Mathematical Development of the Adjoint Circuit Method

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International Journal of Contemporary Research and Review CrossRef DOI: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 ISSN 0976 – 4852 September, 2017|Volume 08|Issue 09|

Mathematical Development of the Adjoint Circuit Method – A Tutorial J. Ladvánszky Ericsson Telecom Hungary Accepted 2017-08-20; Published 2017-09-09

Abstract: Rigorous mathematical background for the well-known adjoint circuit method has been presented. Keywords: Adjoint circuit, sensitivities, automated computer aided circuit design 1. Introduction: Adjoint circuit method is a well-known tool in automated circuit design. It has been presented first in 1969 [1] when the automation concept entered into circuit design. However, the mentioned paper developed the idea in an intuitive way. Purpose of this paper is to add a possible rigorous mathematical explanation for the adjoint circuit method using matrix calculus [2]. Material in this paper was presented first in 1980 [3] as an internal lecture when the author was a young researcher. Surprisingly, since then no attempt was made in the same direction. 2. Problem statement: Let us consider the nV+nI port circuitin Fig. 1.

Fig. 1. The circuit under investigation International Journal of Contemporary Research and Review, Vol. 8, Issue. 9, Page no: TC 20288-20293 doi: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 Page | 20288

J. Ladvánszky / Mathematical Development of the Adjoint Circuit Method – A Tutorial

Given      

The excitations vGj and iGk at n frequency points The topology of the circuit π Types of the circuit elements Element values for a part of the circuit elements Prescribed characteristics 𝑖̂, ̂ 𝐹𝑗 𝑣 Weight factors 𝑊𝐼𝑗 , 𝑊𝑉𝑘 (real)

To be determined: Missing circuit element values so that the goal function 2

1

2 𝜀 = 2 𝛴𝛺 [𝛴𝑗 𝑊𝐼𝑗 (𝑖𝐹𝑗 − 𝑖̂) + 𝛴𝑘 𝑊𝑉𝑘 (𝑣𝐹𝑘 − 𝑣̂ 𝐹𝑗 𝐹𝑘 ) ]

(1)

has a minimum. Ω denotes the set of frequencies. We point out that applying results from matrix calculus [2], the above problem can be solved. For the solution, the generalized version [5] of theTellegen’s theorem will be used. 3. Differentiation with respect to a matrix [2] and its application: Given the matrix-valued matrix function B (F) where B and F can be of arbitrary size. Derivative of B with respect to F is defined as δB/δF=[δB/δ𝑓𝑖𝑗 ]

(2)

that is a hyper matrix of size ps x r t i f sizes of B and F are p x r and s x t, respectively. Derivative of a matrix product is δAB/δF=δA/δF ( I’xB)+( I’’xA) δB/δF

(3)

where I ’ and I’’ are identity matrices of sizes t and s, respectively and x is the Kronecker product. In the following, we use identity matrices of different sizes. Uniformly we denote all of them by I, independently of size. It is easy to find out the size. Now we apply this to our problem. The simplified version of the object function is written as 1

1

𝜀 = 2 (𝑖𝐹 − 𝑖̂𝐹 )𝑇 𝑊𝐼 (𝑖𝐹 − 𝑖̂𝐹 )∗ + 2 (𝑣𝐹 − 𝑣̂𝐹 )𝑇 𝑊𝑉 (𝑣𝐹 − 𝑣̂𝐹 )∗

(4)

Where we left out the summation with respect to frequency points and the double underline format rices. T and the star stands for the matrix transpose and complex conjugate.If the column matrix of the element values to be determined is denoted by p (real and positive), then the truncated Taylor series of the object function is 𝛿𝜀

𝜀(𝑝) ≈ 𝜀(𝑝0 ) + 𝛿𝑝 (𝑝 − 𝑝0 )

(5)

Where 𝑝0 is the starting value and p is another value in the small neighbour hood of 𝑝0 . Thus an iterative process to determine the element values is International Journal of Contemporary Research and Review, Vol. 8, Issue. 9, Page no: TC 20287-20293 doi: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 Page | 20289

J. Ladvánszky / Mathematical Development of the Adjoint Circuit Method – A Tutorial 𝛿𝜀

𝜀(𝑝𝑖+1 ) = 𝜀 (𝑝𝑖 ) + 𝛿𝑝 (𝑝𝑖+1 − 𝑝𝑖 ) and

𝛿𝜀 𝛿𝑝

can be evaluate dusing (3,4). The expression for

(6) 𝛿𝜀 𝛿𝑝

will contain

𝛿𝑖𝐹 𝛿𝑝

and

𝛿𝑣𝐹 𝛿𝑝

. These are called as

sensitivities. Our problem has been reduced to the determination of the sensitivities. For doing this, an efficient tool is theTellegen’s theorem. 4.

Tellegen’s theorem [4, 5]:

Let π’and π ’’two circuits of the same topology. Column matrices of the currents and voltages in the first and these cond circuits are denoted by i’ and v’’, respectively. Tellegen’s theorem [4] says that the total virtual power P is zero: ∗𝑇

P=𝑖 ′ 𝑣 ′′

(7)

P=0

(8)

Generalized version of the Tellegen’s theorem [5] says that the statement remains valid for the linear combination of the currents and voltages: When i’ and v’’are replaced by λ′ 𝑖 ′and λ′′ 𝑣 ′ ′respectively., λ’, λ’’ are square matrices so that the Kirchhoff laws are satisfied: 𝐵 𝜆′′ 𝑣 ′′ = 0

(9)

𝛴𝑘 (𝜆′ 𝑖 ′ )𝑘 = 0

(10)

B is the loop-branch matrix.

5.

Application of theTellegen’s theorem for sensitivity calculations:

Let H and H be circuits of the same topology. Currents and voltages are denoted by i, v, φ and ψ, respectively. Kirchhoff laws are satisfied. Given v𝐺 , 𝑖𝐺 and circuit elements of H.

𝛿𝑣𝐹 𝛿𝑝

and

𝛿𝑖𝐹 𝛿𝑝

should be

determined. First we arrange the order of branches in the following way, voltage and the current generators, finally the branches corresponding to the investigated circuit elements (p): 𝑣𝐺 𝑣𝐹 𝑣𝑃 𝑖𝐹 𝑖𝐺 𝑖𝑃

(11) (12)

𝜓𝐺 𝜓𝐹 𝜓𝑃

(13)

𝜑𝐹 𝜑𝐺 𝜑𝑃

(14)

Applying Tellegen’s theorem: 𝑣𝐺∗𝑇 𝜑𝐹 + 𝑣𝐹∗𝑇 𝜑𝐺 + 𝑣𝑃∗𝑇 𝜑𝑃 = 0 International Journal of Contemporary Research and Review, Vol. 8, Issue. 9, Page no: TC 20287-20293 doi: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 Page | 20290

(15)

J. Ladvánszky / Mathematical Development of the Adjoint Circuit Method – A Tutorial

𝜓𝐺∗𝑇 𝑖𝐹 + 𝜓𝐹∗𝑇 𝑖𝐺 + 𝜓𝑃∗𝑇 𝑖𝑃 = 0

(16)

Now we change the circuit elements p. Generators are not changed. Eq. (3) is applied. (𝐼 𝑥 𝑣𝐺∗𝑇 )

𝛿𝜑𝐹

(𝐼 𝑥 𝜓𝐺∗𝑇 )

𝛿𝑖𝐹

+

𝛿𝑝

𝛿𝑝

𝛿𝑣𝐹∗𝑇 𝛿𝑝 ∗𝑇 𝛿𝜓𝐹

+

𝛿𝑝

(𝐼 𝑥 𝜑𝐺 ) + (𝐼 𝑥 𝑣𝑃∗𝑇 ) (𝐼 𝑥 𝑖𝐺 ) + (𝐼 𝑥 𝜓𝑃∗𝑇 )

𝛿𝜑𝑃 𝛿𝑝

𝛿𝑖𝑃 𝛿𝑝

∗𝑇 𝛿𝑣𝑃

+

+

𝛿𝑝

∗𝑇 𝛿𝜓𝑃

𝛿𝑝

(𝐼 𝑥 𝜑𝑃 ) = 0

(17)

(𝐼 𝑥 𝑖𝑃 ) = 0

(18)

Elements of Hare not changed, thus the corresponding terms are omitted from (17, 18): 𝛿𝑣𝐹∗𝑇 𝛿𝑝

( 𝐼 𝑥 𝜑𝐺 ) +

(𝐼 𝑥 𝜓𝐺∗𝑇 )

𝛿𝑖𝐹 𝛿𝑝

∗𝑇 𝛿𝑣𝑃

𝛿𝑝

( 𝐼 𝑥 𝜑𝑃 ) = 0

(19)

𝛿𝑖

+ (𝐼 𝑥 𝜓𝑃∗𝑇 ) 𝛿𝑝𝑃 = 0

(20)

Sub tracting (20) from (19): 𝛿𝑣𝐹∗𝑇 𝛿𝑝

( 𝐼 𝑥 𝜑𝐺 ) +

∗𝑇 𝛿𝑣𝑃

𝛿𝑝

(𝐼 𝑥 𝜑𝑃 ) − (𝐼 𝑥 𝜓𝐺∗𝑇 )

𝛿𝑖𝐹 𝛿𝑝

𝛿𝑖

− (𝐼 𝑥 𝜓𝑃∗𝑇 ) 𝛿𝑝𝑃 = 0

(21)

Now we distinguish current and voltage controlled branches by separating voltages and currents: 𝑣1 𝜑1 𝜓 𝑖 𝑣𝑃 = [𝑣 ] 𝑖𝑃 = [ 1 ] 𝜓𝑃 = [ 1 ] 𝜑𝑃 = [𝜑 ] 𝑖2 𝜓2 2 2 ′ 𝑣1 [ 𝑖 ] = [𝑍 ′ 2 𝐾

[

′′ 𝜓1 ] = [ 𝑍 ′′ 𝜑2 𝐾

(22)

𝐻′ ] [ 𝑖1 ] 𝑌 ′ 𝑣2

(23)

𝐻′′ ] [ 𝜑1 ] 𝑌 ′′ 𝜓2

(24)

Now (21) is transformed: 𝛿𝑣𝐹∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑𝐺 ) − (𝐼 𝑥 𝜓𝐺∗𝑇 )

𝛿𝑖𝐹 𝛿𝑝

=−

𝛿𝑖

∗𝑇 𝛿𝑣𝑃

𝛿𝑝

𝛿𝑖

(𝐼 𝑥 𝜓1∗𝑇 ) 𝛿𝑝1 + (𝐼 𝑥 𝜓2∗𝑇 ) 𝛿𝑝2 = − −

𝛿𝑣1∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑1 ) −

𝛿𝑖2∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑𝑃 ) + (𝐼 𝑥 𝜓𝑃∗𝑇 ) 𝛿𝑣1∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑1 ) −

(𝐼 𝑥 (−𝜓2 )) −

𝛿𝑣2∗𝑇 𝛿𝑝

𝛿𝑖1∗𝑇 𝛿𝑝

𝛿𝑖𝑃 𝛿𝑝

= −

( 𝐼 𝑥 𝜑2 ) +

𝛿𝑣1∗𝑇 𝛿𝑝 𝛿𝑖1∗𝑇 𝛿𝑝

(𝐼 𝑥 (−𝜓1 )) −

(𝐼 𝑥 𝜑1 ) − (𝐼 𝑥 𝜓1 ) +

𝛿𝑣2∗𝑇 𝛿𝑝

𝛿𝑣2∗𝑇 𝛿𝑝 𝛿𝑖2∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑2 ) + (𝐼 𝑥 𝜓2 ) =

(𝐼 𝑥 𝜑2 )

(25)

(23, 24) are substituted into (25): 𝛿𝑣𝐹∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑𝐺 ) − (𝐼 𝑥 𝜓𝐺∗𝑇 )

′ 𝑖 ∗𝑇 𝜕 − 𝜕𝑝 ([ 1 ] ) [(𝐼 𝑥 [ 𝑍 ′ 𝑣2 𝐾

𝛿𝑖𝐹 𝛿𝑝

𝑖 ∗𝑇 ′ 𝜕 = − 𝜕𝑝 ([ 1 ] [ 𝑍 ′ 𝑣2 𝐾

∗𝑇 ∗𝑇 𝐻′ ] ) (𝐼 𝑥 [ 𝜑1 ]) − 𝜕 [ 𝑖1 ] (𝐼 𝑥 [−𝜓1 ]) = −𝜓2 𝜕𝑝 𝑣2 𝜑2 𝑌′

∗𝑇 ∗𝑇 𝐻′ ] ) (𝐼 𝑥 [ 𝜑1 ]) + (𝐼 𝑥 [−𝜓1 ])] − (𝐼 𝑥 [ 𝑖1 ] ) 𝜕 [ 𝑍 ′ −𝜓2 𝜕𝑝 𝐾 ′ 𝑣2 𝜑2 𝑌′

𝐻′ ] 𝑌′

∗𝑇

𝜑1 (𝐼 𝑥 [−𝜓 ]) 2 (26)

International Journal of Contemporary Research and Review, Vol. 8, Issue. 9, Page no: TC 20287-20293 doi: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 Page | 20291

J. Ladvánszky / Mathematical Development of the Adjoint Circuit Method – A Tutorial

Sensitivity computation is simplified if the following termin Eq. (26) is zero: ′ (𝐼 𝑥 [ 𝑍 ′ 𝐾

∗𝑇 𝐻′ ] ) (𝐼 𝑥 [ 𝜑1 ]) + (𝐼 𝑥 [−𝜓1 ]) = 0 −𝜓2 𝜑2 𝑌′ ′

[𝑍 ′ 𝐾

𝐻′ ] 𝑌′

∗𝑇

(27)

𝜑1 −𝜓 [−𝜓 ] + [ 1 ] = 0 𝜑2 2

′ 𝜓 [ 1] = [ 𝑍 ′ 𝜑2 −𝐾

−𝐻′ ] 𝑌′

∗𝑇

(28)

𝜑1 [𝜓 ] 2

(29)

In the literature, this is defined as the adjoint circuit [6]. Quoderat demonstrandum. Then the remaining terms from (26) are: 𝛿𝑣𝐹∗𝑇 𝛿𝑝

(𝐼 𝑥 𝜑𝐺 ) − (𝐼 𝑥 𝜓𝐺∗𝑇 )

𝛿𝑖𝐹 𝛿𝑝

𝑖1 ∗𝑇 𝜕 𝑍 ′ = − (𝐼 𝑥 [ ] ) 𝜕𝑝 [ ′ 𝑣2 𝐾

𝐻′ ] 𝑌′

∗𝑇

𝜑1 (𝐼 𝑥 [−𝜓 ]) 2

(30)

The original problem was to minimize𝜀 in (4): 1

1

𝜀 = 2 (𝑖𝐹 − 𝑖̂𝐹 )𝑇 𝑊𝐼 (𝑖𝐹 − 𝑖̂𝐹 )∗ + 2 (𝑣𝐹 − 𝑣̂𝐹 )𝑇 𝑊𝑉 (𝑣𝐹 − 𝑣̂𝐹 )∗ 𝛿𝜀

= 𝑅𝑒[(𝐼 𝑥 (𝑖𝐹 − 𝑖̂𝐹 )𝑇 𝑊𝐼 𝛿𝑝

∗ 𝛿𝑖𝐹

𝛿𝑝

+

𝛿𝑣𝐹𝑇 𝛿𝑝

(31)

(𝐼 𝑥 𝑊𝑉 (𝑣𝐹 − 𝑣̂𝐹 )∗ )]

(32)

Taking the real part of (30) and conjugating the left side argument: 𝑖 ∗𝑇 𝜕 ′ 𝛿𝑣 𝑇 𝛿𝑖 ∗ 𝑅𝑒 [ 𝛿𝑝𝐹 (𝐼 𝑥 𝜑𝐺∗ ) − (𝐼 𝑥 𝜓𝐺𝑇 ) 𝛿𝑝𝐹 ] = −𝑅𝑒 [(𝐼 𝑥 [ 1 ] ) 𝜕𝑝 [ 𝑍 ′ 𝑣2 𝐾

𝐻′ ] 𝑌′

∗𝑇

𝜑1 (𝐼 𝑥 [−𝜓 ])] 2

(33)

Now we compare (32) and (33). If the excitations of the adjoint circuit are chosen as: 𝜓𝐺𝑇 = (𝑖𝐹 − 𝑖̂𝐹 )𝑇 𝑊𝐼 𝜑𝐺∗ = −𝑊𝑉 (𝑣𝐹 − 𝑣̂𝐹 )∗

(34) (35)

Then the sensitivities can be calculated from: 𝑖 ∗𝑇 𝜕 ′ [(𝐼 𝑥 [ 1 ] ) [ 𝑍 ′ = 𝑅𝑒 𝛿𝑝 𝜕𝑝 𝐾 𝑣2 𝛿𝜀

𝐻′ ] 𝑌′

∗𝑇

𝜑1 (𝐼 𝑥 [−𝜓 ])] 2

(36)

For example, in a resistive branch, directly from (36): 𝛿𝜀 𝛿𝑅

= 𝑅𝑒(𝑖𝑅∗ 𝜑𝑅 )

(37)

Similarly, in an inductive, conductive or capacitive branch: 𝛿𝜀 𝛿𝐿 𝛿𝜀 𝛿𝐺

= 𝑅𝑒(𝑖𝐿∗ 𝑗𝜔𝜑𝐿 )

(38)

= −𝑅𝑒(𝑣𝐺∗ 𝜓𝐺 )

(39)

International Journal of Contemporary Research and Review, Vol. 8, Issue. 9, Page no: TC 20287-20293 doi: http://dx.doi.org/10.15520/ijcrr/2017/8/09/316 Page | 20292

J. Ladvánszky / Mathematical Development of the Adjoint Circuit Method – A Tutorial 𝛿𝜀 𝛿𝐶

= 𝑅𝑒(𝑣𝐶∗ 𝑗𝜔𝜓𝐶 )

(40)

The task is to analyze both the original and the adjoint circuits, with the excitations 𝑣𝐺 , 𝑖𝐺 , Eq. (34, 35), and then from the branch voltage and current solutions, calculate the sensitivities fromEq. (37-40). Then to it erate as it is given in (6). 6. Conclusions: A detailed mathematical development of the well-known adjoint circuit method has been presented. Our key results are the following. a. b. c. d.

Pointing out that a possible mathematical basis for the adjoint circuit method is matrix calculus. Setting the object function in a well-treatable form, Eq. (1) Expressing the ex citations in a proper way, Eqs. 34-35 Showing that the mathematical results are the same as those previously published [1] (in Eq.(29, 37-40)

7. Acknowledgments: This work was possible to be created in the out standing research atmosphere of the Ericsson Telecom Hungary. Many thanks for our director V. Beskid and our manager B. Kovács. Scientific support from T. Berceli and E. Simonyi is greatly acknowledged. 8. References: 1. S. W. Director, R.Rohrer: „ The generalized adjointnet work and network sensitivities”, IEEE Transactions on Circuit Theory, Volume: 16, Issue: 3, 1969, pp. 318 - 323 2. J. W. Brewer: „Kronecker products and matrix calculus in system theory”, IEEE Trans. on CAS, Vol. CAS-25, No. 9, pp. 772-780 3. J. Ladvánszky: „ The adjointnet work method – Derivation based on matrix calculus”, oral presentation, Research Institute for Telecommunications, 1980 4. B. D. H. Tellegen: “A general network theorem, with applications”, Philips Res. Rept., 7, 259-269, 1952 5. P. Penfield, Jr., R. Spence, S. Duinker: „A generalized form of the Tellegen’stheorem”, IEEE Trans. on CT, Vol. CT-17, No. 3, pp. 302-305 6. L. O. Chua, C. A.Desoer, E. S. Kuh: „Linear and non linear circuits”, McGraw-Hill, 1987

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