PROFESSOR‘S NOTES
vers 1.1
24.1 ACTIVE FILTERS AND FREQUENCY PROFILES:
All circuits have a frequency response characterized by the mix of reactive components and active devices. In some
instances, we merely analyze the frequency character of the circuit, and assess it to see if it will accomplish our signal
processing requirement. But in other instances we desire to command the circuit‘s frequency behavior, and define
the frequency character, or 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
These are represented by figure 24.1–1.
|T|
(dB)
|T|
(dB)
low–pass
high–pass
log10 f
|T|
(dB)
log10 f
|T|
(dB)
band–pass
band–stop
log10 f
log10 f
Figure 24.1–1: Basic frequency profiles
Intermediate– and high–frequency profiles use small components, capacitances in pF, inductances in H. Components are small and compact. Profiles are readily accomplished by means of judiciously–constructed RLC networks.
The principal concern in high–frequency profiling is the effect of the circuit parasitics on the poles and zeros. Parasitics include leakage paths, wiring inductances, and fringing capacitances. Frequency profiles in excess of 500 MHz
may use resonant cavities or artificial crystals to define their frequency character.
240
At lower frequencies, typically associated with audio systems, biological interface systems, or feedback control systems, components may be large and cumbersome, and therefore techniques have been developed in which active drivers are used to replace components or reconfigure a circuit into one which may be cast into integrated–circuit form.
These types of circuits are called active filters. An active filter is a frequency–responsive network driven by one or
more active drivers. The typical driver is an opamp, or one of its cousins. The frequency–responsive network consists
of resistances and frequency–active components. Typically, the active driver is used to eliminate one of the more
cumbersome types of frequency–active components, such as the inductances.
For these profiles, one or more characteristic frequencies are usually necessary to define the stop– and pass–band
edges. Edge and amplitude definitions are represented by figure 24.1–2, for the four basic profiles
')(*'
')(*'
%$&
%$&
!"#
6 87:9;,<>=?
#
+,-
!
6 7A@> B,C=?
')(*'
')(5'
%$&
%$&
4 0
.- .-0
#-3
!3
-
6
D 7AE?
+F
!,1
#21
+,
-
!3
#-3
#21
/-
!,1 +,-
6 +7AEG-
Figure 24–1.2. Edge and amplitude definition of the four basic frequency profiles.
Note that for bandpass and bandstop profiles, we must provide parameters to characterize both edges, unless the profiles are symmetric. For frequency profiles of order greater than 2, the pass–band will (usually) include a ”ripple”
amplitude, as illustrated by figure 24–1.3.
241
Figure 24.1–3. Example of amplitude definition: The 4th–order Chebyshev filter profile.
A great deal of attention has been given to active filter techniques, and so many filter types and constructs are identified
in terms of names or descriptions. We will not attempt to survey all the different types, except to say that there are
usually enough options to satisfy normal profiling and signal–handling requirements. In most instances these will be
tabulated, and we may take table values, load and go, and construct the circuit, provided we have enough knowledge
and perspective so that we may make a wise choice of technique.
Low–pass and high–pass filter types, such as Chebyshev, Thompson, Elliptic, Butterworth, etc, are standard and well–
characterized. They are based on the roots of polynomials and strict mathematical conditions. Normalized versions of
these type filters, for a variety of circuit representations, are compiled in tables associated with particular types of common active ladders.
Biquadratic Circuits:
One of the most basic frequency profiles is the class of functions in which the transfer function T(s) is a ratio of two
quadratics in s (= j ), otherwise called a biquadratic form. The general from of the biquadratic function is:
T
K
N 2s 2 N 1s
s 2 s 0 Q
N0
(24.1–1)
2
0
Table 24.1–1 shows the type of magnitude response that results for different numerator functions. Note that the denominator function is always of the form:
D(s)
s2
s
0
Q
2
0
(24.1–2)
We sometimes call equation (24.1–2) the characteristic function, since it defines a frequency 0 that is characteristic
of the peak, and bandpass factor Q, known as the quality factor of the quadratic function. The ”peak” occurs naturally
for the bandpass function, for which N2 = N0 = 0, and N1 0
242
Table 24.1–1: Characteristics of the biquadratic pass functions
Type
Coefficients
|T| response
(Normalized) function
N0
low–pass
0
T
s2
N2 = 0, N1 = 0
2
0
s
Q
0
2
0
N2
high–pass
0
T
s
N1 = 0, N0 = 0
s 2
2 s
0
Q
2
0
N1
band–pass
0
T
s2
N2 = 0, N0 = 0
s
s
Q
Q 0
0
2
0
N1 = 0
band–stop
T
N2
0, N0
s2
0
s2
s
2
0
Q
0
2
0
N1
all–pass
0
s2
s2
T
N2
0, N0
0
s
s
0
Q
Q
0
2
0
2
0
Note that if we make the choice N1 = 0/Q,
T
then we have, for the bandpass function,
s2
s
s
0
0
Q
Q 2
0
(24.1–3)
which is a normalized form of the bandpass function since |T| = 1 when = 0 . This is a convenient form to illustrate
the meaning of the quality–factor Q. Its response is identified by figure 24.1–3.
243
Figure 24.1–3. Bi–quadratic bandpass function
If we analyze equation (24.1–3) to find the frequency at which |T| solutions:
0
2Q
0
1 2Q
1
2 (= 3dB level), we find that there are four
4Q 2
(24.1–4)
Of these solutions, the only ones which are greater than zero are:
1
0
2Q
0
2Q
1 4Q 2
and
2
0
2Q
0
2Q
1 4Q 2
The difference, , represents the resonance width at 3dB, and will be:
2
1
0
Q
so that the quality factor Q represents the ”sharpness” of the quadratic resonance peak, as
Q 0
(24.1–5)
The resonance peak also manifests itself for the low–pass and high–pass quadratic functions when Q sketches in table 24.1–1 indicate this behavior.
244
1
2 . The
EXAMPLE 24.1–1: We usually have interest in single–amplifier biquadratic circuits since they yield relatively simple building blocks for use in series profiles. The Sallen–Key circuit shown by figure 24.1–4 is such an example.
C2
Figure 24.1–4: Sallen–Key single–amplifier biquad. This is a biquadratic low–pass circuit.
Nodal analysis at v1 and v+ gives:
v 1(G 1
G2
v (G 2
sC 2)
sC 1)
v OsC 1
v 1G 2
v IG 1
v G2
0
0
Using vo = K v+ , ( where K = 1 + RB /RA ), gives:
vO
vI
GG
K 1 2
C 1C 2 s
2
G 2(1 K)
C2
s
G1
C1
G2
G 1G 2
C 1C 2 (24.1–6)
Typically we let C1 = C2 = C and G1 = G2 = G, in which case we get:
vO
vI
where 0
s
2
K
s 0(3
2
0
K)
2
0
(24.1–7)
= G/C = 1/RC. Note that (24.1–7 is of the low–pass form.
For this case we see that Q = 1/(3 – K), and it is necessary that the feedback ratio RB /RA < 2 for stability.
Since the ratio G2 /C2 is consistent throughout equation (24.1–6), we may taper the Sallen–Key biquad by selecting
C2 = C1 = C and G2 = G1 = G. This modification does not change the characteristic frequency 0 , but will
change the form of the expression for quality factor to ”tapered form”
Q
1 (2
a
K)
245
(24.1–8)
EXAMPLE 24.1–2 A more general circuit form, the 2–integrator loop, can produce most, if not all of the basic biquadratic functions. For this reason, it also may be called a state–variable filter. The generalized two–integrator loop
is shown by figure 24.1–5.
+1/Q
–1
vi
0
n2
s
0
v1
v0
v2
s
0
s v0
2
0
v
2
s 0
Figure 24.1–5: Two–integrator loop, general schematic.
where the integrator component usually is the inverting ”Miller” integrator,
s
0
=
The basic feedback circuit consists of two integrator elements and one inverter–summing element in a series loop.
Analysis of the circuit of figure 24.1–5 shows that the output of the summing circuit will be:
v0
1
Q
for which, collecting like terms in v0 and vi .
v 0 s2
0
s v0
s
2
0
2
v0
n 2v i
(24.1–9)
0
Q s 20 n 2s 2v i
(24.1–10)
Resolving equation (24.1–10), we see that the transfer function from vi to vo with then be of a high–pass form:
v0
vi
n s2
2
s2 s Q 0
(24.1–11)
2
0
From equation (24.1–10) and the relationship between stages we see that:
v1
vi
s
0
v0
vi
bandpass
v2
2
0
v0
s 2 v i lowpass
which is why we identify this type of circuit as a state–variable filter, since it provides the three basic biquadratic functions, low–pass, high–pass, and bandpass. Other biquadratic functions, such as the notch and the all–pass can be
created from sums of these functions, usually by means of an extra summing element.
For example, if we add an extra amplifer to perform the sum
v3
then the function
v
v
1 1
1
v i n 2Q v i 1 n 2Q v i 2
s2 0
s2 s Q
0
is created. This is the notch, or band–stop function.
v3
vi
246
2
0
A circuit implementation of the two–integrator loop is shown by figure 24.1–6. The implementation gives results
which are the same as equation (24.1–9), with
R1
R2 R3 1 R
1 R2
R1
1
1
Q R2 R3
R1 R3
n2
(24.1–12a)
(24.1–12b)
Figure 24.1–6. Two–integrator loop – circuit implementation.
Variations of the two–integrator loop exist. For example, the Tow–Thomas state–variable form is shown by figure
24.1–7. In this implementation of the circuit, a simple inverter circuit is used to achieve the polarity necessary for
the feedback function, and one of the integrators is replaced by a lossy integrator, for stability. Nodal analysis gives:
v2 G
4
v 3 sC 2 v 2
(24.1–13)
and
v2
G
G
3
2
G 1 sC 1 v 1 G1 sC 1 v 2 Figure 24.1–7: The Tow–Thomas state–variable filter
247
Resolving the transfer function in terms of v2’ and v1 , (eliminating v2 ) we get
v2
v1
s2
G 4G 3 C 1C 2
sG 1 C 1 G 2G 4 C 1C 2
(24.1–14)
which is low–pass. At node v2 , using equation (24.1–11), we get
v2
v1
which is band–pass.
sG 3 C 1
sG 1 C 1 G 2G 4 C 1C 2
s2
(24.1–15)
The Tow–Thomas circuit is well accepted because it has a good tuning algorithm. The algorithm is as follows:
1. R2 may be adjusted to set 0 .
– Typically, we let C1 = C2 = C and R2 = R4 = R. Then 0 = 1/RC
2. R1 can then be adjusted to define Q without changing 0 . Then Q = R1 /R.
3. R3 can then be adjusted to define amplitude without affecting either Q or 0 . Then |Apeak | = R1 /R3
We can add a fourth opamp as an inverter/summing stage, such that
v4
then, using equation (24.1–13), we get
v2
v1
(v 1
s2
v 2)
v1 1
v2
v1
s(G 1 G 3) C 1 G 2G 4 C 1C 2
sG 1 C 1 G 2G 4 C 1C 2
s
2
(24.1–15)
If G1 = G3 , then equation (24.1–15) is of the form of a notch filter function. If G3 = 2G1 , then equation (24.1–15)
is of the form of an all–pass function.
24.2 TRANSFORMATION AND RESCALING OF FREQUENCY PROFILES:
In defining a frequency profile, it is usually appropriate to mark a frequency about which the rest of the profile will
fall. This frequency typically will be at a symmetry point, a corner, a peak, or a valley. However, this frequency may
be elected from anywhere within the profile, since it is only used as a reference point.
For those profiles that have been standardized, for which placement of poles and zeros are defined according to a specific mathematical criterion, tables are available, just a matter of tracking them down. These tables may either identify
the specific poles and zeros, or if a particular type circuit construct is used, may tabulate values of components. Values
are tabulated in normalized form, i.e. for a reference frequency 0 = 1 r/s, and it is up to the user to rescale values
according to his/her need.
In identifying the rescaling process we will direct our attention primarily to RC circuits since it is possible to recast
most of our circuits, regardless of complexity, into RC form by use of active filter techniques. The process of rescaling
is relatively simple if only one type of frequency–dependent component is present, which in this case would be capacitances. For RC circuits the frequency profile is defined in terms of the poles and zeros that are given entirely by RC
time constants.
Using normal network techniques, a transfer function T(s) for any given circuit is constructed in terms of either admittances or impedances, as indicated by figure 24.2–1. It can be a simple ratio, or it can be a complex mess, but the
frequency characteristics are entirely the result of the frequency characteristics of these basic admittance (or impedance) terms. Therefore we rescale the response T( ) by merely rescaling the individual frequency–active terms, using
a uniform rescaling process.
248
For example, consider the simple, single–time constant RC low–pass circuit indicated by figure 24.2–1a. We let the
reference frequency be the 3dB corner frequency = 1/RC. And for the normalized case, for which 0 = 1 r/s, the
we might have ’normalized’ values of R and C to be R = 1 and C = 1 F.
(a)
(b)
R0 = 1
R1 = 1
C0 = 1
= 1 r/s
0
T
C1 = 1 F
1
G0
sC 0
= 1 Mr/s
YC = 0C0 = 1C1
T
G0
G1
sC 1
G1
1 Mr/s
1 r/s
Figure 24.2–1 Frequency rescaling: Frequency–dependent components (in this case, capacitances) are rescaled to yield the same response at a new frequency.
We see that the same profile is created at a new frequency 1 , as indicated by figure 24.2–1(b) if we just rescale the
frequency–dependent component C such that it has the same admittance behavior at the new corner frequency as it
did at 0 . If we increase the new frequency by a factor of 10, the capacitative admittance will increase by a factor
of 10, just as it did when the corner was at 0 = 1 r/s.
By requiring that the admittance be the same at the new reference frequency
0 we then have
1
which will produce the same profile at new frequency
C1
1
0
1
as it was at the old reference frequency
(24.2–1)
C0
as it did at
0,
provided C1 is rescaled by the factor
0/ 1.
However, since = 1/RC we also realize that the frequency corner could also have been rescaled by an appropriate
rescaling of the resistance(s). This rescaling can only be accomplished independently of the capacitance rescaling
by mathematically removing the frequency–dependence from the capacitances and passing it to the resistances. This
mathematics can be accomplished by examining the transfer function, and noting that if we rewrite the transfer function of the RC low–pass function, and multiply it by the factor s/s, we can lift the frequency dependence from the
capacitances, e.g.
T(s)
1 sC 1
1 sC 1
G1
G1
sC 1
R1
s
s
1 C1
1 C1
sR 1
With transfer function and components thus changed, we may rescale to yet a new frequency
2
R2
1
R1
2
by means of
(24.2–2)
where R1 = R0 . Equation (24.2–2) rescales only the resistance. Capacitances does not rescale during this process,
so that C2 = C1 .
What is evident from this two–step process is that we may rescale both the frequency–active components (capacitances) and the frequency–passive components (resistances) independently of one another by passing through an intermediate rescaling frequency 1 . The entire transformation takes us from an intiial frequency 0 to a final frequency
249
2 , with the sole purpose of frequency 1 being to rescale one type of component, usually capacitances, to a more
convenient magnitude scale, such as F, as indicated by figure 24.2–1. Continuing the example given with figure
24.2–1, we may desire to have the same STC low–pass profile fall at 2 = 2 kr/s, for which we would rescale the value
of resistance, R0 , by equation (24.2–2) to the new value:
R2
1
2
R1
2
10 6
10 3
1
500
while the capacitance has (already been) rescaled to:
C2
C1
0
1
C0
1
10 6
1F
1 F
We can, of course, take this same process and apply it to a circuit in which we have many resistances and capacitances,
for which the entire set of resistances and capacitances will be rescaled according to an initial frequency 0 , a final
frequency f , and an intermediate frequency M . The rescaling algorithm for RC circuits, in general, may be taken
from equations (24–2–1) and (24.2–2), as:
Cf
0
M
Ci
Rf
M
f
Ri
(24.2–3)
where {Ci , Ri } are the initial values of the set of the set of resistances and capacitances, and {Cf , Rf } are the final values,
and where M is the intermediate rescaling frequency. As indicated by the example, the rescaling frequency M is
usually chosen to rescale one type of component, usually capacitances, to a more convenient scale value. It may be
defined by taking one of the capacitances and defining a final value for this capacitance, in which case M will be
given by:
M
0
C i1
C f1
(24.2–4)
where Ci1 and Cf1 represent the initial and final values of the selected capacitance.
24.3 THE RC–CR TRANSFORMATION
As we investigate the set of pass functions, we find that the fundamental profile is low–pass. We can derive other pass
functions by means of frequency transformations.
One of the frequency transformations is a simple complementary transformation. Low– and high–pass functions are
complementary about their 3dB corners, inasmuch as the 3dB corner represents the frequency at which the reactive
impedance is equal to the non–reactive impedance. We can see the complementary behavior by comparing the RC
(=single–time–constant) (STC) circuit for low– and high–pass circuit forms:
C
R
R
C
Figure 24.3–1: The RC–CR transformation.
250
Note that when |1/j C| = R, then the transfer function becomes
v2
v1
1
2
1 p
1
2
(24.3–1)
2
1
where p1 = 1/RC. This frequency represents the ”corner” frequency of the Bode’ magnitude plot, a reference frequency at which the transfer function has the value –3 dB. As we increase the frequency , the magnitude of the transfer
function will change, corresponding to whether the reactive impedance decreases the ratio (low–pass) or increases
it (hi–pass).
RC–CR conversion, in which a low–pass function is transformed to a high–pass function with the same corner, must
follow the impedance effects, referenced to that point at which they are equal, which is the 3dB level. If we desire
to invert the impedance effects, we must change the gender of the components centered at the frequency at which the
effects balance. This transformation is achieved when the impedance of each reactive component is equal to the impedance of an equivalent non–reactive component, and conversely. The transformation must then occur for:
C R
1
k
k
(24.3–2)
Ck
3dB
(24.3–3)
1
Rk
3dB
where R’, C’ represent the replacement values of resistance and capacitance, respectively.
24.4 RLC LADDER CIRCUITS
As indicated by section 24.3 we find that it is appropriate to base our circuits on the low–pass frequency profile and then
apply a transformation to take us to another profile. If we examine low–pass profiles in the general sense, we perceive
low–pass filters as being those which suppress all frequencies above a ’band–edge’. After we have surveyed at some of
the basic filters, e.g. one with a single resistance and capacitance, we find that the band–edge may be relatively ”soft”.
Many times we would like the cut–off edge to be much more abrupt, or to have other features, such as uniform phase–
shift of pulse signals above and below the edge.
We get more ’abrupt’ cut–off if we add more poles, via additional reactive components. Rolloff is accomplished by
the increase of the order of the rolloff profile. The rolloff is approximately (20 dB/decade order ).
As an example we consider the 5th–order Chebyshev profile, which has a very abrupt cutoff, on the order of 100 dB/decade, accomplished by placing poles and zeros according to the Chebychev functions. We will not undertake any of
the discipline and entertainment associated with network analysis of the circuit, but will assume that tabulated results
for placement of poles and zeros, as ordained by many previous analyses of this profile, are reasonable and accurate.
The Chebyshev profile and many other profiles can be straightforwardly implemented by means of the doubly–terminated RLC ladder, as shown by figure 24.4–1. The figure shows a 5th–order ladder. Note that it has five (5) frequency–
active components. And since the inductances are in series and the capacitances in parallel, we should recognize that
this configuration is of low–pass form. By judicious choice of component values, poles and zeros may be defined such
that this circuit is a 5th–order Chebyshev profile.
R1
L4
L2
C3
L6
C5
Figure 24.4–1. Example of (5th–order) doubly–terminated ladder.
251
R7
It might be noted that RLC ladder circuits are favored for frequency profiling since the circuit is relatively insensitive
to small variations of component values, a statement that we will not prove, but merely accept these types of circuits
as a robust and are a convenient baseline. For active filter implementation we prefer that some component types, usually the inductances, be converted to equivalent active forms. As we will see, the doubly–terminated ladder lends itself
readily to component transformations, and to transformations into other type pass functions,
As indicated by tables 24.4–1, and 24.4–2, doubly–terminated RLC low–pass filter profiles are readily available in
tabulated form. The tables are invariably based on a normalized corner frequency of 0 = 1 r/s.
Table 24.4–1: Table of doubly–terminated RLC ladder values for normalized Butterworth low–pass response.
1
L4
L2
C1
C3
Ln
C5
n
2
3
4
5
6
7
C1
1.414
1.000
0.7654
0.6180
0.5176
0.4450
L2
1.414
2.000
1.848
1.618
1.414
1.247
C3
L4
1.000
1.848
2.000
1.932
1.802
0.7654
1.618
1.932
2.000
0.6180
1.414 0.5176
1.802 1.247
0.4450
8
0.3902
1.111
1.663
1.962
1.962
1.663
1.111 0.3902
9
10
0.3473
0.3129
1.000 1.532
0.9080 1.414
1.879
1.782
2.000
1.975
1.879
1.975
1.532
1.782
n
L1
C4
L5
C6
C2
L3
1
C5
L6
1
L3
L1
C2
L7
L8
C9
L10
1.000 0.3473
1.414 0.9080 0.3129
C8
L9
Ln
C4
252
C7
1
C10
Table 24.4–2: Table of doubly–terminated RLC ladder values for normalized Chebyshev low–pass response.
1
L4
L2
C1
n
C1
L2
C5
C3
L4
C3
C5
L6
R2
C7
L8
R2
(A) Ripple = 0.1 dB
2
3
4
5
6
7
0.84304
1.03156
1.10879
1.14681
1.16811
1.18118
0.62201
1.14740
1.30618
1.37121
1.40397
1.42281
1.03156
1.77035
1.97500
2.05621
2.09667
0.81807
1.37121
1.51709
1.57340
1.14681
1.90280
2.09667
0.86184
1.42281
1.18118
8
1.18975
1.43465
2.11990
1.60101
2.16995
1.58408
1.94447
0.73781
1.00000
0.73781
1.00000
0.73781
1.00000
0.87781
0.73781
(B) Ripple = 0.5 dB
3
1.5963
1.0967
1.5963
5
7
1.7058
1.7373
1.2296
1.2582
2.5408
2.6383
1.2296
1.3443
1.7058
2.6383
2.1349
3.0936
1.00000
1.2582
1.00000
1.00000
1.7373
(C) Ripple = 1.0 dB
1.00000
3
5
7
2.0236
2.1349
2.1666
0.9941
1.0911
1.1115
2.0236
3.0009
3.0936
1.0911
1.1735
n
L1
C2
L3
C4
1
1.1115
L5
C6
L3
L1
C2
L7
C8
Ln
C4
253
1.00000
1.00000
2.1666
R2
R2
24.5 RLC:CRD TRANSFORMATIONS:
Circuits such as that shown by figure 24.4–1 can be implemented in active form using only resistances and capacitances by executing a mathematical transformation of the circuit into one in which each type component has an equivalent. If we desire to transform an RL circuit into an RC equivalent, then it is necessary to multiply the the numerator
N(s) and the denominator D(s) of the transfer function by 1/s. This process is represented by figure 24.5–1.
(a)
L
T
R
transformation:
T
R
R
(b)
R
R
sL
sL
1 s
1 s
0
= R/L
0
= 1/R’C’
R s
R s L
R’
C’
T
1 sC 1 sC R R 1/C’
L R’
Figure 24.5–1 Transformation process in which the LR circuit is transformed into an equivalent RC circuit.
In this process we note that the L is transformed into an R’ and the R is transformed into a C’. The same characteristic
frequency 0 results provided that the magnitude of R’ is the same as the magnitude of L and the magnitude of C’ is
the same as the magnitude of 1/R. This type transformation is of the form RL:CR
But when we have RLC circuits, the transformation must include all three type components and we cannot eliminate
the inductance L unless we define a new component derived from capacitance C. The technique is called the
RLC:CRD transformation and is much like the RL:CR transformation, except C is transformed into an active equivalent = D. Otherwise for the transformation R is transformed into an equivalent C, L is transformed into an equivalent
R.
For the RLC:CRD transformation we imply that R C, L R, and C D. The transformed circuit is represented
by figure 24.5–2.
C1
R4
R2
D3
R6
D5
C7
Figure 24.5–2. Same circuit as figure 24.4–1, (RLC doubly–terminated ladder), transformed into CRD form.
If the same transformation as used for the LR:RC in which numerator and denominator components are are multiplied
by 1/s is applied to and RLC circuit, then each capacitative impedance must be transformed into a component of frequency behavior:
1 sC 1 s 1 s 2C 1 s 2D 254
The strange–looking component, D, therefore depends on frequency as the square of s. This response is of the form of a
frequency–dependent negative resistance. It is naturally an active component, and may be implemented by means of
the GIN (generalized immitance network) circuit, as shown by figure 24.5–3 (below).
Z IN
! #"
Figure 24.5–3. GIN configured as frequency dependent negative resistance (FDNR)
Evaluate this thing and you will find that the GIN component has an admittance proportional to the square of the frequency. The magnitude of its admittance is
|Z D|
$&%('
2
D
$&%('
2
C 2C 6R 5
This component will have the measure of a negative resistance with magnitude proportional to '
2.
RESCALING: As indicated by previous sections, tabulated frequency profiles, such as the Chebyshev are always
given for ’normalized’ at frequency ' 0 = 1 r/s and must be rescaled to the frequency of interest. For RLC:CRD transformations we might note that the resistances will be transformed into capacitances. In prototyping of circuits, we
often select a given value of capacitance, particularly if it is used more than once, and let it determine the scaling factor.
It is convenient, so let us so do. Capacitance magnitudes will be transformed from a starting value of resistance according to C’ = 1/R. As capacitances, rescaling will take place after the transformation, for which
'
M
)
1
Ri
$*'
Cf
f
where ' M is the intermediate scaling frequency and ' f is the desired (final) characteristic frequency of the given circuit. Using this value of scaling frequency all capacitances will scale according to
Cf
$ ''
M
f
1
Ri
(24.5–1)
Equation (25.5–1) may be used to get values for all of the capacitance in figure 24.4–2. Note that we must use ' f , the
frequency in rad/sec, since the profiles are always normalized in terms of the unity radian frequency ' 0 = 1 r/s.
Since L + R , all inductances will be transformed and rescaled to resistances. Frequency rescaling must take place
before before they are transformed into resistances, for which:
Rf
$ ''
0
M
Li
255
(24.5–2)
Equation (24.4–2) may be used to get values for all rescaled resistances represented in figure 24.4–3.
The FDNR magnitude must rescale both before and after transformation. Before transformation, the magnitude rescales as ( 0/ M). After transformation, the magnitude rescales as ( M/ f)2. The rescaled value of D is then
2
Df
0
M
Ci
M
f
(24.5–3)
You might take note that the rescaling of the FDNR using equation (24.5–3) is equivalent to transforming its internal
components C2 , C6 , and R5, in terms of transformations that are the same as those of equations (24.5–1) and (24.5–2).
Usually it is of benefit to elect these internal capacitances to be of the same value as those external to the FDNR.
********************************************************************************************
EXAMPLE 24.4–1 RLC:CRD Transformation for an LP5C1 profile.
Let us develop the RLC:CRD transformation for a typical case: 5th–order low–pass single–amplifier Chebyshev configuration with 1 dB ripple. The values for the R’s, L’s and C’s shown in figure 24.5–4, were taken from table 24.4–2(C)
and correspond the the 5th–order Chebyshev profile with 1 dB ripple for which the break frequency = 1 r/s. Since these
component values are defined in terms of characteristic frequency 0 = 1 r/s, they are in unit measures of Farads, Henries, and ohms, representing unwieldy sizes of components, at best. The circuit must be renormalized (rescaled) to
a characteristic frequency f and components rescaled to more realistic magnitudes.
1.0
3.0009
2.1349
1.0911
2.1349
1.0911
1.0
Figure 24.5–4: 5th–order 1 dB Chebychev doubly–terminated ladder.
If we make a direct RLC:CRD transformation, without changing the characteristic frequency, we end up with a circuit
of the form shown by figure 24.5–5.
1.0
3.0009
2.1349
2.1349
v2
v1
RA = 33
1.0
Figure 24.5–5: Normalized (
0
= 1 r/s ) FDNR implementation of figure 24.5–4
256
RB = 40
Note that we must include resistances RA and RB to accommodate low frequencies and provide a DC path for the non–
inverting terminals of the opamps. The values selected are such that R >> 0 C and such that
v2
v1
0.5
RB
RA
R2
R4
(24.4–5)
RB
so that the transfer gain is 0.5 when f << f0 .
Now we must rescale the values of capacitance and resistance so that the same profile is located at the designated characteristic frequency. Our process is greatly simplified by the fact that the table values are based on an equal–termination resistance configuration. It is often convenient to choose a final value of capacitance, from which we can define
the scaling factor. It is usually also worthwhile to let this capacitance value be used everywhere within the circuit,
particularly is we intend to build it, rather than just entertain ourselves with the mathematics. And it is often convenient
to develop a spreadsheet to detail the renormalization process and the rescaled values of the components, and this is
shown by figure 24.5–7.
In this example we chose to define the profile with break frequency at f0 = 10 kHz and to define the circuit such that a
final value of capacitance C = 1050 pF was used everywhere. This value of capacitance was used because we had a box
of about 10,000 capacitances of this particular size on hand. The results, after making the necessary rescaling transformations using equations (24.5–1), (24.5–2), and (24.5–3), are shown by figure 24.5–6.
1050
32.36
45.49
32.36
v2
v1
1050
1050
500
10
10
1050
10
10
606
16.54
16.54
1050
1050
Figure 24.5–6: FDNR implementation of figure 24.5–4 for corner frequency 10kHz and all capacitances
selected as 1050 pF. All resistance values are in k .
257
Figure 24.5–7: Spreadsheet matrix for the 5th–order low–pass Chebyshev response
In the calculation matrix we did not identify the (intermediate) scaling frequency M . In this case it is defined by our
choice of capacitance value, C = 1050 pF, for which, according to equation (24.5–1), we have:
M
C f fR 1
1050pF
(2 10 4)
1
.066mr s
where R1 = 1 , this case.
If everything has been done properly, a SPICE analysis of this circuit will look something like that shown by figure
24.5–8
258
Figure 24.5–8. 5th–order low–pass Chebyshev response
24.6: BANDPASS IMPLEMENTATIONS FROM LOW–PASS RLC LADDER
Because of its relative insensitivity to component variations, the doubly–terminated RLC ladder is favored as a means
of defining higher–order polynomial profiles. The 3rd–order doubly–terminated RLC ladder is represented by figure
24.6–1
R1
L2
L4
C3
R5
Figure 24.6–1. Example of 3rd–order doubly–terminated ladder.
259
From the placement of components, it’s evident that this ladder has low–pass behavior, since the inductances pass low
frequencies and the capacitance will block low frequencies.
Almost all standard frequency profiles are defined in terms of low–pass forms. In order to transform the RLC ladder
from low–pass into a bandpass filter we have to shift the frequency origin. The transformation that will achieve this
shift is illustrated by figure 24.6–2.
L
BW
L
C
BW
C
BW
L 20
BW
C 20
Figure 24.6–2. Transformation necessary for low–pass to band–pass.
where 0 is the characteristic (center) frequency of the bandpass profile, and BW is the 3dB pass–band width, as illustrated by figure 24.6–3
|T|
BW
0
Figure 24.6–3. Band–pass characteristics
The LP–BP transformation effectively shifts the point which we call ”zero frequency” to some reference frequency 0 .
The RLC ladder becomes a little more complicated but, in our favor, a fairly standard equivalent circuit technique is
available for realizing circuits of this or any RLC type form. The technique called leapfrog simulation of a ladder, and
is illustrated by figure 24.6–4.
260
Y1
Y3
+
V2
–
V1
V1
+
V4
–
Z2
–TY1
Y5
Z4
TZ2
Z6
–TY3
RLC Ladder
V6
TZ4
–TY5
–V2
TZ6
–V6
V4
Leapfrog equivalent
Figure 24.6–4. Leapfrog techniques for simulation of a (RLC) ladder
For a ladder construction, as indicated by figure 24.6–4a, network analysis gives us the set of equations:
I1
V2
I3
V4
I5
Y 1(V 1
Z 2(I 1
Y 3(V 2
Z 4(I 3
Y 5(V 4
(24.6–1a)
(24.6–1b)
(24.6–1c)
(24.6–1d)
(24.6–1e)
V 2)
I 3)
V 4)
I 5)
V 6)
Each of these equations represents a transfer function relationship, and can be expressed as such:
V I1
V2
V I3
V4
V I5
T Y1(V 1
T Z2(V I1
T Y3(V 2
T Z4(V I3
T Y5(V 4
V 2)
V I3)
V 4)
V I5)
V 6)
(24.6–2a)
(24.6–2b)
(24.6–2c)
(24.6–2d)
(24.6–2e)
In each case we see that the right–hand side of these transfer functions makes use of a simple difference of two inputs.
To our great satisfaction we see that each of these inputs is available on the left–hand side, with exception of only V1 ,
which is THE input.
If we instead write the difference as addition of a negative, then we may find it to be advantageous to set up our set of
equations accordingly. For example:
V2
T Z2[(
V I1)
V I3]
261
(24.6–3b)
provides us one of the inputs (–V2 ) that is needed in equation (24.6–2a).
Now we see that equation (24.6–3b) needs a term (–VI1 ). Therefore we might rewrite (24.6–2a) as:
(
V I1)
T Y1[V 1
(
V 2)]
(24.6–3a)
If we continue with this approach through the whole list of transfer functions then we will have the set:
V I1 V2 V I3 V4 V I5 T Y1[V 1 (
T Z2[( V I1)
T Y3[( V 2)
T Z4[V I3 (
T Y5[V 4 (
V 2)]
V I3]
V 4]
V I5)]
V 6)]
(24.6–3a)
(24.6–3b)
(24.6–3c)
(24.6–3d)
(24.6–3e)
This set of equations is exactly the equivalent of the circuit of figure 24.6–4b. Note that the transfer functions alternate
in sign as we progress through the circuit.
The only aspect of leapfrog circuits that creates any complication is the signs of each of the impedance/admittance
terms TZk or TYk , respectively, in the string. In some cases, inverting stages must be inserted into the leapfrog string to
ensure correct sign.
The form of the RLC ladder when converted to bandpass, is of the form shown by figure 24.6–5.
R1
VI
VO
TY1
TY3
R5
TZ2
Figure 24.6–5. 3rd–order RLC ladder converted to bandpass form
The components enclosed by the dashed lines each represent an impedance or admittance term that can be realized
by a biquadratic circuit. The second–order function for each of these component groupings is identified by Figure
24.6–6. Note that, for the leapfrog realization, the series RLC string must relate to an admittance term whereas the
parallel RLC, (LC if R not present) string must relate to an impedance term.
262
L
R
C
Y
(1 C)s
(1 RC)s 1 LC
s2
(1 L)s
(R L)s 1 LC
Figure 24.6–6a. RLC series – admittance function
C
L
Z
R
s2
Figure 24.6–6b. RLC parallel – impedance function
C
L
Z
(1 C)s
s2 1 LC
Figure 24.6–6c. LC parallel lossless – impedance function
Figure 24.6–6. Subcircuits and equivalent (second–order) impedance/admittance functions.
All of these RLC strings can be realized by an active circuit, with transfer function suitable for the leapfrog realization
by use of the Delyannis–Friend single–amplifier biquad. The normalized Friend circuit ( 0 = 1 r/s) and equivalent
transfer function(s) are shown by figure 24.6–7. The two configurations presented are those suitable to the impedance/
admittance functions of figures 24.6–6a and 24.6–6c.
T(s)
s2
2kQs
(1 Q)s 1
Figure 24.6–7a. Friend circuit and transfer function needed for RLC realizations
T(s)
k (1 2Q 2) Q s
s2 1
Figure 24.6–7b. Friend circuit and transfer function needed for LC (lossless) realizations.
Figure 24.6–7. Delyannis–Friend biquadratic forms appropriate for realization of RLC circuits.
263
If the input of either of these configurations is also used as a summing point, then the modification shown by figure
24.6–8 must be made to the input node of the circuit. The transfer function is not affected as long as the parallel combination of the set of input resistances add up to R = 1 (or G = 1).
T(s)
s2
2kQs
(1 Q)s 1
Figure 24.6–8. Friend circuit with summing at input
Using the Friend circuit to realize impedance and admittance functions, the bandpass 3rd–order ladder given by figure
24.6–5 can be implemented in leapfrog form, as represented by figure 24.6–9.
! , &
!#"%$'&
!#,-&
!./0"1, &32
:
!#"%$ &
!4"%$
:
()$ &+*
!#,
*
!4"%$87
()$
*
*
!.5/6, 2
*
!4"%$
*
*
()$ 79*
!#,%7
!.5/6,)7 2
"%$
*
!4"%$ 7
*
:
/6;=<>&
/?;=A
B CED%F :GIH C-J8K+LM
*
/?;@<@7
Figure 24.6–9. Leapfrog Implementation of 3rd–order Chebychev bandpass
Note that the required impedance characteristics can be selected by appropriate choices of k1 , k2 , and k3 , and Q1 , Q2 ,
Q3. Since Q1 , Q2 , Q3 all define capacitances, it is convenient to to choose Q1 = Q2 = Q3 , which will make all capacitances equal. Note that the inverter/sum stage has arbitrary resistance values, the only requirement being that they
be equal.
An example of a bandpass filter for f0 = 10 kHz and BW = 5 kHz is represented by figure 24.6–12. The bandpass filter
is developed from the 3rd–order 1–dB ripple low–pass Chebyshev doubly–terminated ladder shown by figure 24.6–10.
and its bandpass realization, shown by figure 24.6–11.
264
1
2.0236
2.0236
1
0.9941
Figure 24.6–10. 3rd–order 1–dB Chebyshev doubly–terminated RLC ladder
VI
1.0
4.0472
R1
L2
4.0472
0.2471
C2
L4
0.2471
C4
VO
0.503
1.9882
L3
C3
R5
1.0
Figure 24.6–11. Bandpass realization of the normalized 3rd–order 1–dB Chebyshev doubly–terminated ladder. For these values 0 = 1 r/s and BW = 5kHz/10Khz = 0.5. Resistances are in ohms and capacitances are in Farads. This realization was achieved by the component transformation shown by figure 24.6–2.
ANALYSIS: When we compare transfer functions in figure 24.6–7 to the circuit transfer functions in figure 24.6–6,
we see that Q1 = L2 /R1 = 4.0472/1.0 = 4.0472. Since L4 = L2 and R5 = R1 , then Q4 also = 4.0472.
Comparing numerators for the transfer function of figure 24.6–7b to the impedance function of figure 24.6–6c, as
needed to realize stage TZ2 , we see that
2Q 22) Q 2
1 C
k 2(1
(24.6–4)
We have flexibility of electing k2 and then finding Q2 , or electing Q2 and then finding k2 . As noted earlier, it is of
considerable convenience to the qualification process to let Q2 = Q1 = Q4 = 4.0472, since this choice will give us a
circuit for which all capacitances are equal. Using this choice for Q2, then
1 C3
1
k2
0.0603
1.9882 8.341
(1 2Q 22) Q 2
with the choices of Q made equal, all capacitances will be equal. Furthermore, specification of Q will also let us specify several of the resistances.
Other resistances are determined by requiring that numerator of the transfer function for figure 24.6–7a be equal to
the admittance numerator of figure 24.6–6a. This circuit is needed to realize stage TY1 . Stage TY3 has identical requirements except that it does not need to be used as a summing point. This requirement gives
2Q 1k 1
1 L2
1 4.0472
(24.6–5)
Since Q1 = 4.0472, we get k1 = 0.0305. Likewise we get k3 = 0.0305. Now we have defined all of the values of k
and Q, which is sufficient to define all of normalized values of resistance and capacitance. The leapfrog circuit, with
normalized values of resistance and capacitance, is shown by figure 24.6–12.
265
32.85
0.1235
32.85
0.1235
65.52
0.1235
16.61
0.1235
65.52
0.1235
32.85
65.52
0.1235
1.065
1.064
1.03
1
32.76
Figure 24.6–12. Example normalized circuit values for C3 bandpass profile,
0
= 1, BW = 0.5.
Note that a summing inverter is embedded in between TY1 and TZ2 to accommodate the necessary sign change. Note
that resistance r is arbitrary, and we do not need to rescale it to some inconvenient value.
The last step is to rescale capacitances and resistances such that the circuit is implemented at the required characteristic
frequency. RC rescaling, when set by a final value of capacitance Cf , is accomplished by determining an intermediatescaling frequency M such that
M
(C i C f)
(24.6–6)
0
Since the circuit is developed in terms of 0 = 1 r/s, then the intermediate frequency is just M = Ci /Cf. In the example
case, for which we have elected Cf = 1050 pF from initial value Ci = 0.1234 F, we find M = 0.1175 Gr/s. The resistance values will then scale according to
Rf
R i(
M
(24.6–7)
)
f
where f is the final desired frequency. If we apply this rescaling to the values in figure 24.6–12 we end up with the
final design for an f0 = 10 kHz and BW = 5 kHz, as given by figure 24.6–13. For convenience, we let r = 10 k since
it is not necessary to scale this value.
61.4
1050
61.4
1050
122.8
1050
1050
1050
31.0
122.8
61.4
1.99
1.99
1050
122.8
1.93
1.87
61.4
Figure 24.6–13. EXAMPLE FINAL DESIGN: Bandpass implementation for 3rd–order 1–dB Chebyshev
doubly–terminated ladder when f0 = 10 kHz and BW = 5 kHz. Since Q1 = Q2 = Q3 , the scaling factor was
chosen such that all capacitances would be 1050 pF. All resistance values are in k .
266
RECIPE FOR ANALYSIS:
1. Starting with figure 24.6–10, execute an LP–BP transformation and develop values for a figure of the form
of 24.6–11.
2. From figure 24.6–11, develop the appropriate values for Q , Q2 , Q3 and k1 , k2 , k3 , such that all capacitances
will be equal. Using these values implement the normalized design, of the form of figure 24.6–12.
3. Choose a value of capacitance. Then determine scaling frequency
define the final circuit, such as that given by figure 24.6–13.
m
and resistance values necessary to
Create a table showing the renormalization process and the rescaled values of the components. You may find that a
spreadsheet is useful for executing this process. In the spreadsheet which I created, I arranged all of the final values of
resistances and capacitances to fall in a column, which could then be exported and printed for use as an input file to
PSPICE. An example of a spreadsheet generator is shown by figure 24.6–14.
Figure 24.6–14. Spreadsheet generator. The values shown are those used to develop the circuits shown by
figures 24.6–11, 24.6–12 and 24.6–13.
267
4. Check analysis with SPICE. For the BP3C1 form, the output is shown by figure 24.6–15
Figure 24.6–15. Typical SPICE output for 3C1 bandpass filter. Opamps were assumed to be ideal.
268
269