Professor notes
CHAPTER 17. LINEAR ACTIVE FILTERS AND FREQUENCY PROFILES
17.1 INTRODUCTION
All circuits have a frequency response characterized by the mix of reactive components and active
devices and their intrinsic time constants. In some instances we merely analyze the frequency character
of the circuit and assess it to see if it will accomplish our signal processing needs. But in other instances
we desire to tailor the circuit’s frequency behavior and define the frequency profile of the circuit.
In most instances we define frequency profiles in terms of amplitude ‘pass’ functions. There are four
basic types:
(1)
(2)
(3)
(4)
low-pass
high-pass
band-pass
band-stop
Unless we elect to get cute and use non-linear components, we should expect that time constants will
relate to linear capacitances and inductances. Intermediate and RF profiles relate to capacitances in pF or
fF and inductances in H. Profiles are also either accomplished or defined by means of RLC networks
and by the placement of poles and zeros. High RF profiles have need to consider the effect of circuit
parasitics such as leakage paths, wiring inductances, and fringing capacitances. Frequency profiles in the
UHF range (300MHz to 3GHz) or higher may use microstrips, resonant cavities or artificial crystals to
define their frequency character.
At lower frequencies, typically associated with audio systems or biological interface systems components
may be large and therefore cumbersome. Therefore techniques have been developed in which
amplification elements are used to reconfigure the requirements. These circuit topologies are usually
called active filters. An active filter is a frequency-responsive network driven by one or more active
drivers. The typical driver typically is an opamp or one of its cousins. Often the circuit can then be cast
in an integrated-circuit form in which the only frequency-dependent components are capacitances.
The integrated circuit active filter is also appropriate to RF frequencies as a means to either avoid or to
accommodate wiring capacitances and inductances.
A frequency profile is generically defined by a transfer function T(s) = |T(s)| ej(s) in the s-domain, for
which s  j The s-domain function T(s) is the same as that used for Laplace transforms. Graphical
representations of |T(s)| and (s) are called the Bode magnitude and Bode phase plots, respectively, after
Hendrik Bode (1905 – 1982).
The order of the profile T(s) is defined by the number of frequency-dependent components. All profiles
will have a numerator N(s) and a denominator D(s),
N ( s)
(17.1-1)
D( s)
for which the numerator zeros are called zeros and the denominator zeros are called poles. Poles are
associated with roll-off of the frequency profile. Zeros will have the effect of ‘roll-upward’. All transfer
T ( s) 
1
Professor notes
functions have the same number of zeros as they do poles. The effects of poles are almost always visible.
The effects of zeros may be less visible since many zeros fall either at zero or at infinity.
A great deal of attention has been given to the art of frequency profiling, for which placement of poles
and zeros is of critical importance to the character of the profile. Thus many profile constructs will be
identified in terms of names of their benefactor or in terms of mathematical descriptors. We will not
attempt to survey all of the different types except to say that there are enough to satisfy most of the
discriminating engineering requirements. Usually the different types are tabulated.
Therefore we can assume the tables values for profiles of interest and then load and go, provided we
know how to rescale and convert to the frequencies of interest and have enough perspective to make a
judicious choice.
Bilinear circuits: The bilinear circuits are the simplest category of frequency profile. The transfer
function T(s) is a ratio of two linear functions, otherwise called a bilinear form. The general form for the
bilinear function is
T ( s)  K
sz
s p
 K'
1 s z
1 s p
(17.1-2)
Table 17.1-2 shows the magnitude response for the two single-time-constant types of bilinear response
(low-pass and high-pass) and their accompanying phase responses. The roll-off slope is of the form of
–20dB/decade (which is the same as a ratio of -1/1). The rolloff slope on the log-log plot is linear, since
for for  > 10p, the magnitude of the denominator becomes approximately linear, i.e.
D( s)  1   p    p
as  >> p
2
Table 17.1-1(a). Bode magnate and phase response for low-pass profile
Table 17.1-1(b). Bode magnate and phase response for high-pass profile
2
Professor notes
Bode magnitude plots are usually represented as log-log scales with the frequency axis of the form of a
decade scale and the amplitude axis of the form of a magnitude scale in dB. The zeros are usually at
either infinity or zero (which is typical) and beyond the range of a finite Bode plot.
Note that, for a more pronounced roll-off, higher-order transfer functions are required. For each pole the
roll-off becomes approximately n   20 dB / dec  , where n = number of poles, assuming that there are no
intervening zeros.
We have stated the case for |T(s)| roll-off in terms of the low-pass profile (Table 24-1.1), which is the
inevitable fact of almost all electronic circuits. The higher the frequency the more difficulty the signal
has for passing through a network, or it is more readily shunted to ground, and for which
T (s  )  0 .
Biquadratic circuits: The simplest categories of frequency profiles are (1) first-order forms and (2)
second-order forms. The second-order are profiles are called biquadratic forms. The biquadratic form
defines a complete set of the pass-band profiles and are represented by Table 17.1-3. The general form of
the biquadratic function is
T ( s)  K
n2 s 2  n1 s  n0
s 2  s  0 Q   02
 K'
1 s a  s2 b
(17.1-3)
1  s  0 Q  s 2  02
Table 17.1-3 shows the types of magnitude response for different numerator functions. For convenience
and definition of the key profile parameters, the denominator function is always of the form
D(s)  s 2  s  0 Q   02
D( s)  1  s  0 Q  s 2  02
or
(17.1-4)
The quadratic denominator form (17.1-4) is the characteristic equation for the frequency response since it
relates to characteristic frequency 0 = 2f0 and quality factor Q. The bandpass function, for which n2 =
0 and n0 = 0, will always have its peak magnitude centered at f0 and will be symmetric about f0 on the
logarithmic frequency scale. The lowpass and highpass functions will show a profile peak if Q  1 2 ,
although for Q < 2 it is not very pronounced.
Type
(Amplitude normalized)
transfer function
(a) Band-pass
T ( s) 
s 0 Q
s  s  0 Q   02
2
(Q = 4 represented)
3
Magnitude Profile
Professor notes
(b) Low-pass
T ( s) 
 02
s 2  s  0 Q   02
(Q = 4 represented)
(c) High-pass
T ( s) 
s2
s 2  s  0 Q   02
(Q = 4 represented)
(d) Band-stop
T ( s) 
s 2  s  0 Q '   02
s 2  s  0 Q   02
(Q = 4 represented)
(e) All-pass
T ( s) 
s 2  s  0 Q   02
s 2  s  0 Q   02
(Q = 4 represented)
Table 17.1-2. Amplitude normalized biquadratic magnitude profiles. By amplitude normalized
we mean that the amplitude maximum = 1.0.
Biquadratic circuit profiles usually serve as a benchmark for their higher-order cousins.
The profile for the profile 17.1-3(e) (all-pass) bears some explanation. It is actually a phase-shift circuit
for which phase changes as a function of frequency but the amplitude does not. Since it is of secondorder it will shift the phase by = 2 x 90o = 180o from one side of the resonance to the other.
The normalized form of the bandpass function, for which n1 = 0/Q is a means to identify the reason why
Q is called the quality factor of the profile. Since the bandpass function is a means to select and pass a
given frequency it is appropriate to identify the bandwidth for the profile  = 2 – 1 defined in terms
of the 3-dB levels ( T  1 2 ) , also known as the half-power levels as represented by figure 17.1-3.
4
Professor notes
Figure 17.1-1. (Normalized) biquadratic bandpass function
If we do the mathematics on the normalized bandpass equation (given in Table 17.1-3(b) then at the levels
for which T  1 2 we end up with four solutions:
 
0
2Q

0
2Q
1  4Q 2
(17.1-5)
Of these solutions the only ones that are greater than zero are
1  
0
2Q

0
2Q
1  4Q 2
for which
   2  1 
and
2  
0
2Q

0
2Q
1  4Q 2
0
(17.1-6a)
Q
which represents the selectivity characteristic of the resonance peak, i.e.
Q
0


f0
f
(17.1-6b)
Otherwise the emphasis is that Q defines a magnitude factor for the low-pass profile:
Q
T (   0 )
(17.1-7)
T (  0)
which was examined more carefully in chapter 16.
As represented by figure 17.1-2 the low-pass and high-pass profiles of primary interest are (1) Q =
1 2 , for which the profile is maximally flat and (2) Q > 4 , for which the resonance peak begins to
dominate.
5
Professor notes
Figure 17.1-2: Low-pass profile for several values of Q, i.e. Q = {0.5, 0.707, 2, 4}. The resonance peak
is approximately at f0 for Q > 4 .
There is more to the story, however, when we extend the reach to specific applications. They are
categorized according to their properties as follows:
(1) Butterworth profiles
(2) Chebyshev profiles
(3) Bessel (Thomson)
= maximally flat in the pass-band
= maximum rolloff at the pass-band corner
= maximum linearity in the phase response and minimum
overshoot for impulse signals
And there are others which may accept ripple in both pass-band and stop-band in order to achieve better
quality response for one of the above categories. But each of the three categories cited above the
biquadratic response may be related to Q as follows:
(1) Butterworth:
(2) Chebyshev:
(3) Bessel:
Q=1 2
Q = 1.0 (3dB ripple )
Q = 0.5
The Chebyshev profile is a little more artistic than suggested by these simple relationships, since it may
be either type (1), with ripple in the pass-band, or type (2) with ripple in the stop-band, or (3) elliptic
(Cauer), with ripple in both pass-band and stop-band. It gets its name from the mathematics of
Chebyshev polynomials. In the s-plane its poles will pattern in the form of an ellipse.
The ripple for the Chebyshev profile may also be of different relative magnitudes. As might be expected
for the biquadratic form, a Q > 1.0 is also a citation for more ripple in the pass band and the benefit of a
greater rolloff at the break point. The circuit designer then takes on the role of an artist in mathematics
and circuit electronics.
17.2 NORMALIZED FORMS and FREQUENCY RESCALING
Many circuit topologies have been analyzed, addressed, profiled and tabulated by generations of electrical
engineering slaves. Tabulated forms are normalized to a characteristic frequency 0 =1.0 r/s or to a
6
Professor notes
feature frequency 1 =1.0 r/s. The normalized form of the topology is typically the one to which
rescaling techniques of different modality are applied.
Frequency rescaling is the analytical process of rescaling the components of a circuit topology to one that
will yield an exact copy of the profile at an another frequency scale. Frequency rescaling can also be used
to transform the circuit into an equivalent topology with the same profile, in which case it is called a
frequency transformation.
Circuit profiles is defined in terms of a set of zeros in the numerator N(s) terms and another set of zeros in
the denominator D(s) defined as poles because of their effect on the profile. Poles and zeros will each
have a relationship to a time constant. And if the time constants are rescaled to a new s’ =j’ plane then
they will have the exact same profile if    1   '   2 . Frequency response is defined by time
constants (duh!). For the simplest case, for which all of the frequency dependent components in a circuit
are capacitances then 1 = R1C1 and 2 = R2C2, and the s and s’ planes will relate to one another by
R1C1   ' R2 C 2
(17.2-1)
where we are identifying that s = j and s’ = j’.
So if we do not change resistance, i.e. R1 = R2, then (17.2-1) reduces to
C1   ' C 2
(17.2-2a)
and we can shift the profile from the s to the s’ plane by merely rescaling the capacitance according to
C2 

C1
'
(17.2-2b)
For example if the s’ domain is a rescale of 1000 s, for which ’ = 1000, then the capacitances must be
1000 times smaller. A 1.0mF capacitance would become a 1.0F capacitance.
If we do not change capacitance, i.e. C1 = C2, then
R1   ' R2
(17.2-3a)
and we can shift the profile from the s to the s’ plane by rescaling the resistance according to
R2 

R1
'
(17.2-3b)
For example if the s’ domain is 1000s, for which ’ = 100, the resistances must be 1000 times smaller
and the circuit 1000 times faster.
Actually we would most likely change both R and C, in a process called RC rescaling. This process is
accomplished by a two-step procedure. (1) We keep R unchanged, i.e. R1 = R2, and rescale the
capacitances to a new frequency M, i.e.
7
Professor notes
C1  M C2
then
M 
C1

C2
(17.2-4a)
for which R1C1  M R1C2
Then (2) we hold C2 constant and rescale the resistances, for which
 M R1C 2   ' R3C 2
for which
R3 
M
R1
'
(17.2-4b)
Now both R and C are changed and the profile is translated into the s’ plane (usually at a higher
frequency).
We can do this process in a different order if we so desire. But usually we rescale the capacitance to an
elected value C2 and then let the resistance follow. Equations (17.2-4a) and (17.2-4b) represented the RC
rescaling algorithm:
M 
and
C init
 init
C final
R final 
(17.2-5a)
M
Rinit
 final
(17.2-5b)
We define M = magnitude rescaling frequency . The init and final are the initial and final frequencies,
respectively. The mathematics is further simplified if the initial frequency init = 1.0 r/s, i.e. associated
with a normalized profile.
We can do the same process with inductances except that we have to acknowledge that the inductive time
constant is   L / R and the equivalent to equation (17.2-1) is
 L1 R1   ' L2 R2
(17.2-6)
We don’t usually rescale according to inductances.
A diagrammatic representation of RC rescaling using equations (17.2-5) is shown by figure 17.2-1.
8
Professor notes
Figure 17.2-1 RC rescaling steps as represented by alternative paths. The upper path is usually
the path of choice and matches equations (17.2-5).
There are several good candidates for RC rescaling that we will check out in later sections of this chapter.
For topologies that include inductances the rescaling process is not quite as simple. For example, the
RLC biquadratic circuits will have
 02 
1
LC

R 1
1
1



L RC  L  C
(17.2-8)
So the resistances do not rescale with a rescaling of both L and C. A different set of paths has to be
followed in order to retain the profile, which includes an impedance magnitude rescaling. The rescaling
algorithm is then
R2  k M R1
(17.2-9(a)
C 2  C1 k M k f

(17.2-9(b)
L2  L1 k M k f

(17.2-9(c)
Quadratic rescaling must therefore be done in terms of either a component translation which retains the
simplicity of equations (17.2-1) or (17.2-7) or additional paths must be followed. Since the translation of
is the same as an equivalent circuit form, then it invites usage of amplifier-driven components and the
reconfiguration of the topologies into an active-filter form.
9
Professor notes
17.3 BIQUADRATIC CIRCUIT TOPOLOGIES
There are a number of RC topologies that we might invite to the table for the biquadratic menu. Almost
all of them employ one or more gain elements of the form of an ideal opamp, which certainly has its
caveats if it turn real. But otherwise provides a reasonable platform for applications in which the standard
biquad profiles suffice.
For example the Sallen-Key circuit topology shown by figure 17.3-1 is a benchmark standard. It is
simple, requires only one amplifier component, and the Q of the circuit is separately tunable relative to
the characteristic frequency 0.
Figure 17.3-1 Sallen-Key single-amplifier biquad.
Nodal analysis at v1 and v+ gives
v1 (G1  G 2  sC 1 )  vo sC 1  v s G1  v  G 2  0
v  (G 2  sC 2 )  v1G 2  0
Using v o  Kv  , where K  1  Rb Ra gives
T (s) 
G G
vo
 K  1 2
vs
 C1C 2



 2
 G 2 1  K )  G1  G 2

 s  s
C2
C1


 G1G 2 
 

 C1C 2 
(17.3-1)
Typically we let C1 = C2 = C and G1 = G2 = G for which
T ( s) 
v0
K 02
 2
v s s  s 0 3  K    02
(17.3-2)
where  0  G C  1 / RC . Note that (23.3-2) is of the low-pass form, with Q = 1/(3 – K) . So Q can be
tuned by means of an adjustment of the ration of Rb/Ra according to
Q  1 2  R B / R A 
(17.3-3)
Since the ratio G2/C2 is consistent throughout equation (17.3-1) we may taper the Sallen-Key biquad by
selecting C2 = aC1 = aC and G2 = aG1 = aG . This modification does not change the characteristic
frequency 0, but will change the form of the expression for quality factor to the form
10
Professor notes
Q  1 1  a  Rb / Ra 
(17.3-4)
In particular, if the gain element is set up for K = 2 (for which Rb/Ra = 1) as shown by figure 17.3.2 and Q
is defined by the tapering factor, then the circuit simplifies to Q  1 a
Figure 17.3-2. Design B of the Sallen-Key for tuneable Q.
Note that we have set the design in normalized form (0 = 1.0r/s), to which we can apply the RC rescaling
algorithm (17.2-5) as shown by figure 17.3-3. In this realization you might note that the parameter
rescaling forms have made usage of the approximation 1 2  0.16 .
Figure 17.3-3. Design B of the Sallen-Key with RC-rescaling to 10kHz and Q stepped through
several values. Also take note that the zero-frequency gain K = 2.0.
Another form of the Sallen-Key topology is the Saraga design shown by figure 17.3-4.
Figure 17.3-4. Saraga low-sensitivity design for the Sallen-Key single-amplifier biquad
11
Professor notes
For the Saraga design, R1 = 1/Q. This topology would then be rescaled via RC rescaling to the frequency
domain of interest.
We can use any number of active components in a circuit, and sometimes it is of practical benefit to do so
as represented by the two-integrator loop of figure 17.3-5. It has three output points, each of which
differs by the factor –0/s .
Figure 17.3-5. Two-integrator loop.
The factor –0/s is generated by the ‘Miller’ integrator
which is a straightforward and simple circuit. The two-integrator loop consists of two Miller integators
and one inverter-summing circuit as shown by figure 17.3-5, for which the output of the summing circuit
will be
vo 
2
1  0 

v o  20 v o  n 2 v I
Q s 
s
(17.3-5)
Collecting like terms and resolving the transfer function from vI to vo yields high-pass transfer function
THP ( s) 
vo
n2 s 2
 2
v I s  s 0 / Q   02
(17.3-6)
From this equation and the relationship between stages
n 2 0 s
v1 v o   0 

 
 2
vI vI  s 
s  s 0 / Q   02
 TBP (s )
(17.3-7a)
bandpass
n2 02
v 2 vo   0 

 
 2
v I v  s 
s  s 0 / Q   02
 TLP (s )
(17.3-7b)
lowpass
12
Professor notes
and so two-integrator loops are also called state-variable filters since they will provide the three
basic functions of low-pass, high-pass, and bandpass.
The two-integrator loop most often deployed is the Tow-Thomas, or ring-of-three topology
shown by figure 17.3-6. One of the integrators is a lossy integrator, and it is this choice that
gives the circuit its operational benefit.
Figure 17.3-6. Tow-Thomas (ring-of-three) two integrator topology
Nodal analysis at the input of opamp U1 gives
0  v s G3  v 2 G4  v1 G1  sC1 
(17.3-8a)
and since U3 is a unity-gain inverter and U2 is a Miller integrator then
v2  
G2
v1
sC2
(17.3-8b)
Solving for v1 we get
 s G3 C1
v1
 2
v s s  s G1 C1  G2 G4 C1C 2
 TBP (s )
(17.3-9)
bandpass
 TLP (s )
(17.3-10)
lowpass
and from equation (17.3-8b) we get
 G2 G3 C1C 2
v2
 2
v s s  s G1 C1  G2 G4 C1C 2
Typically we would (1) choose C1 = C2 = C and R2 = R4 = R, for which  0  1 / RC . Then (2)
choose Q = R1/R . Then (3) choose amplitude k = R1/R3 . Using this tuning algorithm, the profile
parameters 0, Q, and k are separately adjusted in the order (1) thru (3).
13
Professor notes
The tuning simplicity of the Tow-Thomas topology is even more evident when it is put in
normalized form, for which we would let C1 = C2 = 1.0F and R2 = R4 = 1.0.Then R1 = Q and R3
= Q/k, and the topology is of the form shown by figure 17.3-7, which also includes an extra
summing amplifier for the generation of the bandstop function.
Figure 17.3-7. Tow-Thomas topology in normalized form configured with extra
summing amplifier U4 for the generation of bandstop function
In normalized form equations (17.3-9) and (17.3-10) will be
v1
 sk / Q
 2
vs s  s / Q  1
 TBP (s )
(17.3-11a)
bandpass
v2
 k /Q
 2
vs s  s / Q  1
 TLP (s )
(17.3-11b)
lowpass
The bandstop output is provided by summing opamp U4 for which we have an extra output node
v4 = vs + v2 and for which
v4 vs v2


vs vs vs
1
 sk / Q
2
s  s / Q 1

s 2  s (1  k ) / Q  1
s2  s / Q 1
(17.3-12)
Equation 17.3-12 is of the form of a bandstop function. Its behavior for a simulation in which k
= 0.4 and Q = 4 is shown by figure 17.3-8. If we choose values such that k = 1 then the
bandstop function becomes a band-reject form.
14
Professor notes
Figure 17.3-8. Tow-Thomas simulation using tuning algorithm and normalized form as rescaled
to 1.0kHz. Low-pass and bandstop outputs are shown. Peak amplitude of the low-pass is 0.4 and
the minimum value of the bandstop is 0.6. Note that the Tow-Thomas topology acts as a filter and
not as a resonator (i.e. no outputs have greater magnitude than vs).
If we choose k = 2, then a special case biquadratic function occurs at v4 with transfer function
T (s) 
s2  s / Q 1
s2  s / Q 1
(17.3-13)
This function is of the form of an all-pass function and serves only to produce a phase shift. The
phase shift = -90o at the characteristic frequency f0 = 20.
17.4 BIQUADRATIC FILTER FUNCTIONS
If we factor the biquadratic denominator D(s)  s 2  s  0 Q   02 the roots are
s
0 
1  j 4Q 2  1 
2Q 

and
s
0 
1  j 4Q 2  1 
2Q 

(17.4-1)
for which it is evident that the roots are complex when Q > 0.5.
The s-plane is complex and these roots are located in the left (negative) half of the s-plane. As
the parameters 0 and Q,are varied a root-locus plot of these roots will result, as represented by
figure 17.1-1.
15
Professor notes
Figure 17.4-1. Root-locus plot for the poles of a biquadratic function. This figure is a
modified copy Figure 16.5-1 with more emphasis toward the role of Q.
In particular, we can see that the root-locus plot for 0 = constant is a circle with radius 0. If
we determine the magnitude of either of the roots identified by equation 17.4-1, we find that
s
0
2Q


 1  4Q 2  1
 0
(17.4-2a)
and that different values of Q correspond to different values of the angle , as shown by figure
17.4-1, and for which
cos  
0
(17.4-2b)
2Q
We often use the roots and root loci to identify different profiles according to location of the
poles and zeros. For example the fourth-order lowpass Butterworth profile, which will have a
roll-off of 4 x (-20dB/dec) is of the form
H (s) 
k
1  s /  0 4
(17.4-3a)
s   0 e j (  2k ) / 2n   0 e jk
(17.4-3b)
and will have roots at
for n = 4, and where k = 0, 1, 2 … n. And the equation looks like the same entertainment you used to
have with the mathematics of complex variables.
If n = 2, then the two roots that fall in the left half plane are at k = (3/4) and (5/4), both of which relate
to the angle = 45o, for which Q = 1 2 . So the maximally flat profile by another name is the
Butterworth profile.
16
Professor notes
If n = 4, then the four roots that fall in the left half plane are at k = (5/8), (7/8), (9/8), (11/8). These
correspond to two values of Q = 1/(2cos(22.5o)) = 0.541, and Q = 1/(2cos(67.5o)) =1.307, and can be
accomplished by a couple of Sallen-Key topologies in cascade, as shown by figure 17.4-2.
Figure 17.4-2 Use of Sallen-key topologies in cascade to generate 4th-order (maximally-flat)
Butterworth response. Take note that the roll-off slope is –80dB/decade (same as 4 x –20dB/dec)
The root-locus plots are also of advantage in the analysis of profiles that makes use of special
mathematical functions to define the profile and which will have a sharper cut-off at the break frequency
fc of the profile. A fairly popular class of sharp cutoff functions is based on Chebyshev polynomials
according to transfer function
H ( s) 
k
1   cos C n ( /  c ) 
2
2

(17.4-4)
2

where the function C n ( x)  n cos 1  /  c is a Chebyshev polynomial, and is probably familiar to you
only if you are well-versed in special mathematical functions. The factor  is called the ripple factor and
causes a ripple in the pass-band. As represented by figure 17.4-3, which shows a 3dB ripple
corresponding to  = 1. Cutoff frequency is indicated by f c  2   c
Figure 17.4-3. 4th-order Chebyshev profile with  = 1.
17
Professor notes
And if we indulge in the mathematics we would find that the choice of 01 = 0.951c , Q1 = 5.59 and 02
= 0.443c , Q2 = 1.075 for two Sallen-Key biquads in series will given us the –3dB Chebyshev 4th-order
profile, as shown by figure 17.4-4.
Figure 17.4-3. Sallen-Key implementation of 4th-order Chebyshev profile with  = 1. The accompanying
trace is that for the 4th-order Butterworth profile.
A source of biquadratic topologies with names and provenance is located under the Texas Instruments
Split-supply Analog Filter expert. It not only offers topologies for the standard biquadratic forms but
deploys them according to the more common filter properties (i.e. Butterworth, Bessel and Chebyshev).
17.5 DOUBLY-TERMINATED RLC LADDERS and TRANSFORMATIONS
The section on biquadratic forms is a small slice of the collection of frequency profiles that are
standardized. For applications and realizations beyond the biquadratic there are two means by
which we may glean higher-order profiles: (1) poles and zeros defined by polynomials (e.g.
Butterworth, Chebyshev, Bessel) and (2) RLC tables. The polynomials are probably more
mathematically satisfying but the tables are more convenient. It is not uncommon to define the
rescaling and transformation algorithms in terms of a doubly-terminated RLC topology, such as
is represented by figure 17.5-1. A subset of the tables are shown by figure 17.5-2 and credit for
them belongs to one of signature resources for analog filter design (M.E.Van Valkenburgh,
Analog Filter Design, HRW, 1982. Unfortunately much of its content is not in softcopy and so
the tables have to be included in hardcopy.
Figure 17.5-1. Doubly-terminated RLC ladder topologies (equivalent forms)
18
Professor notes
Figure 17.5-2. Doubly-terminated RLC ladder tables (this is only a subset). The strip index at
the top beginning with L1 is associated with the RLC ladder above the tables and the and the
strip index at the bottom beginning with C1 is associated with the RLC ladder below the tables.
19
Professor notes
Direct simulations using the RLC ladder topologies and table values will yield a normalized lowpass profile normalized. The categories (or polynomial types) are listed above the tables. All of
the values are in Ohms, Farads, or Henrys, depending on the component. The rest of the story is
that the tables are the touchstone for a profile of any of the generic types, low-pass, high-pass,
band-pass, band-stop. Profiles are realized by means of (1) transformation and (2) frequency
rescaling.
The mathematics of making a transformation is entertaining, usually with the idea that a transfer
function T(s) is a function of impedance/admittance ratios and the fact that if we transform them
in the same way, the ratio will remain the same. Add a translation to this context and a low-pass
can be translated into an equivalent form with two shoulders instead of one, if we regard the lowpass as a one-shoulder profile centered about zero. All that is needed is to shift the zero and the
other shoulder emerges.
But the greater benefit of the process is that an RLC circuit can be translated into a form that
employs impedance-converter circuits with opamps to eliminate the bulky inductances. And
that takes the process into a form that can be micro-miniaturized since a VLSI chip can literally
contain dozens of opamps.
The active-filter context (using opamps) is then resolved as an R-C circuit, which can be
translated to whatever frequency characteristic is needed using the algorithm of figure 17.2-1 or
its equations (17.2-5). And it surely makes life more simple than the mathematical divinations.
For example take a look at a transformation called the RLC:CRD transformation, so-called
because the R is recast as a C, the L is recast as an R and the C is recast as an impedanceconverted element (= D) called a frequency-converted negative resistance (FDNR). The concept
is illustrated by Figure 17.5-3.
Figure 17.5-3. RLC:CRD transformation
The technique is also called the Bruton transformation and is primarily a mathematical
manipulation, as represented by equation (17.5-2)
Figure 17.5-3(a) has transfer function
T ( s) 
v2
1 sC

v1 R  sL  1 sC
(17.5-1)
And figure 17.5-3(b) has transfer function
20
Professor notes
v2
1 s 2C
T ( s) 

v1 R s  L  1 s 2 C
1 s2D

1 sC '  R '  1 s 2 D
(17.5-2)
Where (17.5-2) is (17.5-1) with both numerator and denominator multiplied by (1/s).
And the change of labeling does not bother the computation.
But the new component is an opamp construct derived from the Fleige topology, as represented
by figure 17.5-3.
Figure 17.5-3. (a) Generalized impedance converter (GIC) topology and (b) translation into
FDNR ( = D component)
Analysis of figure (17.5-3(a) is a matter of nodal analysis at each of the three nodes labeled v0.
They are all virtually connected because of the nullator action of the opamps. Eliminating v1 and
v2 from the resulting three equations gives
Y1Y3Y5  Y2Y4Y6  0
(17.5-2)
Which can be solved for Yin = -Y1 as
Yin 
Y2Y4Y6
Y3Y5
(17.5-3)
With the component choices indicted by figure 17.5-3(b) and making the choice R3 = R4 we have
the desired FDNR behavior
Yin  s 2 C 2 C 6 R5  s 2 D
(17.5-4)
where the magnitude of D is then  C 2 C 6 R5 .
21
Professor notes
The process is just a bit of mathematics. But the mathematics more easily accomplished if
developed on a simulation platform as represented in previous section of this chapter. The
process is illustrated by figures 17.5-4 and 17.5-5
Figure 17.5-4(a) Doubly-terminated RLC topology for a5th-order Chebyshev with 1.0dB ripple
in the passband. Note that the parameter values are the number values from Chebyshev(C) of
figure 17.5-2.
Figure 17.5-4(b) Simulation results. Note that the breakpoint is f0 = (1/2) x 1.0 = 160 mHz.
So this profile is the normalized form.
Figure 17.5-5(a) Same as figure 17.5-4(a) except implemented in CRD form. The L’s are
replaced by R’s, the R’s are replaced by C’s, and the capacitances are replaced by FDNR’s.
22
Professor notes
The parameterization in figure 17.5-5(a) is doing all of the mathematics grunt work, to include
the frequency rescaling. If you look at the very bottom of the figure you will see that the
capacitance has been chosen to be 1.0 F. And then the rescaling algorithm according to
equations (17.2-5) is located in the parameter list in the lower left corner. Notice that the 1.0r/s
normalized frequency is translated into 0.16 since the simulation is always in Hz. The units are
omitted since it is the mathematics that carries the process along. The scaling factor shown as
kR is applied to all resistances, exactly as prescribed by equation (17.2-5(b)).
The diagram is a little busy, but you should take note of a few simplifications that are used and
are not uncommon.
Due to the RLC:CRD transformation the capacitances which are derived directly from the table
relate to the R = 1.0 value at the front end. The other capacitances all relate to this one. But if
we elect to relate them to this one capacitance with C = 1.0F as equal values, then the FDNR
value of D from equation (17.5-4) becomes
D = C2 C6 R5 = 1.0 x 1.0 x R5
= R5
(17.5-5)
So it is not uncommon to deploy the CRD representation in the form of an equal capacitance
realization.
And that lets the rescaling be a relatively simple process, as indicated by the parametric usage in
the lower right corner of figure 17.5-5(a).
For the final frequency (labeled as fc in the figure), the rescaled CRD realization gives the
frequency profile result as shown by figure 17.5-5(b).
Figure 17.5-5(b) Simulation results. Both normalized profile and rescaled profile at fC = 10
kHz are shown as a comparison to confirm that the transformation/rescaling does realize that
same profile but at a different frequency.
23
Professor notes
There are a few extra wrinkles that are necessary which the mathematics does not represent. The
capacitances that replaced the terminations (at each end) leave the circuit components floating
relative to ground, and therefore must be accompanied by leakage resistances (shown as 999M
values). The circuit simulator will also abort and give an error this effect, since its numerical
processes use increments (also called Newton-Raphson iteration) to deal with the signals and
non-linear components.
The RLC-CRD process is a generic form of active-filter design. Others that may be considered
without diving into too much mathematics are ‘Leap-Frog’ designs, which use a clever
philosophy to carry the RLC ladder into a form that readily adapts itself to the low-pass to bandpass (LP:BP) transformation. But in the interest of keeping the very broad topic of frequency
profiling compacted it will be omitted in this text. Regrets.
24
Professor notes
PORTFOLIO and SUMMARY
RC frequency rescaling
M 
C init
 init
C final
R final 
and
M
Rinit
 final
Sallen-Key biquad (lowpass ) = tuneable Q
0 = 1/RC
Q  1 1  a  R f R x 
where a = tapering factor
Tow-Thomas biquad = biquad states for (1) low-pass, (2) band-pass, (3) band-stop, (4) all-pass
0 = 1/RC = 1.0 r/s
normalized
Normalized values:
C1 = C2 = 1.0F
R2 = R4 = 1.0
R1 = Q
R3 = Q/k
TBS (s) 
v S  v1 v4 s 2  k / Q  1


vs
vs s 2  s / Q  1
TBP (s) 
v1
 sk / Q
 2
vs s  s / Q  1
TLP (s) 
v2
 k /Q
 2
vs s  s / Q  1
TAP(s) = TBS(s) when k = 1
Butterworth profiles
= maximally flat in the pass-band
Chebyshev profiles
= maximum rolloff at the pass-band corner
Bessel (Thomson)
= maximum linearity in the phase response, minimum overshoot for impulse
signals, minimum delay of signals
25
Professor notes
Simulations: 5th-order low-pass. Profiles and pulse response
Chebyshev (1.0dB)
Butterworth
Bessel
26