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Sensitivity minimization in active RC networks
c
Sensitivity Minimi.@ion and M.
by A. ROSENBLUM
in Active RC Networks*
s. GHAUSI
Department of Electrical Engineering New York Uniuersi(v, University Heights, Bronx, .New York ABSTRACT:
A general technique
structure
that has degrees of freedom
RC structures
have independent
of the procedure is applied
is presented
to minimize
It is shown that the minimization
RC networks.
to:
sensitivity
procedure
in choosing
the design parameters.
parameters that can be arbitrarily
is quite general. In particular, in this paper (1) fixed
circuit
in the design of active
can be applied to any active RC Since most active
chosen, the applicability
the minimization technique
con$gurations where the circuit
elements
are optimally
decomposition, i.e. synthesis procedures that employ an arbitrary Q(s) where the optimal roots of Q(s) are determined.
chosen and (2) RC:-RC polynomial
To date this technique respect to multiparameter the multiparameter tolerances, restrictions
is the only sensitivity)
sensitivity
component tolerance on circuit
design
procedure
that optimally
the roots of Q(s). In addition,
and the minimization
determines
it is demonstrated
(with that
procedure accounts for component
correlations, the frequency
range of interest
and practical
element values.
I. Introduction
In the design of active RC networks it is important to consider the effects of the element variations on the desired network response. Configurations that are extremely sensitive to component tolerances are of little practical value since they may be unstable or unable to realize the design problem adequately. For those networks that are considered relatively insensitive to element variations, the sensitivity performance may be improved by optimally choosing the circuit components. In this paper a technique to minimize multiparameter sensitivity of active RC networks based on a general measure previously presented (1)is described. The procedure is based on a nonlinear programming algorithm and is implemented for examples where the circuit configuration is given and the elements are optimally chosen, and where a synthesis procedure is employed involving an arbitrary polynomial Q(s) with simple zeros on the negative real axis and the optimal roots of Q(s) are determined. This procedure enables the designer to establish a sensitivity figure of merit for a configuration to be utilized as a basis of comparison or to obtain an optimal set of design parameters. II. Sensitivity
as a Function
of the Nominal
Element
Values
In most active RC synthesis techniques there are degrees of freedom in choosing the element values, i.e. there are generally more elements than * This research was supported by the National Science Foundation Grant GK 18465.
95
A. Rosenblum and M. S. Ghausi design equations. In this section it will be demonstrated how the independent 4 parameters can be chosen to minimize a sensitivity measure. Example Consider the Sallen and Key (2) network shown in Fig. 1. The transfer function for this example is expressed in terms of the element values as T=
KG, G, S, S, s2+s(GlS1+G2S1+G2SZ(1-K))+GIG,S,S,’
(1)
FIG. 1. Sallen and Key network (SK).
where G, = l/Ri, Si = l/C*, i = 1,2 and K is the gain of the voltagecontrolled voltage source active element. To synthesize a normalized Butterworth two-pole transfer function, namely T=
lo
s2+&qs+
(2)
1
the design equations are KG,G,S,S, G,S,+G,S,+G,S,(1-K) G,S,G,S, Consider the classical sensitivity namely
= 10,
(3a)
= ,/2,
(3b)
= 1.
(3c)
with respect to the gain evaluated at s = j,
K aT = l+G2S2K z s,j --Jr
exl-) = Ji
(4)
From Eq. (4) it is apparent that this measure of sensitivity is a function of the nominal values for K, G, and S,. Examination of the design equations [Eqs. (3a, c)], however, yields a value of K as 10,but the G,S, product may be arbitrarily chosen. Thus, the sensitivity measure in Eq. (4) is a function of the product G, S, and to minimize this expression requires G, S2 to be chosen as small as possible. To consider this approach from a practical viewpoint, however, certain component restrictions will constrain the choice of G2S2. For example, if the maximum RC product is l/b, then the minimum GS product is 6. From Eq. (3b) this implies a constraint on G,S,, namely G,S, = JZ-&
96
+(K-1)G2S2>b. 2
(5)
2
Journal of The Franklin Institute
Sensitivity Minimization
in Active RC Networks
Therefore, the minimum value of the measure in Eq. (4) is determined by the smallest value of G, S, that satisfies the restriction in Eq. (5) or the constraint G,S,>b. ZZZ. Minimkation
of the Multiparameter
Measure
of Sensitivity
The example in Section II with its relatively simple sensitivity measure illustrates the approach subsequently developed in this section, namely: (1) Determine the general sensitivity measure as a function of the independent element values. (2) Determine the constraints that the independent parameters must satisfy. (3) Choose the optimal parameter values that minimize the sensitivity measured and satisfy the parameter constraints. For the sensitivity measure to be practical all parameter variations must be accounted for along with the component tolerances. A general multiparameter statistical sensitivity measure (1) is utilized in this paper. This sensitivity measure accounts for all component tolerances and component tolerance correlations. Note that the multiparameter sensitivity measures of Schoeffler (3) and Kuo and Goldstein (4) are special cases of the general measure (5). The sensitivity measure is defined as follows :
(6) where x is the parameter vector, E the expected value, T the transfer function and AT the variation in the transfer function due to the component tolerances Ax. If the element variations are small enough to utilize a linear approximation, then Eq. (6) can be expressed as Wadl’Pd dw, (7) s 01 where P = E[Ax Axt] is the covariance matrix (k x k) of the component tolerances, * denotes complex conjugate, k is the number of elements in the network and the ith component of the vector d is (xi/T) aTlax+ If the transfer function T(ju) is represented as M(x) =
and the following
vectors are defined: a = [a,. . .aJ,
V, T = [aT/aa,.
. . aT/aaJ,
b = [b,. . .bJ, 0, T = [aT/ab,.
. . aT/abJ,
then the expression in Eq. (7) can be written as
Vol. 294,
No. 2, August
1972
97
A. Rosenlhm
and M. S. Ghausi
where G,=
[ :lrc)]
and
G,=
(10)
[ ~~~~r~~]. kx(n+l)
kx(n+l)
The matrices C, and cl, are computed directly from the coefficients of the transfer function described symbolically in terms of the element values. For a given transfer function, the terms V,T and 0, T are fixed and independent of the parameter values. This is particularly useful in expressing the measure as an explicit function of the parameter x, an important aspect in the minimization procedure resulting in rapid convergence. The minimization procedure reduces to solving a nonlinear programming problem, namely minimize the sensitivity measure which is a nonlinear function of the independent parameters subject to nonlinear constraints. To illustrate this point, consider the example in Section II. Utilizing the transfer function in Eq. (1)the C1 and C, matrices are computed ast
0 c,= [
0
G,hG,S,
%S,
0
G,S,+G,Ss(l-K)
G,S,G,S,
0
G,S, +G,S,
G,S,G,S,
0
'%&(1--K) - KG, S,
GI&G!,S2 0 I
0
and
CZ =
0
KG,S,G,S,
0
0
KG,S,G,S,
0
0
KG,S,G,S,
0
0
KG,S,G,S,
L 0
0
KG,S,G,S,
3
1 ,
where the parameter vector x is taken as x = [G, G, S, S, K]l. Defining the product G, S, = y and using the design equations [Eq. (3a, b and c)], the elements of the matrices C, and Cs are written as functions of y, namely
Ci=[i$fi]
andC&=[[i
i].
(11)
Since some of the elements of the C1 matrix are nonlinear functions of y, in view of the measure expression in Eq. (9), the measure is a nonlinear function of y. The constraint equation [Eq. (lo)] is also a nonlinear function of y, namely 1/2---/y+gy>b. (12) t This computation is presented in detail in Ref. (1).
98
Journal of The Franklin
Institute
Sensitivity Minimization
in Active RC Networks
A nonlinear programming algorithm developed by Fiasco and McCormick (6,7) was utilized to implement the minimization procedure (see Appendix 1). The technique is general and can be applied to fixed network configurations and/or synthesis procedures that employ an arbitrary polynomial Q(s) to choose the optimal element values. The inputs to the program are generated from th.e C,, Cs and P matrices and the constraint equations. This is illustrated in the next section where several applications are presented. IV. Optimal Design Values for Fixed Network
Configurations
In this section the minimization procedure (described in Appendix 1) is employed to choose the optimal circuit element values for two Sallen and Key networks. In these examples the maximum RC product is taken as 10, and the normalized parameter variations are assumed independent and are taken with variances of lo-* (i.e. E[(Ax&J2] = 10e4). Example 1 The first illustration is the minimization of the example in Section II. The network configuration shown in Fig. 1 is used to synthesize the desired Butterworth transfer function given in Eq. (2). The synthesis equations are Eqs. (3a, b, c), and the constraint is denoted in Eq. (5) where b = O-1. The matrices Cr and C, are expressed as functions of the independent parameter (y = G, X2) in Eq. (11) along with the constraint in Eq. (12). The minimization procedure yields an optimal value of y as 0.268 and the design parameters as G,S, = 3.74 and G,S, = 0.1. It is noted that the optimal value occurs on the boundary of the constrained region, namely J2 - l/y + 9y = 0.1 [equality portion of Eq. (12)]. Example 2
FIG. 2. Sallen and Key network.
The network configuration (2) for this example symbolic transfer functions is given by
is shown in Fig. 2. The
sKG, 8, T = ~2+~[QIX1+G3X2+G~:3~+G2X1(1-K),+G3X1Gl~2+G,~,~2G2
Vol. 294, No. 2, August1972
(13)
99
A. Rosen&m
and M. 8. CThausi
and the transfer function to be synthesized is T=
1*5tS
s2-+-o’ls+
(14)
1’
Comparing Eqs. (13) and (14), the design equations are expressed as KG,S, G,S1+G,S2+G,S,+G2S,(1-K) Q,S,G,S2+G,S,G2S2
= 1.5,
(15a)
= 0.1,
(1510)
= 1.
(15c)
The matrices Cl and C,, defined in Eq. (lo), are computed -0 0 c,=
GIG, S,
GI S,
O
0
‘32&(1-K)
G2
4
%S,+%S,
GzS,(l--K)+G,S,+G,S,
1
(16)
1
@d’2
0
- KG,S,
0
82
82G2
1
0
as
and
c, =
0
1.5
0
0
0 _ 0
0
0
0
0
1.5
0
0
0
0
0
1.5
0
(16)
’
where the parameter vector is x = [G, G, G3 S, S, Kit. Examining the design equations [(Eqs. (15a, b, c)], it is noted that the elements of the CI matrix in Eq. (16) can be expressed in terms of the gain K and the product G,S,. For example, consider the (3, 2) term of C,, namely G, S, + G, S,. From Eq. (15b) this expression can be written as G,S,+G,S,
= O.l-GISS,+(K-l)G,S,.
(17)
G,S, = 1.5/K.
(19)
Now, from Eq. (15a)
Substituting
Eq. (18) into Eq. (15~) and solving for G, S, ,yields G,S, = l/(G&)
- 1*5/K.
(19)
Utilizing Eqs. (18) and (19) in Eq. (17), the (3, 2) term of C,, i.e. CI(3, 2), is expressed as C,(3,2) = - 1.4 -I-(K - l)/(G,S,).
100
PO)
Journal
of The Franklin
Institute
Sensitivity _Minimixation in Active RC Networks Defining #,S, = y the remaining expressed in terms of K and y as
1*5y/K
15/K
-0
0 c,=
elements of the C, matrix
(1 -K)
16/K)
(l/y-
1 1
0.1-y
0
.
(21)
1
Y
0
-
1 - 15y/K
-1.4+(K-1)/y
O 0
are similarly
0
1.5 -K/y
Thus, in view of Eqs. (16) and (21) and the measure definition in Eq. (9), the network’s sensitivity is a function of the independent parameters y and K. Here, as in the previous example, the restrictions on the RC product impose constraints upon the independent parameters (y and K). If Gt Si 2 0.1, then from Eq. (19) it is apparent that l/y-15/K>O.l.
(22)
Another constraint on y and K occurs from examination for G, S, in Eq. (17) utilizing Eqs. (18) and (19), yields
of Eq. (17). Solving
G,S, = -1’4+(K-1)/y-y. Imposing the condition
(23)
G, S, > O-1 results in (K-1)/y-y>
1.5.
(24)
Thus, the constraints on the independent parameters y and K are determined from Eqs. (22) and (24). The minimization procedure produces the following optimal values for the independent parameters K = 2.58
and
y = G,S, = 0.714.
For these values the design parameters are GsS, = 0.1
and
G,S, = O-814.
It is noted as in the last problem that the minimal values of K and y occur on the boundary of the constraint region, namely, when (K - I)/ y-y = 1.5 [from Eq. (24)]. V. Optimal
Roots
of Q(s)
for
an Arbitrary
Synthesis
Polynomial
Many active RC synthesis procedures involve choosing an arbitrary polynomial Q(s) with simple roots on the negative real axis. In this section two examples are presented to illustrate the application of the minimization technique (described in Appendix 1) for choosing the roots of Q(s) to minimize the network’s sensitivity. It should be noted that the method is quite general and applicable to all such synthesis schemes, namely the RC:-RC decomposition method. The configuration (8) to be synthesized is shown in
Vol.294,No.Z,August1972
6
101
A. Rosenblum and M. X. Ghm.mi Fig. 3. The general transfer function for this network is expressed as (25) where Y3 = Y, = 1 and the operational amplifiers are assumed ideal infinite gain. The maximum RC product is assumed 10 and the element variations are assumed independent with variances of 10-4.
F‘IG. 3. A general network realization configuration.
Example 3 In this example the following transfer function is to be realized using the general configuration in Fig. 3 ;
T_hTo_
w
1
1‘
sq-&4s+
The choice of Q(S) for this illustration is s +a where (T> 0. (Note that the degree of Q(s) must be greater than or equal to maximum degree of N(s) or D(s) minus 1.) Dividing the numerator (N(s)) and d enominator (D(s)) by Q(s), the transfer function in Eq. (26) is written as T= Now consider the numerator, N(s)
---=-
W+o) v+&3s+l)/(s-to)’ namely N(s)/&(s) written as 1
=---
S
1
(28)
SW)
s+a Q(s) 8s-U S(S+a)=a-p*
Comparing Eq. (28) with the numerator
YI=
102
l/CT
and
of Eq. (25), it is apparent that Yz=M.
s
(29)
u
Journal of The Franklin
Institute
Sensitivity Minimization In a similar fashion, the denominator
in Active RC Networks
of Eq. (27), D(s)/&(s),
D(s2 = s2+J(2)s+l = s l+s@-“)+l s+u s(s + u) Q(s) [
=s+I,u+(+-l~o)~ s+u
is written as
I (30)
.
Were, in order to equate the terms in Eq. (30) to the admittances in the denominator of Eq. (25), the term J2 - (J- l/a must be examined. For 0 > 0 this term is always negative. Thus, in Eq. (25) the admittances Y, and Y, are Y,=s+l/a
and
Y4=
((3-t L/o-42)s s+o
(31)
.
In view of Eqs. (29) and (31) the admittances in Eq. (25) can be expressed symbolically in terms of element values as follows: r, = G,,
Y,= G,s/(si-G,S,), y3= G,, Y,= d&+G,, Y4= G4s/(s+G4S4),Y5= G5,
(32)
where 8, = l/o,
G, = l/a,
G3= 1,
G4 = of
G, = 1,
ce=
G,S, = u, l/42,
G,S, = u, G, = l/u.
1,
Substituting the relationships in Eq. (32) into Eq. (25) the transfer function can be expressed symbolically in terms of the element values as
(s-tG&U W%G1Q~--a,Wd+G,GsG,fW,)
(33)
T=(s+G2S2)(s2G2+~(GgGsSs+GBG4S4-G4G5S6)~G3G4S4G6S6) The transfer function in Eq. (33) can be simplified by cancelling and zero* since G, X2 = G, S, = u and writing it as
the pole
s+G,G,S,-G,G,S,)+G,G,G,S,S,
(34)
T=s2G,+s(G3G,S,+G,G4S4-G4G5S,)+G,G4S4G6S6 The equations to generate the elements of the C, and C, matrices determined by equating the coefficients of Eqs. (34) and (26), namely = l/u-l/u
b, = G,G,S,-G,G,S, b, = GlG,G2S2S, a2 = G,=
= 0,
= 1,
1,
(35)
a, = G3G6S,+-G3G4S,-G4G,S, a, = G,G,S,G,S,
are
= 1.
= l/u+-u---/u+J2
= ,I% i
* Neglecting these terms does not have a significant effect on the measure. In this example at the optimal value of cr the measure is 1.10 x 10e3 omitting the pole-zero cancelling terms and 1.05 x 10-s including them.
Vol. 294, No. 2, August 1972
103
A. Rosen&m
and M. S. Ghazcsi
Here, the elements of the Cl and C, matrices which are functions of the root of Q(s) (i.e. c) can be determined by inspection. In particular, if the parameter vector is taken as x = [G, G, G, G4 G, G6 S, S, SJ, then the third row of the C, matrix G, adjaG, is written as G3(a3/aG,) = [l Ia,+G,G,S,il]
= [1ia+l/oi1].
(36)
The remaining elements can be similarly expressed. The minimization procedure applied to this example for wi = 0 and o2 = 1 yields an optimal choice of u as 1.2. The measure value is 1.1 x 10U3. The element values are determined from the relationships in Eq. (32). Now consider an alternate choice of Q(s) for this problem as Q(s) = (s+oJ (s +o,) where us> ur. Proceeding as before, the polynomials N(s)/&(s) and D(s)/&(s) can be expressed as follows : 1
I NW = Q(s) (s+u~)(s+u~)
D(s) -= Q(s)
SH
(37a)
= GZ+s+cr,-s+cP,’
s2+J(2)s+1 (s+u,)
8G
1
SE
SF
(s+u2) = cr,++-+’
(37b)
where G:=
’
Hz-l
4J~’
u2(u2 - 01)’ E=
u;-&qu,+1 GJ‘2 - 4
’
$7 = -__-. +11o%+l %h - 4
Comparing Eqs. (37a) and (37b) with Eq. (25), the admittances and Y, are identified as y, = -L+sc: 01 u2
s+u2’
y, = l+sF % (J2
Proceeding
Y,, Yz, Y6
ya=g> 1
(38)
Y+sE s+u2’
s+u,
1
as before, the transfer function is determined T = b2s2+blsi-b. a,s2+a,s+ao’
as
(39)
where b, = G,G,+G,G,-G,G,, b, = G,S,G,G,+G,S,G,G,+G,G,G,S,-G,G,G,S,, b, = G1 G, G, 82 G, 84, a2 = G,G,i-G,G,-G,G,,
a1 = G,G,G,S,+G,G,G,S,+G,G,S,G,-G,S,G,G,, a, = G,S,G3G,S,G,
104
Journal of The Franklin Institute
Sensitivity Minimizaticrn in Active RC Networlcs and x=[G,G,GGGGGGSSSS]~ 3456782468
(40)
and G, = llaroz, G,
=
VII+2
‘37
=
l/~,ol,
G,f& =
G, = U[~Aoz -
G,=
41,
G, = E,
G, = 1,
41,
G, = P, Q4S4
(~2,
=
1,
G,S, =
02,
G,S, = CT,.
01,
The minimization procedure for w1 = 0 and w2 = 1 yields optimal values of or = 0.94 and a2 = 3.3 with a measure value of 1.7 x 10-3. Comparing this result with the case for Q(s) = s+ cr, it is seen that choosing a second-order synthesis polynomial Q(s) yields poorer results. Example 4 The configuration in Fig. 3 is used in this example to realize a three-pole Butterworth transfer function, namely 1
T=
(41)
s3+2s~+2s+l'
Here, Q(s) is chosen as
Q(s)= (s+ud(s+4, where aZ > ul. The numerator
decomposition 1
N(s) ---=
N(s)/&(s)
1
_
Q(s) (S+d(S+%)
I
(42)
can be expressed as
_____--
siruz(u2
a
S/[%(U2-
s+u,
(Ulf32)
41.
(4‘3)
s+u,
Comparing Eq. (43) with the numerator in Eq. (25), it is apparent that +
YI = 1
s/[u2(u2
-
+)I
s+u,
Ul%
and
y = S/[T(%--al)l 2
SfUl
Similarly, the denominator
decomposition
yields
;++Es
D(s)_1
Q(s)
(44)
*
__ s+u1
UlU2
(45)
G2+s’
where
f(d
P= -
01 (a2
-
E = Ul)
’
fb2) U2(Q2
-
Cl)
and f(uJ
Vol. 294, No. 2, August 7
1972
= - u: f 20; - 2cr,+ 1
for i = 1,2.
105
A. Rosenbhm
and M. X. fCY%musi
In order to determine the admittances Y6 and Y4 in the denominator of (25), different ranges of al and u2 must be chosen (as shown in Table I) sincef(a,)>Ofor O
TABLE I Admittance assignments Case
F
E
k6
k,
.I__Fs
1. o,
s+u,
u,>l
(b) ul< 1, o,a
1 -Es
s+u, The optimization procedure must be applied separately to each case in Table I since each case will yield a different network configuration. Symbolic element values are assigned as in Eq. (32). For example, consider Case 2, namely
r, = G,, (46) Y, = G,s+G,, where G, = l/u,uz,
G,=
G, = l/CuAuz- 41,
G, = l/u,uz,
G, = l/[~l(uz -
G, = -E,
41,
1,
G3= 1,
a,&
= 01,
G, = -P,
G,&
=
G,&
= ~1,
G,X, = u2, c,=
1.
(52,
The results of the optimization procedure are summarized in Table II(a), (b) and (c) for different frequency ranges. It is noted that Case 2 yields the smallest minimum value. The output for Case 2 is found in Appendix 1. The last column in Table II is Schoeffler’s measure of sensitivity (3) namely, (47) where + is a special case of the measure in Eq. (9) [see Ref. (5)] and is generated at s = j to compare the results in Table II with those found in a
106
Journal
of The Franklin
Institute
Sensitivity Minimization
in Active RC Network8
correspondence by Aumman (9). Note that the orders of magnitude difference between the measure M(x) and 4 in Table II are accounted for by component tolerances of 1O-4 and the step size of 10-l used in the numerical integration. Multiplying 4 by 1O-5 will generate a number that has the same order of magnitude as the measure value. TABLE Summary
of sensitivity
Case
(a)w,=S;
I II III
0.52 0.84 1
I II III
0.6 0.81 1
II
0.5
II
analysis for example 4
co,=1 1 1*81 1.85 (b) w,=O;w,= 1 1.84 1.8 (c) w1 = 0; oJ2= 1.75
MM 1.63 x 10-Z 0.718 x 10-a 0.756 x 10-3 1 4.10 x IO-3 2.33 x 1O-3 2.43 x 1O-3 0.1 0.12 x 10-S
4L7.1 165 37 50 117 37 51
t M(x), minimum measure value. The existence of a minimum + is demonstrated for the synthesis of a three-pole normalized Butterworth filter [Eq. (41)], utilizing an arbitrary synthesis polynomial Q(s) = (s + al) (s + 0.J. Aumman treats the decomposition in the form of Y,-K,Y, (47) y3-&y4 and demonstrates a minimum of &j) = 22.6 at or = 0.85 and CT~ = 2.2. The decomposition in Eq. (47) is not general enough to accommodate the synthesis structure of this example, i.e. the sensitivity effects of the conductances G, and G, in Eq. (25) must be ignored. Thus, if G, and G, are not taken as components of the parameter vector, the measure (M(x)) has a minimum at 'Tl = 0.85 and g2 = 1.9 for w1 = 0.8 and w2 = 1. For these values of o1 and u2 and ignoring the sensitivity effects of G, and G,, 4 has a value of 23.5. Thus, these results are consistent with those presented by Aumman. The significance of the results presented in Table II is exemplified by considering the same synthesis procedure [Eq. (25)] and arbitrarily choosing values for a1 and u2. For example, if o1 = O-5 and a, = 1, the measure value for w1 = 0.8 and wa = 1 is 1.63 x 10-3. From Table II(a) (lines 1 and 2) it is seen that this is over 100 per cent higher than the optimal value. An even more extreme case is noted for o1 = 0.1, (TV= 0.3, w1 = 0.8 and wa = 1. Here, the measure value is 1.4 x 10-l, more than 100 times greater than the optimal value. Hence, from this example, it is apparent that the sensitivity of the network response can be critioally affected by the choice of the roots of the polynomial Q(s).
Vol.294,No.2,August1972
107
A. Rosenblum and M. S. Ghausi Conclusion
In this paper an optimization procedure to minimize a general measure of sensitivity is developed. It is seen that the minimization procedure accounts for the component tolerances, a frequency range of interest and practical restrictions on the circuit element values. The examples presented demonstrate the generality of the procedure. The technique can be implemented to determine the optimal network elements for a given circuit configuration (Section IV). The minimization procedure can also be utilized to determine the optimal roots of the synthesis polynomial Q(s) (Section V). To date, this is the only general sensitivity minimization procedure that determines the optimal roots of Q(s). The technique evolves as a practical design tool. It enables the circuit designer to account for sensitivity in the design procedure by obtaining a figure of merit for sensitivity of the network response and/or obtain an optimal set of design parameters.
Appendix
1
The procedure for minimizing the sensitivity measure in Eq. (9) is described. The technique is based upon a nonlinear programming algorithm developed by Fiacco and McCormick (6, 7). First, the nonlinear programming problem must be explicitly stated. Let the vector z be defined as the independent parameter vector, the measure in Eq. (9) denoted as &f(z) and the constraints be expressed as g,(z) > 0 for i = 1,2, . . ., m, where it is assumed that there are m constraints on the independent parameter vector z. To illustrate, consider Example 2 in Section IV. Here, the independent parameters are y and K. The measure is expressed in terms of the independent parameters through Eqs. (16) and (9) and from Eqs. (22) and (24) the constraints are g,(z) = l/y- 1*5/K-0*1 and g,(z) = (K - 1)/y-y1.5 where z = [y, Kit. The nonlinear programming problem is stated as follows : Minimize
M(z).
Subject
to g,(z)>0
for i = 1, 2, . . . . m.
The algorithm developed by Fiacco and McCormick treats the constrained minimization problems as a sequence of unconstrained minimization where the sequence converges to the solution of the constrained problem.* The algorithm defines a function of the form
m,
Tk.)= M(z)
+G31w)~
(A.1)
where rk > 0 and rlc z;$_, l/gi(z) is a penalty function. The sequence of unconstrained functions {P(z, rk)} is generated by choosing a strictly monotonic decreasing sequence {TV}. The choice of the initial value rI, convergence criteria and other aspects of the computational algorithm can be found in Ref. (7). A flowchart in Fig. 4 describes the logic of the algorithm. The essential computational aspect of the procedure is determining the minimum value for the unconstrained function P(z, TV). The principal numerical techniques used * Convergence to a global minimum gi(z) are concave functions of 2.
108
will occur only if M(z)
is a convex
function
and
Journal of The Franklin Institute
Xensitivity Minimization
in Active RC Network
0t
In
Select initial point. zO i Select initial value Of ‘x Lj )
I Determine
minimum
4
____“f_r‘k~k)_ F3.z >‘x) = M(z)+r,
&z,
Estimate PMIN for G+r
FIG.
4. Fiacco and McCormick nonlinear programming
algorithm.
to minimize an unconstrained function are first- and second-order gradient techniques that can be summarized respectively as follows :
%I = Z,_l-evz~(Zn-l,rk)
(A.2)
and z* =
!a,_1
-
efwd
v, P(z,-~, u
where
v, P(Z,_l, PP __
az; :
El=
Tk) = [aP/az,,
. ., aP/az,]t
. ..
.
(Hessian matrix-(q
x y))
PP [
G
...
and IJis the number of components of the independent parameter vector z. In the first-order methods, the nth point Z~ is obtained from the n- 1 point z,+ by descending the gradient of P evaluated at z,_~. 0 is the distance along the gradient that
Vol. 294,No. 2, August 1972
A. Rosenblum and M. 23.Ghausi is traveled. The second-order method obtains the next point by moving down a vector that is the gradient of P mapped by the inverse of the Hessian matrix (second partial derivatives of P). The first-order method requires only first derivatives; i.e. V, P, but converges too slowly for minimizing each of a sequence of P functions. The second-order method, on the other hand, converges rapidly, but it requires a knowledge of second-order derivatives, which is usually difllcult to obtain. Since the sensitivity measure in Eq. (9) is expressed explicitly in terms of the independent parameters, first and second derivatives can be obtained; i.e. the matrices C, and C, in Eq. (9) are explicit functions of the independent parameters. A flowchart depicting the minimization of the unconstrained object function P(z, rk) is given in Fig. 5. The INPUT subroutine provides the matrices C,, C, in Eq. (9) and
Yes
Q-Is
out
“+-ll
0
+,
“PF-Penalty
function (5
% I/gi(z)
;=I FXG. 5. Minimization
of lmconstrained
1
object function P(z, re) ]Eq. (A.l)].
all first and second derivative information explicitly in terms of the independent parameters required to compute the measure, the gradient of the measure (subroutine GRAD2), and the Hessian of the measure (subroutine SEC2). In addition, the subroutine INPUT contains the penalty flmction (TV2 l/gi(z)), the gradient, and the Hessian of the penalty function. A sample output* is summarized in Tables III and IV. The iterations for the main algorithm are tabulated in Table III and the iterations for the minimization of the initial unconstrained object function (P(z, TV)) are summarized in Table IV. * Note : This is Example 3 in Section V for the measure taken over a wider frequency range (wi = 0, w2 = 3). This example was chosen because it illustrates the convergence of the technique for a relatively large number of unconstrained P(z, ra) functions.
110
Jowml of The Franklin
Institute
Sensitivity MirGmization in Active RC Network8 1
TABLE III
.
Minimization
Iteration 1 2 3 4 5 6 7
of Example
P(z, re) x 104 (unconstrained function)
M(z) x 104 (sensitivity
10,464 1394 643 491 447 434 427
2093 1127 544 456 433 428 427
measure)
3 in Section V
Penalty function (r, Z1
l/g*(z))
8371 268 99 35 14 6 0.3
r>;x 104
e1
02
1105 138 17.3 2.1 0.27 0.03 0.0042
0.85 0.45 0.74 0.92 0.981 0.996 0.999
2.2 9.66 6.5 5.6 5.75 5.75 5.73
TABLE IV Unconstrained minimization of P(z, rk) in Example
3 of Section V for rk = r1 = 8371
Iteration
P(z, rlc)x lo4
Convergence criteria
1 2 3 4
10,464 3778 3292 3261
2507 263 26 0.06
0.85 0.45 0.49 0.45
2.2 4.89 8.86 9.59
References Rosenblum and M. S. Ghausi, “Multiparameter sensitivity in active RC networks”, IEEE Trans. Circuit Theory Special Issue on Digital and Active FiZters, Vol. CT-l& No. 6, pp. 592-599, Nov. 1971, also New York University Technical Report No. 400-206, July 1970. Sallen and Key, “A practical method of designing RC active networks”, IRE Trans. Circwit Theory, Vol. CT-2, Mar. 1955. J. D. Schoeffler, “The synthesis of minimum sensitivity networks”, Trans. IEEE, Vol. CT-II, pp. 271-288, June 1964. A. J. Goldstein and F. F. Kuo, “Multiparameter sensitivity”, IRE Trans. C+rcuit Theory, Vol. CT-S, No. 2, pp. 177-178, June 1961. A. Rosenblum, “Multiparameter sensitivity and sensitivity minimization in active RC networks”, Ph.D. thesis, New York Univ., Apr. 1971. A. V. Fiaoco and G. P. McCormick, “Programming under nonlinear constraints by unconstrained minimization: A primal dual method”; Research Analysis Corp., RACTP-96, Bethesda, Md., Sept., 1963. A. V. Fiacco and G. P. McCormick, “Computational algorithm for the sequential unconstrained minimization technique for nonlinear programming”, Management Sci., 10, pp. 601-617, 1964. W. P. Lovering, “Analog Computer simulation of transfer functions”, Proc. IEEE, Vol. 53, No. 3, pp. 306307, Mar. 1965. H. Aumman, “Sensitivity minimum in active transfer fun&ion synthesis”, Proc. IEEE, Vol. 58, No. 4, pp. 595-596, Apr. 1970.
(1) A.
(2) (3) (4) (5) (6)
(7)
(8) (9)
vol. 204, No. 2, August1978
111
Sensitivity Minimi.@ion and M.
by A. ROSENBLUM
in Active RC Networks*
s. GHAUSI
Department of Electrical Engineering New York Uniuersi(v, University Heights, Bronx, .New York ABSTRACT:
A general technique
structure
that has degrees of freedom
RC structures
have independent
of the procedure is applied
is presented
to minimize
It is shown that the minimization
RC networks.
to:
sensitivity
procedure
in choosing
the design parameters.
parameters that can be arbitrarily
is quite general. In particular, in this paper (1) fixed
circuit
in the design of active
can be applied to any active RC Since most active
chosen, the applicability
the minimization technique
con$gurations where the circuit
elements
are optimally
decomposition, i.e. synthesis procedures that employ an arbitrary Q(s) where the optimal roots of Q(s) are determined.
chosen and (2) RC:-RC polynomial
To date this technique respect to multiparameter the multiparameter tolerances, restrictions
is the only sensitivity)
sensitivity
component tolerance on circuit
design
procedure
that optimally
the roots of Q(s). In addition,
and the minimization
determines
it is demonstrated
(with that
procedure accounts for component
correlations, the frequency
range of interest
and practical
element values.
I. Introduction
In the design of active RC networks it is important to consider the effects of the element variations on the desired network response. Configurations that are extremely sensitive to component tolerances are of little practical value since they may be unstable or unable to realize the design problem adequately. For those networks that are considered relatively insensitive to element variations, the sensitivity performance may be improved by optimally choosing the circuit components. In this paper a technique to minimize multiparameter sensitivity of active RC networks based on a general measure previously presented (1)is described. The procedure is based on a nonlinear programming algorithm and is implemented for examples where the circuit configuration is given and the elements are optimally chosen, and where a synthesis procedure is employed involving an arbitrary polynomial Q(s) with simple zeros on the negative real axis and the optimal roots of Q(s) are determined. This procedure enables the designer to establish a sensitivity figure of merit for a configuration to be utilized as a basis of comparison or to obtain an optimal set of design parameters. II. Sensitivity
as a Function
of the Nominal
Element
Values
In most active RC synthesis techniques there are degrees of freedom in choosing the element values, i.e. there are generally more elements than * This research was supported by the National Science Foundation Grant GK 18465.
95
A. Rosenblum and M. S. Ghausi design equations. In this section it will be demonstrated how the independent 4 parameters can be chosen to minimize a sensitivity measure. Example Consider the Sallen and Key (2) network shown in Fig. 1. The transfer function for this example is expressed in terms of the element values as T=
KG, G, S, S, s2+s(GlS1+G2S1+G2SZ(1-K))+GIG,S,S,’
(1)
FIG. 1. Sallen and Key network (SK).
where G, = l/Ri, Si = l/C*, i = 1,2 and K is the gain of the voltagecontrolled voltage source active element. To synthesize a normalized Butterworth two-pole transfer function, namely T=
lo
s2+&qs+
(2)
1
the design equations are KG,G,S,S, G,S,+G,S,+G,S,(1-K) G,S,G,S, Consider the classical sensitivity namely
= 10,
(3a)
= ,/2,
(3b)
= 1.
(3c)
with respect to the gain evaluated at s = j,
K aT = l+G2S2K z s,j --Jr
exl-) = Ji
(4)
From Eq. (4) it is apparent that this measure of sensitivity is a function of the nominal values for K, G, and S,. Examination of the design equations [Eqs. (3a, c)], however, yields a value of K as 10,but the G,S, product may be arbitrarily chosen. Thus, the sensitivity measure in Eq. (4) is a function of the product G, S, and to minimize this expression requires G, S2 to be chosen as small as possible. To consider this approach from a practical viewpoint, however, certain component restrictions will constrain the choice of G2S2. For example, if the maximum RC product is l/b, then the minimum GS product is 6. From Eq. (3b) this implies a constraint on G,S,, namely G,S, = JZ-&
96
+(K-1)G2S2>b. 2
(5)
2
Journal of The Franklin Institute
Sensitivity Minimization
in Active RC Networks
Therefore, the minimum value of the measure in Eq. (4) is determined by the smallest value of G, S, that satisfies the restriction in Eq. (5) or the constraint G,S,>b. ZZZ. Minimkation
of the Multiparameter
Measure
of Sensitivity
The example in Section II with its relatively simple sensitivity measure illustrates the approach subsequently developed in this section, namely: (1) Determine the general sensitivity measure as a function of the independent element values. (2) Determine the constraints that the independent parameters must satisfy. (3) Choose the optimal parameter values that minimize the sensitivity measured and satisfy the parameter constraints. For the sensitivity measure to be practical all parameter variations must be accounted for along with the component tolerances. A general multiparameter statistical sensitivity measure (1) is utilized in this paper. This sensitivity measure accounts for all component tolerances and component tolerance correlations. Note that the multiparameter sensitivity measures of Schoeffler (3) and Kuo and Goldstein (4) are special cases of the general measure (5). The sensitivity measure is defined as follows :
(6) where x is the parameter vector, E the expected value, T the transfer function and AT the variation in the transfer function due to the component tolerances Ax. If the element variations are small enough to utilize a linear approximation, then Eq. (6) can be expressed as Wadl’Pd dw, (7) s 01 where P = E[Ax Axt] is the covariance matrix (k x k) of the component tolerances, * denotes complex conjugate, k is the number of elements in the network and the ith component of the vector d is (xi/T) aTlax+ If the transfer function T(ju) is represented as M(x) =
and the following
vectors are defined: a = [a,. . .aJ,
V, T = [aT/aa,.
. . aT/aaJ,
b = [b,. . .bJ, 0, T = [aT/ab,.
. . aT/abJ,
then the expression in Eq. (7) can be written as
Vol. 294,
No. 2, August
1972
97
A. Rosenlhm
and M. S. Ghausi
where G,=
[ :lrc)]
and
G,=
(10)
[ ~~~~r~~]. kx(n+l)
kx(n+l)
The matrices C, and cl, are computed directly from the coefficients of the transfer function described symbolically in terms of the element values. For a given transfer function, the terms V,T and 0, T are fixed and independent of the parameter values. This is particularly useful in expressing the measure as an explicit function of the parameter x, an important aspect in the minimization procedure resulting in rapid convergence. The minimization procedure reduces to solving a nonlinear programming problem, namely minimize the sensitivity measure which is a nonlinear function of the independent parameters subject to nonlinear constraints. To illustrate this point, consider the example in Section II. Utilizing the transfer function in Eq. (1)the C1 and C, matrices are computed ast
0 c,= [
0
G,hG,S,
%S,
0
G,S,+G,Ss(l-K)
G,S,G,S,
0
G,S, +G,S,
G,S,G,S,
0
'%&(1--K) - KG, S,
GI&G!,S2 0 I
0
and
CZ =
0
KG,S,G,S,
0
0
KG,S,G,S,
0
0
KG,S,G,S,
0
0
KG,S,G,S,
L 0
0
KG,S,G,S,
3
1 ,
where the parameter vector x is taken as x = [G, G, S, S, K]l. Defining the product G, S, = y and using the design equations [Eq. (3a, b and c)], the elements of the matrices C, and Cs are written as functions of y, namely
Ci=[i$fi]
andC&=[[i
i].
(11)
Since some of the elements of the C1 matrix are nonlinear functions of y, in view of the measure expression in Eq. (9), the measure is a nonlinear function of y. The constraint equation [Eq. (lo)] is also a nonlinear function of y, namely 1/2---/y+gy>b. (12) t This computation is presented in detail in Ref. (1).
98
Journal of The Franklin
Institute
Sensitivity Minimization
in Active RC Networks
A nonlinear programming algorithm developed by Fiasco and McCormick (6,7) was utilized to implement the minimization procedure (see Appendix 1). The technique is general and can be applied to fixed network configurations and/or synthesis procedures that employ an arbitrary polynomial Q(s) to choose the optimal element values. The inputs to the program are generated from th.e C,, Cs and P matrices and the constraint equations. This is illustrated in the next section where several applications are presented. IV. Optimal Design Values for Fixed Network
Configurations
In this section the minimization procedure (described in Appendix 1) is employed to choose the optimal circuit element values for two Sallen and Key networks. In these examples the maximum RC product is taken as 10, and the normalized parameter variations are assumed independent and are taken with variances of lo-* (i.e. E[(Ax&J2] = 10e4). Example 1 The first illustration is the minimization of the example in Section II. The network configuration shown in Fig. 1 is used to synthesize the desired Butterworth transfer function given in Eq. (2). The synthesis equations are Eqs. (3a, b, c), and the constraint is denoted in Eq. (5) where b = O-1. The matrices Cr and C, are expressed as functions of the independent parameter (y = G, X2) in Eq. (11) along with the constraint in Eq. (12). The minimization procedure yields an optimal value of y as 0.268 and the design parameters as G,S, = 3.74 and G,S, = 0.1. It is noted that the optimal value occurs on the boundary of the constrained region, namely J2 - l/y + 9y = 0.1 [equality portion of Eq. (12)]. Example 2
FIG. 2. Sallen and Key network.
The network configuration (2) for this example symbolic transfer functions is given by
is shown in Fig. 2. The
sKG, 8, T = ~2+~[QIX1+G3X2+G~:3~+G2X1(1-K),+G3X1Gl~2+G,~,~2G2
Vol. 294, No. 2, August1972
(13)
99
A. Rosen&m
and M. 8. CThausi
and the transfer function to be synthesized is T=
1*5tS
s2-+-o’ls+
(14)
1’
Comparing Eqs. (13) and (14), the design equations are expressed as KG,S, G,S1+G,S2+G,S,+G2S,(1-K) Q,S,G,S2+G,S,G2S2
= 1.5,
(15a)
= 0.1,
(1510)
= 1.
(15c)
The matrices Cl and C,, defined in Eq. (lo), are computed -0 0 c,=
GIG, S,
GI S,
O
0
‘32&(1-K)
G2
4
%S,+%S,
GzS,(l--K)+G,S,+G,S,
1
(16)
1
@d’2
0
- KG,S,
0
82
82G2
1
0
as
and
c, =
0
1.5
0
0
0 _ 0
0
0
0
0
1.5
0
0
0
0
0
1.5
0
(16)
’
where the parameter vector is x = [G, G, G3 S, S, Kit. Examining the design equations [(Eqs. (15a, b, c)], it is noted that the elements of the CI matrix in Eq. (16) can be expressed in terms of the gain K and the product G,S,. For example, consider the (3, 2) term of C,, namely G, S, + G, S,. From Eq. (15b) this expression can be written as G,S,+G,S,
= O.l-GISS,+(K-l)G,S,.
(17)
G,S, = 1.5/K.
(19)
Now, from Eq. (15a)
Substituting
Eq. (18) into Eq. (15~) and solving for G, S, ,yields G,S, = l/(G&)
- 1*5/K.
(19)
Utilizing Eqs. (18) and (19) in Eq. (17), the (3, 2) term of C,, i.e. CI(3, 2), is expressed as C,(3,2) = - 1.4 -I-(K - l)/(G,S,).
100
PO)
Journal
of The Franklin
Institute
Sensitivity _Minimixation in Active RC Networks Defining #,S, = y the remaining expressed in terms of K and y as
1*5y/K
15/K
-0
0 c,=
elements of the C, matrix
(1 -K)
16/K)
(l/y-
1 1
0.1-y
0
.
(21)
1
Y
0
-
1 - 15y/K
-1.4+(K-1)/y
O 0
are similarly
0
1.5 -K/y
Thus, in view of Eqs. (16) and (21) and the measure definition in Eq. (9), the network’s sensitivity is a function of the independent parameters y and K. Here, as in the previous example, the restrictions on the RC product impose constraints upon the independent parameters (y and K). If Gt Si 2 0.1, then from Eq. (19) it is apparent that l/y-15/K>O.l.
(22)
Another constraint on y and K occurs from examination for G, S, in Eq. (17) utilizing Eqs. (18) and (19), yields
of Eq. (17). Solving
G,S, = -1’4+(K-1)/y-y. Imposing the condition
(23)
G, S, > O-1 results in (K-1)/y-y>
1.5.
(24)
Thus, the constraints on the independent parameters y and K are determined from Eqs. (22) and (24). The minimization procedure produces the following optimal values for the independent parameters K = 2.58
and
y = G,S, = 0.714.
For these values the design parameters are GsS, = 0.1
and
G,S, = O-814.
It is noted as in the last problem that the minimal values of K and y occur on the boundary of the constraint region, namely, when (K - I)/ y-y = 1.5 [from Eq. (24)]. V. Optimal
Roots
of Q(s)
for
an Arbitrary
Synthesis
Polynomial
Many active RC synthesis procedures involve choosing an arbitrary polynomial Q(s) with simple roots on the negative real axis. In this section two examples are presented to illustrate the application of the minimization technique (described in Appendix 1) for choosing the roots of Q(s) to minimize the network’s sensitivity. It should be noted that the method is quite general and applicable to all such synthesis schemes, namely the RC:-RC decomposition method. The configuration (8) to be synthesized is shown in
Vol.294,No.Z,August1972
6
101
A. Rosenblum and M. X. Ghm.mi Fig. 3. The general transfer function for this network is expressed as (25) where Y3 = Y, = 1 and the operational amplifiers are assumed ideal infinite gain. The maximum RC product is assumed 10 and the element variations are assumed independent with variances of 10-4.
F‘IG. 3. A general network realization configuration.
Example 3 In this example the following transfer function is to be realized using the general configuration in Fig. 3 ;
T_hTo_
w
1
1‘
sq-&4s+
The choice of Q(S) for this illustration is s +a where (T> 0. (Note that the degree of Q(s) must be greater than or equal to maximum degree of N(s) or D(s) minus 1.) Dividing the numerator (N(s)) and d enominator (D(s)) by Q(s), the transfer function in Eq. (26) is written as T= Now consider the numerator, N(s)
---=-
W+o) v+&3s+l)/(s-to)’ namely N(s)/&(s) written as 1
=---
S
1
(28)
SW)
s+a Q(s) 8s-U S(S+a)=a-p*
Comparing Eq. (28) with the numerator
YI=
102
l/CT
and
of Eq. (25), it is apparent that Yz=M.
s
(29)
u
Journal of The Franklin
Institute
Sensitivity Minimization In a similar fashion, the denominator
in Active RC Networks
of Eq. (27), D(s)/&(s),
D(s2 = s2+J(2)s+l = s l+s@-“)+l s+u s(s + u) Q(s) [
=s+I,u+(+-l~o)~ s+u
is written as
I (30)
.
Were, in order to equate the terms in Eq. (30) to the admittances in the denominator of Eq. (25), the term J2 - (J- l/a must be examined. For 0 > 0 this term is always negative. Thus, in Eq. (25) the admittances Y, and Y, are Y,=s+l/a
and
Y4=
((3-t L/o-42)s s+o
(31)
.
In view of Eqs. (29) and (31) the admittances in Eq. (25) can be expressed symbolically in terms of element values as follows: r, = G,,
Y,= G,s/(si-G,S,), y3= G,, Y,= d&+G,, Y4= G4s/(s+G4S4),Y5= G5,
(32)
where 8, = l/o,
G, = l/a,
G3= 1,
G4 = of
G, = 1,
ce=
G,S, = u, l/42,
G,S, = u, G, = l/u.
1,
Substituting the relationships in Eq. (32) into Eq. (25) the transfer function can be expressed symbolically in terms of the element values as
(s-tG&U W%G1Q~--a,Wd+G,GsG,fW,)
(33)
T=(s+G2S2)(s2G2+~(GgGsSs+GBG4S4-G4G5S6)~G3G4S4G6S6) The transfer function in Eq. (33) can be simplified by cancelling and zero* since G, X2 = G, S, = u and writing it as
the pole
s+G,G,S,-G,G,S,)+G,G,G,S,S,
(34)
T=s2G,+s(G3G,S,+G,G4S4-G4G5S,)+G,G4S4G6S6 The equations to generate the elements of the C, and C, matrices determined by equating the coefficients of Eqs. (34) and (26), namely = l/u-l/u
b, = G,G,S,-G,G,S, b, = GlG,G2S2S, a2 = G,=
= 0,
= 1,
1,
(35)
a, = G3G6S,+-G3G4S,-G4G,S, a, = G,G,S,G,S,
are
= 1.
= l/u+-u---/u+J2
= ,I% i
* Neglecting these terms does not have a significant effect on the measure. In this example at the optimal value of cr the measure is 1.10 x 10e3 omitting the pole-zero cancelling terms and 1.05 x 10-s including them.
Vol. 294, No. 2, August 1972
103
A. Rosen&m
and M. S. Ghazcsi
Here, the elements of the Cl and C, matrices which are functions of the root of Q(s) (i.e. c) can be determined by inspection. In particular, if the parameter vector is taken as x = [G, G, G, G4 G, G6 S, S, SJ, then the third row of the C, matrix G, adjaG, is written as G3(a3/aG,) = [l Ia,+G,G,S,il]
= [1ia+l/oi1].
(36)
The remaining elements can be similarly expressed. The minimization procedure applied to this example for wi = 0 and o2 = 1 yields an optimal choice of u as 1.2. The measure value is 1.1 x 10U3. The element values are determined from the relationships in Eq. (32). Now consider an alternate choice of Q(s) for this problem as Q(s) = (s+oJ (s +o,) where us> ur. Proceeding as before, the polynomials N(s)/&(s) and D(s)/&(s) can be expressed as follows : 1
I NW = Q(s) (s+u~)(s+u~)
D(s) -= Q(s)
SH
(37a)
= GZ+s+cr,-s+cP,’
s2+J(2)s+1 (s+u,)
8G
1
SE
SF
(s+u2) = cr,++-+’
(37b)
where G:=
’
Hz-l
4J~’
u2(u2 - 01)’ E=
u;-&qu,+1 GJ‘2 - 4
’
$7 = -__-. +11o%+l %h - 4
Comparing Eqs. (37a) and (37b) with Eq. (25), the admittances and Y, are identified as y, = -L+sc: 01 u2
s+u2’
y, = l+sF % (J2
Proceeding
Y,, Yz, Y6
ya=g> 1
(38)
Y+sE s+u2’
s+u,
1
as before, the transfer function is determined T = b2s2+blsi-b. a,s2+a,s+ao’
as
(39)
where b, = G,G,+G,G,-G,G,, b, = G,S,G,G,+G,S,G,G,+G,G,G,S,-G,G,G,S,, b, = G1 G, G, 82 G, 84, a2 = G,G,i-G,G,-G,G,,
a1 = G,G,G,S,+G,G,G,S,+G,G,S,G,-G,S,G,G,, a, = G,S,G3G,S,G,
104
Journal of The Franklin Institute
Sensitivity Minimizaticrn in Active RC Networlcs and x=[G,G,GGGGGGSSSS]~ 3456782468
(40)
and G, = llaroz, G,
=
VII+2
‘37
=
l/~,ol,
G,f& =
G, = U[~Aoz -
G,=
41,
G, = E,
G, = 1,
41,
G, = P, Q4S4
(~2,
=
1,
G,S, =
02,
G,S, = CT,.
01,
The minimization procedure for w1 = 0 and w2 = 1 yields optimal values of or = 0.94 and a2 = 3.3 with a measure value of 1.7 x 10-3. Comparing this result with the case for Q(s) = s+ cr, it is seen that choosing a second-order synthesis polynomial Q(s) yields poorer results. Example 4 The configuration in Fig. 3 is used in this example to realize a three-pole Butterworth transfer function, namely 1
T=
(41)
s3+2s~+2s+l'
Here, Q(s) is chosen as
Q(s)= (s+ud(s+4, where aZ > ul. The numerator
decomposition 1
N(s) ---=
N(s)/&(s)
1
_
Q(s) (S+d(S+%)
I
(42)
can be expressed as
_____--
siruz(u2
a
S/[%(U2-
s+u,
(Ulf32)
41.
(4‘3)
s+u,
Comparing Eq. (43) with the numerator in Eq. (25), it is apparent that +
YI = 1
s/[u2(u2
-
+)I
s+u,
Ul%
and
y = S/[T(%--al)l 2
SfUl
Similarly, the denominator
decomposition
yields
;++Es
D(s)_1
Q(s)
(44)
*
__ s+u1
UlU2
(45)
G2+s’
where
f(d
P= -
01 (a2
-
E = Ul)
’
fb2) U2(Q2
-
Cl)
and f(uJ
Vol. 294, No. 2, August 7
1972
= - u: f 20; - 2cr,+ 1
for i = 1,2.
105
A. Rosenbhm
and M. X. fCY%musi
In order to determine the admittances Y6 and Y4 in the denominator of (25), different ranges of al and u2 must be chosen (as shown in Table I) sincef(a,)>Ofor O
TABLE I Admittance assignments Case
F
E
k6
k,
.I__Fs
1. o,
s+u,
u,>l
(b) ul< 1, o,a
1 -Es
s+u, The optimization procedure must be applied separately to each case in Table I since each case will yield a different network configuration. Symbolic element values are assigned as in Eq. (32). For example, consider Case 2, namely
r, = G,, (46) Y, = G,s+G,, where G, = l/u,uz,
G,=
G, = l/CuAuz- 41,
G, = l/u,uz,
G, = l/[~l(uz -
G, = -E,
41,
1,
G3= 1,
a,&
= 01,
G, = -P,
G,&
=
G,&
= ~1,
G,X, = u2, c,=
1.
(52,
The results of the optimization procedure are summarized in Table II(a), (b) and (c) for different frequency ranges. It is noted that Case 2 yields the smallest minimum value. The output for Case 2 is found in Appendix 1. The last column in Table II is Schoeffler’s measure of sensitivity (3) namely, (47) where + is a special case of the measure in Eq. (9) [see Ref. (5)] and is generated at s = j to compare the results in Table II with those found in a
106
Journal
of The Franklin
Institute
Sensitivity Minimization
in Active RC Network8
correspondence by Aumman (9). Note that the orders of magnitude difference between the measure M(x) and 4 in Table II are accounted for by component tolerances of 1O-4 and the step size of 10-l used in the numerical integration. Multiplying 4 by 1O-5 will generate a number that has the same order of magnitude as the measure value. TABLE Summary
of sensitivity
Case
(a)w,=S;
I II III
0.52 0.84 1
I II III
0.6 0.81 1
II
0.5
II
analysis for example 4
co,=1 1 1*81 1.85 (b) w,=O;w,= 1 1.84 1.8 (c) w1 = 0; oJ2= 1.75
MM 1.63 x 10-Z 0.718 x 10-a 0.756 x 10-3 1 4.10 x IO-3 2.33 x 1O-3 2.43 x 1O-3 0.1 0.12 x 10-S
4L7.1 165 37 50 117 37 51
t M(x), minimum measure value. The existence of a minimum + is demonstrated for the synthesis of a three-pole normalized Butterworth filter [Eq. (41)], utilizing an arbitrary synthesis polynomial Q(s) = (s + al) (s + 0.J. Aumman treats the decomposition in the form of Y,-K,Y, (47) y3-&y4 and demonstrates a minimum of &j) = 22.6 at or = 0.85 and CT~ = 2.2. The decomposition in Eq. (47) is not general enough to accommodate the synthesis structure of this example, i.e. the sensitivity effects of the conductances G, and G, in Eq. (25) must be ignored. Thus, if G, and G, are not taken as components of the parameter vector, the measure (M(x)) has a minimum at 'Tl = 0.85 and g2 = 1.9 for w1 = 0.8 and w2 = 1. For these values of o1 and u2 and ignoring the sensitivity effects of G, and G,, 4 has a value of 23.5. Thus, these results are consistent with those presented by Aumman. The significance of the results presented in Table II is exemplified by considering the same synthesis procedure [Eq. (25)] and arbitrarily choosing values for a1 and u2. For example, if o1 = O-5 and a, = 1, the measure value for w1 = 0.8 and wa = 1 is 1.63 x 10-3. From Table II(a) (lines 1 and 2) it is seen that this is over 100 per cent higher than the optimal value. An even more extreme case is noted for o1 = 0.1, (TV= 0.3, w1 = 0.8 and wa = 1. Here, the measure value is 1.4 x 10-l, more than 100 times greater than the optimal value. Hence, from this example, it is apparent that the sensitivity of the network response can be critioally affected by the choice of the roots of the polynomial Q(s).
Vol.294,No.2,August1972
107
A. Rosenblum and M. S. Ghausi Conclusion
In this paper an optimization procedure to minimize a general measure of sensitivity is developed. It is seen that the minimization procedure accounts for the component tolerances, a frequency range of interest and practical restrictions on the circuit element values. The examples presented demonstrate the generality of the procedure. The technique can be implemented to determine the optimal network elements for a given circuit configuration (Section IV). The minimization procedure can also be utilized to determine the optimal roots of the synthesis polynomial Q(s) (Section V). To date, this is the only general sensitivity minimization procedure that determines the optimal roots of Q(s). The technique evolves as a practical design tool. It enables the circuit designer to account for sensitivity in the design procedure by obtaining a figure of merit for sensitivity of the network response and/or obtain an optimal set of design parameters.
Appendix
1
The procedure for minimizing the sensitivity measure in Eq. (9) is described. The technique is based upon a nonlinear programming algorithm developed by Fiacco and McCormick (6, 7). First, the nonlinear programming problem must be explicitly stated. Let the vector z be defined as the independent parameter vector, the measure in Eq. (9) denoted as &f(z) and the constraints be expressed as g,(z) > 0 for i = 1,2, . . ., m, where it is assumed that there are m constraints on the independent parameter vector z. To illustrate, consider Example 2 in Section IV. Here, the independent parameters are y and K. The measure is expressed in terms of the independent parameters through Eqs. (16) and (9) and from Eqs. (22) and (24) the constraints are g,(z) = l/y- 1*5/K-0*1 and g,(z) = (K - 1)/y-y1.5 where z = [y, Kit. The nonlinear programming problem is stated as follows : Minimize
M(z).
Subject
to g,(z)>0
for i = 1, 2, . . . . m.
The algorithm developed by Fiacco and McCormick treats the constrained minimization problems as a sequence of unconstrained minimization where the sequence converges to the solution of the constrained problem.* The algorithm defines a function of the form
m,
Tk.)= M(z)
+G31w)~
(A.1)
where rk > 0 and rlc z;$_, l/gi(z) is a penalty function. The sequence of unconstrained functions {P(z, rk)} is generated by choosing a strictly monotonic decreasing sequence {TV}. The choice of the initial value rI, convergence criteria and other aspects of the computational algorithm can be found in Ref. (7). A flowchart in Fig. 4 describes the logic of the algorithm. The essential computational aspect of the procedure is determining the minimum value for the unconstrained function P(z, TV). The principal numerical techniques used * Convergence to a global minimum gi(z) are concave functions of 2.
108
will occur only if M(z)
is a convex
function
and
Journal of The Franklin Institute
Xensitivity Minimization
in Active RC Network
0t
In
Select initial point. zO i Select initial value Of ‘x Lj )
I Determine
minimum
4
____“f_r‘k~k)_ F3.z >‘x) = M(z)+r,
&z,
Estimate PMIN for G+r
FIG.
4. Fiacco and McCormick nonlinear programming
algorithm.
to minimize an unconstrained function are first- and second-order gradient techniques that can be summarized respectively as follows :
%I = Z,_l-evz~(Zn-l,rk)
(A.2)
and z* =
!a,_1
-
efwd
v, P(z,-~, u
where
v, P(Z,_l, PP __
az; :
El=
Tk) = [aP/az,,
. ., aP/az,]t
. ..
.
(Hessian matrix-(q
x y))
PP [
G
...
and IJis the number of components of the independent parameter vector z. In the first-order methods, the nth point Z~ is obtained from the n- 1 point z,+ by descending the gradient of P evaluated at z,_~. 0 is the distance along the gradient that
Vol. 294,No. 2, August 1972
A. Rosenblum and M. 23.Ghausi is traveled. The second-order method obtains the next point by moving down a vector that is the gradient of P mapped by the inverse of the Hessian matrix (second partial derivatives of P). The first-order method requires only first derivatives; i.e. V, P, but converges too slowly for minimizing each of a sequence of P functions. The second-order method, on the other hand, converges rapidly, but it requires a knowledge of second-order derivatives, which is usually difllcult to obtain. Since the sensitivity measure in Eq. (9) is expressed explicitly in terms of the independent parameters, first and second derivatives can be obtained; i.e. the matrices C, and C, in Eq. (9) are explicit functions of the independent parameters. A flowchart depicting the minimization of the unconstrained object function P(z, rk) is given in Fig. 5. The INPUT subroutine provides the matrices C,, C, in Eq. (9) and
Yes
Q-Is
out
“+-ll
0
+,
“PF-Penalty
function (5
% I/gi(z)
;=I FXG. 5. Minimization
of lmconstrained
1
object function P(z, re) ]Eq. (A.l)].
all first and second derivative information explicitly in terms of the independent parameters required to compute the measure, the gradient of the measure (subroutine GRAD2), and the Hessian of the measure (subroutine SEC2). In addition, the subroutine INPUT contains the penalty flmction (TV2 l/gi(z)), the gradient, and the Hessian of the penalty function. A sample output* is summarized in Tables III and IV. The iterations for the main algorithm are tabulated in Table III and the iterations for the minimization of the initial unconstrained object function (P(z, TV)) are summarized in Table IV. * Note : This is Example 3 in Section V for the measure taken over a wider frequency range (wi = 0, w2 = 3). This example was chosen because it illustrates the convergence of the technique for a relatively large number of unconstrained P(z, ra) functions.
110
Jowml of The Franklin
Institute
Sensitivity MirGmization in Active RC Network8 1
TABLE III
.
Minimization
Iteration 1 2 3 4 5 6 7
of Example
P(z, re) x 104 (unconstrained function)
M(z) x 104 (sensitivity
10,464 1394 643 491 447 434 427
2093 1127 544 456 433 428 427
measure)
3 in Section V
Penalty function (r, Z1
l/g*(z))
8371 268 99 35 14 6 0.3
r>;x 104
e1
02
1105 138 17.3 2.1 0.27 0.03 0.0042
0.85 0.45 0.74 0.92 0.981 0.996 0.999
2.2 9.66 6.5 5.6 5.75 5.75 5.73
TABLE IV Unconstrained minimization of P(z, rk) in Example
3 of Section V for rk = r1 = 8371
Iteration
P(z, rlc)x lo4
Convergence criteria
1 2 3 4
10,464 3778 3292 3261
2507 263 26 0.06
0.85 0.45 0.49 0.45
2.2 4.89 8.86 9.59
References Rosenblum and M. S. Ghausi, “Multiparameter sensitivity in active RC networks”, IEEE Trans. Circuit Theory Special Issue on Digital and Active FiZters, Vol. CT-l& No. 6, pp. 592-599, Nov. 1971, also New York University Technical Report No. 400-206, July 1970. Sallen and Key, “A practical method of designing RC active networks”, IRE Trans. Circwit Theory, Vol. CT-2, Mar. 1955. J. D. Schoeffler, “The synthesis of minimum sensitivity networks”, Trans. IEEE, Vol. CT-II, pp. 271-288, June 1964. A. J. Goldstein and F. F. Kuo, “Multiparameter sensitivity”, IRE Trans. C+rcuit Theory, Vol. CT-S, No. 2, pp. 177-178, June 1961. A. Rosenblum, “Multiparameter sensitivity and sensitivity minimization in active RC networks”, Ph.D. thesis, New York Univ., Apr. 1971. A. V. Fiaoco and G. P. McCormick, “Programming under nonlinear constraints by unconstrained minimization: A primal dual method”; Research Analysis Corp., RACTP-96, Bethesda, Md., Sept., 1963. A. V. Fiacco and G. P. McCormick, “Computational algorithm for the sequential unconstrained minimization technique for nonlinear programming”, Management Sci., 10, pp. 601-617, 1964. W. P. Lovering, “Analog Computer simulation of transfer functions”, Proc. IEEE, Vol. 53, No. 3, pp. 306307, Mar. 1965. H. Aumman, “Sensitivity minimum in active transfer fun&ion synthesis”, Proc. IEEE, Vol. 58, No. 4, pp. 595-596, Apr. 1970.
(1) A.
(2) (3) (4) (5) (6)
(7)
(8) (9)
vol. 204, No. 2, August1978
111