On the Calculation of Nonlinear AC Circuits

G. E. Pukhov, S. P. Amosova

The question considered below is the calculation of electrical circuits with nonlinear elements such that their presence does not lead to a noticeable deviation of the circuit operating mode from sinusoidal; it is shown that, when the volt- and phase-ampere characteristics of the elements are available, the calculation can be carried out by the method of iteration.

Unlike the calculation methods proposed in [1—5], the iteration method makes it possible, in the case of convergence of the process, to calculate a circuit of arbitrarily complex configuration and containing any number of nonlinear elements.

1. Let the circuit contain \(n\) nonlinear elements, the properties of each of which are specified by volt- and phase-ampere characteristics:

\[ U_k = U_k(I_k), \]

\[ \varphi_k = \varphi_k(I_k), \tag{1.1} \]

where \(U_k\) and \(I_k\) are the rms values of the voltage and current of the \(k\)-th nonlinear element, and \(\varphi_k\) is the phase-shift angle between the sinusoids \(u_k\) and \(i_k\).

The characteristics (1.1) for a given value of the current \(I_{k(s)}\) determine the complex impedance of the nonlinear element:

\[ z_{k(s)} = z_{k(s)} e^{i\varphi_{k(s)}}, \tag{1.2} \]

where

\[ z_{k(s)} = \frac{U_{k(s)}}{I_{k(s)}} = z_{k(s)}(I_{k(s)}), \]

\[ \varphi_{k(s)} = \varphi_{k(s)}(I_{k(s)}). \]

The circuit mode corresponding to the currents \(\dot I_{k(s)}\) can be described by the system of equations

\[ \dot I_{k(s)} = \sum_{l=1}^{m} Y_{kl(s)} \dot{\mathcal E}_l, \tag{1.3} \]

where \(m\) is the number of sources with emf \(\dot{\mathcal E}_l\) \((l = 1,2\, m)\).

The coefficients of equations (1.3) depend on the currents \(I_{k(s)}\), i.e.

\[ Y_{kl(s)} = Y_{kl(s)}(Z_{1(s)}, Z_{2(s)}, \ldots Z_{n(s)}). \tag{1.4} \]

Therefore the determination of the currents \(I_k\) by the method of iteration may be carried out on the basis of the calculation equations:

\[ \dot I_{k,s+1} = \sum_{l=1}^{m} Y_{kl(s)} \dot{\mathcal E}_l, \qquad \begin{gathered} (k = 1,2,\ldots n)\\ (s = 0,1,2,\ldots). \end{gathered} \tag{1.5} \]

Assigning the absolute values of the currents \(I_{k(0)}\), from the characteristics (1.1) and expressions (1.2) one finds the complex impedances \(Z_{k(0)}\) of the nonlinear ...

elements. Substituting the latter into equations (1.5), one finds the currents \(\dot I_{k(1)}\). From the absolute values of these currents the new values of the impedance complexes \(Z_{k(1)}\), etc., are determined.

The calculation is carried out until the currents \(\dot I_{k(s+1)}\) and \(\dot I_{k(s)}\) become almost identical, i.e. \(\dot I_{k(s+1)} \simeq \dot I_{k(s)}\). As calculations show, the iteration process converges well for such nonlinear elements whose resistances decrease as the current passing through them increases. When the resistances increase, however, it is advisable to transform expressions (1.5) into the equations

\[ \dot U_{k(s+1)}=\sum_{l=1}^{m} a_{kl(s)} \dot{\mathcal E}_{l}, \qquad \begin{gathered} (k=1,2\ldots n)\\ (s=0,1,2\ldots), \end{gathered} \tag{1.6} \]

in which \(\dot U_k\) are the voltages on the nonlinear elements.

In those cases where some of the elements have characteristics of the first kind and others of the second, the calculation equations should be transformed to the form:

\[ \left| \begin{array}{c} \dot U_{1(s+1)}\\ \dot U_{2(s+1)}\\ \cdot\\ \dot U_{q(s+1)}\\ \dot I_{(q+1)(s+1)}\\ \cdot\\ \dot I_{n(s+1)} \end{array} \right| = c_{(s)} \left| \begin{array}{c} \dot{\mathcal E}_{1}\\ \dot{\mathcal E}_{2}\\ \cdot\\ \dot{\mathcal E}_{m} \end{array} \right| \]

The first \(q\) rows refer to nonlinear elements with resistances increasing as the currents increase, and the remaining \(n-q\) rows to elements with decreasing resistances.

The number of calculation equations can, of course, be reduced if the given circuit is first transformed into a simpler one, as indicated in [5].

1. Let us give examples of calculations of several circuits.

Example 1. The circuit shown in Fig. 1 and connected to the alternating voltage \(\dot U=120\) V consists of an active resistance \(R=25\) ohms, an inductance coil with \(Z_0=20+j45\) ohms, and a nonlinear element \(R_n\), whose volt-ampere characteristic is shown in Fig. 2.\(^{1}\)

It is required to determine the current in the nonlinear element \(\dot I_n\). Since the resistance of the nonlinear element increases as the current increases, it is advisable to find the voltage on it. This voltage is evidently equal to:

\[ \dot U_n= \frac{\dot U Z_0 R_n}{R_n R+RZ_0+R_n Z_0}. \tag{2.1} \]

Consequently, the calculation equation for \(U_n\) has the form:

\[ \dot U_{n(s+1)}= \frac{\dot U Z_0 R_{n(s)}}{R R_{n(s)}+RZ_0+R_{n(s)}Z_0}. \tag{2.2} \]

\(^{1}\) The data of Example 1 are borrowed from the work of R. A. Voronov [3].

Substituting the numerical values of the quantities, we obtain:^[The calculations were carried out on a slide rule.]

\[ \dot U_{H(s+1)}= \frac{120(20+j45)R_{H(s)}}{25R_{H(s)}+25(20+j45)+R_{H(s)}(20+j45)} \ \text{V}. \tag{2.3} \]

Fig. 1

The calculation according to this formula is given in Table 1.

Table 1.

Fig. 2

Thus, as a result, we obtain

\[ \dot U_H = 64.3/14^\circ 20' \ \text{V}, \]

\[ \dot I_H = 1.555/14^\circ 20' \ \text{A}. \]

Example 2. Let us consider the calculation of a circuit containing two nonlinear elements (Fig. 3)\(^1\).

Both nonlinear elements have identical volt-ampere characteristics (Fig. 2). The supply voltage is \(\dot U = 120\ \mathrm{V}\), and the resistance \(Z_0 = 20 + j45\ \Omega\).

Fig. 3

Let us determine the currents \(\dot I_1\) and \(\dot I_2\).

Putting \(|R_1|=\dfrac{|U_1|}{|I_1|}\) and \(|R_2|=\dfrac{|U_2|}{|I_2|}\), we obtain expressions for the voltages on the nonlinear elements:

\[ \dot U_1 = \frac{\dot U (R_2 + Z_0) R_1} {R_1R_2 + Z_0(R_1 + R_2)} , \tag{2.4} \]

\[ \dot U_2 = \frac{\dot U R_2 Z_0} {R_1R_2 + Z_0(R_1 + R_2)} . \tag{2.5} \]

The following expressions serve as the calculation equations:

\[ \dot U_{1(s+1)} = \frac{120\,[R_{2(s)} + (20 + j45)]\,R_{1(s)}} {R_{1(s)}R_{2(s)} + (20 + j45)(R_{1(s)} + R_{2(s)})} . \tag{2.6} \]

\[ \dot U_{2(s+1)} = \frac{120(20 + j45)R_{2(s)}} {R_{1(s)}R_{2(s)} + (20 + j45)(R_{1(s)} + R_{2(s)})} . \tag{2.7} \]

The calculation according to these formulas is given in Table 2.

Table 2

\(^1\) The data of the example are borrowed from the work of R. A. Voronov [3].

From the table we see that the required voltages and currents may be taken as equal to:

\[ \dot U_1 = 84.8 \angle -7^\circ 20' \ \mathrm{V}, \qquad \dot I_1 = 1.72 \angle -7^\circ 20' \ \mathrm{A} \]

\[ \dot U_2 = 37.6 \angle 16^\circ 40' \ \mathrm{V}, \qquad \dot I_2 = 1.26 \angle 16^\circ 40' \ \mathrm{A}. \]

Example 3. The circuit shown in Fig. 4 contains an alternating-voltage source \(\dot E_0 = 140 \ \mathrm{V}\), active resistances \(r_3 = 191 \ \Omega\), \(r_4 = 224 \ \Omega\), \(r_5 = 490 \ \Omega\), and two nonlinear elements, whose volt- and phase-ampere characteristics are presented in Figs. 5 and 6. The characteristics were taken under sinusoidal voltages.

Fig. 4

The state of the circuit may be described by the system of equations:

\[ \dot I_1 = Y_{11}\dot U_1 + Y_{12}\dot U_2 + Y_{10}, \]

\[ \dot I_2 = Y_{21}\dot U_1 + Y_{22}\dot U_2 + Y_{20}. \tag{2.8} \]

For the linear part of the circuit, the coefficients are determined in the usual way:

\[ Y_{11} = -0.00651 \ \Omega^{-1}, \qquad Y_{21} = -0.00204 \ \Omega^{-1}, \]

\[ Y_{12} = -0.0021 \ \Omega^{-1}, \qquad Y_{22} = -0.00729 \ \Omega^{-1}, \]

\[ Y_{10} = -0.00655 \ \dot E_0 \ \mathrm{A}, \qquad Y_{20} = 0.00729 \ \dot E_0 \ \mathrm{A}. \tag{2.9} \]

Using the expressions

\[ \dot U_1 = Z_1 I_1, \]

\[ \dot U_2 = Z_2 I_2, \tag{2.10} \]

Fig. 5

Fig. 6

we solve the system of equations (2.8) with respect to the currents $\dot I_1$ and $\dot I_2$

\[ \dot I_1= \frac{(1-Y_{22}Z_2)Y_{10}+Y_{12}Z_2Y_{20}} {(1-Y_{11}Z_1)(1-Y_{22}Z_2)-Y_{12}Z_2Y_{21}Z_1}, \tag{2.11} \]

\[ \dot I_2= \frac{(1-Y_{11}Z_1)Y_{20}+Y_{21}Z_1Y_{10}} {(1-Y_{11}Z_1)(1-Y_{22}Z_2)-Y_{12}Z_2Y_{21}Z_1}. \tag{2.12} \]

The calculation equations are (2.13, 2.14)

\[ \dot I_{1(s+1)}= \frac{140\left(0.00655+Z_{2(s)}\,3.24\cdot10^{-5}\right)} {1+0.00729Z_{2(s)}+0.00651Z_{1(s)}+4.312\cdot10^{-5}Z_{1(s)}Z_{2(s)}}\,a \tag{2.13} \]

\[ \dot I_{2(s+1)}= \frac{140\left(0.00729+3.404\cdot10^{-5}Z_{1(s)}\right)} {1+0.00729Z_{2(s)}+0.00651Z_{1(s)}+4.312\cdot10^{-5}Z_{1(s)}Z_{2(s)}}\,a. \tag{2.14} \]

The calculation by these formulas is given in Table 3.

Table 3

The convergence of the process is clearly illustrated by the graph of the dependence of the currents $I_1$ and $I_2$ on the number of approximations $s$, shown in Fig. 7. It may be assumed that the currents are equal to: $I_1=0.344\angle -49^\circ a$ and $I_2=0.288\angle -64^\circ a$.

These results were verified experimentally. The experimental data are:

\(I_1 = 0.33\,a;\ I_2 = 0.28\,a.\)

Fig. 7

REFERENCES

1. Voronov R. A., Ponomareva G. F. Circular diagrams in the study of nonlinear circuits. Elektrichestvo, No. 12, 1951.

2. Voronov R. A. A grapho-analytical method for constructing the characteristics of nonlinear circuits of alternating and direct currents. Proceedings of TЭMIIT, issue XVI, 1950.

3. Voronov R. A. Calculation of circuits with nonlinear elements by the method of corrections. Elektrichestvo, No. 11, 1952.

4. Pukhov G. E. On the question of calculating an electrical circuit with one nonlinear element under established sinusoidal conditions. Izv. TPI, vol. 72, 1952.

5. Pukhov G. E., Amosova S. P. Transformations of nonlinear circuits under established sinusoidal conditions. Izv. TPI, vol. 72, 1952.