Adjoint Method in Network Analysis

1. Adjoint Method in Network Analysis Dr. Janusz A. Starzyk

2. 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

3. 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.

4. 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)

5. 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)

6. 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

7. = Adjoint Method - example • Find sensitivity of Vout with respect to G4. From KCL: • System equations TX = W are C2=1 G1=1 + + v4 E=1 Vout G3=1 - G4=4 -

8. Adjoint Method - example • If we use s = 1 then the solution for X is calculate therefore

9. 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)

10. 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

11. 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

12. = 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

13. C2=1 G1=1 + + E=1 Vout G3=1 Cp - G4=4 - = 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

14. 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.

15. R 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

16. 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).

17. 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.

18. C2=1 G1=1 + + E=1 Vout G3=1 - G4=4 - Noise Analysis Using Adjoint Vectors - example Example: Calculate the signal-to-noise ratio for the output voltage. Ignore noise due to op-amp.

19. 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:

20. Noise Analysis Using Adjoint Vectors - example • and • Thus the total noise signal is:

21. Noise Analysis Using Adjoint Vectors - example • We can replace by with to obtain • and the signal to noise ratio is computed from:

22. 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

23. Questions?