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Adjoint Method in Network Analysis
Dr. Janusz A. Starzyk
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Outline
-- Definition of Sensitivities
-- Derivatives of Linear Algebraic Systems
-- Adjoint Method
-- Adjoint Analysis in Electrical Networks
-- Consideration of Parasitic Elements
-- Solution of Linear Systems using the Adjoint Vector
-- Noise Analysis Using the Adjoint Vector
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Sensitivity
Normalized sensitivity of a function F w.r.t. parameter
Two semi-normalized sensitivities are discussed when either F or h is zero
and
F can be a network function, its pole or zero, quality factor, resonant frequency, etc., while
h can be component value, frequency s, operating temperature humidity, etc.
h
F
F
h
hln
FlnS F
h
h
Fh
h
FS Fh
ln h
F
Fh
FS Fh
1ln
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Derivatives of Linear Algebraic Systems
Consider a linear system
(i) TX = W
where T and W are, in general case, functions of parameters h. Differentiate (i) with respect to a single parameter hi
We are interested in derivatives of the response vector, so we can get
(ii)
hi
WX
hi
T
hi
XT
hi
WX
hi
TT
hi
X 1
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Adjoint Method
Very often, the output function is a linear combination of the components of X
(iii)
where d is a constant (selector) vector. We will compute using the so called adjoint method.
From (ii) and (iii) we will get
Let us define an adjoint vector to get
(iv)
Xd T
hi /
hi
WX
hi
TTd
hi1T
1TTa TdX
hi
WX
hi
TX
hi
Ta
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Adjoint Method
From its definition, the adjoint vector can be obtained by solving
(v)
Note that solution of this system can be obtained based on LU factorization of the original system - thus saving computations, since
1TTa TdX dXT aT
TTTT LULUT
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Adjoint Method - example
Find sensitivity of Vout with respect to G4.
From KCL:
System equations TX = W are
C2=1
E=1G3=1
G1=1
G4=4
Vout
+
-
+
-
343
221
GGG
sCsCG
out
4
v
v
0
EG1
=
0
1
15
12 4
outv
v
v4 0
0
4344
4241
out
out
vvGvG
vvsCEvG
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Adjoint Method - example
If we use s = 1 then the solution for X is
calculate
therefore
3/5
3/1
0
1
25
11
3
10
1
15
121
4
outv
v
0
0,
01
00
4343
221
44G
Wand
GGG
sCsCG
GG
T
3
1
0
3
53
1
01
00
44 G
wX
G
T
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Adjoint Method - example
Since Vout =[0 1] X, we get d = [0 1]T, and compute the adjoint vector from
so
and the output derivative is obtained from equation (iv)
dXT aT
1
0
11
52
2
1a
a
x
x
3/2
3/5
1
0
21
51
3
1
2
1
a
a
x
x
9
2
3/1
03/23/5
444
G
WX
G
TX
G
v Taout
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Adjoint Analysis in Electrical Networks
Adjoint analysis is extremely simple in electrical networks and have the following features:
1. Derivative to a source is simple, since in this case
and
where eK is defined as a unit vector:
and the output derivative w.r.t. source is
0h
T
ji eeh
W
0...001..00K
T
Ke
ai
ajji
Ta xxeeXh
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Adjoint Analysis in Electrical Networks
2. Derivative to a component is also simple, since each component value appears in at most 4 locations in matrix T
so
and the derivative of the output function is found as
hh
hh
j
T
iLKtionertin sec
10 orisvwhereeeeesh
T TLKji
v
LKaj
ai
vTLKji
Tav xxxxsXeeeeXsh
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In the previously analyzed network we had:
and
Thus to find the derivative we need to calculate
- only a single multiplication
Adjoint Analysis in Electrical Networks - example
4G
vout
9
2
3
1
3
212
0
4
xxsG
v aout
3
53
1
X
3
23
5
aX
343
221
GGG
sCsCG
out
4
v
v
0
EG1
=
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Adjoint Analysis in Electrical Networks
3. Derivative to parasitic elements can be calculated without additional analysis. We can use the same vectors X and Xa,
since the nominal value of a parasitic is zero. Suppose that we want to find a derivative with respect to a
parasitic capacitance CP shown in the same system, then
343
221
GGG
sCsCG
out
4
v
v
0
EG1
=
C2=1
E=1G3=1
G1=1
G4=4
Vout
+
-
+
-Cp
9
5
3
1
3
511
ssxxs
Ca
P
considering parasitic location
and there is no need to repeat the circuit analysis
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Finding a response of a network with different right hand side vectors is easy using the adjoint vectors.
– Consider a system with different r.h.s. vectors:
– (vi)
– we have– (vii)
– so all i can be obtained with a single analysis of the adjoint system
– this is a significant improvement comparing to repeating forward
and backward substitutions for each vector Wi.
Solution of Linear Systems using the Adjoint Vector
miXd
WTX
i
T
i
ii ...,,2,1
i
Tai
1Ti
1Ti WXWTdWTd
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Noise analysis is always performed with the use of linearized network model because amplitudes involved are extremely small.
– To illustrate how the adjoint analysis can be used in estimation of the noise signal let us consider thermal noise of a resistive element described by an independent current
source in parallel with noiseless resistor.
Noise Analysis Using the Adjoint Vector
R
RfkTin 4
where k Boltzmann's constantT temperature in Kelvinsf frequency bandwidth
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We assume that noise sources are random and uncorrelated.
The mean-square value of the output noise energy is
– where is the output signal due to the i-th noise source.
Since the noise sources are uncorrelated, we cannot use superposition.
Instead the linear circuit has to be analyzed with different noise sources as excitations (different r.h.s. vectors in system equations).
Noise Analysis Using the Adjoint Vector
22
2
2
1
2... n
mnnn VVVV
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We can use equation (vi) to perform noise analysis very efficiently. We will get
(viii)
– where is the output signal due to the i-th noise source.
Since contains at most two entries
then only one subtraction and one multiplication are needed for each noise source.
Noise Analysis Using the Adjoint Vector
ni
Tani WXv
niW
nii
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Noise Analysis Using Adjoint Vectors - example
Example:
Calculate the signal-to-noise ratio for the output voltage. Ignore noise due to op-amp.
C2=1
E=1G3=1
G1=1
G4=4
Vout
+
-
+
-
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The adjoint vector was found in the previous example.
Using (viii) we have the nominal output
The same equation is used to obtain noise outputs:
Noise Analysis Using Adjoint Vectors - example
3/2
3/5
2
1a
a
x
x
0
1
3
2
3
500 WxV
Ta
nn
nTan ii
WXV 11
11 3
5
03
2
3
5
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and
Thus the total noise signal is:
Noise Analysis Using Adjoint Vectors - example
nn
n
nn
n
ii
V
ii
V
44
4
33
3
3
20
3
2
3
5
3
20
3
2
3
5
2
4
2
3
2
1
2
9
4
9
4
9
25 nnnn iiiV
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We can replace by with to obtain
and the signal to noise ratio is computed from:
Noise Analysis Using Adjoint Vectors - example
nii iGfTk 4
ii R
G1
fkTfkT
GGGfkTV n
20)16425(49
1
)4425(49
1431
2
203
50
fkTV
Vn
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Adjoint method is an efficient numerical technique Adjoint vector can be used used to calculate output
derivatives to various circuit parameters Adjoint vector can be used to find a response of a
network with different right hand side vectors Sensitivity analysis, circuit optimization and noise
analysis can benefit from this approach
Summary
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Questions?