On the Calculation of an Electrical Circuit with One Nonlinear Element in Steady-State Sinusoidal Operation
Abstract Generated abstract
This note addresses the calculation of an electrical circuit containing a single inertialess nonlinear element under assumed steady-state sinusoidal conditions. In contrast to a prior graphical method based on treating the circuit as a linear four-terminal network with a nonlinear load and using circular diagrams, the author reformulates the problem as two series-connected two-terminal networks: a linear equivalent network and the nonlinear element. From the vector diagram, a direct relation is derived between the no-load electromotive force, the current, the linear-network impedance parameters, and the nonlinear element’s voltage-current and phase-current characteristics. The proposed construction yields the equivalent circuit characteristic, permits determination of the nonlinear element current, voltage, and phase angle for a given source voltage, and extends the treatment to arbitrary phase-current characteristics and circuits with multiple energy sources.
Full Text
On the Problem of Calculating an Electrical Circuit with One Nonlinear Element in a Steady-State Sinusoidal Regime
G. E. Pukhov
1. In the work of R. A. Voronov and G. F. Ponomareva [1], a graphical method is considered for calculating an electrical circuit with one inertialess nonlinear element. The authors make the assumption that the impedance angle of the nonlinear element is constant \((\varphi_n=\mathrm{const})\). The second assumption is the supposition that all currents and voltages of the circuit are sinusoidal. The authors regard the entire circuit as a linear four-terminal network whose load is the nonlinear element, and investigate the circuit by constructing circular diagrams of the four-terminal network together with the volt-ampere characteristic of the nonlinear element.
The method includes interesting graphical constructions and, undoubtedly, solves the problem posed. However, in our opinion, this problem can be solved more simply, without recourse to the theory of circular diagrams, and moreover for a somewhat more general case, which we shall consider below.
2. Let the nonlinear element be specified by two characteristics: the volt-ampere characteristic \(U_n=U_n(I)\) and the phase-ampere characteristic \(\varphi_n=\varphi_n(I)\), where \(\varphi_n\) is the phase-shift angle between the sinusoids of voltage and current of the element.
If the circuit contains only one nonlinear element, then, naturally, it should be considered as the load of a two-terminal network, not of a four-terminal network. Then, from the vector diagram of the resulting equivalent circuit, consisting of two series-connected two-terminal networks—a linear one and a nonlinear one—it is easy to obtain the dependence
\[ U_0=\left|\sqrt{\left\{U_n(I)+z_l I\cos\left[\varphi_l-\varphi_n(I)\right]\right\}^2 +\left\{z_l I\sin\left[\varphi_l-\varphi_n(I)\right]\right\}^2}\right|. \tag{1} \]
Here
\[ U_0 \]
is the e.m.f. of the linear two-terminal network, equal to the no-load voltage,
\[ z_l \ \text{and}\ \varphi_l \]
are the resistance and resistance angle of the linear two-terminal network,
\[ I \]
is the rms value of the current of the nonlinear element.
Using the volt- and phase-ampere characteristics of the nonlinear element, from equation (1) it is easy to construct the dependence \(U_0=U_0(I)\), i.e., the volt-ampere characteristic of the equivalent circuit. Having the latter, for a given value of \(U_0\) we find the current of the nonlinear element \(I\), and then its voltage \(U_n\) and angle \(\varphi_n\). The further determination of currents and voltages in the linear part of the circuit presents no difficulty, since it can be carried out by any of the known methods.
The method set forth, being simpler than the method using circular diagrams of a four-terminal network, makes it possible to take into account any form of the phase-ampere characteristic \(\varphi_n=\varphi_n(I)\), and not only \(\varphi_n=\mathrm{const}\). Moreover, this method is obviously applicable to circuits with many energy sources.
References
- Voronov R. A. and Ponomareva G. F.—Circular diagrams in the investigation of nonlinear circuits, Electricity, No. 12, 1951.