Transformations of Nonlinear Circuits in the Steady-State Sinusoidal Regime

G. E. Pukhov and S. P. Amosova

Introduction

Below we consider a graph-analytic method for calculating electrical circuits with so-called inertial nonlinear elements (thermal resistances, etc.), whose presence does not lead to the appearance of sufficiently noticeable harmonics of currents and voltages, as a result of which the circuit regime may be regarded as quite close to sinusoidal. The idea of the method essentially coincides with the idea of the transformation (transfiguration) method used in calculations of linear circuits.

We shall describe the properties of the nonlinear two-terminal networks constituting the circuit by means of two characteristics: the volt-ampere characteristic \(U = U(I)\) and the phase-ampere characteristic \(\varphi = \varphi(I)\), where \(U\) and \(I\) are the effective (rms) values of the voltage and current of the given two-terminal network, and \(\varphi\) is the phase-shift angle between them. The characteristics \(U(I)\) and \(\varphi(I)\) can be obtained experimentally.

The regime of a circuit with constant currents and voltages may be considered as sinusoidal under the condition that the period of the sinusoids tends to infinity. In this case the properties of the two-terminal network are determined by only one volt-ampere characteristic.

1. Series Connection of Nonlinear Two-Terminal Networks

We shall agree to measure the phase angles \(\varphi_1, \varphi_2 \ldots \varphi_n\) from the current vector to the voltage vectors of the two-terminal networks (Figs. 1 and 2), and to regard the angles as positive if the voltage leads the current in phase (in Fig. 2, \(\varphi_1\) is positive, while the others are negative). From the vector diagram we see that the volt-ampere and phase-ampere characteristics are expressed in terms of the characteristics of the corresponding two-terminal networks as follows:

Fig. 1

\[ U(I)= \sqrt{ \left[\sum_{k=1}^{n} U_k(I)\cos\varphi_k(I)\right]^2 + \left[\sum_{k=1}^{n} U_k(I)\sin\varphi_k(I)\right]^2 } \tag{1.1} \]

\[ \varphi(I)=\operatorname{arctg} \frac{\displaystyle\sum_{k=1}^{n} U_k(I)\sin\varphi_k(I)} {\displaystyle\sum_{k=1}^{n} U_k(I)\cos\varphi_k(I)}, \tag{1.2} \]

where \(U_k(I)\) and \(\varphi_k(I)\) are the voltage and phase-voltmeter characteristics of the \(k\)-th two-terminal network.

The calculation of a circuit consisting of nonlinear two-terminal networks connected in series, on the basis of the dependences given above, is carried out in the following order:

1. Having the voltage and phase-voltmeter characteristics of the two-terminal networks making up the circuit, by formulas (1.1) and (1.2) one constructs the voltage and phase-voltmeter characteristics of the equivalent two-terminal network, \(U(I)\) and \(\varphi(I)\).

Fig. 2

Fig. 3

1. Knowing the voltage applied to the circuit, from the characteristics \(U(I)\) and \(\varphi(I)\) it is easy to find the circuit current and the phase-shift angle between the current and the external voltage.

Fig. 4

Fig. 5

1. From the current value found in item 2 and from the voltage and phase-voltmeter characteristics, the voltages across the component two-terminal networks and the phase-shift angles between the voltages and the circuit current are found.

In the case of a circuit with constant currents and voltages, expression (1.1) is simplified and takes the form

\[ U(I)=\sum_{k=1}^{n} U_k(I), \tag{1.3} \]

since the angles \(\varphi_k=0\).

This dependence can be read as follows: in a circuit with constant currents and voltages, the volt-ampere characteristics under series connection of two-terminal networks are added in the direction in which the voltages are measured.

Fig. 6

Fig. 7

As an example of the calculation of a circuit consisting of series-connected nonlinear two-terminal networks, the calculation is given for the circuit shown in Fig. 3.

As the nonlinear elements for this circuit, an ordinary incandescent lamp was used, with the characteristic presented in Fig. 4, and a coil with a steel core, whose volt-ampere and phase-current characteristics are given in Fig. 5.

Figure 6 shows the vector diagram of the circuit under consideration. In this case

\[ U_0(I)=\sqrt{\left[U_r(I)+U_L(I)\cos\varphi_L(I)\right]^2+\left[U_L(I)\sin\varphi_L(I)\right]^2}, \tag{1.4} \]

\[ \varphi_0(I)=\operatorname{arctg} \frac{U_L(I)\sin\varphi_L(I)} {U_r(I)+U_L(I)\cos\varphi_L(I)}. \tag{1.5} \]

By assigning arbitrary values to the currents, from the dependences of Figs. 4 and 5 we find the corresponding values of \(U_r(I)\), \(U_L(I)\), and \(\varphi_L(I)\).

Table 1 illustrates the course of the calculation.

Table 1

Calculation of a circuit with a series connection of an incandescent lamp
and a coil with a steel core

As a result of the calculation we obtain the volt-ampere characteristic $U_0(I)$ and the phase–volt-ampere characteristic $\varphi_0(I)$ of the entire circuit. Experimental verification of the calculation gave a satisfactory result (Fig. 7).

2. Parallel Connection of Nonlinear Two-Terminal Networks

From the vector diagram of the circuit (Fig. 8) we see that the volt- and phase–volt-ampere characteristics of the equivalent two-terminal network are expressed in the following way through the characteristics of the component two-terminal networks:

\[ I(U)=\sqrt{\left[\sum_{k=1}^{n} I_k(U)\cos\varphi_k(U)\right]^2+ \left[\sum_{k=1}^{n} I_k(U)\sin\varphi_k(U)\right]^2}, \tag{2.1} \]

\[ \varphi(U)=\operatorname{arctg} \frac{\sum_{k=1}^{n} I_k(U)\sin\varphi_k(U)} {\sum_{k=1}^{n} I_k(U)\cos\varphi_k(U)} , \tag{2.2} \]

Fig. 8

where $I_k(U)$ is the volt-ampere characteristic of the $k$-th two-terminal network, read in the direction of the current reference, and $\varphi_k(U)$ is the characteristic obtained from the volt- and phase–volt-ampere characteristics by the recalculation $\varphi_k(U)=\varphi_k[U_k(I)]$. The procedure for transforming a circuit consisting of parallel-connected two-terminal networks is similar to the transformations formulated above for a series connection, except that instead of the characteristics $U_k(I)$ and $\varphi_k(I)$ one must operate with the characteristics $I_k(U)$ and $\varphi_k(U)$.

For direct currents and voltages, expression (2.1) takes the form:

\[ I(U)=\sum_{k=1}^{n} I_k(U),\quad (k=1,2\ldots n), \tag{2.3} \]

which can be read as follows: in a circuit with constant currents and voltages, the volt-ampere characteristics under parallel connection are added in the direction of the current reference.

3. Circuits Consisting of Series- and Parallel-Connected Nonlinear Two-Terminal Elements

Complex circuits that can be represented as consisting only of series- and parallel-connected two-terminal elements can be studied with the aid of the dependences given above. In transformations it becomes necessary to pass from the characteristics \(U(I)\) to \(I(U)\) and from \(\varphi(I)\) to \(\varphi(U)\).

The first transformation practically does not have to be carried out, since one and the same volt-ampere characteristic in the coordinates \((U,I)\) will be the characteristic \(U(I)\), while in the coordinates \((I,U)\) it will be the characteristic \(I(U)\).

The transition from the characteristic \(\varphi(I)\) to \(\varphi(U)\) and back is carried out with the aid of the volt-ampere characteristic \(U(I)\). For example, by assigning values of the current \(I\), one can from the graphs \(U(I)\) and \(\varphi(I)\) find the corresponding values of the voltages and angles \(\varphi\), knowing which it is easy to construct the characteristic \(\varphi(U)\). Of course, all transformations connected with passing from one set of characteristics to another require no analytical computations and are performed purely graphically.

4. Complex Circuits with Two Nonlinear Elements

For the study of circuits in which transformations of a delta into a star and back are required, as well as of a multi-arm star into a polygon, the dependences given above are insufficient. Sometimes a complex circuit with identical nonlinear elements can be replaced by an unbranched circuit containing only one nonlinear element.

An example of the calculation of such a complex circuit may be the calculation of the circuit of a bridge voltage stabilizer (Fig. 9), using as nonlinear elements \(r_{\ell 1}\) and \(r_{\ell 3}\) identical incandescent lamps with the characteristics shown in Fig. 10. The resistances \(r_2 = 210.5\ \Omega\); \(r_4 = 205\ \Omega\); \(r_5 = 222\ \Omega\) are linear.

Fig. 9

The circuit under consideration can be replaced by the circuit shown in Fig. 11, and then by the circuit given in Fig. 12. The basic equation

of the four-terminal network, taking into account the condition \(U_1(I_1)=U_3(I_3)\), is written in the following form:

\[ U_3(I_3)=A_{11}U_1(I_1)+A_{12}I_1+A_{10}, \tag{4.1} \]

or

\[ U_1(I_1)=A_{11}U_1(I_1)+A_{12}I_1+A_{10}, \tag{4.2} \]

whence

\[ U_1(I_1)-\left(\frac{A_{12}}{1-A_{11}}\right)I_1=\frac{A_{10}}{1-A_{11}}. \tag{4.3} \]

Fig. 10

Let us denote

\[ R=\frac{A_{12}}{A_{11}-1} \tag{4.4} \]

and

\[ \mathcal{E}=\frac{A_{10}}{1-A_{11}}. \tag{4.5} \]

Fig. 11  Fig. 12

Then equation (4.3) corresponds to the circuit shown in Fig. 13. From the circuit of Fig. 11, by ordinary methods, the coefficients of the four-terminal network can be determined:

\[ A_{11}=-\left(\frac{r_5}{r_2}+1\right), \tag{4.6} \]

\[ A_{12}=r_5, \tag{4.7} \]

\[ A_{10}=U_0\left(1-\frac{r_5}{r_2}\right). \tag{4.8} \]

For our example,

\[ A_{11}=-2.054;\qquad A_{12}=-222\ \Omega;\qquad A_{10}=U_0\cdot 2.054\ \mathrm{V}. \]

Knowing the coefficients of the four-terminal network, we determine the numerical value of \(R\) and \(\mathcal{E}(I)\),

\[ R=72.8\ \Omega;\qquad \mathcal{E}(I)=U_0(I)\cdot 0.673. \]

The further calculation is reduced to considering the circuit shown in Fig. 13, with one nonlinear element. From the characteristics \(U_1(I_1)\) and \(U_R(I_1)\), the dependence \(\mathcal{E}(I_1)\) is constructed; knowing it, it is easy to pass to the values of \(U_0(I)\).

Fig. 13

Let us find the relation between the voltage \(U_0\) applied to the circuit and \(U_5\), the voltage across the bridge diagonal. From the circuit in Fig. 11 it follows that

\[ U_5(I_1)=U_0(I_1)-2U_1(I_1). \tag{4.9} \]

Fig. 14

Figure 14 presents the calculated and experimental characteristics of the stabilizer.

If a complex electric circuit contains two dissimilar nonlinear elements, the calculation circuit can always be reduced to the form shown in Fig. 15. The relation between the voltages and currents of nonlinear elements 1 and 2 is given by the equations of an active four-terminal network

\[ \left. \begin{aligned} \dot U_1&=A_{11}\dot U_2+A_{12}\dot I_2+A_{10}\\ \dot I_1&=A_{21}\dot U_2+A_{22}\dot I_2+A_{20} \end{aligned} \right\} \tag{4.10} \]

Table 2

Calculation of a nonlinear bridge circuit

Let us denote by $U_1(I_1)$ and $\varphi_1(I)$ the volt-ampere and phase-volt-ampere characteristics of the element connected at the input of the four-terminal network, and by $U_2(I_2)$ and $\varphi_2(I_2)$ the same quantities for the element at the output of the four-terminal network. We shall take the vector $\dot I_2$ as the reference vector, that is, we set $\dot I_2=\dot I_2$.

Fig. 15

Equations (4.10) can now be rewritten in the form:

\[ \left. \begin{aligned} U_1(I_1)\,\bigl|\alpha_1(I_2)\bigr| &= A_{11}U_2(I_2)\,\bigl|\varphi_2(I_2)\bigr|+A_{12}I_2+A_{10},\\ I_1(I_2)\,\bigl|\beta_1(I_2)\bigr| &= A_{21}U_2(I_2)\,\bigl|\varphi_2(I_2)\bigr|+A_{22}I_2+A_{20}. \end{aligned} \right\} \tag{4.11} \]

These equations make it possible to construct the volt- and phase-volt-ampere characteristics at the input of the four-terminal network by trial. Having these characteristics and $U_1(I_1)$ and $\varphi_1(I_1)$, we find the actual values of $U_1$ and $\varphi_1$, and then the remaining currents and voltages.

In addition to the method considered, another method may also be proposed for calculating complex electric circuits with two nonlinear elements. The given circuit is replaced by an active four-terminal network. The coefficients $A_{ik}$ of the latter are calculated. The active four-terminal network, in turn, as is known, can be represented by a T-shaped equivalent circuit (Fig. 16), whose parameters are

\[ \left. \begin{aligned} Z_a&=\frac{A_{11}-1}{A_{21}},\\[4pt] Z_b&=\frac{A_{22}-1}{A_{21}},\\[4pt] Y_v&=A_{21},\\ \mathcal{E}_a&=A_{10}-(Z_a-Z_v)A_{20},\\ \mathcal{E}_b&=-Z_vA_{20}. \end{aligned} \right\} \tag{4.12} \]

The circuit (Fig. 16) can be calculated by the trial method according to the dependences given in §§ 1 and 2 for circuits consisting of series- and parallel-connected nonlinear two-terminal networks.

Fig. 16

5. Conclusion

The graph-analytical method considered here for calculating electrical circuits with inertial nonlinear elements gives quite satisfactory results for simple and branched electrical circuits in the case where the presence of nonlinear elements in the electrical circuit does not lead to a noticeable violation of the sinusoidal character of the voltages and currents under consideration in the circuit. The studies of ferroresonant circuits carried out by the authors do not give an acceptable agreement of the theoretical calculation with experiment. For circuits with several nonlinear elements, the method considered in § 4 may be proposed; only, for the solution it will be necessary to invoke the theory of the linear multi-terminal network.

REFERENCES

1. Voronov R. A. A graph-analytical method for constructing the characteristics of nonlinear circuits of alternating and direct currents, Proceedings of GEI, issue XVI, 1950.

2. Voronov R. A., Polyakarev G. F. Circular diagrams in the study of nonlinear circuits, Elektro, No. 12, 1951.

3. Pukhov G. E. On the question of calculating an electrical circuit with one nonlinear element in a steady-state sinusoidal regime, Izvestiya TPI, vol. 72, 1952.

4. Zelyakh E. V. Fundamentals of the General Theory of Linear Electrical Circuits, Publishing House of the Academy of Sciences of the USSR, 1951.