On the Influence of Distortions in the Waveform of the Load Current of AC Sources on the Magnitude of the Output Voltage

A. N. Obergan

(Recommended by the scientific and technical seminar of the Department of Information and Measurement Technology)

The analysis of operating modes, vector diagrams of voltages and currents, external characteristics, and other parameters of transformers, various types of electromechanical converters, and synchronous and asynchronous generators, as presented in fundamental and periodical literature, is, for the purpose of simplification, constructed mainly for cases of sinusoidal variation of the load current. With such a simplification, the change in the output voltage of AC generators is due only to the armature reaction from the fundamental harmonic of the load current, as well as to the voltage drop across the active and reactive resistance of the stator from the sinusoidal load current.

In particular, as applied to synchronous generators, the relative change in generator voltage when a linear inductive-active load is connected is calculated from the expression

\[ \Delta U = \frac{U - U_{\mathrm{н}}}{U_{\mathrm{н}}}\cdot 100\%, \]

where \(U_{\mathrm{н}}\) is the nominal no-load voltage of the generator;
\(U\) is the generator voltage at nominal speed, a specified rms value of the load current, and a definite \(\cos\varphi\).

In analyzing unbalanced operating modes of three-phase systems by the method of symmetrical components, the inductive-active load resistance is represented as linear. In practice, very often nonlinear inductive-active resistances are used as loads, i.e., such AC resistances in which clearly expressed hysteresis, magnetic saturation, and the eddy-current effect are observed; or time-varying resistances are used as the load.

When AC power sources are loaded by such resistances, in the phase and line currents there may appear, in the general case, the entire spectrum of harmonics. The waveform of the phase and line currents, and consequently the nature of the load reaction, will be determined by the volt-ampere characteristic of the load.

A theoretical analysis of the influence of the load-current waveform on the output parameters of the power source (in particular, on the output voltage) presents considerable difficulties, since the relation between the output voltage of AC generators and the load current cannot be expressed strictly analytically.

The change in generator voltage \(\Delta U\) due to the load current is deter-

...is determined from the external characteristic \(U=f(I)\) and the vector diagram of voltages.

To assess the influence of distortion of the load-current waveform on the change in the output voltage of ac generators, the nonlinear-distortion factor is sometimes used,

\[ K_{\text{нu}}=\frac{\sqrt{\sum_{k=2}^{k=\infty} I_k^2}}{I_1}, \tag{1} \]

where \(I_k\) is the rms value of the \(k\)-th harmonic;
\(I_1\) is the rms value of the first harmonic of the current.

As is seen from formula (1), the nonlinear-distortion factor does not take into account the phase shifts of the higher harmonics of the load current caused by the presence of hysteresis, magnetic saturation, and the eddy-current effect in nonlinear inductive-active resistances.

It is advisable to evaluate the influence of distortion of the load-current waveform on the output parameters of ac sources, taking into account phase distortions of the higher harmonics, by means of the coefficient of the formula

\[ K_{\phi}=-\frac{I}{I_{\text{cp}}} \tag{2} \]

or the amplitude factor

\[ K_a=\frac{I_a}{I}, \tag{3} \]

where \(I\) is the rms value of the load current;
\(I_{\text{cp}}\) is the mean value of the load current;
\(I_a\) is the amplitude value of the load current.

It is known that \(I_{\text{cp}}\) and \(I_a\) depend on the ratio of the phase shifts of the higher harmonics of the load current and make it possible to give a more complete characterization of distortions of the current waveform.

From the analysis of circuits with nonlinear parameters it is known that the greatest distortions in the current waveform of an active-inductive circuit are introduced by the third harmonic of the current curve. In this respect one might suppose that three-phase systems compare favorably with single-phase systems, since in symmetrical three-phase systems the third harmonic of the current is absent.

An analysis of practical circuits and operating modes of a large group of synchronous generators, inductor generators, single-arm rotating and static converters for special purposes with power up to 5 kW shows that, in the overwhelming majority of cases, the load of three-phase generators is unbalanced because of the presence in them of special voltage- and frequency-regulation circuits connected to one of the phases of the output voltage. In many cases, different sections of the regulation circuit, having different complex impedances, are supplied from different phases of the output voltage.

The load parameters specified by the technical conditions for the indicated ac sources are, as a rule, limited by power, current, and \(\cos\varphi\), each of which is specified with a certain tolerance (no stricter than \(\pm 5\%\)). In rare cases the nonlinear-distortion factor is specified within limits of up to \(10 \div 15\%\).

At the same time, in all cases without exception, a sufficiently strict change in the output voltage is specified for the nominal operating modes of the load.

In practice, there are cases in which the specified change in generator voltage is exceeded while the standards for load current and pow-

…the quantity \(P\), caused by the nonidentity of the nonlinearity of the inductive-active load at the consumer’s site and at the manufacturer’s test bench.

The absence of simple methods for measuring the form factors \(K_{\phi}\) and the amplitude factor \(K_a\) makes it difficult to standardize the influence of distortion of the load-current waveform on the output voltage of alternating-current generators, and to bring the loads at the manufacturer’s and the consumer’s sites into correspondence.

The instrument developed by the authors for measuring the form factor of current and voltage [3] makes it possible to analyze the operation of alternating-current sources working into any nonlinear load, and also to establish standardized deviations of the load-current waveform from a sinusoidal one in order to ensure normal operating conditions of alternating-current sources.

Figure 1 shows a circuit for investigating the influence of changes in the load-current waveform on the output voltage of a single-phase generator with the power of a \(K_{\phi}\)-meter, and Fig. 2 shows that for a three-phase generator with an unbalanced load.

Fig. 1

In investigating the operation of generators with a nonlinear inductive-active load, the load current was set equal to the rated value, and the active power of the load, measured by an electrodynamic wattmeter, was kept constant.

If the supply-source voltage is sinusoidal, then the wattmeter measures the active power of the first harmonic of the load current even when

Fig. 2

nonsinusoidal load current. In this case the power factor, determined, as a rule, from the relation

\[ \cos \varphi=\frac{P_a}{UI}, \tag{4} \]

is the power factor of the first harmonic.

Owing to the reaction of the load on the generator output voltage, the voltage waveform is also distorted, and the wattmeter measures the sum of the active powers of all harmonics, i.e.,

\[ P=U_1 I_1 \cos \varphi_1+U_3 I_3 \cos \varphi_3+U_5 I_5 \cos \varphi_5+\ldots \tag{5} \]

In this case the equivalent \(\cos \varphi_{\mathrm{e}}\) will be determined by the phase shifts of the higher harmonics and will differ from \(\cos \varphi\), determined by formula (4).

Fig. 3

Figure 3 presents plots of the dependence of the output voltage of an inductor generator on the form factor of the load-current curve (solid lines) and on the nonlinear-distortion factor (dashed lines), obtained experimentally according to the circuit of Fig. 1 at two different fixed values of active power \(P\) and a specified load current \(I\).

From the plots it is evident that a change in the waveform of the load current leads to a substantial change in the generator voltage.

Similar results were obtained in the study of asymmetrical operating modes of a three-phase synchronous generator. If one takes into account that an instrument for measuring the current form factor has a sensitivity of the order of \(0.1\%\)/division, while an instrument for measuring the nonlinear-distortion factor has no more than \(1\%\)/division, then one may conclude that it is expedient to use the current form factor for standardizing distortions of the load current.

It should also be noted that, in studying generators, nonlinear inductive-active resistances without pronounced hysteresis were used as the load. When loads with hysteresis are used, the phase shifts of the higher harmonics will be taken into account by the \(K_\phi\)-method, whereas the nonlinear-distortion meter is insensitive to the phase relationships between harmonics. This circumstance once again emphasizes the advisability of standardizing distortions of the load-current waveform by means of the current form factor.

References

1. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, GEI, 1963.
2. L. A. Bessonov. Nonlinear Electrical Circuits. VZEI Publishing House, 1958.
3. I. G. Leshchenko, A. N. Obergan, V. I. Panov. Compensation methods for measuring the form factor of current and voltage. Izv. TPI, vol. 212, 1969.