters, 2nd ed., Pitman and Sons, 1948.
6. G. A. Campbell, “Physical Theory of the Electric Wave-Filter,” Bell System Technical Journal, vol. 1, no.
2, pp. 1-32, Nov. 1922. See also his U. S. Patent #1,227,113, May 22, 1917.
7. “Spulen- und Kondensatorleitungen” (Inductor and Capacitor Lines), Archiv für Electrotechnik, vol. 8,
July 1919.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 2 of 38
EE414 Handout #14: Spring 2001
Campbell’s colleague, Otto J. Zobel, published a much-referenced extension of Campbell’s work, but which was still limited to filters derived from transmission line ideas.8 In
the developments of subsequent decades one sees an evolving understanding of how
closely one may approach in practice the theoretical ideal of a perfectly flat passband, constant group delay, and an infinitely steep transition to an infinitely attenuating stopband.
Conscious acknowledgment that this theoretical ideal is unattainable leads to the important idea that one must settle for approximations. Some of the more important, practical
and well-defined of these approximations are the Butterworth, Chebyshev and Cauer
(elliptical) filter types we’ll study in this chapter.
Shortly after WWII, the subject of filter design advanced at an accelerated pace. Investigation into methods for accommodating finite-Q elements in lumped filters offered hope for
improved predictability and accuracy. In the microwave domain, filter topologies based
directly on lumped prototypes came to be supplemented by ones that exploit, rather than
ignore, distributed effects. Many of these are readily implemented in microstrip form, and
are the ultimate focus of this chapter.
The advent of transistors assured that the size of active devices no longer dominated that
of a circuit. Numerous active filter topologies evolved to respond to a growing demand for
miniaturization, replacing bulky passive inductor-capacitor circuits in many instances.
Aside from enabling dramatic size reductions, some active filters are also electronically
tunable. However, these attributes do come at a price: active filters consume power, suffer
from nonlinearity and noise, and possess diminished upper operational frequencies
because of the need to realize gain elements with well-controlled characteristics at high
frequencies. These tradeoffs become increasingly serious as microwave frequencies are
approached. This statement should not be taken to mean that microwave active filters can
never be made to work well enough for some applications (because successful examples
certainly abound), but it remains true that the best filters at such frequencies continue to be
passive implementations. It is for this reason that this chapter considers passive filters
exclusively.
The arrival of transistors also coincided with (and helped drive) a rapidly decreasing cost
of computation.9 No longer limited to considering only straightforward analytical solutions, theorists were free to pose the filter approximation problem much more generally,
e.g., “Place the poles and zeros of a network to minimize the mean-square error (or maximum error, or some other performance metric) in a particular frequency interval, relative
to an ideal response template.” Of great practical importance is that a numerical approach
readily accommodates lossy inductors and capacitors, something that is difficult with earlier analytical approaches. The resulting filters are optimum in the sense that one cannot
do better (as evaluated by whatever design criteria were imposed in the first place) for a
given filter order. The tradeoff is that the resulting design may not be as easily understood
8. O. J. Zobel, “Theory and Design of Uniform and Composite Electric Wave-filters,” Bell System Technical Journal, vol. 2, no. 1, pp. 1-46, Jan. 1923.
9. Regrettably, space limitations force us to neglect here the fascinating story of electrolytic tanks and other
analog computers used to design filters based on potential theory.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 3 of 38
EE414 Handout #14: Spring 2001
as those based on analytical approaches. In many cases, element values are best obtained
from tables that summarize the results of extensive computations.
The same philosophical approach of numerical optimization is also how most modern
microwave filters are designed. And again, solutions for the more complex types are best
extracted from tables. The main purpose here is to provide an intuitive explanation for
how these filters work, leaving many of the mathematical details to published theoretical
treatments.
3.0 Filters from Transmission Lines
We start with the “electric wave filters” of Campbell, Zobel and Wagner. As mentioned,
these derive from lumped approximations to transmission lines, so we begin by examining
such “artificial” lines to see how a limited bandwidth arises.
3.1 Constant-k filters
For convenience, we repeat here some of the calculations from the chapter on distributed
systems. Recall that we first consider the driving point impedance, Zin, of the following
infinite ladder network:
FIGURE 1. Infinite ladder network as artificial line
Z
A
Z
Y
Zin
Z
B
C
Y
Z
Y
In this drawing, the resistor symbols represent generalized impedances (Z) and admittances (Y).
To simplify the derivation, it is helpful to note that the impedance looking to the right of
point A also equals Zin, this being an infinite network. We can then collapse the infinite
network into a much simpler finite one:
FIGURE 2. Conversion of infinite line into finite network
Z
Zin
Y
Zin
Solving for the Zin of this simple network yields:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 4 of 38
EE414 Handout #14: Spring 2001
2
Z in
Z ± Z + 4 ( Z ⁄ Y)
Z
4
=
=
1± 1+
,
2
2
ZY
(1)
where one would generally disallow negative values and thus choose only the sum solution.
As a specific (but typical) case, consider a low-pass filter in which Y = jωC and Z = jωL.
Then, the input impedance of the infinite artificial line is:
Z in =
jωL
4
.
1± 1−
2
2
ω LC
(2)
At very low frequencies, the factor under the radical is negative and large in magnitude,
making the term within the brackets almost purely imaginary. The overall Zin in that frequency range is therefore largely real, with
Z in ≈
Z
=
Y
L
= k.
C
(3)
Because the ratio Z/Y is a constant here, such filters are often known as constant-k filters.10
As long as the input impedance has a real component, nonzero average power will couple
into the line from the source. Above some particular frequency, however, the input impedance becomes purely imaginary, as can be seen from inspection of Eqn. 2. Under this condition, no real power can be delivered to the network, and the filter thus attenuates
heavily.11 For self-evident reasons the frequency at which the input impedance becomes
purely imaginary is called the cutoff frequency which, for this low-pass filter example, is
given by:
ωh =
2
.
(4)
LC
Any practical filter must employ a finite number of sections, of course, leading to the
question of the relevancy of any analysis that assumes an infinite number of sections. Intuitively, it seems reasonable that a “sufficiently large” number of sections would lead to
acceptable agreement. Based on lumped network theory, we also expect the ultimate rate
of rolloff to be determined by the filter order, and hence by the number of sections to
which the network is truncated (we’ll have more to say on this subject later). The greater
10. Campbell used the symbol k in precisely this context, but it was Zobel (op. cit.) who apparently first
used the actual term “constant-k.”
11. Attenuation without dissipative elements might initially seem intuitively unpalatable. However, consider
that a filter might also operate by reflecting energy, rather than by dissipating it. That is, a filter can function
by producing a purposeful impedance mismatch over some band of frequencies. In fact, many filter design
approaches are based directly on manipulation of the reflection coefficient as a function of frequency.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 5 of 38
EE414 Handout #14: Spring 2001
the number of sections, the greater the rate of rolloff. As we’ll see, there is also some (but
practically limited) flexibility in the choice of Z and Y, permitting a certain level of tradeoff among passband, transition band, and stopband characteristics. However, it remains
true that one limitation of filters based on artificial-line concepts is the inability to specify
these characteristics in detail, if at all. Note, for example, the conspicuous absence of any
discussion about how the filter behaves near cutoff. We don’t know if the transition from
passband to stopband is gradual or abrupt, monotonic or oscillatory. We also don’t know
the precise shape of the passband. Finally, we don’t have any guide how to modify the
transition shape should we find it unsatisfactory. As we’ll see, these shortcomings lead us
to consider other filter design approaches.
Once the filter order is chosen (by whatever means), the next problem is one of termination. Note that the foregoing analysis assumes that the filter is terminated in an impedance
that behaves as described by Eqn. 2. That is, our putative finite filter must be terminated in
the impedance produced by the prototype infinite ladder network: It must have a real
impedance at low frequencies, then become purely imaginary above the cutoff frequency.
Stated another way, rigorous satisfaction of the criteria implied by Eqn. 2 absurdly
requires that we supply a load element which itself is the filter we desire! We should therefore not be too surprised to discover that a practical realization involves compromises, all
intimately related to the hopeless task of mimicking the impedance behavior of an infinite
structure with a finite one. For example the near-universal choice is to terminate the following with a simple resistance R equal to k:
FIGURE 3. Low-pass constant-k filter example using two cascaded T-sections
L/2
C/2
L/2
C
L/2
L/2
L/2
C
L/2
C/2
A source with a Thévenin resistance also of value k is assumed to drive this filter. Note
that this example uses two complete T-sections (shown in the boundaries), with a half-section placed on each end. Termination in half sections is the traditional way to construct
such filters. The series-connected inductors, shown individually to identify clearly the separate contributions of the unit T-sections, are combined into a single inductance in practice. Alternatively one may implement the filter with π-sections. With those, one uses
terminating half-sections that are mirror images of the ones shown. The choice of which
implementation to employ is often determined in practice by the nature of the parasitics
that dominate the input and output interfaces. If these parasitics are primarily capacitive in
nature, the T-section implementation shown is favored, since the parasitics may be
absorbed into the capacitances at the ends of the filter. Similarly, inductive parasitics are
most readily accommodated by a filter using internal π-sections.
The design equations for this filter are readily derived from combining Eqn. 3 and Eqn. 4:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 6 of 38
EE414 Handout #14: Spring 2001
2 1
C = ,
ω R
h
(5)
2
L = R.
ω
(6)
and
h
Thus, once one specifies the characteristic impedance, R, the desired cutoff frequency, and
the total number of sections, the filter design is complete.
Regrettably, deducing the number of sections required is a bit of a cut and try affair in
practice. There are equations that can provide guidance, but they are either cumbersome or
inaccurate enough that one typically increases the number of sections until simulations
reveal that the filter behaves as desired. Furthermore the unsophisticated termination of a
simple resistance leads to degradation of important filter characteristics, often resulting in
a hard-to-predict insertion loss and passband flatness, as well as reduced stopband attenuation (relative to predictions based on true, infinite-length lines). These difficulties are
apparent from an inspection of the following table, which shows the attenuation at the cutoff frequency, as well as the –3dB and –6dB bandwidths (expressed as a fraction of the
cutoff frequency), of constant-k filters (both T- and π-implementations) as a function of
order. In the table, n is the number of complete T- (or π-) sections in the central core of the
filter. The filter order is therefore 2n + 3.
TABLE 1. Characteristics of ideal constant-k filters
n
Attenuation
at cutoff
frequency
(dB)
Normalized
–3dB
Bandwidth
Normalized
–6dB
Bandwidth
Normalized
–60dB
Bandwidth
Normalized
–10dB S11
Bandwidth
0
3.0
1.000
1.201
10.000
0.693
1
7.0
0.911
0.980
3.050
0.810
2
10.0
0.934
0.963
1.887
0.695
3
12.3
0.954
0.969
1.486
0.773
4
14.2
0.967
0.976
1.302
0.696
5
15.7
0.976
0.981
1.203
0.756
Note that the attenuation at the nominal cutoff frequency, as well as the bandwidth, are
both dependent on the number of filter sections. Further note that the cutoff frequency (as
computed by Eqn. 4) equals the –3dB bandwidth only for n = 0, and is as much as 10%
beyond the –3dB bandwidth in the worst case. In critical applications, the cutoff frequency
target may have to be altered accordingly to achieve a specified bandwidth.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 7 of 38
EE414 Handout #14: Spring 2001
The following figure is a frequency response plot for a constant-k filter which consists of
five full sections, and a terminating half-section on each end. Note that the frequency axis
is linear, not logarithmic:
FIGURE 4. Response of six-section low pass constant-k filter (n = 5)
Aside from the ripple evident in the figure, it is also unfortunate that the bandwidth over
which the return loss exceeds 10dB is typically only ~70-80% of the cutoff frequency. A
considerable improvement in performance is possible by using filter sections whose
impedance behavior better approximates a constant resistance over a broader frequency
range. One example, developed by Zobel, uses “m-derived” networks, either as terminating structures, or as filter sections:
FIGURE 5. Low-pass m-derived filter using cascaded T-sections
L1/2
L1/2
2L2
C/2
L1/2
L1/2
L2
C
L1/2
L1/2
L2
C
2L2
C/2
As with the prototype constant-k filter of Figure 5, this structure is both driven and terminated with a resistance of value k ohms. The m-derived filter, which itself is a constant-k
structure, is best understood by noting that the prototype constant-k filter previously analyzed has a response that generally attenuates more strongly as the cutoff frequency ω1 is
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 8 of 38
EE414 Handout #14: Spring 2001
approached. At small fractions of the cutoff frequency, the response is fairly flat, so it
should seem reasonable that increasing the cutoff frequency to some value ω2 should produce a more constant response within the original bandwidth ω1. The first step in designing an m-derived filter, then, consists simply of increasing the cutoff frequency of a
prototype constant-k filter. In the absence of inductor L2, we see that scaling the values of
L1 and C each, say, by a factor m (with m ranging from 0 to 1) increases the cutoff frequency by a factor of 1/m, from a value ω1, to ω2 = ω1/m. The characteristic impedance
remains unchanged at k because the ratio of L1 to C is unaffected by this scaling.
Now to restore the original cutoff frequency, add an inductance L2 to produce a series resonance with C. At the resonant frequency, this series arm presents a short circuit, creating
a notch in the filter’s transmission. If this notch is placed at the right frequency (just a bit
above the desired cutoff frequency), the filter’s cutoff frequency can be brought back
down to ω1. However, be aware that the filter response does pop back up above the notch
frequency (where the resonant branch then looks like a simple inductance). This characteristic needs to be taken into account when using the m-derived filter.
An alternative to a series resonance in the shunt arm of each filter section is a parallel resonance in the series arm(s) of a filter section. Both types of m-derived filters will provide
the same behavior. The choice of topology in practice is often determined by which implementation uses more easily realized components, or which more gracefully accommodates
parasitic elements.
Following a procedure exactly analogous to that used in determining the cutoff frequency
of ordinary constant-k filters, we find that the cutoff frequency of an m-derived filter may
be expressed as
ω1 =
2 ( R ⁄ L1 )
4
L2
L1
.
(7)
+1
To remove L1 from the equation, note that the cutoff frequency may also be expressed as
ω1 =
2m
,
(8)
L1 C
while the characteristic impedance is given by
R =
L1
C
.
(9)
Combining these last three equations allows us to solve for L2:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 9 of 38
EE414 Handout #14: Spring 2001
2
L2 =
(1 − m ) R
.
2mω 1
(10)
Solving Eqn. 8 and Eqn. 9 for L1 and C yields
2m 1
C =
ω R
(11)
2m
L1 =
R.
ω
(12)
1
and
1
Use of the foregoing equations requires that the designer have an idea of what value of m
is desirable. As m approaches unity the response exhibits a monotonic rolloff (and therefore an increasing passband error), while passband peaking increases as m approaches
zero. In practice a rather narrow range is encountered (say, within 25-30% of 0.5) as a
compromise between these two behaviors, and the parameter m is commonly chosen equal
to 0.6. This value yields a reasonably broad frequency range over which the transmission
magnitude remains roughly constant. The following table enumerates (to more significant
digits than are practically significant) some of the more relevant characteristics of mderived filters, for the specific value of m = 0.6. As with Table 1, the parameter n is the
number of complete T- (or π-) sections used in the filter. The column labeled “minimum
stopband attenuation” gives the worst-case value of attenuation above the transmission
notch frequency, where the filter response pops back up.
TABLE 2. Characteristics of ideal m-derived filters (m = 0.6)
n
Attenuation
at cutoff
frequency
(dB)
Normalized
–3dB
Bandwidth
Normalized
–6dB
Bandwidth
Normalized
–10dB S11
Bandwidth
Minimum
stopband
attenuation
(dB)
0
1.34
1.031
1.063
0.965
8.21
1
3.87
0.993
1.013
0.956
21.24
2
6.27
0.988
0.999
0.969
34.25
3
8.30
0.989
0.996
0.979
47.09
4
10.00
0.991
0.995
0.954
59.81
5
11.44
0.993
0.996
0.961
72.43
Note that the cutoff frequency and –3dB bandwidth are much more nearly equal than for
the prototype constant-k case (the worst-case difference here is about 3%). The bandwidth
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 10 of 38
EE414 Handout #14: Spring 2001
over which the return loss exceeds 10dB is also a much greater fraction of the cutoff frequency (above 95%, in fact). Note also that the minimum stopband attenuation increases
by about 12-13dB per increment in n for this range of values.
The following figure illustrates how the use of m-derived sections can improve the magnitude response (note that the vertical axis now spans 80dB, rather than 50dB):
FIGURE 6. Frequency response of six-section m-derived low pass filter (m = 0.6, n = 5)
Compared with Figure 4, this response shows significantly less passband ripple, as well as
a much faster transition to stopband, owing to the stopband notch.
On the frequency scale of Figure 6, the characteristic notch is invisible, as is the poppingup of the response at higher frequencies. The following plot shows these features more
clearly:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 11 of 38
EE414 Handout #14: Spring 2001
FIGURE 7. Frequency response of m-derived low pass filter, plotted over wider range
Aside from the potential for improved flatness over the passband, the notches that are
inherent in m-derived filters can be used to null out interfering signals at a specific frequency (or frequencies, if sections with differing values of m are used). Later, we will see
that judiciously distributed notches can be combined with passband ripple to produce what
are known as elliptic or Cauer filters whose responses exhibit even more dramatic transitions from passband to stopband.
If the precise location of a notch is of importance, it is helpful to know that the frequency
of the null ω∞ is related to m as follows:
ω∞
ω1
=
1
,
1−m
(13)
2
so that the value of m needed to produce a notch at a specified frequency ω∞ is
ω1
1−
.
ω
2
m =
(14)
∞
A value of 0.6 for m corresponds to a notch frequency that is a factor of 1.25 times the cutoff frequency.
Table 3 summarizes the design of constant-k and m-derived low pass filters. Component
values (again, to many more digits than are practically relevant) are for the specific case of
a termination (and source) resistance of 50Ω and a cutoff frequency of 1GHz. The left two
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 12 of 38
EE414 Handout #14: Spring 2001
columns are for the simple constant-k case, and the last three columns give values for the
specific m-derived case where m = 0.6.
TABLE 3. Component values for 1GHz constant-k and m-derived filters (Z = 50Ω m = 0.6)
L
15.9155nH
C
6.3662pF
L1
9.5493nH
C
3.8197pF
L2
4.2441nH
For filters with a cutoff frequency other than 1GHz, simply multiply all component values
by the ratio of 1GHz to the desired cutoff frequency. For a different characteristic impedance, multiply all component impedances by the ratio of the desired impedance to 50Ω.
One may also combine ordinary constant-k and m-derived sections because the individual
sections for both are constant-k in nature. Such a composite filter may be desirable, for
example, to effect a compromise between flatness and the production of notches at specific
frequencies. Unfortunately, design of such a filter is very much an ad hoc affair. One simply mixes and matches sections as seems sensible, then simulates to verify if the design
indeed functions satisfactorily.
3.1.1 High-pass, bandpass and bandstop shapes
At least in principle, a high-pass constant-k filter is readily constructed from the low pass
constant-k prototype simply by swapping the positions of the inductors and capacitors; the
values remain the same. Thus one may design, say, a 1GHz constant-k low-pass filter
using the values of Table 3, then interchange the Ls and Cs to synthesize a 1GHz highpass filter.
The reason for the qualifier “at least in principle” is that high-pass filters typically exhibit
serious deviations from desired behavior. These deviations motivate microwave filter
designers to avoid high-pass filters based on lower frequency prototypes. Although there
are many ways – too numerous to mention, in fact – in which a practical filter of any kind
can fall short of expectations, perhaps the following lumped high pass filter example will
suffice to illustrate the general nature of the problem. Specifically, consider:
FIGURE 8. High pass filter?
Every practical inductor is shunted by some capacitance, and thus exhibits a resonance of
its own. Above the resonant frequency, the “inductor” actually appears as a capacitance.
Similarly every practical capacitance has in series with it some inductance. Above the cor-
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 13 of 38
EE414 Handout #14: Spring 2001
responding series resonance, the capacitor actually appears inductive. Hence, at sufficiently high frequencies, our high pass filter actually acts as a low pass filter.
A complementary effect afflicts low pass filters where, at high frequencies, it is possible
for the response to pop back up.
As a practical workaround, it is traditional to employ a bandpass filter with a sufficiently
wide passband to approximate the desired filter shape. Of course, that solution presupposes knowledge of how to construct bandpass filters. Fortunately, the constant-k structure
works here, too (we’ll later examine alternative bandpass implementations as well). As a
general strategy for deriving a bandpass filter from a low-pass prototype, replace the
inductance of a low-pass prototype with a series LC combination, and the capacitance with
a parallel LC combination:
FIGURE 9. Bandpass constant-k filter example using two cascaded T-sections
2L2
L1
C1
L1
C1
C2/2
L2
C2
L2
L1
C2
C1
2L2
C2/2
Unlike our previous figures, the individual T-sections are not shown, in order to simplify
the schematic.
Note that this structure continues to exhibit the correct qualitative behavior even if inductors ultimately become capacitors and vice-versa. This property is fundamental to the
potentially reduced sensitivity of this topology to parasitic effects.
The formula for the inductance L1 of the series resonator is the same as that for the inductance in the prototype low pass filter, except that the bandwidth (defined by the difference
between the upper and lower cutoff frequencies) replaces the cutoff frequency. The capacitance C1 is then chosen to produce a series resonance at the center frequency (defined
here as the geometric mean of the two cutoff frequencies12). Hence
L1 =
2
R
( ω2 − ω1 )
(15)
and
12. In some of the literature, it is unfortunately left unclear as to what sort of mean should be used. For the
common case of small fractional bandwidths, this ambiguity is acceptable, for there is then little difference
between an arithmetic and geometric mean. Practical component tolerances make insignificant such minor
differences. However, the discrepancy grows with the fractional bandwidth, and the error can become quite
noticeable at large fractional bandwidths if the arithmetic mean is used.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 14 of 38
EE414 Handout #14: Spring 2001
C1 =
( ω2 − ω1 ) 1
.
2
R
2ω
(16)
0
Similarly, the equation for the capacitance of the low-pass prototype is modified for the
bandpass case by replacing the cutoff frequency with the bandwidth. The resonating
inductance is again chosen to produce a resonance at the center frequency:
C2 =
2
1
( ω2 − ω1 ) R
(17)
and
L2 =
( ω2 − ω1)
2
R.
(18)
2ω 0
Values for a constant-k bandpass filter with cutoff frequencies of 950MHz and 1.05GHz
(corresponding to a center frequency of approximately 998.75MHz) are given in the following table:
TABLE 4. Component values for a 100MHz bandwidth, constant-k bandpass filter at 1GHz
L1
C1
159.15nH
0.15955pF
L2
0.39888nH
C2
63.662pF
As is its low-pass counterpart, the bandpass filter is terminated in half-sections. Each halfsection consists of components of value L1/2, 2C1, 2L2 and C2/2. The resulting filters have
the same characteristics enumerated in Table 1, where the bandwidth normalizations continue to be performed, sensibly enough, with respect to the bandwidth, rather than the center frequency.
The following figure shows the frequency response of a bandpass filter derived from a
low-pass constant-k filter with six sections (n = 5). The design bandwidth is 100MHz, centered at 1GHz. Not surprisingly, the behavior at the passband edges resembles that of the
low-pass prototype.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 15 of 38
EE414 Handout #14: Spring 2001
FIGURE 10. Frequency response for bandpass filter derived from six-section constant-k (n = 5)
For a different bandwidth, multiply C1 and L2 by the ratio of the new bandwidth to
100MHz, and reduce L1 and C2 by the same factor. For a different center frequency,
reduce C1 and L2 each by the square of the ratio of the new center frequency to 1GHz.
Finally, for a different characteristic impedance, increase the impedance of all four components by the ratio of the new impedance to 50Ω.
The bandpass filter can be converted into a bandstop (also known as a band-reject) filter
simply by swapping the positions of the series and parallel resonators. As in the conversion from low pass to high pass, the values remain unchanged.
From the tables and examples given, it is clear that the constant-k and m-derived filters are
extremely simple to design, since they consist of identical iterated sections (plus a terminating half-section on each end). This simplicity is precisely their greatest attribute. In
exchange for this ease of design, however, the foregoing procedures neglect certain details
(such as passband ripple) because they do not incorporate any specific constraints on
response shape. It is clear from the tables, for example, that the cutoff frequency doesn’t
correspond to a certain fixed attenuation value, such as –6dB, and monotonicity is far from
guaranteed. Stopband behavior is similarly mysterious. Finally, because the source and
load terminations are assumed equal in value, any necessary impedance transformations
have to be provided separately. It should seem reasonable, however, to expect that a more
advanced synthesis technique might, on occasion, accommodate impedance transformation as a natural accompaniment to the filtering operation. Shortcomings such as these
explain why there are alternative filter design approaches. Because the relative merits of
these alternatives are best appreciated after identification and definition of key filter performance metrics, we now consider a brief sidebar and introduce these parameters.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 16 of 38
EE414 Handout #14: Spring 2001
4.0 Filter Classifications and Specifications
Filters may be classified broadly by their general response shapes – e.g., low-pass, bandpass, band reject and high-pass – and further subdivided according to bandwidth, shape
factor (or skirt selectivity), and amount of ripple (in either the phase or magnitude
response, and in either the passband, stopband, or both). This subdivision is an acknowledgment that ideal, brickwall filter shapes are simply unrealizable (not merely impractical). Different approaches to approximating ideal characteristics result in different
tradeoffs, and the consequences of these compromises require characterization.
Bandwidth is perhaps the most basic descriptive parameter, and is conventionally defined
using –3dB points in the response. However, it is important to recognize that 3dB is quite
an arbitrary choice (there is nothing fundamental about the half-power point, after all), and
we will use other bandwidth definitions that may be more appropriate from time to time. It
is certainly an incomplete specification, because there are infinitely many filter shapes that
share a common –3dB bandwidth. Shape factor is an attempt to convey some information
about the filter’s response at frequencies well removed from the –3dB point. It is defined
as the ratio of bandwidths measured at two different attenuation values (i.e., values at two
different points on the skirt). As an arbitrary example a “6/60” shape factor specification is
defined as the bandwidth at –60dB attenuation, divided by the bandwidth at –6dB attenuation:
FIGURE 11. Illustration of 6/60 shape factor
S 6 ⁄ 60 ≡
∆ω 60
∆ω 6
∆ω6
–6dB
∆ω60
–60dB
Clearly from the definition of shape factor, values approaching unity imply response
shapes that approach infinitely steep transitions from passband to stopband. A single-pole
lowpass filter (or a standard single-LC bandpass resonator) has a 6/60 shape factor of
roughly 600, a value generally regarded as pathetically large.13 This trio of numbers is
easily remembered, though, because of the decimal progression.
13. The actual number is closer to 577, but has less of a mnemonic value than 600.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 17 of 38
EE414 Handout #14: Spring 2001
Because the relevance of a given shape factor depends very much on context, there cannot
be a single, universally relevant definition. Thus although 6/60 happens to be a common
one, other specifications are often encountered.
As stated earlier, the inability of practical filters to provide perfectly flat passbands and
infinitely steep transitions to infinitely attenuating stopbands implies that we must always
accept approximations to the ideal. In the best case we have the opportunity to quantify
and specify bounds on the approximation error. The traditional way of doing so is to specify the following parameters:
FIGURE 12. General filter response template (shown for the low pass case)
|H(jω)|2
1
Passband
Transition
Band
1/(1+ε2)
Stopband
1/A2
ωp
ωs
Note that the square of the magnitude is plotted in the figure, rather than the magnitude
itself, because it is proportional to power gain. This convention isn’t universally followed,
but it is quite common because of the RF engineer’s typical preoccupation with power
gain.
Note also the pervasiveness of reciprocal quantities on the vertical axis. This annoying
feature is avoided by plotting attenuation, rather than gain, as a function of frequency,
explaining why many treatments present data in precisely that manner.
Note further that the filter response template accommodates some amount of variation
within the passband (whose upper limit is denoted ωp), with a maximum permitted deviation of 1/(1+ε2). Additionally, a finite transition between the passband and stopband
(whose lower frequency limit is denoted ωs) is also permitted, with a minimum allowed
power attenuation of A2 in the stopband. Specification of these parameters thus allows the
design of real filters. We now consider several important classes of approximations which
make use of these parameters.
5.0 Modern Filters: Common Approximations
The constant-k filter’s limitations ultimately derive from a synthesis procedure which
ignores the control over filter response afforded by direct manipulation of the pole (and
zero) locations. This limitation is a natural consequence of the transmission-line theoretical basis for constant-k filters; because transmission lines are infinite-order systems, conFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 18 of 38
EE414 Handout #14: Spring 2001
sideration of pole locations there would be unnatural, and in any case would lead to
numerous analytical difficulties.
However if one no longer insists on treating filters from a transmission line viewpoint,
these difficulties disappear (but are replaced by new ones). Additional, and highly powerful, techniques then may be brought to bear on the filter analysis and synthesis problem. In
this section, we underscore this point by following a synthesis procedure not possible with
the constant-k filter: starting from a specification of a desired frequency response, compute
a corresponding pole-zero constellation, and then synthesize a lumped network that exhibits the prescribed characteristics.
5.1 Butterworth filters
Some applications are entirely intolerant of ripple, limiting the number of options for
response shape. As do all practical filters, the Butterworth seeks to approximate the ideal
rectangular brickwall shape. The Butterworth filter’s monotonic response shape minimizes the approximation error in the vicinity of zero frequency by maximizing the number
of derivatives whose value is zero there. For a filter of order n, that maximum number happens to be 2n – 1. As the filter order approaches infinity, the filter shape progressively
approximates better the ideal brickwall shape.
A natural, but potentially undesirable consequence of a design philosophy which places
greater importance on the approximation error at low frequencies is that the error grows as
the cutoff frequency is approached. If this characteristic is indeed undesirable, one must
seek shapes other than the Butterworth. Some of these alternatives are discussed in subsequent sections.14
The Butterworth’s response magnitude (squared) as a function of frequency is given for
the low pass case by the following expression:
H ( jω )
2
=
1
ω 2n
1+
ω
c
,
(19)
where ωc is the frequency at which the power gain has dropped to 0.5.15 The parameter n
is the order of the filter, and equals the number of independent energy storage elements, as
well as the power of ω with which the response magnitude ultimately rolls off. From the
equation, it is straightforward to conclude that the response is indeed monotonic.
14. As will be discussed later, one of these alternatives, the Type II Chebyshev, actually achieves better
passband flatness than the Butterworth (making it flatter than maximally flat), by permitting stopband ripple,
while preserving a monotonic passband response.
15. Although not rigorously correct (because of the possibility of unequal input and output impedances), we
will frequently use the term “power gain” interchangeably with the more cumbersome “response magnitude
squared.”
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 19 of 38
EE414 Handout #14: Spring 2001
In designing a Butterworth filter, one may specify ωc directly, but to maintain consistency
with the template of Figure 12, let us derive ωc from the other filter parameters. First we
may express the power gain at the passband and stopband edges as follows:
1
1+ε
2
=
1
2n
ωp
1+
ω
(20)
c
and
1
A
2
=
1
ω s 2n
1+
ω
.
(21)
c
Solving these two equations for the required filter order, n, yields
2
n =
ln( ε ⁄ A − 1 )
ωp
ln
ω
.
(22)
s
Thus, once the attenuation at the passband edge, minimum attenuation at the stopband
edge, and the frequencies of those edges are specified, the required filter order is immediately determined. Because Eqn. 22 generally yields non-integer values, one must choose
the next higher integer as the filter order. In that case, the resulting filter will exhibit characteristics that are superior to those originally sought. One way to use the “surplus” performance is to retain the original ωp, in which case the filter will exhibit greater
attenuation at ωs than required. Alternatively, one may instead retain the original ωs, in
which case the filter exhibits smaller attenuation (i.e. smaller error) at the passband edge
than originally targeted. Or, one may elect a strategy that is intermediate between these
two choices.
Pursuing the strategy of retaining the originally sought performance at the passband edge,
the –3dB corner frequency ωc may be computed from the foregoing equations as
ωc =
ωp
ε
1⁄n
.
(23)
Alternatively, Eqn. 21 may be solved for a (generally different) ωc derived from a specification of ωs.
Because its approximation error is very small near DC, the Butterworth shape is also
described as maximally flat.16 However it is important to recognize that maximally flat
does not imply perfectly flat.17 Rather, it implies the flattest passband that can be
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 20 of 38
EE414 Handout #14: Spring 2001
achieved, subject to the constraint of monotonicity (later, we will see that it is possible to
have an even flatter passband response if we are willing to permit ripple in the stopband).
As a design example, let us continue the exercise that we began with the constant-k topology. We now have the ability to specify more filter parameters than in that case, so we’ll
do so. Here, arbitrarily allow a 1dB loss (gain of 0.794) at the passband edge of 1GHz, and
require a 30dB factor of minimum attenuation at a 3GHz stopband edge.
From the passband specification, we find that ε is approximately 0.5088. From the stopband specification, we see that A2 is 1000. As a result, the minimum filter order required to
meet the specifications is
n =
ln( 0.5088 ⁄ 999 )
≈ 3.76 ,
1
ln
3
(24)
which we round upward to four. Choosing to meet precisely the specification at the passband edge, we find that the corresponding value of ωc is approximately
ωc =
ωp
ε
1⁄n
≈ 7.44Grps .
(25)
From this point, we would typically consult a table of component values for a fourth-order
filter, scaling the values for the desired cutoff frequency (and, possibly, impedance
level).18 As a final check, it is always wise to simulate the proposed filter, just to make
sure that no computational errors (or typographical errors – some published tables have
incorrect entries!) have corrupted the design. In demanding applications, simulation is
also valuable for assessing the sensitivities of the filter to practical variations in component values, or to other imperfections (such as finite Q or parasitics).
5.2 Chebyshev (equiripple or minimax) filters
Although monotonicity certainly has an esthetic appeal, insisting on it constrains other
valuable filter shape properties. These include the steepness of transitions from passband
to stopband, as well as the stopband attenuation for a given filter order. Alternative filters,
based on non-monotonic frequency response, are named after the folks who invented
them, or who developed the underlying mathematics. The Chebyshev filter, an example of
16. This term was evidently introduced by V. D. Landon in his paper, “Cascade Amplifiers with Maximal
Flatness,” RCA Review, vol. 5, pp. 347-362, January 1941. Coining of the term thus follows by more than a
decade Butterworth’s own exposition of the subject in “On the Theory of Filter Amplifiers,” Wireless Engr.,
vol. 7, pp. 536-541, Oct. 1930. Although others published similar results earlier, Butterworth and maximal
flatness are now seemingly linked forever.
17. In this way, “maximally flat” is used a bit like “creme filling” in describing the ingredients of an Oreo
cookie; it means something a little different from how it initially sounds.
18. Later, we will present a synthesis method that allows computation of component values directly.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 21 of 38
EE414 Handout #14: Spring 2001
the latter, allows a reduction in filter order precisely by relaxing the constraint of monotonicity.19 In contrast with the Butterworth approximation, which is preoccupied with minimizing error at low frequencies, the Chebyshev minimizes the maximum approximation
error (relative to the ideal brickwall shape) throughout the entire passband. The resulting
minimax response shape thus exhibits some ripple, the amount of which may be specified
by the designer. For a given order, the Chebyshev filter shape offers a more dramatic transition from passband to stopband than a Butterworth offers. The steepness of the transition
is also a function of the passband ripple one allows; the greater the permissible ripple, the
steeper the transition.
A consequence of minimizing the maximum error is that the ripples of a Chebyshev
response are all of equal amplitude. A rigorous proof of the minimax optimality of an
equiripple shape is surprisingly involved, so we won’t attempt one here. However, it
should seem intuitively reasonable that equiripple behavior would be optimal in the minimax sense for, if any one error peak were larger than the others, a better approximation
could probably be produced by reducing it, at the cost of increasing the size of one or
more of the others. Such tradings-off would proceed until all error peaks were equal,
because nothing would then be left to trade for anything else.
Similar advantages also accrue if the stopband, rather than the passband, is allowed to
exhibit ripple. The inverse Chebyshev filter (also known as a Type II Chebyshev filter) is
based on this idea, and actually combines a flatter-than-Butterworth passband with an
equiripple stopband.
To understand how the simple act of allowing either stopband or passband ripple provides
these advantages, we need to review the properties of a complex pole pair. First recall that
one standard (and perfectly general) form for the transfer function of such a pair is:
1
H ( s) =
s
2
,
(26)
2ζs
+
+1
2
ωn ωn
where ωn is the distance to the poles from the origin, and ζ (zeta) is the damping ratio:
19. Pafnuti L’vovich Chebyshev (1821-1894) did no work on filters at all. In fact he developed his equations
during a study of mechanical linkages used in steam engines (see his posthumously published “Théorie des
mécanismes connus sous le nom de parallélogrammes,” (Theory of mechanisms known under the name of
parallelograms)), Oeuvres, vol. I, St. Petersburg, 1899. “Parallelograms” translate rotary motion into an
approximation of rectilinear motion. By the way, the spelling of his name here is just one of many possible
transliterations of Pafnutiy L¡voviq Qeb[wev.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 22 of 38
EE414 Handout #14: Spring 2001
FIGURE 13. Two-pole constellation
jω n 1 − ζ
2
ωn
–ζωn
A zero damping ratio corresponds to purely imaginary poles, and a damping ratio of unity
corresponds to a pair of poles coincident on the real axis. The former condition applies to
an oscillator, and the latter defines critical damping. Above a damping ratio of unity the
two poles split, with one moving toward the origin, the other toward minus infinity, all the
while remaining on the real axis. Whatever the value of damping, the frequency ωn always
equals the geometric mean of the pole frequencies.
The property that is most relevant to the subject of filter design is the dependency of the
frequency response shape on the damping ratio. While it is true that all zero-free two pole
systems have a frequency response that ultimately rolls off as ω-2, the frequency response
magnitude, and the slope, in the vicinity of the peak are very much functions of the damping ratio, increasing as ζ decreases (Figure 14b). For damping ratios above 1 ⁄ 2 , the frequency response exhibits no peaking. Below that value of ζ, peaking increases without
bound as the damping ratio approaches zero. For small values of ζ, the peak gain is
inversely proportional to damping ratio. Stated alternatively, lower damping ratios lead to
greater ultimate attenuation, relative to the peak gain, and to slopes that are normally associated with higher (and perhaps much higher) order systems.
Now consider ways a filter might exploit this ζ-dependent behavior. Specifically, suppose
we use a second-order section to improve the magnitude characteristics of a single-pole
filter. If we arrange for the peak of the second-order response to compensate (boost) the
response of the first-order section beyond where the latter has begun a significant rolloff,
the frequency range over which the magnitude of the cascade remains roughly constant
can be increased. At the same time, the rolloff beyond the compensation point can exhibit
a rather high initial slope, providing an improved transition from passband to stopband.
Clearly, additional sections may be used to effect even larger improvements, with each
added section possessing progressively smaller damping ratios. This latter requirement
stems from the need to provide larger boosts to compensate for ever larger attenuations.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 23 of 38
EE414 Handout #14: Spring 2001
FIGURE 14. Step- and frequency-response of second-order lowpass transfer characteristic
Vout
time
a) Step response
ζ decreasing
|H(jω)|
1
–2
log ω
b) Frequency response
Having developed this understanding, we may revisit the Butterworth and Chebyshev
approximations. The Butterworth condition results when the poles of the transfer characteristic are arranged so that the modest amount of frequency response peaking of a complex pole pair offsets, to a certain extent, the rolloff of any real pole present. The resulting
combination exhibits roughly flat transmission magnitude over a broader frequency range
than that of either the real pole or complex pair alone. The result is that all of the poles lie
on a semicircle in the s-plane, distributed as if there were twice as many poles disposed at
equal angles along the circumference, the right half-plane poles being ignored.20 A thirdFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 24 of 38
EE414 Handout #14: Spring 2001
order Butterworth, for example, has a single pole on the real axis, and a complex conjugate pair at 60° angles with the real axis. The distance from the origin to the poles is the
3dB cutoff frequency.
FIGURE 15. Pole constellation for third order Butterworth low pass filter
ωc
The element values, normalized to a 1Ω impedance level, and to a 1 rad/sec passband
edge, for an nth-order Butterworth low-pass filter are given by the following set of equations:
g0 = 1
(27)
and
k
g k = 2 sin
( 2 − 1) π
2
n
,
(28)
where k ranges from 1 to n, and
gn + 1 = 1 .
(29)
Conversion into a bandpass filter is easily achieved using the same transformations used in
the constant-k case.
The Chebyshev filter goes further by allowing passband (or stopband) ripple. Continuing
with our third-order example, the response of the real pole is allowed to drop below the
low-frequency value by some specified amount (the permissible ripple) before the complex pair’s peaking is permitted to bring the response back up. The damping ratio of the
complex pair must be lower than that in the Butterworth case to produce enough additional peaking to compensate for the greater attenuation of the real pole. A side effect of
20. Okay, perhaps it isn’t quite “intuitively obvious,” it is true, but finding the roots of Eqn. 19 to discover
the factoid about Butterworth poles lying on a circle isn’t all that bad.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 25 of 38
EE414 Handout #14: Spring 2001
this lower damping is that there is a more dramatic rolloff beyond the cutoff frequency. In
this manner the Chebyshev filter permits the designer to trade passband flatness for better
stopband attenuation.
Although it is even less intuitively obvious, the poles of a Chebyshev low pass filter are
located along a (semi)ellipse, remarkably with imaginary parts that are equal to those of a
corresponding Butterworth low pass filter.21 Increasing the eccentricity of the ellipse
increases the ripple.
Inverse Chebyshev filters have poles located at the reciprocals of the “normal” Chebyshev,
and purely imaginary zeros distributed in some complicated fashion. The resulting polezero constellation roughly resembles the Greek letter Ω rotated counter-clockwise by 90°.
FIGURE 16. Third order Butterworth and Chebyshev low pass filter pole constellations
Mathematically, the Chebyshev response is of the general form
H ( jω )
2
1
=
1+ε
2
2
Cn
ω
ω
,
(30)
p
where ωp once again is the frequency at which the response magnitude squared has
dropped to a value
1
1+ε
2
,
(31)
as in the Butterworth case. For self-evident reasons ε is known as the ripple parameter, and
is specified by the designer. The function Cn(x) is known as a Chebyshev polynomial of
21. There are many references that provide excellent derivations of the Butterworth and Chebyshev conditions. A particularly enlightening derivation may be found in chapters 12 and 13 of R. W. Hamming’s volume, Digital Filters, Prentice-Hall, 2nd ed., 1983.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 26 of 38
EE414 Handout #14: Spring 2001
order n. The most relevant property of such polynomials is that they oscillate between –1
and +1 as the argument x varies over the same interval. Outside of this interval the magnitude grows rapidly (as xn in fact). The filter’s (power) response thus varies between 1 and
1/(1+ ε2) as the frequency increases from DC to ωp. That entire frequency interval is often
called the ripple passband, and the parameter ωp the ripple bandwidth (or ripple cutoff frequency). In general the ripple passband differs from the more conventional –3dB bandwidth.
There are a couple of ways of generating Chebyshev polynomials algorithmically. One is
through a recursion formula,
C n ( x ) = 2xC n − 1 ( x ) − C n − 2 ( x ) ,
(32)
where C0 = 1 and C1 = x (just to get you started). As can be seen from the recursion formula, the leading coefficient of Chebyshev polynomials is 2n-1, a fact we shall use later in
comparing Chebyshev and Butterworth polynomials.
Another method for generating the Chebyshev polynomials is in terms of some trigonometric functions, from which the oscillation between –1 and +1 (for |x| < 1) is directly
deduced:
−1
C n ( x ) = cos ( ncos x ) for |x| < 1 .
(33)
For arguments larger than unity, the formula changes a little bit:
−1
C n ( x ) = cosh ( ncosh x ) for |x| > 1 .
(34)
Although it is probably far from obvious at this point, these functions are likely familiar to
you as Lissajous figures, formed and displayed when sinewaves drive both the vertical and
horizontal deflection plates of an oscilloscope. That is, suppose that the horizontal deflection plates are driven by a signal
x = cos t ,
(35)
so that
−1
t = cos x .
(36)
Further suppose that the vertical plates are simultaneously driven by a signal
y = cos nt .
(37)
Substituting Eqn. 36 into Eqn. 37 to remove the time parameter yields
−1
y = cos ( ncos x ) ,
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
(38)
Page 27 of 38
EE414 Handout #14: Spring 2001
which is seen to be the same as Eqn. 33. That is, what’s displayed on an oscilloscope
driven in this fashion is in fact the Chebyshev polynomial for that order n, for values of |x|
up to one. Over that interval the function displayed looks very much like a sinusoid
sketched on a piece of paper, wrapped around a cylinder, and then viewed from a distance.
A few Chebyshev polynomials are sketched crudely in the following figure, and expressions for the first ten Chebyshev polynomials are listed in Table 5:
FIGURE 17. Rough sketches of some Chebyshev polynomials
+1
+1
–1
+1
–1
+1
–1
+1
–1
–1
C1
–1
C2
+1
C3
+1
–1
+1
–1
+1
+1
+1
–1
+1
–1
–1
–1
C4
C5
C6
TABLE 5. First ten Chebyshev polynomials
Order, n
Filters
Polynomial
0
1
1
x
2
2x2 – 1
3
4x3 – 3x
4
8x4 – 8x2 + 1
5
16x5 – 20x3 + 5x
6
32x6 – 48x4 + 18x2 – 1
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 28 of 38
EE414 Handout #14: Spring 2001
TABLE 5. First ten Chebyshev polynomials
Order, n
Polynomial
7
64x7 – 112x5 + 56x3 – 7x
8
128x8 – 256x6 + 160x4 – 32x2 + 1
9
256x9 – 576x7 + 432x5 – 120x3 + 9x
From the foregoing equations, we may derive an expression for the filter order required to
satisfy the specified constraints:
2 1
A − ε
.
ω
−1 s
cosh
ω
cosh
n =
−1
(39)
p
As with the Butterworth case, the order as computed by Eqn. 39 should be rounded
upward to the next integer value. Again, the resulting “excess” performance can be used to
improve some combination of passband and stopband characteristics.
One way in which the Chebyshev is superior to a Butterworth is in the ultimate stopband
attenuation provided. At high frequencies, the Butterworth provides an attenuation that is
approximately
ω
A j
ω
c
2
ω 2n
≈ .
ω
(40)
c
Compare that asymptotic behavior with that of a Chebyshev (with ε = 1 so that the –3dB
frequency of the Butterworth corresponds to the passband edge of the Chebyshev):22
ω
A j
ω
c
2
≈2
ω 2n
.
ω
c
2n − 2
(41)
Clearly the Chebyshev filter offers higher ultimate attenuation, by an amount equal to
3dB(2n – 2), for a given order. As a specific example, a 7th-order Chebyshev ultimately
provides 36dB more stopband attenuation than does a 7th order Butterworth.
As another comparison, the relationship between the poles of a Butterworth and those of a
Chebyshev of the same order can be put on a quantitative basis by normalizing the two filters to have precisely the same –3dB bandwidth. It also may be shown (but not by us) that
the –3dB bandwidth of a Chebyshev may be reasonably well approximated by23
22. This comparison should not mislead you into thinking that such large ripple values are commonly used.
In fact, Chebyshev filters are typically designed with ripple values below 1dB.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 29 of 38
EE414 Handout #14: Spring 2001
1
−1 1
cosh sinh .
n
ε
(42)
Since the diameter of a Butterworth’s circular pole constellation is the –3dB bandwidth,
we normalize the Chebyshev’s ellipse to have a major axis defined by Eqn. 42. The imaginary parts of the poles of a Chebyshev filter are the same as for the Butterworth, while the
real parts of the Butterworth prototype are merely scaled by the factor
−1 1
1
tanh sinh
n
ε
(43)
to yield the real parts of the poles of a Chebyshev filter. Thus design of a Chebyshev filter
is quite straightforward because it requires only a prototype Butterworth, and it’s trivial to
design the latter. Clearly, the Butterworth may be considered merely a special case of a
Chebyshev, one for which the ripple parameter is zero.
There is one subtlety that requires discussion, however, and this concerns the source and
termination impedances of a passive Chebyshev filter. From both the sketches and equations, it’s clear that only odd-order Chebyshev polynomials have a zero value for zero
arguments. Hence, the DC value of the filter transfer function will be unity for such polynomials (that is, the passband hits its ripple extremum at some frequency above DC). For
even-order Chebyshev filters, however, the filter’s transfer function starts off at a ripple
extremum, with a DC power transmission value of 1/(1+ ε2), implying a termination resistance that is less than the source resistance. If, as is usually the case, such an impedance
transformation is undesired, one must either use only odd-order Chebyshev filters, or add
an impedance transformer to an even-order Chebyshev filter. As the former is less complex, it is the near universal choice to use only odd-order Chebyshev realizations in practice.
Finally, recognize that the elliptical pole distribution implies that the ratio of the imaginary
to real parts of the poles, and hence the Qs of the poles, are higher for Chebyshevs than for
Butterworths of the same order. As a result, Chebyshev filters are more strongly affected
by the finite Q of practical components. The problem increases rapidly in severity as the
order of the filter increases. This important practical issue must be kept in mind when
choosing a filter type.
Element values for the Chebyshev filter are given by the following sequence of equations.
First, compute four auxiliary quantities, whose significance may initially seem mysterious:24
L Ar
β = ln coth
,
17.372
(44)
23. See, for example, M. E. Van Valkenburg, Introduction to Modern Network Synthesis, Wiley, 1960, pp.
380-381.
24. After close examination, these remain mysterious. Sorry. At least I can tell you that the 17.372 factor is
40log10e, as if that helps.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 30 of 38
EE414 Handout #14: Spring 2001
where LAr is in dB;
β
γ = sinh
2
n
;
(45)
k
( 2 − 1) π
a k = sin
2
n
;
(46)
and
2
2 kπ
b k = γ + sin .
n
(47)
Once the values of the auxiliary parameters are known, the following equations yield the
normalized element values:
g0 = 1 ;
g1 =
gk =
2a 1
γ
(48)
;
4a k − 1 a k
bk − 1 gk − 1
(49)
;
(50)
g n + 1 = 1 for n odd ;
(51)
2 β
g n + 1 = coth for n even .
4
(52)
and
5.3 Type II (Inverse) Chebyshev filters
We have alluded several times to the possibility of realizing a flatter-than-maximally flat
transfer characteristic. The Type II (also known as an inverse or reciprocal) Chebyshev filter achieves such flatness by permitting ripple in the stopband, while continuing to insist
on passband monotonicity.
The Type II filter derives from the Type I (ordinary) Chebyshev through a pair of simple
transformations. In the first step, the Type I Chebyshev response is simply subtracted from
unity, leading to the conversion of a low-pass filter into a high-pass one. Note that the
resulting response is monotonic in the new passband. The second step replaces ω by 1/ω.
Since high frequencies are thus mapped into low ones, and vice-versa, this second transformation converts the filter shape back into a low-pass response, but in a way that
exchanges the ripple at low frequencies with ripple at high frequencies. This transformaFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 31 of 38
EE414 Handout #14: Spring 2001
tion thus restores a monotonic passband, and also happens to map the Type I passband
edge into the stopband edge. Furthermore the larger the permissible stopband ripple, the
flatter the passband response.
Mathematically, these transformations ultimately result in the following power response
for a Type II filter:
H ( jω )
2
ω
2 2 p
ε Cn
ω
1
= 1−
=
.
ω
ω
2 2 p
2 2 p
1 + ε Cn
1 + ε Cn
ω
ω
(53)
Although the Type II filter is not encountered as often as the Butterworth, its relative rarity
should not be taken to imply a corresponding lack of utility. Despite the superior flatness
provided by the inverse Chebyshev, it appears that, for purely cultural reasons, the Butterworth filter continues to dominate in those applications where passband uniformity is
allegedly prized.
5.4 Elliptic (Cauer) filters
Having seen that allowing ripple in the passband or stopband confers desirable attributes,
perhaps it should not be surprising that the elliptic or Cauer filter further improves transition steepness by allowing ripple in both the passband and stopband simultaneously.25 Just
as a complex pole pair provides peaking, a complex zero pair provides notching. We’ve
seen this behavior already, where the purely imaginary zeros of an m-derived filter provide
notches of infinite depth. Cauer filters exploit this notching to create a dramatic transition
from passband to stopband, at the expense of a stopband response that bounces back up
beyond the notch frequency (again, just as in an m-derived filter, and for the same reasons). The name elliptic comes from the appearance of elliptic functions in the mathematics, and should not be confused with the elliptic pole distribution of a Chebyshev filter.
Elliptic filters have the following power transmission behavior:
H ( jω )
2
1
=
1+ε
2
2
Un
ω
ω
,
(54)
c
where Un (x) is a Jacobian elliptic function of order n:26
25. These are also sometimes known as Darlington or Zolotarev filters. Sidney Darlington (of “Darlington
pair” fame in bipolar circuits) made major contributions in the field of network synthesis. Igor Ivanovich
Zolotarev independently studied Chebyshev functions a decade or so before Chebyshev did.
26. These are named for the mathematician Karl Gustav Jacob Jacobi (1804-1851), who began studying
these functions in the 1820s, at the start of his career.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 32 of 38
EE414 Handout #14: Spring 2001
x
Un ( x) =
∫
0
dy
2
.
(55)
2 2
(1 − y ) (1 − n y )
These functions are messy enough that quantitative information about them is generally
presented in tabular form (but not here, though; see, e.g., the oft-cited work by G.W. Spencely and R. M. Spencely, “Smithsonian Elliptic Function Tables,” Publication 3863,
Smithsonian Institution, Washington, D. C., 1947). Suffice it to say that, just as Chebyshev
polynomials do, these elliptic functions oscillate within narrow limits for arguments |x|
smaller than unity, and rapidly grow in magnitude for arguments outside of that range.
However, unlike Chebyshev polynomials, whose magnitudes grow monotonically outside
of that range, these elliptic functions oscillate in some fashion between infinity and a specified finite value. Hence the filter response exhibits stopband ripples, with a finite number
of frequencies at which the filter transmission is (ideally) zero. The following figure shows
crude sketches of the first several Jacobian elliptic functions, from which this behavior
may be discerned:
FIGURE 18. Rough sketches of some Jacobian elliptic functions
+1
+1
–1
+1
+1
–1
–1
+1
+1
–1
+1
–1
Filters
–1
–1
+1
–1
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 33 of 38
EE414 Handout #14: Spring 2001
The attenuation poles correspond to transmission zeros (notches), in the proximity of
which the filter response changes rapidly. Thus, perhaps you can see how permitting such
ripples in the stopband allows for a much more dramatic transition from passband to stopband, and thus allows one to combine the attributes of ordinary and inverse Chebyshev filters.
Wilhelm Cauer is the inventor whose deep physical insights (and intimate familiarity both
with the notches of Zobel’s m-derived filters, and with elliptic functions in general)
allowed him first to recognize that this additional degree of freedom existed, and then to
exploit it, even though he did not offer a formal mathematical proof of the correctness of
his ideas.27 At a time when minimizing component count was an obsession, Cauer was
able to use fewer inductors than the best filters that were then in use. According to lore,
publication of his patent reportedly sent Bell Labs engineers and mathematicians scurrying off to the New York City Public Library to bone up on the then-obscure (okay, stillobscure) literature on elliptic functions.28
6.0 Coupled Resonator Bandpass Filters
Up to now we’ve focused mainly on low pass filters, having derived other filter shapes
from low pass prototypes. It is worthwhile to develop additional insights, however, so that
we don’t always have to return to the low pass case whenever we wish to design, say, a
bandpass filter. This freedom, in turn, allows us to analyze and synthesize filter types that
are not readily related to lumped networks at all.
We’ve seen that the poles, say, of a “good” filter aren’t all coincident; they’re distributed
in some manner. Viewed from a broad perspective, then, the goal of filter design is to distribute the transfer function poles and zeros in some manner to achieve a desired response
shape. This important idea is the basis for essentially all lumped filters, bandpass or otherwise. A particularly simple way to synthesize bandpass filters with a variety of response
shapes is to exploit the mode splitting that occurs when two or more resonant systems
interact. That is, when two identical resonators are connected together in some fashion, the
poles of the resulting coupled system generally differ from those of the resonators in isolation. By controlling the degree of interaction (coupling) the pole locations can be adjusted
to produce a desired response shape.
27. Cauer (1900-1945) became familiar with elliptic functions while studying at the University of Göttingen
with the mathematician David Hilbert. Hilbert was as well known for his absentmindedness as for his mathematics. Once he suddenly asked a close friend, physicist James Franck, “Is your wife as mean as mine?”
Though taken aback, Franck managed to respond, “Why, what has she done?” Hilbert answered, “I discovered today that my wife does not give me an egg for breakfast. Heaven only knows how long this has been
going on.”
It is unfortunate that stories about Cauer are not as lighthearted. Tragically, he was shot to death during the
Soviet occupation of Berlin in the closing days of WWII, in a manner sadly reminiscent of the death of
Archimedes (see http://www-ft.ee.tu-berlin.de/geschichte/th_nachr.htm).
28. M. E. Van Valkenburg, Analog Filter Design, Harcourt Brace Jovanovich, 1982, p. 379.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 34 of 38
EE414 Handout #14: Spring 2001
To illustrate this important idea, consider two simple LC resonators whose inductors are
magnetically coupled to each other. Such a system may be modeled by representing the
coupled inductors with a transformer. The transformer in turn is modeled as a T-connection of three inductors:
FIGURE 19. Coupled LC resonators
L–M
L–M
C
M
C
k =
M
L
The inductance L is that which is present in each resonator in isolation. The mutual inductance M is a fraction of L, and depends on the magnitude of the coupling. The latter is captured in the coupling coefficient k, which ranges from zero to unity as the flux linkage of
the magnetic fields of the two inductors increases from zero to 100%.
To find the resonant frequencies of the resulting 4th-order system29 one can always
employ a brute-force approach: Find the transfer function (first, one needs to define the
input and output terminals), then solve for the roots of the denominator polynomial. This
method is quite general, but also quite cumbersome, particularly for networks of order
higher than two or three. Here, the network happens to be symmetrical, a situation that
almost demands exploitation to simplify analysis by bypassing uninspired routes to the
answer.
First recall what poles are. Yes, they are the roots of the denominator of the transfer function, but a deeper significance is that they are the natural frequencies of a network. That is,
if the system is given some initial energy, the evolution of the system state in the absence
of any further input takes place with characteristic frequencies whose values are those of
the poles. Cleverly chosen initial conditions may excite only a subset of all possible
modes at a time, thus converting a difficult high-order problem into a collection of more
simply solved low-order ones. Very clever (or lucky) choices can even result in the excitation of a single mode at a time.
We may use this understanding to devise a simple method for finding the poles of our coupled resonator system. First, provide a common-mode excitation by depositing, say, an
equal amount of initial charge on the two capacitors. Regardless of what the network does
subsequently, we know by symmetry that the capacitor voltages must evolve the same
way. Because the two capacitor voltages are thus always equal, we may short the capacitors together with impunity, resulting in the following network:
29. Despite there being five energy storage elements in the network, the system is nonetheless of the fourth
order, because not all of the elements are independent. Note, for example, that specifying the currents in two
of the inductors automatically determines that flowing in the third, by Kirchhoff’s current law. Thus, the
three inductors actually contribute only two degrees of freedom, diminishing by one the order of the overall
network.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 35 of 38
EE414 Handout #14: Spring 2001
FIGURE 20. Equivalent network of coupled LC resonators for common-mode initial conditions
(L – M)/2
M
2C
The common-mode resonant frequency is thus that of a simple parallel LC network:
ω cm =
1
( 1 − k) L
+ kL 2C
2
=
1
( 1 + k ) LC
.
(56)
There are two conjugate imaginary poles of this frequency, so we only need to find the
other two poles of this fourth-order network.
Since a common-mode initial condition is so fruitful in discovering two of the poles, it
seems reasonable to try a differential initial condition next. Specifically, if one capacitor
voltage is initially made equal to some voltage V, and the other to –V, (anti)symmetry
allows to assert that, however the system state evolves from this initial condition, it must
do so in a manner that guarantees zero voltage across the common shunt inductance of
value M. Consequently, no current flows through it, and the shunt inductance may be
removed (either by open- or short-circuiting it; both actions will lead to the same answer).
Removing that inductance yields the following differential-mode resonant frequency:
ω dm =
1
1
=
( 1 − k ) LC
C
[ 2 ( 1 − k) L]
2
.
(57)
Now that we’ve found the pole frequencies, let’s see what intuition may be extracted from
the exercise. First consider extremely loose coupling, i.e., values of k very near zero. In
that situation the two mode frequencies are nearly the same, because we have two nearly
independent and identical tanks. As k increases, however, one resonant frequency
decreases, while the other increases; mode splitting occurs. The stronger the coupling, the
wider the separation in resonant frequencies.
As an illustration that mode-splitting is an extremely general consequence of coupling resonators together, consider the use of capacitive coupling:
FIGURE 21. Capacitively coupled resonators
Cc
C–Cc
Filters
L
L
C–Cc
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 36 of 38
EE414 Handout #14: Spring 2001
Here, the individual resonator capacitances are arbitrarily expressed as a function of the
coupling capacitance. One could just as well have labeled the resonator capacitances simply as C, but the choice shown simplifies the analytical expressions somewhat, as will be
seen.
Following an approach analogous to that used to analyze the magnetically coupled case,
we find that the two mode frequencies are given by:
ω cm =
1
( C − Cc ) L
=
1
(58)
( 1 − k ) LC
and
ω dm =
1
( C + Cc ) L
=
1
( 1 + k ) LC
.
(59)
For these equations, an explicit expression for the coupling coefficient, k, is found to be
Cc
k =
C
.
(60)
As with the magnetic case, the coupling coefficient cannot exceed unity (if negative element values are disallowed) when expressed in this manner. As we can see, both magnetic
and capacitive coupling give rise to the same splitting of modes. This mechanism is so
general that it explains a host of phenomena, such as the formation of energy bands in solids (here, the initially identical mode frequencies – energy levels – of free atoms split as
the atoms are brought closer together to form a solid).
From Eqn. 58 and Eqn. 59, it should be clear that one may use a measurement of the two
mode frequencies to determine k experimentally. Indeed, for small values of coupling, the
difference in mode frequencies (normalized to their geometric mean) is approximately
equal to k.
The mode splitting that accompanies coupling permits placement of poles to produce
response shapes such as Butterworth and Chebyshev. Simply adding an input and output
port to the basic structure of Figure 21, for example, readily produces a bandpass filter
which may be extended to any number of stages:
FIGURE 22. Coupled resonator filter
R
C1
Filters
Cc1
L1
R
L2
C2
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 37 of 38
EE414 Handout #14: Spring 2001
From the analysis of mode splitting, it should seem reasonable that small amounts of coupling (small values of coupling capacitance in this particular example) produce narrowband filters, and that relatively large amounts of coupling produce broadband filters. When
this idea is placed on a quantitative basis, it is possible to express the bandpass filter
design problem entirely in terms of coupling coefficients, uncoupled resonant frequencies,
and tank loading. This reformulation in terms of invariant parameters (e.g., resonant frequency, impedance levels, bandwidth) facilitates the design of microstrip filters where,
owing to their distributed nature, it is not always possible to identify individual lumped
elements.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 38 of 38
Filters
1.0 Introduction
The subject of filter design is so vast that we have to abandon any hope of doing justice to
it as a subset of a textbook. Indeed, even though we have chosen to present this material
over two chapters, the limited aim here is to focus on important qualitative ideas and practical information about filters, instead of attempt a comprehensive review of all possible
filter types and supply complete mathematical details of their underlying theory. For those
interested in the rigor that we will tragically neglect, we will be sure to provide pointers to
the relevant literature. And for those who would rather ignore the small amount of rigor
that we do provide, the reader is invited to skip directly to the appendices, which summarize filter design information in “cookbook” form.
Although our planar focus would normally imply a discussion limited to microstrip implementations, many such filters derive directly from lower frequency lumped prototypes.
Because so many key concepts may be understood by studying those prototypes, we will
follow a roughly historical path and begin with a discussion of lumped filter design. It is
definitely the case that certain fundamental insights are universal, and it is these that will
be emphasized in this chapter, despite differences in implementation details between
lumped and distributed realizations.
Only passive filters will be considered here, partly to limit the length of the chapter to
something manageable. Another reason is that, compared to passive filters, active filters
generally suffer from higher noise and nonlinearity, limited operational frequency range,
higher power consumption, and relatively high sensitivity to parameter variations, particularly at the GHz frequencies with which we are concerned in this textbook.
2.0 Background
2.1 A quick history
The use of frequency selective circuits certainly dates back at least to the earliest research
on electromagnetic waves. In his classic experiments of 1887-1888 Hertz himself used
dipole and loop antennas (ring resonators) to clean up the spectrum generated by his spark
gap apparatus and thereby impart a small measure of selectivity to his primitive receivers.
Wireless pioneer Sir Oliver Lodge of the U.K. coined the term “syntony” to describe the
action of tuned circuits, showing a conscious appreciation of the value of such tuning,
despite the hopelessly broadband nature of spark signals.1 At nearly the same time, Nikola
Tesla and Guglielmo Marconi developed tuned circuits (Marconi’s patent #7777 was so
1. See H. Aitken’s excellent book, Syntony and Spark, Princeton, 1987, for a technically detailed and fascinating account of early work in wireless.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 1 of 38
EE414 Handout #14: Spring 2001
valuable that it became the subject of bitter and protracted litigation)2 for the specific purpose of rejecting unwanted signals, anticipating the advent of sinusoidal carrier based
communications.
Despite that foundation, however, modern filter theory does not trace directly back to
those early efforts in wireless. Rather the roots go back even further in time: it is research
into the properties of transmission lines for telegraphy and telephony that primarily inform
early filter theory. In 1854 William Thomson (who would later become Lord Kelvin), carried out the first analysis of a transmission line, considering only the line’s distributed
resistance and capacitance. His work, inspired by what was to be the 3000-kilometer
Atlantic Cable Project, established a relationship between practical transmission rates and
line parameters. A bit over 20 years later, Oliver Heaviside and others augmented Kelvin’s
analysis by including distributed inductance, thereby extending greatly the frequency
range over which transmission line behavior could be described accurately.3 Following up
on one particular implication of Heaviside’s work, both George Ashley Campbell of the
American Bell Company and Michael Idvorsky Pupin of Columbia University suggested
around 1900 the insertion of lumped inductances at regularly spaced intervals along telephone transmission lines to reduce dispersion (the smearing out of pulses).4 This suggestion is relevant to the filter story because Heaviside recognized that a lumped line differs
from a continuous one in possessing a definite cutoff frequency. Campbell and Pupin provided design guidelines for guaranteeing a certain minimum bandwidth.5
In true engineering fashion, the apparent liability of a lumped line’s limited bandwidth
was quickly turned into an asset, and thus was established the main evolutionary branch of
filter design. The first published formalism is Campbell’s, whose classic 1922 paper
describes in fuller detail ideas he had developed and patented during WWI.6 Karl Willy
Wagner also developed these ideas at about the same time, but German military authorities
delayed publication, giving Campbell priority.7 It is now acknowledged that credit should
be shared by these two pioneers, who independently and nearly simultaneously hit upon
the same great idea.
2. The U.S. Supreme Court eventually ruled it invalid (in 1943) because of prior work by Lodge, Tesla and
others.
3. For additional background on this story, see Paul J. Nahin’s excellent book, Oliver Heaviside: Sage in
Solitude, IEEE Press, 1987.
4. As with many key ideas of great commercial import, a legal battle erupted over this one. It is a matter of
record that the Bell System was already experimenting with loading coils developed by Campbell well
before publication of Pupin’s 1900 paper. Pupin’s self-promotional abilities were superior, though, and he
was able to obtain a patent nonetheless. He eventually earned royalties of over $400,000 from Campbell’s
employer (at a time when there was no U.S. income tax) for his “invention.” To add to the insult, Pupin’s
Pulitzer-prize winning autobiography of 1924 shamefully fails to acknowledge Campbell and Heaviside.
5. A. T. Starr, Electric Circuits and Wave Filters, 2nd ed., Pitman and Sons, 1948.
6. G. A. Campbell, “Physical Theory of the Electric Wave-Filter,” Bell System Technical Journal, vol. 1, no.
2, pp. 1-32, Nov. 1922. See also his U. S. Patent #1,227,113, May 22, 1917.
7. “Spulen- und Kondensatorleitungen” (Inductor and Capacitor Lines), Archiv für Electrotechnik, vol. 8,
July 1919.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 2 of 38
EE414 Handout #14: Spring 2001
Campbell’s colleague, Otto J. Zobel, published a much-referenced extension of Campbell’s work, but which was still limited to filters derived from transmission line ideas.8 In
the developments of subsequent decades one sees an evolving understanding of how
closely one may approach in practice the theoretical ideal of a perfectly flat passband, constant group delay, and an infinitely steep transition to an infinitely attenuating stopband.
Conscious acknowledgment that this theoretical ideal is unattainable leads to the important idea that one must settle for approximations. Some of the more important, practical
and well-defined of these approximations are the Butterworth, Chebyshev and Cauer
(elliptical) filter types we’ll study in this chapter.
Shortly after WWII, the subject of filter design advanced at an accelerated pace. Investigation into methods for accommodating finite-Q elements in lumped filters offered hope for
improved predictability and accuracy. In the microwave domain, filter topologies based
directly on lumped prototypes came to be supplemented by ones that exploit, rather than
ignore, distributed effects. Many of these are readily implemented in microstrip form, and
are the ultimate focus of this chapter.
The advent of transistors assured that the size of active devices no longer dominated that
of a circuit. Numerous active filter topologies evolved to respond to a growing demand for
miniaturization, replacing bulky passive inductor-capacitor circuits in many instances.
Aside from enabling dramatic size reductions, some active filters are also electronically
tunable. However, these attributes do come at a price: active filters consume power, suffer
from nonlinearity and noise, and possess diminished upper operational frequencies
because of the need to realize gain elements with well-controlled characteristics at high
frequencies. These tradeoffs become increasingly serious as microwave frequencies are
approached. This statement should not be taken to mean that microwave active filters can
never be made to work well enough for some applications (because successful examples
certainly abound), but it remains true that the best filters at such frequencies continue to be
passive implementations. It is for this reason that this chapter considers passive filters
exclusively.
The arrival of transistors also coincided with (and helped drive) a rapidly decreasing cost
of computation.9 No longer limited to considering only straightforward analytical solutions, theorists were free to pose the filter approximation problem much more generally,
e.g., “Place the poles and zeros of a network to minimize the mean-square error (or maximum error, or some other performance metric) in a particular frequency interval, relative
to an ideal response template.” Of great practical importance is that a numerical approach
readily accommodates lossy inductors and capacitors, something that is difficult with earlier analytical approaches. The resulting filters are optimum in the sense that one cannot
do better (as evaluated by whatever design criteria were imposed in the first place) for a
given filter order. The tradeoff is that the resulting design may not be as easily understood
8. O. J. Zobel, “Theory and Design of Uniform and Composite Electric Wave-filters,” Bell System Technical Journal, vol. 2, no. 1, pp. 1-46, Jan. 1923.
9. Regrettably, space limitations force us to neglect here the fascinating story of electrolytic tanks and other
analog computers used to design filters based on potential theory.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 3 of 38
EE414 Handout #14: Spring 2001
as those based on analytical approaches. In many cases, element values are best obtained
from tables that summarize the results of extensive computations.
The same philosophical approach of numerical optimization is also how most modern
microwave filters are designed. And again, solutions for the more complex types are best
extracted from tables. The main purpose here is to provide an intuitive explanation for
how these filters work, leaving many of the mathematical details to published theoretical
treatments.
3.0 Filters from Transmission Lines
We start with the “electric wave filters” of Campbell, Zobel and Wagner. As mentioned,
these derive from lumped approximations to transmission lines, so we begin by examining
such “artificial” lines to see how a limited bandwidth arises.
3.1 Constant-k filters
For convenience, we repeat here some of the calculations from the chapter on distributed
systems. Recall that we first consider the driving point impedance, Zin, of the following
infinite ladder network:
FIGURE 1. Infinite ladder network as artificial line
Z
A
Z
Y
Zin
Z
B
C
Y
Z
Y
In this drawing, the resistor symbols represent generalized impedances (Z) and admittances (Y).
To simplify the derivation, it is helpful to note that the impedance looking to the right of
point A also equals Zin, this being an infinite network. We can then collapse the infinite
network into a much simpler finite one:
FIGURE 2. Conversion of infinite line into finite network
Z
Zin
Y
Zin
Solving for the Zin of this simple network yields:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 4 of 38
EE414 Handout #14: Spring 2001
2
Z in
Z ± Z + 4 ( Z ⁄ Y)
Z
4
=
=
1± 1+
,
2
2
ZY
(1)
where one would generally disallow negative values and thus choose only the sum solution.
As a specific (but typical) case, consider a low-pass filter in which Y = jωC and Z = jωL.
Then, the input impedance of the infinite artificial line is:
Z in =
jωL
4
.
1± 1−
2
2
ω LC
(2)
At very low frequencies, the factor under the radical is negative and large in magnitude,
making the term within the brackets almost purely imaginary. The overall Zin in that frequency range is therefore largely real, with
Z in ≈
Z
=
Y
L
= k.
C
(3)
Because the ratio Z/Y is a constant here, such filters are often known as constant-k filters.10
As long as the input impedance has a real component, nonzero average power will couple
into the line from the source. Above some particular frequency, however, the input impedance becomes purely imaginary, as can be seen from inspection of Eqn. 2. Under this condition, no real power can be delivered to the network, and the filter thus attenuates
heavily.11 For self-evident reasons the frequency at which the input impedance becomes
purely imaginary is called the cutoff frequency which, for this low-pass filter example, is
given by:
ωh =
2
.
(4)
LC
Any practical filter must employ a finite number of sections, of course, leading to the
question of the relevancy of any analysis that assumes an infinite number of sections. Intuitively, it seems reasonable that a “sufficiently large” number of sections would lead to
acceptable agreement. Based on lumped network theory, we also expect the ultimate rate
of rolloff to be determined by the filter order, and hence by the number of sections to
which the network is truncated (we’ll have more to say on this subject later). The greater
10. Campbell used the symbol k in precisely this context, but it was Zobel (op. cit.) who apparently first
used the actual term “constant-k.”
11. Attenuation without dissipative elements might initially seem intuitively unpalatable. However, consider
that a filter might also operate by reflecting energy, rather than by dissipating it. That is, a filter can function
by producing a purposeful impedance mismatch over some band of frequencies. In fact, many filter design
approaches are based directly on manipulation of the reflection coefficient as a function of frequency.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 5 of 38
EE414 Handout #14: Spring 2001
the number of sections, the greater the rate of rolloff. As we’ll see, there is also some (but
practically limited) flexibility in the choice of Z and Y, permitting a certain level of tradeoff among passband, transition band, and stopband characteristics. However, it remains
true that one limitation of filters based on artificial-line concepts is the inability to specify
these characteristics in detail, if at all. Note, for example, the conspicuous absence of any
discussion about how the filter behaves near cutoff. We don’t know if the transition from
passband to stopband is gradual or abrupt, monotonic or oscillatory. We also don’t know
the precise shape of the passband. Finally, we don’t have any guide how to modify the
transition shape should we find it unsatisfactory. As we’ll see, these shortcomings lead us
to consider other filter design approaches.
Once the filter order is chosen (by whatever means), the next problem is one of termination. Note that the foregoing analysis assumes that the filter is terminated in an impedance
that behaves as described by Eqn. 2. That is, our putative finite filter must be terminated in
the impedance produced by the prototype infinite ladder network: It must have a real
impedance at low frequencies, then become purely imaginary above the cutoff frequency.
Stated another way, rigorous satisfaction of the criteria implied by Eqn. 2 absurdly
requires that we supply a load element which itself is the filter we desire! We should therefore not be too surprised to discover that a practical realization involves compromises, all
intimately related to the hopeless task of mimicking the impedance behavior of an infinite
structure with a finite one. For example the near-universal choice is to terminate the following with a simple resistance R equal to k:
FIGURE 3. Low-pass constant-k filter example using two cascaded T-sections
L/2
C/2
L/2
C
L/2
L/2
L/2
C
L/2
C/2
A source with a Thévenin resistance also of value k is assumed to drive this filter. Note
that this example uses two complete T-sections (shown in the boundaries), with a half-section placed on each end. Termination in half sections is the traditional way to construct
such filters. The series-connected inductors, shown individually to identify clearly the separate contributions of the unit T-sections, are combined into a single inductance in practice. Alternatively one may implement the filter with π-sections. With those, one uses
terminating half-sections that are mirror images of the ones shown. The choice of which
implementation to employ is often determined in practice by the nature of the parasitics
that dominate the input and output interfaces. If these parasitics are primarily capacitive in
nature, the T-section implementation shown is favored, since the parasitics may be
absorbed into the capacitances at the ends of the filter. Similarly, inductive parasitics are
most readily accommodated by a filter using internal π-sections.
The design equations for this filter are readily derived from combining Eqn. 3 and Eqn. 4:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 6 of 38
EE414 Handout #14: Spring 2001
2 1
C = ,
ω R
h
(5)
2
L = R.
ω
(6)
and
h
Thus, once one specifies the characteristic impedance, R, the desired cutoff frequency, and
the total number of sections, the filter design is complete.
Regrettably, deducing the number of sections required is a bit of a cut and try affair in
practice. There are equations that can provide guidance, but they are either cumbersome or
inaccurate enough that one typically increases the number of sections until simulations
reveal that the filter behaves as desired. Furthermore the unsophisticated termination of a
simple resistance leads to degradation of important filter characteristics, often resulting in
a hard-to-predict insertion loss and passband flatness, as well as reduced stopband attenuation (relative to predictions based on true, infinite-length lines). These difficulties are
apparent from an inspection of the following table, which shows the attenuation at the cutoff frequency, as well as the –3dB and –6dB bandwidths (expressed as a fraction of the
cutoff frequency), of constant-k filters (both T- and π-implementations) as a function of
order. In the table, n is the number of complete T- (or π-) sections in the central core of the
filter. The filter order is therefore 2n + 3.
TABLE 1. Characteristics of ideal constant-k filters
n
Attenuation
at cutoff
frequency
(dB)
Normalized
–3dB
Bandwidth
Normalized
–6dB
Bandwidth
Normalized
–60dB
Bandwidth
Normalized
–10dB S11
Bandwidth
0
3.0
1.000
1.201
10.000
0.693
1
7.0
0.911
0.980
3.050
0.810
2
10.0
0.934
0.963
1.887
0.695
3
12.3
0.954
0.969
1.486
0.773
4
14.2
0.967
0.976
1.302
0.696
5
15.7
0.976
0.981
1.203
0.756
Note that the attenuation at the nominal cutoff frequency, as well as the bandwidth, are
both dependent on the number of filter sections. Further note that the cutoff frequency (as
computed by Eqn. 4) equals the –3dB bandwidth only for n = 0, and is as much as 10%
beyond the –3dB bandwidth in the worst case. In critical applications, the cutoff frequency
target may have to be altered accordingly to achieve a specified bandwidth.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 7 of 38
EE414 Handout #14: Spring 2001
The following figure is a frequency response plot for a constant-k filter which consists of
five full sections, and a terminating half-section on each end. Note that the frequency axis
is linear, not logarithmic:
FIGURE 4. Response of six-section low pass constant-k filter (n = 5)
Aside from the ripple evident in the figure, it is also unfortunate that the bandwidth over
which the return loss exceeds 10dB is typically only ~70-80% of the cutoff frequency. A
considerable improvement in performance is possible by using filter sections whose
impedance behavior better approximates a constant resistance over a broader frequency
range. One example, developed by Zobel, uses “m-derived” networks, either as terminating structures, or as filter sections:
FIGURE 5. Low-pass m-derived filter using cascaded T-sections
L1/2
L1/2
2L2
C/2
L1/2
L1/2
L2
C
L1/2
L1/2
L2
C
2L2
C/2
As with the prototype constant-k filter of Figure 5, this structure is both driven and terminated with a resistance of value k ohms. The m-derived filter, which itself is a constant-k
structure, is best understood by noting that the prototype constant-k filter previously analyzed has a response that generally attenuates more strongly as the cutoff frequency ω1 is
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 8 of 38
EE414 Handout #14: Spring 2001
approached. At small fractions of the cutoff frequency, the response is fairly flat, so it
should seem reasonable that increasing the cutoff frequency to some value ω2 should produce a more constant response within the original bandwidth ω1. The first step in designing an m-derived filter, then, consists simply of increasing the cutoff frequency of a
prototype constant-k filter. In the absence of inductor L2, we see that scaling the values of
L1 and C each, say, by a factor m (with m ranging from 0 to 1) increases the cutoff frequency by a factor of 1/m, from a value ω1, to ω2 = ω1/m. The characteristic impedance
remains unchanged at k because the ratio of L1 to C is unaffected by this scaling.
Now to restore the original cutoff frequency, add an inductance L2 to produce a series resonance with C. At the resonant frequency, this series arm presents a short circuit, creating
a notch in the filter’s transmission. If this notch is placed at the right frequency (just a bit
above the desired cutoff frequency), the filter’s cutoff frequency can be brought back
down to ω1. However, be aware that the filter response does pop back up above the notch
frequency (where the resonant branch then looks like a simple inductance). This characteristic needs to be taken into account when using the m-derived filter.
An alternative to a series resonance in the shunt arm of each filter section is a parallel resonance in the series arm(s) of a filter section. Both types of m-derived filters will provide
the same behavior. The choice of topology in practice is often determined by which implementation uses more easily realized components, or which more gracefully accommodates
parasitic elements.
Following a procedure exactly analogous to that used in determining the cutoff frequency
of ordinary constant-k filters, we find that the cutoff frequency of an m-derived filter may
be expressed as
ω1 =
2 ( R ⁄ L1 )
4
L2
L1
.
(7)
+1
To remove L1 from the equation, note that the cutoff frequency may also be expressed as
ω1 =
2m
,
(8)
L1 C
while the characteristic impedance is given by
R =
L1
C
.
(9)
Combining these last three equations allows us to solve for L2:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 9 of 38
EE414 Handout #14: Spring 2001
2
L2 =
(1 − m ) R
.
2mω 1
(10)
Solving Eqn. 8 and Eqn. 9 for L1 and C yields
2m 1
C =
ω R
(11)
2m
L1 =
R.
ω
(12)
1
and
1
Use of the foregoing equations requires that the designer have an idea of what value of m
is desirable. As m approaches unity the response exhibits a monotonic rolloff (and therefore an increasing passband error), while passband peaking increases as m approaches
zero. In practice a rather narrow range is encountered (say, within 25-30% of 0.5) as a
compromise between these two behaviors, and the parameter m is commonly chosen equal
to 0.6. This value yields a reasonably broad frequency range over which the transmission
magnitude remains roughly constant. The following table enumerates (to more significant
digits than are practically significant) some of the more relevant characteristics of mderived filters, for the specific value of m = 0.6. As with Table 1, the parameter n is the
number of complete T- (or π-) sections used in the filter. The column labeled “minimum
stopband attenuation” gives the worst-case value of attenuation above the transmission
notch frequency, where the filter response pops back up.
TABLE 2. Characteristics of ideal m-derived filters (m = 0.6)
n
Attenuation
at cutoff
frequency
(dB)
Normalized
–3dB
Bandwidth
Normalized
–6dB
Bandwidth
Normalized
–10dB S11
Bandwidth
Minimum
stopband
attenuation
(dB)
0
1.34
1.031
1.063
0.965
8.21
1
3.87
0.993
1.013
0.956
21.24
2
6.27
0.988
0.999
0.969
34.25
3
8.30
0.989
0.996
0.979
47.09
4
10.00
0.991
0.995
0.954
59.81
5
11.44
0.993
0.996
0.961
72.43
Note that the cutoff frequency and –3dB bandwidth are much more nearly equal than for
the prototype constant-k case (the worst-case difference here is about 3%). The bandwidth
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 10 of 38
EE414 Handout #14: Spring 2001
over which the return loss exceeds 10dB is also a much greater fraction of the cutoff frequency (above 95%, in fact). Note also that the minimum stopband attenuation increases
by about 12-13dB per increment in n for this range of values.
The following figure illustrates how the use of m-derived sections can improve the magnitude response (note that the vertical axis now spans 80dB, rather than 50dB):
FIGURE 6. Frequency response of six-section m-derived low pass filter (m = 0.6, n = 5)
Compared with Figure 4, this response shows significantly less passband ripple, as well as
a much faster transition to stopband, owing to the stopband notch.
On the frequency scale of Figure 6, the characteristic notch is invisible, as is the poppingup of the response at higher frequencies. The following plot shows these features more
clearly:
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 11 of 38
EE414 Handout #14: Spring 2001
FIGURE 7. Frequency response of m-derived low pass filter, plotted over wider range
Aside from the potential for improved flatness over the passband, the notches that are
inherent in m-derived filters can be used to null out interfering signals at a specific frequency (or frequencies, if sections with differing values of m are used). Later, we will see
that judiciously distributed notches can be combined with passband ripple to produce what
are known as elliptic or Cauer filters whose responses exhibit even more dramatic transitions from passband to stopband.
If the precise location of a notch is of importance, it is helpful to know that the frequency
of the null ω∞ is related to m as follows:
ω∞
ω1
=
1
,
1−m
(13)
2
so that the value of m needed to produce a notch at a specified frequency ω∞ is
ω1
1−
.
ω
2
m =
(14)
∞
A value of 0.6 for m corresponds to a notch frequency that is a factor of 1.25 times the cutoff frequency.
Table 3 summarizes the design of constant-k and m-derived low pass filters. Component
values (again, to many more digits than are practically relevant) are for the specific case of
a termination (and source) resistance of 50Ω and a cutoff frequency of 1GHz. The left two
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 12 of 38
EE414 Handout #14: Spring 2001
columns are for the simple constant-k case, and the last three columns give values for the
specific m-derived case where m = 0.6.
TABLE 3. Component values for 1GHz constant-k and m-derived filters (Z = 50Ω m = 0.6)
L
15.9155nH
C
6.3662pF
L1
9.5493nH
C
3.8197pF
L2
4.2441nH
For filters with a cutoff frequency other than 1GHz, simply multiply all component values
by the ratio of 1GHz to the desired cutoff frequency. For a different characteristic impedance, multiply all component impedances by the ratio of the desired impedance to 50Ω.
One may also combine ordinary constant-k and m-derived sections because the individual
sections for both are constant-k in nature. Such a composite filter may be desirable, for
example, to effect a compromise between flatness and the production of notches at specific
frequencies. Unfortunately, design of such a filter is very much an ad hoc affair. One simply mixes and matches sections as seems sensible, then simulates to verify if the design
indeed functions satisfactorily.
3.1.1 High-pass, bandpass and bandstop shapes
At least in principle, a high-pass constant-k filter is readily constructed from the low pass
constant-k prototype simply by swapping the positions of the inductors and capacitors; the
values remain the same. Thus one may design, say, a 1GHz constant-k low-pass filter
using the values of Table 3, then interchange the Ls and Cs to synthesize a 1GHz highpass filter.
The reason for the qualifier “at least in principle” is that high-pass filters typically exhibit
serious deviations from desired behavior. These deviations motivate microwave filter
designers to avoid high-pass filters based on lower frequency prototypes. Although there
are many ways – too numerous to mention, in fact – in which a practical filter of any kind
can fall short of expectations, perhaps the following lumped high pass filter example will
suffice to illustrate the general nature of the problem. Specifically, consider:
FIGURE 8. High pass filter?
Every practical inductor is shunted by some capacitance, and thus exhibits a resonance of
its own. Above the resonant frequency, the “inductor” actually appears as a capacitance.
Similarly every practical capacitance has in series with it some inductance. Above the cor-
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 13 of 38
EE414 Handout #14: Spring 2001
responding series resonance, the capacitor actually appears inductive. Hence, at sufficiently high frequencies, our high pass filter actually acts as a low pass filter.
A complementary effect afflicts low pass filters where, at high frequencies, it is possible
for the response to pop back up.
As a practical workaround, it is traditional to employ a bandpass filter with a sufficiently
wide passband to approximate the desired filter shape. Of course, that solution presupposes knowledge of how to construct bandpass filters. Fortunately, the constant-k structure
works here, too (we’ll later examine alternative bandpass implementations as well). As a
general strategy for deriving a bandpass filter from a low-pass prototype, replace the
inductance of a low-pass prototype with a series LC combination, and the capacitance with
a parallel LC combination:
FIGURE 9. Bandpass constant-k filter example using two cascaded T-sections
2L2
L1
C1
L1
C1
C2/2
L2
C2
L2
L1
C2
C1
2L2
C2/2
Unlike our previous figures, the individual T-sections are not shown, in order to simplify
the schematic.
Note that this structure continues to exhibit the correct qualitative behavior even if inductors ultimately become capacitors and vice-versa. This property is fundamental to the
potentially reduced sensitivity of this topology to parasitic effects.
The formula for the inductance L1 of the series resonator is the same as that for the inductance in the prototype low pass filter, except that the bandwidth (defined by the difference
between the upper and lower cutoff frequencies) replaces the cutoff frequency. The capacitance C1 is then chosen to produce a series resonance at the center frequency (defined
here as the geometric mean of the two cutoff frequencies12). Hence
L1 =
2
R
( ω2 − ω1 )
(15)
and
12. In some of the literature, it is unfortunately left unclear as to what sort of mean should be used. For the
common case of small fractional bandwidths, this ambiguity is acceptable, for there is then little difference
between an arithmetic and geometric mean. Practical component tolerances make insignificant such minor
differences. However, the discrepancy grows with the fractional bandwidth, and the error can become quite
noticeable at large fractional bandwidths if the arithmetic mean is used.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 14 of 38
EE414 Handout #14: Spring 2001
C1 =
( ω2 − ω1 ) 1
.
2
R
2ω
(16)
0
Similarly, the equation for the capacitance of the low-pass prototype is modified for the
bandpass case by replacing the cutoff frequency with the bandwidth. The resonating
inductance is again chosen to produce a resonance at the center frequency:
C2 =
2
1
( ω2 − ω1 ) R
(17)
and
L2 =
( ω2 − ω1)
2
R.
(18)
2ω 0
Values for a constant-k bandpass filter with cutoff frequencies of 950MHz and 1.05GHz
(corresponding to a center frequency of approximately 998.75MHz) are given in the following table:
TABLE 4. Component values for a 100MHz bandwidth, constant-k bandpass filter at 1GHz
L1
C1
159.15nH
0.15955pF
L2
0.39888nH
C2
63.662pF
As is its low-pass counterpart, the bandpass filter is terminated in half-sections. Each halfsection consists of components of value L1/2, 2C1, 2L2 and C2/2. The resulting filters have
the same characteristics enumerated in Table 1, where the bandwidth normalizations continue to be performed, sensibly enough, with respect to the bandwidth, rather than the center frequency.
The following figure shows the frequency response of a bandpass filter derived from a
low-pass constant-k filter with six sections (n = 5). The design bandwidth is 100MHz, centered at 1GHz. Not surprisingly, the behavior at the passband edges resembles that of the
low-pass prototype.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 15 of 38
EE414 Handout #14: Spring 2001
FIGURE 10. Frequency response for bandpass filter derived from six-section constant-k (n = 5)
For a different bandwidth, multiply C1 and L2 by the ratio of the new bandwidth to
100MHz, and reduce L1 and C2 by the same factor. For a different center frequency,
reduce C1 and L2 each by the square of the ratio of the new center frequency to 1GHz.
Finally, for a different characteristic impedance, increase the impedance of all four components by the ratio of the new impedance to 50Ω.
The bandpass filter can be converted into a bandstop (also known as a band-reject) filter
simply by swapping the positions of the series and parallel resonators. As in the conversion from low pass to high pass, the values remain unchanged.
From the tables and examples given, it is clear that the constant-k and m-derived filters are
extremely simple to design, since they consist of identical iterated sections (plus a terminating half-section on each end). This simplicity is precisely their greatest attribute. In
exchange for this ease of design, however, the foregoing procedures neglect certain details
(such as passband ripple) because they do not incorporate any specific constraints on
response shape. It is clear from the tables, for example, that the cutoff frequency doesn’t
correspond to a certain fixed attenuation value, such as –6dB, and monotonicity is far from
guaranteed. Stopband behavior is similarly mysterious. Finally, because the source and
load terminations are assumed equal in value, any necessary impedance transformations
have to be provided separately. It should seem reasonable, however, to expect that a more
advanced synthesis technique might, on occasion, accommodate impedance transformation as a natural accompaniment to the filtering operation. Shortcomings such as these
explain why there are alternative filter design approaches. Because the relative merits of
these alternatives are best appreciated after identification and definition of key filter performance metrics, we now consider a brief sidebar and introduce these parameters.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 16 of 38
EE414 Handout #14: Spring 2001
4.0 Filter Classifications and Specifications
Filters may be classified broadly by their general response shapes – e.g., low-pass, bandpass, band reject and high-pass – and further subdivided according to bandwidth, shape
factor (or skirt selectivity), and amount of ripple (in either the phase or magnitude
response, and in either the passband, stopband, or both). This subdivision is an acknowledgment that ideal, brickwall filter shapes are simply unrealizable (not merely impractical). Different approaches to approximating ideal characteristics result in different
tradeoffs, and the consequences of these compromises require characterization.
Bandwidth is perhaps the most basic descriptive parameter, and is conventionally defined
using –3dB points in the response. However, it is important to recognize that 3dB is quite
an arbitrary choice (there is nothing fundamental about the half-power point, after all), and
we will use other bandwidth definitions that may be more appropriate from time to time. It
is certainly an incomplete specification, because there are infinitely many filter shapes that
share a common –3dB bandwidth. Shape factor is an attempt to convey some information
about the filter’s response at frequencies well removed from the –3dB point. It is defined
as the ratio of bandwidths measured at two different attenuation values (i.e., values at two
different points on the skirt). As an arbitrary example a “6/60” shape factor specification is
defined as the bandwidth at –60dB attenuation, divided by the bandwidth at –6dB attenuation:
FIGURE 11. Illustration of 6/60 shape factor
S 6 ⁄ 60 ≡
∆ω 60
∆ω 6
∆ω6
–6dB
∆ω60
–60dB
Clearly from the definition of shape factor, values approaching unity imply response
shapes that approach infinitely steep transitions from passband to stopband. A single-pole
lowpass filter (or a standard single-LC bandpass resonator) has a 6/60 shape factor of
roughly 600, a value generally regarded as pathetically large.13 This trio of numbers is
easily remembered, though, because of the decimal progression.
13. The actual number is closer to 577, but has less of a mnemonic value than 600.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 17 of 38
EE414 Handout #14: Spring 2001
Because the relevance of a given shape factor depends very much on context, there cannot
be a single, universally relevant definition. Thus although 6/60 happens to be a common
one, other specifications are often encountered.
As stated earlier, the inability of practical filters to provide perfectly flat passbands and
infinitely steep transitions to infinitely attenuating stopbands implies that we must always
accept approximations to the ideal. In the best case we have the opportunity to quantify
and specify bounds on the approximation error. The traditional way of doing so is to specify the following parameters:
FIGURE 12. General filter response template (shown for the low pass case)
|H(jω)|2
1
Passband
Transition
Band
1/(1+ε2)
Stopband
1/A2
ωp
ωs
Note that the square of the magnitude is plotted in the figure, rather than the magnitude
itself, because it is proportional to power gain. This convention isn’t universally followed,
but it is quite common because of the RF engineer’s typical preoccupation with power
gain.
Note also the pervasiveness of reciprocal quantities on the vertical axis. This annoying
feature is avoided by plotting attenuation, rather than gain, as a function of frequency,
explaining why many treatments present data in precisely that manner.
Note further that the filter response template accommodates some amount of variation
within the passband (whose upper limit is denoted ωp), with a maximum permitted deviation of 1/(1+ε2). Additionally, a finite transition between the passband and stopband
(whose lower frequency limit is denoted ωs) is also permitted, with a minimum allowed
power attenuation of A2 in the stopband. Specification of these parameters thus allows the
design of real filters. We now consider several important classes of approximations which
make use of these parameters.
5.0 Modern Filters: Common Approximations
The constant-k filter’s limitations ultimately derive from a synthesis procedure which
ignores the control over filter response afforded by direct manipulation of the pole (and
zero) locations. This limitation is a natural consequence of the transmission-line theoretical basis for constant-k filters; because transmission lines are infinite-order systems, conFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 18 of 38
EE414 Handout #14: Spring 2001
sideration of pole locations there would be unnatural, and in any case would lead to
numerous analytical difficulties.
However if one no longer insists on treating filters from a transmission line viewpoint,
these difficulties disappear (but are replaced by new ones). Additional, and highly powerful, techniques then may be brought to bear on the filter analysis and synthesis problem. In
this section, we underscore this point by following a synthesis procedure not possible with
the constant-k filter: starting from a specification of a desired frequency response, compute
a corresponding pole-zero constellation, and then synthesize a lumped network that exhibits the prescribed characteristics.
5.1 Butterworth filters
Some applications are entirely intolerant of ripple, limiting the number of options for
response shape. As do all practical filters, the Butterworth seeks to approximate the ideal
rectangular brickwall shape. The Butterworth filter’s monotonic response shape minimizes the approximation error in the vicinity of zero frequency by maximizing the number
of derivatives whose value is zero there. For a filter of order n, that maximum number happens to be 2n – 1. As the filter order approaches infinity, the filter shape progressively
approximates better the ideal brickwall shape.
A natural, but potentially undesirable consequence of a design philosophy which places
greater importance on the approximation error at low frequencies is that the error grows as
the cutoff frequency is approached. If this characteristic is indeed undesirable, one must
seek shapes other than the Butterworth. Some of these alternatives are discussed in subsequent sections.14
The Butterworth’s response magnitude (squared) as a function of frequency is given for
the low pass case by the following expression:
H ( jω )
2
=
1
ω 2n
1+
ω
c
,
(19)
where ωc is the frequency at which the power gain has dropped to 0.5.15 The parameter n
is the order of the filter, and equals the number of independent energy storage elements, as
well as the power of ω with which the response magnitude ultimately rolls off. From the
equation, it is straightforward to conclude that the response is indeed monotonic.
14. As will be discussed later, one of these alternatives, the Type II Chebyshev, actually achieves better
passband flatness than the Butterworth (making it flatter than maximally flat), by permitting stopband ripple,
while preserving a monotonic passband response.
15. Although not rigorously correct (because of the possibility of unequal input and output impedances), we
will frequently use the term “power gain” interchangeably with the more cumbersome “response magnitude
squared.”
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 19 of 38
EE414 Handout #14: Spring 2001
In designing a Butterworth filter, one may specify ωc directly, but to maintain consistency
with the template of Figure 12, let us derive ωc from the other filter parameters. First we
may express the power gain at the passband and stopband edges as follows:
1
1+ε
2
=
1
2n
ωp
1+
ω
(20)
c
and
1
A
2
=
1
ω s 2n
1+
ω
.
(21)
c
Solving these two equations for the required filter order, n, yields
2
n =
ln( ε ⁄ A − 1 )
ωp
ln
ω
.
(22)
s
Thus, once the attenuation at the passband edge, minimum attenuation at the stopband
edge, and the frequencies of those edges are specified, the required filter order is immediately determined. Because Eqn. 22 generally yields non-integer values, one must choose
the next higher integer as the filter order. In that case, the resulting filter will exhibit characteristics that are superior to those originally sought. One way to use the “surplus” performance is to retain the original ωp, in which case the filter will exhibit greater
attenuation at ωs than required. Alternatively, one may instead retain the original ωs, in
which case the filter exhibits smaller attenuation (i.e. smaller error) at the passband edge
than originally targeted. Or, one may elect a strategy that is intermediate between these
two choices.
Pursuing the strategy of retaining the originally sought performance at the passband edge,
the –3dB corner frequency ωc may be computed from the foregoing equations as
ωc =
ωp
ε
1⁄n
.
(23)
Alternatively, Eqn. 21 may be solved for a (generally different) ωc derived from a specification of ωs.
Because its approximation error is very small near DC, the Butterworth shape is also
described as maximally flat.16 However it is important to recognize that maximally flat
does not imply perfectly flat.17 Rather, it implies the flattest passband that can be
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 20 of 38
EE414 Handout #14: Spring 2001
achieved, subject to the constraint of monotonicity (later, we will see that it is possible to
have an even flatter passband response if we are willing to permit ripple in the stopband).
As a design example, let us continue the exercise that we began with the constant-k topology. We now have the ability to specify more filter parameters than in that case, so we’ll
do so. Here, arbitrarily allow a 1dB loss (gain of 0.794) at the passband edge of 1GHz, and
require a 30dB factor of minimum attenuation at a 3GHz stopband edge.
From the passband specification, we find that ε is approximately 0.5088. From the stopband specification, we see that A2 is 1000. As a result, the minimum filter order required to
meet the specifications is
n =
ln( 0.5088 ⁄ 999 )
≈ 3.76 ,
1
ln
3
(24)
which we round upward to four. Choosing to meet precisely the specification at the passband edge, we find that the corresponding value of ωc is approximately
ωc =
ωp
ε
1⁄n
≈ 7.44Grps .
(25)
From this point, we would typically consult a table of component values for a fourth-order
filter, scaling the values for the desired cutoff frequency (and, possibly, impedance
level).18 As a final check, it is always wise to simulate the proposed filter, just to make
sure that no computational errors (or typographical errors – some published tables have
incorrect entries!) have corrupted the design. In demanding applications, simulation is
also valuable for assessing the sensitivities of the filter to practical variations in component values, or to other imperfections (such as finite Q or parasitics).
5.2 Chebyshev (equiripple or minimax) filters
Although monotonicity certainly has an esthetic appeal, insisting on it constrains other
valuable filter shape properties. These include the steepness of transitions from passband
to stopband, as well as the stopband attenuation for a given filter order. Alternative filters,
based on non-monotonic frequency response, are named after the folks who invented
them, or who developed the underlying mathematics. The Chebyshev filter, an example of
16. This term was evidently introduced by V. D. Landon in his paper, “Cascade Amplifiers with Maximal
Flatness,” RCA Review, vol. 5, pp. 347-362, January 1941. Coining of the term thus follows by more than a
decade Butterworth’s own exposition of the subject in “On the Theory of Filter Amplifiers,” Wireless Engr.,
vol. 7, pp. 536-541, Oct. 1930. Although others published similar results earlier, Butterworth and maximal
flatness are now seemingly linked forever.
17. In this way, “maximally flat” is used a bit like “creme filling” in describing the ingredients of an Oreo
cookie; it means something a little different from how it initially sounds.
18. Later, we will present a synthesis method that allows computation of component values directly.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 21 of 38
EE414 Handout #14: Spring 2001
the latter, allows a reduction in filter order precisely by relaxing the constraint of monotonicity.19 In contrast with the Butterworth approximation, which is preoccupied with minimizing error at low frequencies, the Chebyshev minimizes the maximum approximation
error (relative to the ideal brickwall shape) throughout the entire passband. The resulting
minimax response shape thus exhibits some ripple, the amount of which may be specified
by the designer. For a given order, the Chebyshev filter shape offers a more dramatic transition from passband to stopband than a Butterworth offers. The steepness of the transition
is also a function of the passband ripple one allows; the greater the permissible ripple, the
steeper the transition.
A consequence of minimizing the maximum error is that the ripples of a Chebyshev
response are all of equal amplitude. A rigorous proof of the minimax optimality of an
equiripple shape is surprisingly involved, so we won’t attempt one here. However, it
should seem intuitively reasonable that equiripple behavior would be optimal in the minimax sense for, if any one error peak were larger than the others, a better approximation
could probably be produced by reducing it, at the cost of increasing the size of one or
more of the others. Such tradings-off would proceed until all error peaks were equal,
because nothing would then be left to trade for anything else.
Similar advantages also accrue if the stopband, rather than the passband, is allowed to
exhibit ripple. The inverse Chebyshev filter (also known as a Type II Chebyshev filter) is
based on this idea, and actually combines a flatter-than-Butterworth passband with an
equiripple stopband.
To understand how the simple act of allowing either stopband or passband ripple provides
these advantages, we need to review the properties of a complex pole pair. First recall that
one standard (and perfectly general) form for the transfer function of such a pair is:
1
H ( s) =
s
2
,
(26)
2ζs
+
+1
2
ωn ωn
where ωn is the distance to the poles from the origin, and ζ (zeta) is the damping ratio:
19. Pafnuti L’vovich Chebyshev (1821-1894) did no work on filters at all. In fact he developed his equations
during a study of mechanical linkages used in steam engines (see his posthumously published “Théorie des
mécanismes connus sous le nom de parallélogrammes,” (Theory of mechanisms known under the name of
parallelograms)), Oeuvres, vol. I, St. Petersburg, 1899. “Parallelograms” translate rotary motion into an
approximation of rectilinear motion. By the way, the spelling of his name here is just one of many possible
transliterations of Pafnutiy L¡voviq Qeb[wev.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 22 of 38
EE414 Handout #14: Spring 2001
FIGURE 13. Two-pole constellation
jω n 1 − ζ
2
ωn
–ζωn
A zero damping ratio corresponds to purely imaginary poles, and a damping ratio of unity
corresponds to a pair of poles coincident on the real axis. The former condition applies to
an oscillator, and the latter defines critical damping. Above a damping ratio of unity the
two poles split, with one moving toward the origin, the other toward minus infinity, all the
while remaining on the real axis. Whatever the value of damping, the frequency ωn always
equals the geometric mean of the pole frequencies.
The property that is most relevant to the subject of filter design is the dependency of the
frequency response shape on the damping ratio. While it is true that all zero-free two pole
systems have a frequency response that ultimately rolls off as ω-2, the frequency response
magnitude, and the slope, in the vicinity of the peak are very much functions of the damping ratio, increasing as ζ decreases (Figure 14b). For damping ratios above 1 ⁄ 2 , the frequency response exhibits no peaking. Below that value of ζ, peaking increases without
bound as the damping ratio approaches zero. For small values of ζ, the peak gain is
inversely proportional to damping ratio. Stated alternatively, lower damping ratios lead to
greater ultimate attenuation, relative to the peak gain, and to slopes that are normally associated with higher (and perhaps much higher) order systems.
Now consider ways a filter might exploit this ζ-dependent behavior. Specifically, suppose
we use a second-order section to improve the magnitude characteristics of a single-pole
filter. If we arrange for the peak of the second-order response to compensate (boost) the
response of the first-order section beyond where the latter has begun a significant rolloff,
the frequency range over which the magnitude of the cascade remains roughly constant
can be increased. At the same time, the rolloff beyond the compensation point can exhibit
a rather high initial slope, providing an improved transition from passband to stopband.
Clearly, additional sections may be used to effect even larger improvements, with each
added section possessing progressively smaller damping ratios. This latter requirement
stems from the need to provide larger boosts to compensate for ever larger attenuations.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 23 of 38
EE414 Handout #14: Spring 2001
FIGURE 14. Step- and frequency-response of second-order lowpass transfer characteristic
Vout
time
a) Step response
ζ decreasing
|H(jω)|
1
–2
log ω
b) Frequency response
Having developed this understanding, we may revisit the Butterworth and Chebyshev
approximations. The Butterworth condition results when the poles of the transfer characteristic are arranged so that the modest amount of frequency response peaking of a complex pole pair offsets, to a certain extent, the rolloff of any real pole present. The resulting
combination exhibits roughly flat transmission magnitude over a broader frequency range
than that of either the real pole or complex pair alone. The result is that all of the poles lie
on a semicircle in the s-plane, distributed as if there were twice as many poles disposed at
equal angles along the circumference, the right half-plane poles being ignored.20 A thirdFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 24 of 38
EE414 Handout #14: Spring 2001
order Butterworth, for example, has a single pole on the real axis, and a complex conjugate pair at 60° angles with the real axis. The distance from the origin to the poles is the
3dB cutoff frequency.
FIGURE 15. Pole constellation for third order Butterworth low pass filter
ωc
The element values, normalized to a 1Ω impedance level, and to a 1 rad/sec passband
edge, for an nth-order Butterworth low-pass filter are given by the following set of equations:
g0 = 1
(27)
and
k
g k = 2 sin
( 2 − 1) π
2
n
,
(28)
where k ranges from 1 to n, and
gn + 1 = 1 .
(29)
Conversion into a bandpass filter is easily achieved using the same transformations used in
the constant-k case.
The Chebyshev filter goes further by allowing passband (or stopband) ripple. Continuing
with our third-order example, the response of the real pole is allowed to drop below the
low-frequency value by some specified amount (the permissible ripple) before the complex pair’s peaking is permitted to bring the response back up. The damping ratio of the
complex pair must be lower than that in the Butterworth case to produce enough additional peaking to compensate for the greater attenuation of the real pole. A side effect of
20. Okay, perhaps it isn’t quite “intuitively obvious,” it is true, but finding the roots of Eqn. 19 to discover
the factoid about Butterworth poles lying on a circle isn’t all that bad.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 25 of 38
EE414 Handout #14: Spring 2001
this lower damping is that there is a more dramatic rolloff beyond the cutoff frequency. In
this manner the Chebyshev filter permits the designer to trade passband flatness for better
stopband attenuation.
Although it is even less intuitively obvious, the poles of a Chebyshev low pass filter are
located along a (semi)ellipse, remarkably with imaginary parts that are equal to those of a
corresponding Butterworth low pass filter.21 Increasing the eccentricity of the ellipse
increases the ripple.
Inverse Chebyshev filters have poles located at the reciprocals of the “normal” Chebyshev,
and purely imaginary zeros distributed in some complicated fashion. The resulting polezero constellation roughly resembles the Greek letter Ω rotated counter-clockwise by 90°.
FIGURE 16. Third order Butterworth and Chebyshev low pass filter pole constellations
Mathematically, the Chebyshev response is of the general form
H ( jω )
2
1
=
1+ε
2
2
Cn
ω
ω
,
(30)
p
where ωp once again is the frequency at which the response magnitude squared has
dropped to a value
1
1+ε
2
,
(31)
as in the Butterworth case. For self-evident reasons ε is known as the ripple parameter, and
is specified by the designer. The function Cn(x) is known as a Chebyshev polynomial of
21. There are many references that provide excellent derivations of the Butterworth and Chebyshev conditions. A particularly enlightening derivation may be found in chapters 12 and 13 of R. W. Hamming’s volume, Digital Filters, Prentice-Hall, 2nd ed., 1983.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 26 of 38
EE414 Handout #14: Spring 2001
order n. The most relevant property of such polynomials is that they oscillate between –1
and +1 as the argument x varies over the same interval. Outside of this interval the magnitude grows rapidly (as xn in fact). The filter’s (power) response thus varies between 1 and
1/(1+ ε2) as the frequency increases from DC to ωp. That entire frequency interval is often
called the ripple passband, and the parameter ωp the ripple bandwidth (or ripple cutoff frequency). In general the ripple passband differs from the more conventional –3dB bandwidth.
There are a couple of ways of generating Chebyshev polynomials algorithmically. One is
through a recursion formula,
C n ( x ) = 2xC n − 1 ( x ) − C n − 2 ( x ) ,
(32)
where C0 = 1 and C1 = x (just to get you started). As can be seen from the recursion formula, the leading coefficient of Chebyshev polynomials is 2n-1, a fact we shall use later in
comparing Chebyshev and Butterworth polynomials.
Another method for generating the Chebyshev polynomials is in terms of some trigonometric functions, from which the oscillation between –1 and +1 (for |x| < 1) is directly
deduced:
−1
C n ( x ) = cos ( ncos x ) for |x| < 1 .
(33)
For arguments larger than unity, the formula changes a little bit:
−1
C n ( x ) = cosh ( ncosh x ) for |x| > 1 .
(34)
Although it is probably far from obvious at this point, these functions are likely familiar to
you as Lissajous figures, formed and displayed when sinewaves drive both the vertical and
horizontal deflection plates of an oscilloscope. That is, suppose that the horizontal deflection plates are driven by a signal
x = cos t ,
(35)
so that
−1
t = cos x .
(36)
Further suppose that the vertical plates are simultaneously driven by a signal
y = cos nt .
(37)
Substituting Eqn. 36 into Eqn. 37 to remove the time parameter yields
−1
y = cos ( ncos x ) ,
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
(38)
Page 27 of 38
EE414 Handout #14: Spring 2001
which is seen to be the same as Eqn. 33. That is, what’s displayed on an oscilloscope
driven in this fashion is in fact the Chebyshev polynomial for that order n, for values of |x|
up to one. Over that interval the function displayed looks very much like a sinusoid
sketched on a piece of paper, wrapped around a cylinder, and then viewed from a distance.
A few Chebyshev polynomials are sketched crudely in the following figure, and expressions for the first ten Chebyshev polynomials are listed in Table 5:
FIGURE 17. Rough sketches of some Chebyshev polynomials
+1
+1
–1
+1
–1
+1
–1
+1
–1
–1
C1
–1
C2
+1
C3
+1
–1
+1
–1
+1
+1
+1
–1
+1
–1
–1
–1
C4
C5
C6
TABLE 5. First ten Chebyshev polynomials
Order, n
Filters
Polynomial
0
1
1
x
2
2x2 – 1
3
4x3 – 3x
4
8x4 – 8x2 + 1
5
16x5 – 20x3 + 5x
6
32x6 – 48x4 + 18x2 – 1
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 28 of 38
EE414 Handout #14: Spring 2001
TABLE 5. First ten Chebyshev polynomials
Order, n
Polynomial
7
64x7 – 112x5 + 56x3 – 7x
8
128x8 – 256x6 + 160x4 – 32x2 + 1
9
256x9 – 576x7 + 432x5 – 120x3 + 9x
From the foregoing equations, we may derive an expression for the filter order required to
satisfy the specified constraints:
2 1
A − ε
.
ω
−1 s
cosh
ω
cosh
n =
−1
(39)
p
As with the Butterworth case, the order as computed by Eqn. 39 should be rounded
upward to the next integer value. Again, the resulting “excess” performance can be used to
improve some combination of passband and stopband characteristics.
One way in which the Chebyshev is superior to a Butterworth is in the ultimate stopband
attenuation provided. At high frequencies, the Butterworth provides an attenuation that is
approximately
ω
A j
ω
c
2
ω 2n
≈ .
ω
(40)
c
Compare that asymptotic behavior with that of a Chebyshev (with ε = 1 so that the –3dB
frequency of the Butterworth corresponds to the passband edge of the Chebyshev):22
ω
A j
ω
c
2
≈2
ω 2n
.
ω
c
2n − 2
(41)
Clearly the Chebyshev filter offers higher ultimate attenuation, by an amount equal to
3dB(2n – 2), for a given order. As a specific example, a 7th-order Chebyshev ultimately
provides 36dB more stopband attenuation than does a 7th order Butterworth.
As another comparison, the relationship between the poles of a Butterworth and those of a
Chebyshev of the same order can be put on a quantitative basis by normalizing the two filters to have precisely the same –3dB bandwidth. It also may be shown (but not by us) that
the –3dB bandwidth of a Chebyshev may be reasonably well approximated by23
22. This comparison should not mislead you into thinking that such large ripple values are commonly used.
In fact, Chebyshev filters are typically designed with ripple values below 1dB.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 29 of 38
EE414 Handout #14: Spring 2001
1
−1 1
cosh sinh .
n
ε
(42)
Since the diameter of a Butterworth’s circular pole constellation is the –3dB bandwidth,
we normalize the Chebyshev’s ellipse to have a major axis defined by Eqn. 42. The imaginary parts of the poles of a Chebyshev filter are the same as for the Butterworth, while the
real parts of the Butterworth prototype are merely scaled by the factor
−1 1
1
tanh sinh
n
ε
(43)
to yield the real parts of the poles of a Chebyshev filter. Thus design of a Chebyshev filter
is quite straightforward because it requires only a prototype Butterworth, and it’s trivial to
design the latter. Clearly, the Butterworth may be considered merely a special case of a
Chebyshev, one for which the ripple parameter is zero.
There is one subtlety that requires discussion, however, and this concerns the source and
termination impedances of a passive Chebyshev filter. From both the sketches and equations, it’s clear that only odd-order Chebyshev polynomials have a zero value for zero
arguments. Hence, the DC value of the filter transfer function will be unity for such polynomials (that is, the passband hits its ripple extremum at some frequency above DC). For
even-order Chebyshev filters, however, the filter’s transfer function starts off at a ripple
extremum, with a DC power transmission value of 1/(1+ ε2), implying a termination resistance that is less than the source resistance. If, as is usually the case, such an impedance
transformation is undesired, one must either use only odd-order Chebyshev filters, or add
an impedance transformer to an even-order Chebyshev filter. As the former is less complex, it is the near universal choice to use only odd-order Chebyshev realizations in practice.
Finally, recognize that the elliptical pole distribution implies that the ratio of the imaginary
to real parts of the poles, and hence the Qs of the poles, are higher for Chebyshevs than for
Butterworths of the same order. As a result, Chebyshev filters are more strongly affected
by the finite Q of practical components. The problem increases rapidly in severity as the
order of the filter increases. This important practical issue must be kept in mind when
choosing a filter type.
Element values for the Chebyshev filter are given by the following sequence of equations.
First, compute four auxiliary quantities, whose significance may initially seem mysterious:24
L Ar
β = ln coth
,
17.372
(44)
23. See, for example, M. E. Van Valkenburg, Introduction to Modern Network Synthesis, Wiley, 1960, pp.
380-381.
24. After close examination, these remain mysterious. Sorry. At least I can tell you that the 17.372 factor is
40log10e, as if that helps.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 30 of 38
EE414 Handout #14: Spring 2001
where LAr is in dB;
β
γ = sinh
2
n
;
(45)
k
( 2 − 1) π
a k = sin
2
n
;
(46)
and
2
2 kπ
b k = γ + sin .
n
(47)
Once the values of the auxiliary parameters are known, the following equations yield the
normalized element values:
g0 = 1 ;
g1 =
gk =
2a 1
γ
(48)
;
4a k − 1 a k
bk − 1 gk − 1
(49)
;
(50)
g n + 1 = 1 for n odd ;
(51)
2 β
g n + 1 = coth for n even .
4
(52)
and
5.3 Type II (Inverse) Chebyshev filters
We have alluded several times to the possibility of realizing a flatter-than-maximally flat
transfer characteristic. The Type II (also known as an inverse or reciprocal) Chebyshev filter achieves such flatness by permitting ripple in the stopband, while continuing to insist
on passband monotonicity.
The Type II filter derives from the Type I (ordinary) Chebyshev through a pair of simple
transformations. In the first step, the Type I Chebyshev response is simply subtracted from
unity, leading to the conversion of a low-pass filter into a high-pass one. Note that the
resulting response is monotonic in the new passband. The second step replaces ω by 1/ω.
Since high frequencies are thus mapped into low ones, and vice-versa, this second transformation converts the filter shape back into a low-pass response, but in a way that
exchanges the ripple at low frequencies with ripple at high frequencies. This transformaFilters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 31 of 38
EE414 Handout #14: Spring 2001
tion thus restores a monotonic passband, and also happens to map the Type I passband
edge into the stopband edge. Furthermore the larger the permissible stopband ripple, the
flatter the passband response.
Mathematically, these transformations ultimately result in the following power response
for a Type II filter:
H ( jω )
2
ω
2 2 p
ε Cn
ω
1
= 1−
=
.
ω
ω
2 2 p
2 2 p
1 + ε Cn
1 + ε Cn
ω
ω
(53)
Although the Type II filter is not encountered as often as the Butterworth, its relative rarity
should not be taken to imply a corresponding lack of utility. Despite the superior flatness
provided by the inverse Chebyshev, it appears that, for purely cultural reasons, the Butterworth filter continues to dominate in those applications where passband uniformity is
allegedly prized.
5.4 Elliptic (Cauer) filters
Having seen that allowing ripple in the passband or stopband confers desirable attributes,
perhaps it should not be surprising that the elliptic or Cauer filter further improves transition steepness by allowing ripple in both the passband and stopband simultaneously.25 Just
as a complex pole pair provides peaking, a complex zero pair provides notching. We’ve
seen this behavior already, where the purely imaginary zeros of an m-derived filter provide
notches of infinite depth. Cauer filters exploit this notching to create a dramatic transition
from passband to stopband, at the expense of a stopband response that bounces back up
beyond the notch frequency (again, just as in an m-derived filter, and for the same reasons). The name elliptic comes from the appearance of elliptic functions in the mathematics, and should not be confused with the elliptic pole distribution of a Chebyshev filter.
Elliptic filters have the following power transmission behavior:
H ( jω )
2
1
=
1+ε
2
2
Un
ω
ω
,
(54)
c
where Un (x) is a Jacobian elliptic function of order n:26
25. These are also sometimes known as Darlington or Zolotarev filters. Sidney Darlington (of “Darlington
pair” fame in bipolar circuits) made major contributions in the field of network synthesis. Igor Ivanovich
Zolotarev independently studied Chebyshev functions a decade or so before Chebyshev did.
26. These are named for the mathematician Karl Gustav Jacob Jacobi (1804-1851), who began studying
these functions in the 1820s, at the start of his career.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 32 of 38
EE414 Handout #14: Spring 2001
x
Un ( x) =
∫
0
dy
2
.
(55)
2 2
(1 − y ) (1 − n y )
These functions are messy enough that quantitative information about them is generally
presented in tabular form (but not here, though; see, e.g., the oft-cited work by G.W. Spencely and R. M. Spencely, “Smithsonian Elliptic Function Tables,” Publication 3863,
Smithsonian Institution, Washington, D. C., 1947). Suffice it to say that, just as Chebyshev
polynomials do, these elliptic functions oscillate within narrow limits for arguments |x|
smaller than unity, and rapidly grow in magnitude for arguments outside of that range.
However, unlike Chebyshev polynomials, whose magnitudes grow monotonically outside
of that range, these elliptic functions oscillate in some fashion between infinity and a specified finite value. Hence the filter response exhibits stopband ripples, with a finite number
of frequencies at which the filter transmission is (ideally) zero. The following figure shows
crude sketches of the first several Jacobian elliptic functions, from which this behavior
may be discerned:
FIGURE 18. Rough sketches of some Jacobian elliptic functions
+1
+1
–1
+1
+1
–1
–1
+1
+1
–1
+1
–1
Filters
–1
–1
+1
–1
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 33 of 38
EE414 Handout #14: Spring 2001
The attenuation poles correspond to transmission zeros (notches), in the proximity of
which the filter response changes rapidly. Thus, perhaps you can see how permitting such
ripples in the stopband allows for a much more dramatic transition from passband to stopband, and thus allows one to combine the attributes of ordinary and inverse Chebyshev filters.
Wilhelm Cauer is the inventor whose deep physical insights (and intimate familiarity both
with the notches of Zobel’s m-derived filters, and with elliptic functions in general)
allowed him first to recognize that this additional degree of freedom existed, and then to
exploit it, even though he did not offer a formal mathematical proof of the correctness of
his ideas.27 At a time when minimizing component count was an obsession, Cauer was
able to use fewer inductors than the best filters that were then in use. According to lore,
publication of his patent reportedly sent Bell Labs engineers and mathematicians scurrying off to the New York City Public Library to bone up on the then-obscure (okay, stillobscure) literature on elliptic functions.28
6.0 Coupled Resonator Bandpass Filters
Up to now we’ve focused mainly on low pass filters, having derived other filter shapes
from low pass prototypes. It is worthwhile to develop additional insights, however, so that
we don’t always have to return to the low pass case whenever we wish to design, say, a
bandpass filter. This freedom, in turn, allows us to analyze and synthesize filter types that
are not readily related to lumped networks at all.
We’ve seen that the poles, say, of a “good” filter aren’t all coincident; they’re distributed
in some manner. Viewed from a broad perspective, then, the goal of filter design is to distribute the transfer function poles and zeros in some manner to achieve a desired response
shape. This important idea is the basis for essentially all lumped filters, bandpass or otherwise. A particularly simple way to synthesize bandpass filters with a variety of response
shapes is to exploit the mode splitting that occurs when two or more resonant systems
interact. That is, when two identical resonators are connected together in some fashion, the
poles of the resulting coupled system generally differ from those of the resonators in isolation. By controlling the degree of interaction (coupling) the pole locations can be adjusted
to produce a desired response shape.
27. Cauer (1900-1945) became familiar with elliptic functions while studying at the University of Göttingen
with the mathematician David Hilbert. Hilbert was as well known for his absentmindedness as for his mathematics. Once he suddenly asked a close friend, physicist James Franck, “Is your wife as mean as mine?”
Though taken aback, Franck managed to respond, “Why, what has she done?” Hilbert answered, “I discovered today that my wife does not give me an egg for breakfast. Heaven only knows how long this has been
going on.”
It is unfortunate that stories about Cauer are not as lighthearted. Tragically, he was shot to death during the
Soviet occupation of Berlin in the closing days of WWII, in a manner sadly reminiscent of the death of
Archimedes (see http://www-ft.ee.tu-berlin.de/geschichte/th_nachr.htm).
28. M. E. Van Valkenburg, Analog Filter Design, Harcourt Brace Jovanovich, 1982, p. 379.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 34 of 38
EE414 Handout #14: Spring 2001
To illustrate this important idea, consider two simple LC resonators whose inductors are
magnetically coupled to each other. Such a system may be modeled by representing the
coupled inductors with a transformer. The transformer in turn is modeled as a T-connection of three inductors:
FIGURE 19. Coupled LC resonators
L–M
L–M
C
M
C
k =
M
L
The inductance L is that which is present in each resonator in isolation. The mutual inductance M is a fraction of L, and depends on the magnitude of the coupling. The latter is captured in the coupling coefficient k, which ranges from zero to unity as the flux linkage of
the magnetic fields of the two inductors increases from zero to 100%.
To find the resonant frequencies of the resulting 4th-order system29 one can always
employ a brute-force approach: Find the transfer function (first, one needs to define the
input and output terminals), then solve for the roots of the denominator polynomial. This
method is quite general, but also quite cumbersome, particularly for networks of order
higher than two or three. Here, the network happens to be symmetrical, a situation that
almost demands exploitation to simplify analysis by bypassing uninspired routes to the
answer.
First recall what poles are. Yes, they are the roots of the denominator of the transfer function, but a deeper significance is that they are the natural frequencies of a network. That is,
if the system is given some initial energy, the evolution of the system state in the absence
of any further input takes place with characteristic frequencies whose values are those of
the poles. Cleverly chosen initial conditions may excite only a subset of all possible
modes at a time, thus converting a difficult high-order problem into a collection of more
simply solved low-order ones. Very clever (or lucky) choices can even result in the excitation of a single mode at a time.
We may use this understanding to devise a simple method for finding the poles of our coupled resonator system. First, provide a common-mode excitation by depositing, say, an
equal amount of initial charge on the two capacitors. Regardless of what the network does
subsequently, we know by symmetry that the capacitor voltages must evolve the same
way. Because the two capacitor voltages are thus always equal, we may short the capacitors together with impunity, resulting in the following network:
29. Despite there being five energy storage elements in the network, the system is nonetheless of the fourth
order, because not all of the elements are independent. Note, for example, that specifying the currents in two
of the inductors automatically determines that flowing in the third, by Kirchhoff’s current law. Thus, the
three inductors actually contribute only two degrees of freedom, diminishing by one the order of the overall
network.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 35 of 38
EE414 Handout #14: Spring 2001
FIGURE 20. Equivalent network of coupled LC resonators for common-mode initial conditions
(L – M)/2
M
2C
The common-mode resonant frequency is thus that of a simple parallel LC network:
ω cm =
1
( 1 − k) L
+ kL 2C
2
=
1
( 1 + k ) LC
.
(56)
There are two conjugate imaginary poles of this frequency, so we only need to find the
other two poles of this fourth-order network.
Since a common-mode initial condition is so fruitful in discovering two of the poles, it
seems reasonable to try a differential initial condition next. Specifically, if one capacitor
voltage is initially made equal to some voltage V, and the other to –V, (anti)symmetry
allows to assert that, however the system state evolves from this initial condition, it must
do so in a manner that guarantees zero voltage across the common shunt inductance of
value M. Consequently, no current flows through it, and the shunt inductance may be
removed (either by open- or short-circuiting it; both actions will lead to the same answer).
Removing that inductance yields the following differential-mode resonant frequency:
ω dm =
1
1
=
( 1 − k ) LC
C
[ 2 ( 1 − k) L]
2
.
(57)
Now that we’ve found the pole frequencies, let’s see what intuition may be extracted from
the exercise. First consider extremely loose coupling, i.e., values of k very near zero. In
that situation the two mode frequencies are nearly the same, because we have two nearly
independent and identical tanks. As k increases, however, one resonant frequency
decreases, while the other increases; mode splitting occurs. The stronger the coupling, the
wider the separation in resonant frequencies.
As an illustration that mode-splitting is an extremely general consequence of coupling resonators together, consider the use of capacitive coupling:
FIGURE 21. Capacitively coupled resonators
Cc
C–Cc
Filters
L
L
C–Cc
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 36 of 38
EE414 Handout #14: Spring 2001
Here, the individual resonator capacitances are arbitrarily expressed as a function of the
coupling capacitance. One could just as well have labeled the resonator capacitances simply as C, but the choice shown simplifies the analytical expressions somewhat, as will be
seen.
Following an approach analogous to that used to analyze the magnetically coupled case,
we find that the two mode frequencies are given by:
ω cm =
1
( C − Cc ) L
=
1
(58)
( 1 − k ) LC
and
ω dm =
1
( C + Cc ) L
=
1
( 1 + k ) LC
.
(59)
For these equations, an explicit expression for the coupling coefficient, k, is found to be
Cc
k =
C
.
(60)
As with the magnetic case, the coupling coefficient cannot exceed unity (if negative element values are disallowed) when expressed in this manner. As we can see, both magnetic
and capacitive coupling give rise to the same splitting of modes. This mechanism is so
general that it explains a host of phenomena, such as the formation of energy bands in solids (here, the initially identical mode frequencies – energy levels – of free atoms split as
the atoms are brought closer together to form a solid).
From Eqn. 58 and Eqn. 59, it should be clear that one may use a measurement of the two
mode frequencies to determine k experimentally. Indeed, for small values of coupling, the
difference in mode frequencies (normalized to their geometric mean) is approximately
equal to k.
The mode splitting that accompanies coupling permits placement of poles to produce
response shapes such as Butterworth and Chebyshev. Simply adding an input and output
port to the basic structure of Figure 21, for example, readily produces a bandpass filter
which may be extended to any number of stages:
FIGURE 22. Coupled resonator filter
R
C1
Filters
Cc1
L1
R
L2
C2
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 37 of 38
EE414 Handout #14: Spring 2001
From the analysis of mode splitting, it should seem reasonable that small amounts of coupling (small values of coupling capacitance in this particular example) produce narrowband filters, and that relatively large amounts of coupling produce broadband filters. When
this idea is placed on a quantitative basis, it is possible to express the bandpass filter
design problem entirely in terms of coupling coefficients, uncoupled resonant frequencies,
and tank loading. This reformulation in terms of invariant parameters (e.g., resonant frequency, impedance levels, bandwidth) facilitates the design of microstrip filters where,
owing to their distributed nature, it is not always possible to identify individual lumped
elements.
Filters
1999 Thomas H. Lee, rev. April 27, 2001; All rights reserved
Page 38 of 38
