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Variational solution of lumped element and distributed electrical circuits

A.A.P. Gibson B.M. Dillon

Indexing terms: Electrical circuits, Lumped elements, Variational methods

Abstract: Electrical circuits exhibit a stationary power property and are solvable via variational methods. An analytic technique based on this pro- perty is enunciated and the relation to classic circuit theorems is discussed. Linear and nonlinear circuit problems are presented as examples with graphically illustrated solutions provided where appropriate. A method of determining the har- monic content of signals in nonlinear circuits is described and the unified solution of a combined distributed field problem with lumped element components is also presented.

1 Introduction

Variational energy methods are readily applied to the governing equations of all the physical sciences. In elec- trical engineering such techniques have previously been used to solve boundary value problems in electrostatics, magnetostatics and electromagnetic field theory [l-31. The variational property of lumped element electrical cir- cuits is also understood [4, 51. A stationary quantity con- structed using the instantaneous power of an electrical network is one approach which is related to the Tellegen theorem [SI. Hammond applied the principles of varia- tional mechanics to deduce circuit parameters from elec- tromagnetic fields quantities [4, 61. A useful property of the variational approach is that it can be used to examine the effect of perturbations and errors on the system vari- ables. For example, second-order derivatives can be employed to study the dependence of the solution on par- ticular unknowns. Upper and lower bound confidence limits can also be deduced for the circuit solution by using complementary variational formulations [4].

The aim in this paper is to describe a stationary power analytic technique for linear and nonlinear electrical cir- cuits. In addition this formulation can be conveniently linked with methods in variational electromagnetics. It requires the formulation of a characteristic equation for the circuit. This equation is usually based on power and is constructed methodically by adding the contributing terms for each branch of the network. As the character- istic equation is a scalar quantity, any confusion with current and voltage polarities is avoided. In this imple- mentation of the variational method the stationary point

0 IEE, 1994 Paper 1205A (S8), first received 20th October 1993 and in revised form 8th March 1994 The authors are with the EEE, UMIST, PO Box 88, Manchester M60 lQD, United Kingdom

IEE Proc.-Sci. Meas. Technol., Vol. 141, No. 5, September 1994

of the characteristic equation coincides with Kirchhoffs laws. Some simple circuit examples with graphical illus- trations are presented here for linear and nonlinear cir- cuits.

The extension of variational methods to circuits is consistent with the approach currently used in many elec- tromagnetic CAD packages. For example, finite element calculations are often based on the stationary property of an energy-related functional [7]. With finite elements, the accuracy of the solution depends on the degree of dis- cretisation. A computationally efficient global solver dealing with both lumped element and distributed electri- cal and magnetic parameters simultaneously is therefore a realistic proposition. The solution of a lumped element capacitor problem with distributed field effects is included here to illustrate this unifying property. The finite element solution of the fields in space is solved in conjunction with circuit nodal voltages. A combined lumped element/Laplace energy functional is used for this purpose.

2

The variational property of electrical circuits can be used to calculate voltages and currents. Additional informa- tion can be generated concerning errors, perturbations and the dependency of system variables on the solution. Tellegen's theorem combines the two Kirchhoff laws in a simple variational form and provides a unique solution for the power distribution [SI. In its simplest form it states that in an isolated network, of n elements, the instantaneous power is zero. This is usually expressed as

Variational solution of electrical circuits

where the values of a are the elements of the network [SI. This equation permits the voltages (currents) of the network to be varied independently of the currents (voltages). This suggests a variational property of electri- cal networks based on power. The stationary power pro- perty is used here to describe a circuit analysis procedure. The stationary point has KCL or KVL as the Euler equation. The formulation described has the advantage that it can be combined with variational methods for solving electromagnetic fields. For any RLC network the time-averaged value of the instantaneous power, taken

The authors would like to thank Prof. P. Hammond of Southampton University, UK, for his helpful comments and encouragement. The SERC are also thanked for their financial support.

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over one cycle of the sinusoidal excitation, is written as the variation

where T is the period of the excitation. In a linear circuit the instantaneous power can be formed in terms of nodal voltages and branch admittances as

N N+l

(3)

where a total of (N + 1) principle nodes in the circuit has been assumed, N of which have unknown nodal voltages P'{f). The ith row of the admittance matrix Y has zero elements except in columns j where a branch exists between nodes i and j . The time-averaged power is obtained using eqn. 2 and its stationary point with respect to each unknown Vi is given by

~n N Ntl

which are a set of N linear equations with N unknowns. These equations correspond to the KCL equations for the circuit. If branch currents were used as unknowns, KVL would have been obtained. This approach can be used to solve any linear RLC network with a sinusoidal excitation. In the special case of DC reactive circuits the total energy stored in the circuit must be used to form the characteristic equation. This energy minimisation approach is described in more detail elsewhere [8].

For nonlinear circuits, the contribution to the instan- taneous power for each component is calculated using the more general form [4]

where v(t) and V,(t) are the nodal voltages. It is assumed here that the nonlinear circuit contains only passive com- ponents. The instantaneous power for a circuit of 'x' resistors, 'y' capacitors and '2' inductors can then be written as

p(t) = 1 P,,(t) + aE,(t)/at + caE,(t)/at (8) Y

The KCL equations can be satisfied at any point in time by finding the stationary point of the instantaneous power with respect to each unknown nodal voltage. Non- linear components in a circuit produce harmonics of the original signal. To find the frequency domain solution, a Fourier series expansion can be used for each unknown nodal voltage

P vir) = a, cos nwt + c a, sin (n - r )o t (9)

where a,, n = 0, ..., p , are the set of unknown Fourier coefficients for V@). The instantaneous power equation then has (N x p) unknown coefficients. Calculating the time-averaged power reduces the complexity of the equa- tion because the sinusoidal terms disappear. The station- ary point of the time-averaged power is found by setting

" = O " = I + 1

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for all ai, coefficients to zero. This results in a set of non- linear equations which can be solved using an iterative scheme.

3

The power dissipated in an electrical circuit is a scalar quantity which can be written in terms of the nodal volt- ages and impedances of the network. This scalar quantity exhibits a stationary value when the circuit's nodal volt- ages correspond to the circuit's exact solution satisfying Kirchhoffs laws. Some examples are described for DC and AC circuits with linear RLC components.

Consider the purely resistive circuit illustrated in Fig. 1. There are three principle nodes, one of which is taken

Variational solution of linear circuits

zon v, 2511 v, 1 x 1

Fig. 1 nodal uoltages V, and V,

Schematic diagram of a resistor network with two unknown

as a reference. The two unknowns are denoted V, and V, . Any classic circuit analysis technique may be invoked to deduce V, and V, . The variational solution starts by gen- erating the power characteristic equation for the electri- cal circuit under scrutiny. Using eqn. 3 and referring to Fig. 1 the characteristic equation is constructed as

(V, - lo), v: (V, - VJ2 P(V1, V,) = 20 + - + ~ 10 25

v2 (V, - 20)2 +>+- 20 15

A minimisation procedure is then invoked to establish the stationary point of this expression. In accordance with eqn. 4 this operation yields

The unique solution to this pair of simultaneous equa- tions is VI = 4.67 V and V, = 9.70 V. This solution can also be illustrated graphically. The variation of power dissipated as a function of the unknown nodal voltages V, and V, is depicted in Fig. 2 as a surface plot with a minimum exhibited at the true solution for the problem. An alternative contour plot is presented in Fig. 3.

Purely resistive circuits display a minimum point at the true solution for the network [4]. In the case of general linear RLC circuits the true solution is stationary

I E E Proc.-Sei. Meas. Technol., Vol. 141, No. 5, September 1994

but not necessarily a minimum power point. This station- ary property of RLC circuits can also be utilised in a variational solution process. Consider the RLC network

Fig. 2 tion ofunknown nodal voltages V, and V,

Surface plot of power dissipnted in circuit of Fig. I as nfunc-

1 1 0~

10 5 -

10 0-

9 5 -

9 0 -

v2

a 5-

8 0 -

7 5-

where phasor notation has been used. This expression is differentiated with respect to the unknown complex vari- able VI and the result is set to zero. This provides the following equation

which has the unique solution of VI equal to (4.12 - j3.53) volts. It is of further interest to investigate the nature of this stationary point. To do this the magni- tude of eqn. 14 has been illustrated as a function of the real and imaginary parts of VI. A stationary saddle point is clearly illustrated in Fig. 5 and the contour representa- tion of this saddle point is depicted in Fig. 6.

Fig. 5 Surface plot ofstationary power pointfor V, in circuit ofFig. 4

701 1 , , I , , , VI

2 0 2 5 3 0 3 5 LO 4 5 5 0 5 5 6 0 6 5 7 0

Fig. 3 tion ofunknown nodal voltages V, and VI

Contour plot of power dissipated in circuit of Fig I as afunc-

illustrated in Fig. 4 where there is one unknown complex voltage denoted VI. A characteristic power equation can be constructed for this circuit using eqn. 3. The three

Re(V1)

Fig. 6 Fig. 4

Contour plot of stationary power point for V, in circuit of

Fig. 4 Schematic diagram of an RLC circuit with one unknown complex nodal voltage V,

branch impedances of the network generate three entries in the characteristic equation as

(VI - lo)* v: v: P( VI) = ~ +-+-

-12 2 +j4

4

The analysis of networks containing nonlinear elements requires special attention. One common approach is to use linearisation with small-signal analysis. If the circuit cannot be linearised then the nonlinear element must be dealt with directly by developing an equation for voltage and current. A number of techniques exist for the analysis of nonlinear circuits including graphical methods and numerical analysis [SI.

For passive nonlinear circuits, the solution may be cal- culated using a variational approach so long as the current-voltage relationship in each component is single-

Variational solution of nonlinear circuits

425 IEE Proc.-Sci. Meas. Technol., Vol. 141, No. 5 , September I994

valued. The instantaneous power is calculated using eqn. 3 and the integrals in eqns. 5-7. An example of a DC resistive circuit with a nonlinear component is illustrated in Fig. 7. The dissipated power can be written as

and 5 as

p(t) = f (vl - 3 + 2 sin + 1 ' V 2 d V

(8-Y1) v P(v,) = 1 osdV + ['v2 dV

5 0 ~ (16) v: = (8 - Vl)2 + -

3

v: = +(v, - 3 + 2 sin ut)' + - 3

2 3 5 6 30

0 v1

Fig. 7 circuit

Plot of power as a function of V, for inset nonlinear resistive

The stationary point of this equation gives

(17)

This is the corresponding KCL equation for the nodal voltage Vl and can be solved to give VI = 2.37 V. The power dissipated in the circuit is plotted against V, in Fig. 7.

The next example is a reactive circuit containing a nonlinear capacitor as shown in Fig. 8. As in the DC case, for reactive circuits the variational property of the stored energy is used. The total stored energy in the circuit is obtained using eqn. 6.

0 01 3

= (VI - 4)2 + 0.5V: - - V:

The stored energy is a minimum at the true solution, so

- -0.OlV: + 3V1 - 8 = 0 aE, a Vl

(19) _-

and hence Vl is 2.69 V. Fig. 8 also illustrates the variation in the stored energy with V,, illustrated in Fig. 9.

An AC resistive circuit is considered next. The instan- taneous power for the circuit can be written using eqns. 3

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Fig. 8 capocitance circuit

Plot of stored energy as a function of V, for inset nonlinear

For a frequency domain solution, the nodal voltage is assumed to be a truncated series

V, = a,, + a , cos wt + a2 cos 2wt

+ a3 sin wt + a4 sin 2wt

Ir 3 v T

I

Fig. 9 Nonlinear resistor circuit with an AC source

which when substituted into eqn. 20 gives an expression for power with the form

2

+ qobmn cos" awt sin" but (22) u , b m , n = l

where k,,, pan and pnbmn represent sets of unknown time- independent coefficients. Each of these coefficients can be evaluated in terms of the Fourier series coefficients a i , by

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substituting eqn. 21 into eqn. 20. Calculating the time- averaged power simplifies eqn. 22 considerably so that

Pa, = ko + %12 + k22 + P i 2 + P 2 2 ) - a42112

a 3 a 2 = 5.5 - 3a0 - a3 ~ - + -

4 3

+ &at + a: + U: + a: + a:)

+ fuo(u: + a: + U: + U:) (23) The stationary point of the time-averaged power with respect to each unknown coefficient gives the following set of equations

ap,, a2 a3 aa2 4

- + a o a 2 - - = 0

a2 a3 + l = 0 - + aoa3 -~ ap,, a3 8%

ap,, a4 aa4

- + aoa , = 0

These equations can be solved using an iterative scheme to give

VI = 1.25 - 0.58 sin wt + 0.05 cos 2wt (25) An alternative approach to solving this simple circuit is to use eqn. 3 to evaluate VI in the time domain and then apply a Fourier transform.

5 Combined lumped element/distributed field

The variational solution of electrical circuits has certain unifying properties, some of which have been discussed. As variational methods are universal then in principle unified lumped element and electromagnetic field prob- lems are possible. As an example the solution of a lumped element capacitor circuit modified by distributed fields will be described. This approach provides a compu- tationally efficient solution to hybrid problems. Finite element analysis of the distributed electrostatic fields is solved simultaneously with the circuit variables. The for- mulation used always gives an upper bound on the exact solution. The finite element procedure is described in detail elsewhere and will be discussed but not reproduced here [7].

It is recalled that the energy functional FL(V) for an electrostatics problem governed by Laplace's equation is given by [7]

problems

F L ( V ) = EIVV12dS (26) b where E is the permittivity of the media. The distributed potential ( V ) must be determined over the cross-sectional surface area (S) . In the case of a combined lumped element/distributed field problem this energy functional must be minimised in conjunction with the circuit energy terms. Consider the circuit of Fig. 10, the cross-section illustrates two lumped element capacitors connected in series on a metallised dielectric substrate. The distributed capacitance of the substrate arrangement will perturb the voltage VI on the central conductor between the two

IEE Proc.-Sci. Meas. Technol., Vol. 141, No. 5, September 1994

capacitors. The combined energy expression governing the entire region has a hybrid form and is given by

F(V) = W &IVVI2 dS + iC(5 - VJ2 + i C V : (27) JS

lumped eiernent capacitor

C

\ metollisation zero volts dielectric

su bstrote

Fig. 10 mounted on a metallised substrate

Cross-sectional diagram of a lumped element capacitor circuit

where C = 25 nF. It is assumed that the distributed fields are uniform across the width ( W ) of the metallisation which has a value W = 2.5 mm. The finite element spatial discretisation of the problem region is schematically illus- trated in Fig. 11. In this arrangement all the nodes in the

i V I i

Fig. 11

finite element mesh on the central conductor are con- strained to have the same unknown potential VI. The lumped element contribution to the system energy is written in terms of this potential VI. A global matrix is constructed by adding this contribution to the appropri- ate main diagonal entry of the finite element matrices. The finite element solver returned a value for Vi = 2.374 V using a mesh of about 88 first-order elements. A contour plot of the potential distribution in the region of one of the lumped element capacitors is shown in Fig. 12.

Finite element discretisation ofsubstrate region in Fig. 10

ov Fig. 12

6 Conclusions

An often overlooked property of electrical circuits is that the nodal voltages and branch currents are arranged so that the circuit exhibits a stationary power quantity. This

Partialfinite element potential plot in vicinity alone capacitor

427

stationary property underpins many of the fundamental theorems and methods in network analysis. The varia- tional solution of electrical networks has been described and some examples have been graphically illustrated. The method has been extended to illustrate the solution of a hybrid distributed field/lumped element circuit problem.

7 References

1 HARRINGTON, R.F.: ‘Field computation by moment methods’

2 MIKHLIN, S.G.: ‘Variational methods in mathematical physics’ (Macmillan, New York, 1968)

(Macmillan, New York, 1964)

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3 BERK, A.D.: ‘Variational principles of electromagnetic resonators and waveguides’, IRE Trans., 1956, AP-4, pp. 104-111

4 HAMMOND, P.: ‘Energy methods in electromagnetism’ (Monographs in Electrical and Electronic Engineering, Oxford Science Publications, Oxford, 1981)

5 PENFIELD, P., SPENCE, R., and DUNIKER, S. : ’Tellegen’s theorem and electrical networks’ (MIT Press, Cambridge, MA, 1970)

6 HAMMOND, P., and PENMAN, J.: ‘Calculation of inductance and capacitance by dual energy principles’, IEE Proc., 1976, 123, (6), pp. 554-559

7 SILVESTER, P.P., and FERRARI, R.L.: ‘Finite element methods for electrical engineers’ (Cambridge University Press, 1990)

8 GIBSON, A.A.P.: ‘Circuit analogy to introduce variational methods’, Int. J. Educ. in Elec. Eng., 1994,31, (2), pp. 144-147

9 MAAS, S.A.: ‘Nonlinear microwave circuits’ (Artech House, 1988)

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