USE OF ENERGY RELATIONS IN THE ANALYSIS OF LINEAR AND NONLINEAR CIRCUITS

E. I. Goldshtein

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In recent years, interest has grown in the energy-based consideration of problems from various fields of engineering. The present work continues a critical examination of energy relations known from the literature [1, 2], with a view to choosing ways of constructing and developing sufficiently universal and reasonably engineering-oriented methods for the analysis and synthesis of nonlinear circuits and devices based on them. Attention is also given to energy relations used in the analysis of linear circuits, since a “strengthening” of the energy approach to the calculation of linear circuits makes it possible to relate Kirchhoff’s laws in a clear way to more general laws of conservation of energy, and ensures continuity in the transition from linear problems to nonlinear ones.

I. Linear Circuits

In [3, 4, 5, 6] the possibility is shown of obtaining Kirchhoff’s laws on the basis of the principle of least action and the Euler–Lagrange equation that follows from it. The results of C. Seeley [3] should be especially noted; he examined in detail the relations between force functions and electromechanical analogues and successfully applied them to the study of electromechanical transducers. However, even in the practice of scientific research, not to mention the educational process, equations of motion in generalized coordinates are used quite insufficiently; this is explained to a certain degree by the historically established tradition of regarding Kirchhoff’s laws as fundamental, and power-balance conditions only as auxiliary, checking relations.

In order to formalize the procedure for obtaining equations according to Kirchhoff’s second law and to relate them better to the general laws of conservation of energy, it seems useful to use the equations of motion in a somewhat modified form of notation.

The generally accepted form of writing the equation of motion is [3]

\[ \frac{d}{dt}\left(\frac{\partial \mathcal{E}_{k}}{\partial \dot q}\right) - \frac{\partial \mathcal{E}_{k}}{\partial q} + \frac{\partial \mathcal{E}_{p}}{\partial q} + \frac{\partial F_R}{\partial \dot q} = N, \tag{1} \]

where \(\mathcal{E}_{k}\) and \(\mathcal{E}_{p}\) are force functions corresponding to the kinetic and potential energy stored in the inductance \(L\) at current \(i\) and the capacitance \(C\) at charge \(q\);

\[ \mathcal{E}_{\mathrm{k}}=\sum \frac{1}{2} L_{\mathrm{k}} i_{\mathrm{k}}^{2}; \qquad \mathcal{E}_{\mathrm{n}}=\sum \frac{1}{2}\frac{q_{\mathrm{k}}^{2}}{C}; \tag{2} \]

\(F_R\) is Rayleigh’s dissipation function, taking into account heat losses in the resistance \(R\);

\[ F_R=\sum \frac{1}{2} R_{\mathrm{k}} i_{\mathrm{k}}^{2}; \tag{3} \]

\(\dot q\) is the time derivative of the generalized coordinate, in our case the current \(i\);

\(N\) is the generalized component of the applied force (in our case, voltage).

If we take into account that

\[ \frac{\partial \mathcal{E}_{\mathrm{k}}}{\partial q}=0; \qquad \frac{d}{dt}\left(\frac{\partial \mathcal{E}_{\mathrm{k}}}{\partial i}\right)=L\frac{di}{dt}=U_L, \]

\[ \frac{\partial \mathcal{E}_{\mathrm{n}}}{\partial q}=\frac{q}{C}=\frac{1}{C}\int i\,dt=U_C; \qquad \frac{\partial F_R}{\partial i}=Ri, \tag{4} \]

and use the expressions for reactive and active powers

\[ Q_{L\mathrm{k}}=I_{\mathrm{k}}^{2}\omega L_{\mathrm{k}}=U_{L\mathrm{k}}I_{\mathrm{k}}; \qquad Q_{C\mathrm{k}}=I_{\mathrm{k}}^{2}\frac{1}{\omega C_{\mathrm{k}}}=U_{C\mathrm{k}}I_{\mathrm{k}}, \]

\[ \sum Q_{\mathrm{H}}=\sum Q_{L\mathrm{k}}-\sum Q_{C\mathrm{k}}; \qquad \sum P_{\mathrm{H}}=\sum I_{\mathrm{k}}^{2}R_{\mathrm{k}}, \]

then equation (1) can be replaced by another one, (6), directly connected with the traditionally well-known condition of balance of active and reactive powers of the source (index “И”) and the load (index “Н”), familiar to every student and, hence, to every engineer:

\[ \frac{\partial S_{\mathrm{И}}}{\partial I_{\mathrm{k}}} = \frac{1}{2}\frac{\partial S_{\mathrm{H}}}{\partial I_{\mathrm{k}}}, \tag{6} \]

where

\[ S_{\mathrm{И}}=\sum E_{\mathrm{k}}I_{\mathrm{k}}\cos\varphi_{\mathrm{k}} +j\sum E_{\mathrm{k}}I_{\mathrm{k}}\sin\varphi_{\mathrm{k}}; \]

\[ S_{\mathrm{H}}=\sum P_{\mathrm{H}}+j\sum Q_{\mathrm{H}}; \]

\(E_{\mathrm{k}}\) and \(I_{\mathrm{k}}\) are the rms values of the voltage and current of the power-supply sources;

\(\varphi\) is the angle between the corresponding emf’s and currents of the sources. Thus, writing the equations of power balance as

\[ P_{\mathrm{И}}=\sum E_{\mathrm{k}}I_{\mathrm{k}}\cos\varphi_{\mathrm{k}}=\sum P_{\mathrm{H}}; \]

\[ Q_{\mathrm{И}}=\sum E_{\mathrm{k}}I_{\mathrm{k}}\sin\varphi_{\mathrm{k}}=\sum Q_{\mathrm{H}}, \tag{7} \]

and reducing the number of unknowns with the aid of Kirchhoff’s first law, it is easy, from (6), to obtain equations according to Kirchhoff’s second law for all loops of the circuit.

The proposed method is applicable in solving problems of calculating linear dc and ac circuits and makes it possible to simplify the transition to the calculation of nonlinear circuits by means of energy relations.

II. Circuits with Nonlinear Reactive Resistances

In [7, 8], for the calculation of an inductive parametric generator (parametron), the energy-balance condition for the magnetic energy \(W_\text{m}(t)\) in a lossless system was used:

\[ \frac{d}{dt}\left[W_\text{m}(t)\right]=0, \tag{8} \]

where

\[ W_\text{m}(t)=0.5\,B(t)\cdot H(t), \]

\(B(t)\) and \(H(t)\) are the instantaneous values of induction and field strength. For the general case, i.e., in the presence of losses in the system, an equation analogous to (8) was obtained in [9] as the equality of the rate of decrease of the total energy of the system to the rate of heat dissipation

\[ -\frac{dW}{dt}=\sum R_k i_k^2. \tag{9} \]

Taking into account the uniqueness of the expansion of functions in a Fourier series, from (8) and (9) one can obtain as many equations as there are power harmonics taken into account in the analysis,

\[ -\frac{dW}{dt}(\kappa\omega t)=\sum R_k i_k^2(\kappa\omega t). \tag{10} \]

A similar approach was used by the author and his collaborators in analyzing frequency repetition and doubling regimes in parametrons, as well as in the study of active compensating smoothing filters, and made it possible to obtain relationships convenient for engineering calculations among inductions, field strengths, and the angles between them.

III. Circuits with Nonlinear Active Resistances

In the analysis of circuits with nonlinear active resistances, in particular with valves, some authors in one form or another use energy relations for instantaneous powers [12, 10, 11, etc.]. However, in the indicated and a number of similar works, the possibilities of decomposing instantaneous power into individual harmonics and into their sine and cosine components are not fully used; this makes it possible to substantially increase the number of equations and to carry out part of the computational transformations irrespective of the form of the nonlinear characteristic and its approximation. In works [14, 13], certain questions of calculating circuits with valves under the proposed approach are considered, and recent studies [15] confirm its universality and fruitfulness, especially for ideal valves. For such conditions it is convenient to use the balance equation for the sine and cosine components, identical in frequency, of the power harmonics at the “input” and “output” of a circuit with ideal valves:

\[ p_{\text{c.in}}(\kappa\omega t)=p_{\text{c.out}}(\kappa\omega t), \]

\[ p_{k.\text{in}}(\kappa\omega t)=p_{k.\text{out}}(\kappa\omega t). \tag{11} \]

It is interesting to note that for lossless reactive systems, in particular for a parametron in the frequency-repetition mode, an equality of reactive powers analogous in form to (11) is valid; however,

In the general case, especially when a constant component of the induction appears (the frequency-doubling regime), the question of the possibility of applying such an approach remains open and requires further investigation.

IV. Law of Conservation of Instantaneous Power

An analysis of the questions considered above shows the particular promise of wide application of the law of conservation of instantaneous power, which has in principle long been known but has been undeservedly little used in scientific research. Special attention is devoted to this question in the monograph Tellegen’s Theorem and Electrical Circuits [16], where it is shown that the sum of the instantaneous powers in all elements of any circuit is equal to zero, independently of the nature of the elements and excitations. It follows from this that equations of the form (8), (10), (11) may be interpreted as special cases of the more general equation (12) for the sum of the harmonics of instantaneous powers of the same frequency in all elements of a circuit,

\[ \sum p_{\mathrm{n}}(k\omega t)=0. \tag{12} \]

Here a special case of equation (12) for \(k=0\) is also the condition, widely used in practice, of the balance of the active (or reactive) powers averaged over a period.

Conclusions

1. The most universal relations are those following from the laws of conservation of energy.

2. In the practice of calculating nonlinear and linear circuits, one should make wider use of calculation relations following from the balance conditions of instantaneous powers.

3. It may be assumed that the creation of a unified methodology for calculating linear and nonlinear circuits, based on relations for energies and powers, is in principle possible.

References

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... Thesis abstracts of the Third All-Union Interuniversity Conference on the Theory and Methods of Calculation of Nonlinear Electrical Circuits and Systems. FAN Publishing House, Tashkent, 1967.

1. E. I. Goldshtein. Use of low-frequency inductive parameters as controlled generators of telemechanics systems. Thesis abstracts of the Twelfth All-Union Conference on Magnetic Elements of Automation and Computer Engineering. FAN Publishing House, Tashkent, 1968.

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6. E. I. Goldshtein, N. I. Kuzebnykh. On the study of energy relations in circuits with bridge rectifiers. Reports of the Second All-Union Seminar on Higher Harmonics. Tomsk, TSU Publishing House, 1971.

7. E. I. Goldshtein, A. F. Terpugova. Invariants of instantaneous power for circuits with ideal active nonlinear resistances. Thesis abstracts of the Fourth All-Union Conference on Nonlinear Circuits. Issue 1, Tashkent, 1971.

8. E. I. Goldshtein, N. I. Kuzebnykh, N. I. Kuzmenko. Use of energy relations in the calculation of circuits with ideal valves. Thesis abstracts of the Fourth All-Union Conference on Nonlinear Circuits. Issue 1, Tashkent, 1971.

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