COMPENSATION OF LOAD REACTIVE ENERGY IN VOLTAGE INVERTERS

V. I. IVANCHURA, S. I. KOROLEV, B. P. SOUSTIN

(Presented by the scientific-technical seminar of the Department of Electrical Equipment, TPI)

For voltage inverters there exist several methods of compensating the reactive energy of the load: by the direct-current source through feedback diodes, by short-circuiting the load, and, for multiphase circuits, by transferring reactive energy from phase to phase. In creating inverter circuits with a regulated output voltage by wide-pulse or pulse-width methods, the choice of compensation method determines the inverter circuit and the type of modulation. For analysis, a circuit of a single-phase inverter with individual commutation and feedback diodes was considered (Fig. 1a). Compensation of the reactive energy of the load is carried out either by the direct-current source or by short-circuiting the load.

Fig. 1.

The following assumptions are made: a) the forward resistance of the thyristors and diodes in the open state is equal to zero, and in the reverse closed state—to infinity—

…ness; b) the DC voltage source has a low internal resistance.

With broad control (Fig. 1b) and compensation of reactive energy by the DC current source, the form of the voltage across the load depends not only on the conduction angle of the power thyristors \(\Theta_1\), but also on \(\cos\varphi_{\mathrm{H}}\) of the load. In addition, there is a limiting value \(\Theta_{1\mathrm{p}}\) as a function of \(\cos\varphi_{\mathrm{H}}\).

The initial equations in operator form are

\[ \frac{U}{p}=RI(p)+LpI(p) \qquad 0<\Theta\leqslant\Theta_1 \]

\[ -\frac{U}{p}=RI(p)+LpI(p)-LI(\Theta_1) \qquad \Theta_1<\Theta\leqslant\frac{\pi}{2}. \tag{1} \]

From the condition \(\Theta_1+\Theta_2=\pi\), the limiting value of the conduction angle of the power thyristors is found; \(\Theta_2\) is the conduction angle of the reverse diodes:

\[ \Theta_{1\mathrm{p}}=\frac{\pi}{2}-\operatorname{tg}\varphi_{\mathrm{H}}\cdot \ln\left(2e^{\Theta\cdot\operatorname{ctg}\varphi}-1\right). \tag{2} \]

A graphical solution of this equation is given in Fig. 2. When the reactive energy of the load is returned to the DC current source up to angles smaller than the limiting value, \(\Theta<\Theta_{1\mathrm{p}}\), the limiting current has a discontinuous character.

The voltage at the inverter output satisfies the condition \(f(\Theta)=f(\Theta+\pi)\) and has only odd harmonics of sinusoidal and cosinusoidal character,

\[ f(\Theta)=\sum_{n=1}^{\infty}\left(an\cdot\cos\Theta\cdot n+bn\cdot\sin\Theta\cdot n\right), \tag{3} \]

where \(n\) is the ordinal number of the harmonic.

Fig. 2.

The amplitude of the first harmonic has the form:

\[ A_1=\frac{2U}{\pi}\sqrt{6-4\cos\Theta_1-4\cos\Theta_2+2\cos(\Theta_1+\Theta_2)}. \]

In the case of compensation of reactive energy by the dc source and by short-circuiting the load (the combined compensation method), the voltage waveform has the form shown in Fig. 1, b; \(\Theta_2\) is absent, and the current in the load is continuous. The amplitude of the first harmonic is

\[ A_1^*=\frac{2U}{\pi}\sqrt{2(1-\cos\Theta_1)}. \]

Figure 3 shows the dependences \(A_1=f(\Theta_1,\Theta_2)\), characterizing the change in the amplitude of the first harmonic for a purely inductive load when the reactive energy is returned to the dc source (curve 1), under combined compensation (curve 2), and the ratio of these amplitudes (curve 3). Curve 4 characterizes the change in the amplitude of the first harmonic under compensation by the dc source at \(\Theta_1=60\) electrical degrees, with \(\Theta_2\) varying from 0 to \(60^\circ\).

Fig. 3

Analyzing the curves, one may conclude that, when regulating the conduction angle of the main thyristors over a wide range, combined compensation of the load reactive energy is preferable. At \(\Theta_1\) greater than 60 electrical degrees, the amplitude of the first harmonic increases when compensation is performed by the dc source. The function for regulating the amplitude of the first harmonic under compensation by the dc source is nonlinear.

When regulating the load voltage according to a sinusoidal law, the two methods of reactive-energy compensation are compared in two types of modulation of a two-stroke ...

\[ t_{\mathrm{uk}}=\frac{T}{2}+\frac{T}{2}\,\mu\sin\Omega_{0}T\cdot k \]

\[ t_{\mathrm{nk}}=-\frac{T}{2}-\frac{T}{2}\,\mu\sin\Omega_{0}\cdot T\cdot k \]

and single-cycle

\[ t_{\mathrm{uk}}=\mu T\left|\sin\Omega_{0}Tk\right| \]

where \(t_{\mathrm{uk}}\) is the duration of the \(k\)-th positive voltage pulse, counted from the origin of coordinates;

\(t_{\mathrm{nk}}\) is the duration of the negative voltage pulse in the \(k\)-th interval.

\(\mu=(0\div1)\) is the modulation depth,

\(T=t_{\mathrm{uk}}+t_{\mathrm{nk}}\) is the modulation period,

\(\Omega_{0}\) is the frequency of the fundamental voltage harmonic.

For clarification, Fig. 4a, b shows voltage graphs as a function of \(\frac{\pi}{n}\cdot k\) for \(\mu=1\) and \(\Omega_{0}T=\frac{\pi}{6}\). In what follows \(\Omega_{0}T=\frac{\pi}{n}\), where \(n\) is the number of intervals in \(\pi\).

It should be noted that two-cycle voltage modulation can be obtained when the reactive energy of the load is compensated by the power source, while single-cycle modulation can be obtained with combined compensation.

Fig. 4

In the case of unipolar modulation, the output voltage contains odd cosinusoidal and sinusoidal terms of the Fourier series.

\[ a_{\mathrm{m}}=\frac{2}{\pi}\int_{0}^{\pi} f(a)\cdot \cos ma\,da \]

\[ b_{\mathrm{m}}=\frac{2}{\pi}\int_{0}^{\pi} f(a)\cdot \sin ma\,da . \tag{4} \]

The expressions for any voltage harmonic are complex and not very suitable for analysis. Therefore, to obtain an analytical dependence of the amplitude of the first harmonic of the output voltage, we shall use an approximate method of integrating expressions (4).

The area \(a, b, c, d\) for any interval is replaced by the algebraic sum of the area of the rectangle \(a, b, c, d\) and the area of the triangle \(bfe\) (Fig. 4, c), with the hypotenuse of the triangle being tangent to the sinusoid at point \(b\)

\[ S_{abcd}=S_{abcd}+S_{bfe} \]

As can be seen from the figure, for each integration interval

\[ \begin{aligned} S_{abcd} &= \mu\,\frac{\pi}{n}\sin^2\frac{\pi}{n}\cdot k,\\ S_{bfe} &= -\frac{1}{2}\left(\mu\frac{\pi}{n}\right)^2\cdot \sin^2\frac{\pi}{n}\cdot k\cdot \cos\frac{\pi}{n}\cdot k, \end{aligned} \tag{5} \]

using (5), taking the amplitude of the rectangular voltage as unity,

\[ b_1= \frac{2}{\pi}\sum_{k=1}^{k=n-1}\sin^2\frac{\pi}{n}\cdot k\cdot \mu\frac{\pi}{n} + \frac{2}{\pi}\sum_{k=1}^{k=n-1}\frac{1}{2}\cos\frac{\pi}{n}\cdot k\cdot \frac{\mu^2\pi^2}{n^2}\sin^2\frac{\pi}{n}\cdot k. \]

After summation,

\[ b_1=\mu+\frac{1}{2}\mu^2\frac{\pi}{n^2}. \tag{6} \]

In an analogous way, the expression for the cosine component of the first harmonic is found

\[ a_1= \frac{2}{\pi}\sum_{k=1}^{k=n-1}\mu\frac{\pi}{n}\sin\frac{\pi}{n}\cdot k\cdot \cos\frac{\pi}{n}\cdot k - \frac{2}{\pi}\sum_{k=1}^{k=n-1}\frac{1}{2}\frac{\mu^2\pi^2}{n^2}\sin^3\frac{\pi}{n}\cdot k, \tag{7} \]

\[ a_1=-\mu^2\frac{\pi}{n^2}. \tag{8} \]

The amplitude of the first harmonic is

\[ A_1=\sqrt{b_1^2+a_1^2}\approx b_1 \]

The phase of the voltage of the first harmonic is

\[ \operatorname{tg}\varphi=\frac{b_1}{a_1} \]

\[ \operatorname{tg}\varphi=-\frac{n^2+\frac{1}{2}\mu\cdot\pi}{\mu\pi}. \]

With bipolar modulation, only odd sinusoidal components will be present in the output-voltage waveform, if the origin of coordinates is shifted to the point \(\frac{\pi}{4\pi}\) (Fig. 4, a). Applying approximate integration, we obtain for the amplitude of the first harmonic

\[ b'_1=\mu\cos\frac{\pi}{2n}+\frac{\pi}{2n}\cdot\mu\cdot\sin\frac{\pi}{2n}. \]

According to the theorem on shifting the origin of coordinates, the phase shift of the first harmonic will be \(\varphi'=-\dfrac{\pi}{2n}\).

On the basis of the expressions obtained, one can find the nonlinear-distortion coefficient

\[ k_{\mathrm{r}}=\sqrt{\frac{U_g^2-U_1^2}{U_1^2}}, \]

where \(U_g\) is the rms value of all harmonics of the output voltage;
\(U_1\) is the rms value of the first harmonic.

For unipolar modulation,

\[ U_g=\sqrt{\frac{1}{\pi}\sum_{k=1}^{n}\mu\frac{\pi}{n}\sin\frac{\pi}{n}\,k} =\sqrt{\frac{\mu}{n}\operatorname{ctg}\frac{\pi}{2n}} \]

\[ k_{\mathrm{r}}'= \sqrt{ \frac{ \frac{1}{n}\operatorname{ctg}\frac{\pi}{2n}\,\mu \left(1+\frac{1}{2}\mu\frac{\pi}{n^2}\right)^2 }{ \frac{1}{2}\mu\left(1+\frac{1}{2}\mu\frac{\pi}{n^2}\right)^2 } } \tag{10} \]

\[ k_{\mathrm{r}\,n\to\infty}=\sqrt{\frac{4}{\pi\mu}-1}. \]

For bipolar modulation,

\[ U_g=1 \]

\[ k_{\mathrm{r}}''= \sqrt{ \frac{ 1-\frac{\mu^2}{2}\left(\cos\frac{\pi}{2n}+\frac{\pi}{2n}\sin\frac{\pi}{2n}\right)^2 }{ \frac{\mu^2}{2}-\left(\cos\frac{\pi}{2n}+\frac{\pi}{2n}\sin\frac{\pi}{2n}\right)^2 } } \tag{11} \]

\[ k_{\mathrm{r}\,n\to\infty}''=\sqrt{\frac{2}{\mu^2}-1}. \tag{12} \]

Figure 5 shows the dependences of \(k_{\mathrm{r}}'\) and \(k_{\mathrm{r}}''\) as functions of the modulation depth for various \(n\).

Fig. 5.

Analyzing (6) and (9), we see that the amplitude of the first harmonic is a nonlinear function of the modulation depth, and the nonlinearity will be smaller the higher the carrier frequency, i.e., the larger \(n\). For large \(n\), the amplitudes of the first harmonics are identical over the entire range of variation and are proportional to \(\mu\).

Conclusions

1. The method of compensating the reactive energy of the load affects the nature of regulation of the inverter voltage and current.

2. The combined method of compensating the reactive energy of the load proves preferable; it provides continuous current under pulse-width control and a smaller harmonic coefficient under pulse-width control and sinusoidal pulse-duration modulation.