Adjoint Method in Network Analysis

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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.

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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)

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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)

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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

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Adjoint Method - example

• Find sensitivity of Vout with respect to G4.
• 
  From KCL
• System equations TX W are

C21
G11


v4
E1
Vout
G31
G44
-
-
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Adjoint Method - example

• If we use s 1 then the solution for X is
• calculate
• therefore

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Adjoint Method - example

• Since Vout 0 1 X, we get d 0 1T, and
  compute the adjoint vector from
• so
• and the output derivative is obtained from
  equation (iv)

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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

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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

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Adjoint Analysis in Electrical Networks - example

• In the previously analyzed network we had
• 
  and
• Thus to find the derivative we need to
  calculate
• - only a single multiplication

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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

considering parasitic location and there is
no need to repeat the circuit analysis
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Solution of Linear Systems using the Adjoint
Vector

• 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.

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Noise Analysis Using the Adjoint Vector

• 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.

where k Boltzmann's constant T temperature in
Kelvins Df frequency bandwidth
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Noise Analysis Using the Adjoint Vector

• 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).

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Noise Analysis Using the Adjoint Vector

• 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.

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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.
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Noise Analysis Using Adjoint Vectors - example

• 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

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Noise Analysis Using Adjoint Vectors - example

• and
• Thus the total noise signal is

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Noise Analysis Using Adjoint Vectors - example

• We can replace by with
  to obtain
• and the signal to noise ratio is computed from

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Summary

• 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

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Questions?