COMPARISON OF TWO METHODS FOR COMPENSATING THE REACTIVE ENERGY OF A LOAD IN CURRENT-FORMING INVERTERS

V. I. Ivanchura, S. I. Korolev, B. P. Soustin

(Presented by the scientific and technical seminar of the Department of Electrical Equipment)

Recently, inverters with a closed-loop system for forming the load current have been finding ever wider application [1, 2]. The current in such an inverter may have any form, in particular a sinusoidal one. With a sinusoidal current, the method of compensating the reactive energy of the load has a significant effect on the qualitative characteristics of the inverter; the characteristics of an inverter should include, first of all, the switching frequency of the power elements \(f_k\) for a specified value of the current reproduction error \(\Delta I\).

Fig. 1.

The load current \(i_{\text{н}}(t)\) oscillates between two sinusoids (Fig. 1)—the upper \(i_3(t) = I_m \sin \omega t + \dfrac{\Delta I}{2}\) and the lower \(i_1(t) = I_m \sin \omega t - \dfrac{\Delta I}{2}\), passing through zero at the instants \(\omega t = 0\) and \(\omega t = \pi\). At the instant \(\omega t = 0\), the transistor or thyristor switches \(T_1\) and \(T_2\) (Fig. 2) are turned on, and the current in the load increases to the value of the current of the upper sinusoid, after which the switches open and the load is connected by diodes \(D_3\) and \(D_4\) to the source in the opposite direction. The current begins to decrease, and when it reaches the value of the current of the lower sinusoid, switches \(T_1\) and \(T_2\) are turned on again, and so on. As can be seen, in the inverter the reactive energy of the load is returned to the DC source.

The coordinates of the points of intersection of the load current with the upper sinusoid are determined from the equations

Fig. 2

The coordinates of the points of encounter with the lower sinusoid are determined from the equations

\[ I_{\mathrm{m}}\sin\omega t+\frac{\Delta I}{2}=i_3(t), \]

\[ \frac{U}{R}+\left[-\frac{U}{R}+I_0(t)\right]e^{-\frac{R}{L}t}=i_{\mathrm{H}}(t). \tag{1} \]

\[ I_{\mathrm{m}}\sin\omega(t+t_2)-\frac{\Delta I}{2}=i_1(t), \]

\[ -\frac{U}{R}+\left[\frac{U}{R}+I_0(t)\right]e^{-\frac{R}{L}t}=i_{\mathrm{H}}(t), \tag{2} \]

To obtain an analytical dependence of the maximum frequency, in addition to generally accepted assumptions (the resistance of open valves is equal to zero, that of closed valves is infinite, and the internal resistance of the power source in the forward and reverse directions is equal to zero), it is assumed that the sinusoid at small angles, measured from the origin of coordinates, is replaced by the tangent, and the exponential at small arguments is determined by the first two terms of its Maclaurin-series expansion.

Fig. 3.

In addition, it is assumed that the switching-frequency period remains constant in the intervals \(0 \div \frac{\pi}{2}\left(\pi \div \frac{3}{2}\pi\right)\) and \(\frac{\pi}{2} \div \pi\left(\frac{3}{2}\pi \div 2\pi\right)\)

and are equal, respectively, to \(T_{1\text{av}}\) and \(T_{2\text{av}}\). The average switching-frequency period is determined in the corresponding intervals at the level \(0.5 I_m\), since the slopes of all the approximating straight lines and, consequently, the switching period are a linear function of the current. The latter assumption is valid when \(f_{\text{sw}} \gg f_{\text{out}}\). The switching frequency is determined by the expression

\[ f_{\text{sw}} = 2\left( \frac{\pi}{2\omega T_{1\text{av}}} + \frac{\pi}{2\omega T_{2\text{av}}} \right) \frac{\omega}{2\pi} = \frac{T_{1\text{av}}+T_{2\text{av}}}{2T_{1\text{av}}T_{2\text{av}}}. \tag{3} \]

The period of the switching frequency, both \(T_{1\text{av}}\) and \(T_{2\text{av}}\), consists of the current rise and fall times, which can be determined as follows.

For the interval \(0 \div \pi/2\), equations (1), taking into account the adopted assumptions, are

\[ i_3(t)=0.5I_m+\frac{\Delta I}{2}+0.866I_m\omega t, \]

\[ i_{\text{н}}(t)= \left(0.5I_m-\frac{\Delta I}{2}\right) + \left( \frac{U}{L} - 0.5I_m\frac{R}{L} + \frac{\Delta I}{2}\frac{R}{L} \right)t. \]

From the condition of equality \(i_3(t)=i_{\text{н}}(t)\), the current rise time in the interval \(\left(0 \div \pi/2\right)\) (Fig. 3) is

\[ t'_2= \frac{\Delta I}{ \frac{U}{L} - \frac{R}{2L}(I_m-\Delta I) - 0.866I_m\omega }. \tag{4} \]

In an analogous manner, the current rise time in the interval \(\left(\pi/2 \div \pi\right)\) is determined:

\[ t'_2= \frac{\Delta I}{ \frac{U}{L} - \frac{R}{2L}(I_m-\Delta I) + 0.866I_m\omega }. \tag{5} \]

The current fall time is determined from equation (2), taking into account the adopted assumptions.

In the interval \(0 \div \pi/2\):

\[ i_1(t)=0.5I_m-\frac{\Delta I}{2}+0.866I_m\omega t, \]

\[ i_{\text{п}}(t)= \left(0.5I_m+\frac{\Delta I}{2}\right) - \left( \frac{U}{L} + 0.5I_m\frac{R}{L} + \frac{\Delta I}{2}\frac{R}{L} \right)t; \]

from the condition of equality \(i_1(t)=i_{\text{п}}(t)\),

\[ t''_1= \frac{\Delta I}{ \frac{U}{L} + (I_m+\Delta I)\frac{R}{2L} + 0.866I_m\omega }. \tag{6} \]

Accordingly, for the interval \(\left(\pi/2 \div \pi\right)\),

\[ t''_2= \frac{\Delta I}{ \frac{U}{L} + \frac{R}{2L}(I_m+\Delta I) - 0.866I_m\omega }. \tag{7} \]

From expressions (3)–(6), neglecting quantities of second order, the approximate value of the switching frequency is determined as

\[ f'_{k\mathrm{av}}=-\frac{R}{2L}\cdot \frac{4k_u^2-3\tan^2\varphi-1}{4k_i k_u}, \tag{8} \]

where

\[ k_u=\frac{U_p}{I_m R} \]

is the coefficient of utilization of the supply-source voltage;

\[ k_i=\frac{\Delta I}{I_m} \]

is the coefficient of error in reproducing the current waveform;

\[ \tan\varphi=\frac{\omega L}{R}. \]

The approximate value of the maximum possible reproducible frequency \(f_{\max \mathrm{pr}}\), for a specified error in reproducing the current waveform, is determined from the condition that the switching period be equal to infinity at the level \(0.5 I_m\):

\[ f_{\max \mathrm{pr}}=-\frac{R}{2L\pi} \sqrt{\frac{4k_u^2-1}{3}} . \tag{9} \]

Since the expression is valid for the case \(f_k \gg f_{\mathrm{out}}\), in order to obtain an expression valid over the entire range of variation of the output frequency, the method of approximating (8) by selecting the simplest functions [3] is used; as a result, the refined value of the switching frequency has the form:

\[ f_{k\mathrm{av}}= \frac{R}{L}\, \frac{1.43\tan^2\varphi}{\sqrt{k_i k_u^2}\,2\pi} \sqrt{\frac{4k_u^2-1}{3}} + \frac{R}{L}\, \frac{4k_u^2-3\tan^2\varphi-1}{8k_i k_u}. \tag{10} \]

The refined value of the maximum reproducible frequency is obtained from (10) under the condition that the switching frequency be equal to

\[ t_{\max}=\frac{-1\pm\sqrt{1+4ab}}{2a}. \tag{11} \]

where

\[ a=\frac{2\pi L}{R} \left( \frac{1.43}{k_u^2\sqrt{k_i}} \sqrt{\frac{4k_u^2-1}{3}} - \frac{3\pi}{4k_i k_u} \right). \]

\[ b=\frac{R}{2L}\, \frac{4k_u^2-1}{4k_u k_i}. \]

The maximum switching frequency at low output frequencies is determined from (10) under the condition \(\omega_{\mathrm{out}}=0\).

\[ f_{k\max}=\frac{R}{2L}\, \frac{4k_u^2-1}{4k_u k_i}. \tag{12} \]

Reactive power can also be compensated by short-circuiting the load at certain instants of time.

Figure 1 shows the dependence of the current variation for this method of compensation, taking into account that the rise and fall of the current occur according to the law

\[ i_{\mathrm{н}}^{*}(t)=\frac{U}{R}+\left(-\frac{U}{R}+I_{0}\right)e^{-\frac{R}{L}t}, \]

\[ i_{\mathrm{н}}^{*}(t)=I_{0}e^{-\frac{R}{L}t}. \tag{13} \]

A characteristic feature of such a mode is the presence of a point \(\Theta_{\mathrm{gr}}\), from which the current, changing from its initial value to zero, does not intersect the lower sinusoid \(i_{1}(t)\). In this case, when the current reaches the value

\[ I=I_{\mathrm{m}}\cos\omega t+\frac{\Delta I}{2}k+\frac{\Delta I}{2} \]

(point A), it is necessary to switch the inverter into the mode of compensation by a dc source. Here \(k=0.1\div 0.2\) characterizes the magnitude by which the load current exceeds the current of the upper sinusoid (Fig. 1).

The assumptions adopted earlier are valid in this case up to \(\Theta_{\mathrm{gr}}\), and the switching frequency can be determined if the average value of the switching period in the interval \(0\div\Theta_{\mathrm{gr}}\) is found:

\[ t_{k}=2\left(\frac{\Theta_{\mathrm{gr}}}{\omega T_{1\mathrm{cp}}^{*}}\right)\frac{\omega}{2\pi} =\frac{\Theta_{\mathrm{gr}}}{\pi T_{1\mathrm{cp}}^{*}}. \tag{14} \]

where

\[ T_{1\mathrm{cp}}=t_{1}^{\prime *}+t_{1}^{\prime\prime *} \]

is the average value of the switching period in the interval \(0\div\Theta_{\mathrm{gr}}\);

\[ t_{1}^{\prime *},\,t_{1}^{\prime\prime *} \]

are, respectively, the times of current rise and fall in the interval \((2\div\Theta_{\mathrm{gr}})\).

In the interval from \(\Theta_{\mathrm{gr}}\) to \(\pi\), the switching frequency is taken to be zero. From the equality \(i_{\mathrm{н}}^{*}(t)=i_{1}(t)\) and (13), taking into account the assumptions, one obtains:

\[ t_{1}^{\prime\prime *}= \frac{\Delta I}{0.866 I_{\mathrm{m}}\omega+0.5 I_{\mathrm{m}}\frac{R}{L}+\frac{\Delta I}{2}\frac{R}{L}}. \]

Taking into account (4), (13), (14), \(t_{1}^{*}=t_{1}^{\prime}\), the approximate value of the switching frequency is determined as:

\[ f_{k\mathrm{cp}}^{*} =\frac{R}{L}\, \frac{\Theta_{\mathrm{gr}}}{\pi}\, \frac{2k_{u}(1.73\tg\varphi+1)-(1.73\tg\varphi+1)^{2}}{4k_{i}k_{u}}. \tag{15} \]

The approximate value of the maximum possible reproducible frequency is determined, as in the preceding case, by

\[ f_{\max\mathrm{pr}}^{*} =\frac{R}{L}\,\frac{2k_{u}-1}{1.73\cdot 2\pi}. \tag{16} \]

By virtue of the assumptions adopted, expression (15) is valid for frequencies \(f_{k}\gg f_{\mathrm{out}}\). On the basis of graphical constructions and analysis of (15), the dependence \(f_{k\mathrm{cp}}^{*}\) can be approximated by the expression \(f_{k\mathrm{cp}}^{*}=f_{\mathrm{n}}+f_{\mathrm{v}}\), where \(f_{\mathrm{n}}\) and \(f_{\mathrm{v}}\) characterize the behavior of \(f_{k}\) at low and high frequencies. The frequency \(f_{\mathrm{n}}\) is obtained analytically and coincides with (15), while the frequency

\[ f_{\mathrm{v}}=\frac{R}{L}\,\frac{2k_{u}-1}{1.73\cdot 2}\, \frac{\pi}{\Theta_{\mathrm{gr}}}\,\tg\varphi\, \frac{1}{k_{u}\sqrt{k_{i}}}, \]

is obtained using the method of selecting elementary functions [—3]. The final expression for the switching frequency has the form

\[ f_{k}^{*} =\frac{R}{L}\, \frac{\Theta_{\mathrm{gr}}}{\pi}\, \frac{2k_{u}(1.73\tg\varphi+1)-(1.73\tg\varphi-1)^{2}}{4k_{i}k_{u}} +\frac{R}{L}\,\frac{2k_{u}-1}{1.73\cdot 2\pi}\times \]

\[ \times\frac{1}{k_{u}\sqrt{k_{i}}}\, \frac{\pi}{\Theta_{\mathrm{gr}}}\,\tg\varphi, \tag{17} \]

Fig. 4.

The value \(\Theta_{\mathrm{gr}}\) may lie in two zones (Fig. 4):

\[ \begin{aligned} &\text{I zone}\quad \Theta' < \Theta_{\mathrm{gr}} < \Theta'',\\ &\text{II zone}\quad \Theta'' < \Theta_{\mathrm{gr}} < \Theta'''. \end{aligned} \]

The first zone is characterized by the fact that line 1 (Fig. 4), approximating the descending current, is tangent to the lower sinusoid

\[ i_1(t)=I_m \sin \Theta-\frac{\Delta I}{2}. \]

As can be seen from Fig. 4, \(\Theta'=\arcsin(1-k_i)\), while \(\Theta''\) is found from the simultaneous solution of the equation for the tangent at the point \(\pi-\frac{k_i}{2}\) to the lower sinusoid and the equation of the upper sinusoid, assuming that

\[ \sin\left(\pi-\frac{k_i}{2}\right)\approx \frac{k_i}{2} \quad\text{and}\quad \cos\left(\pi-\frac{k_i}{2}\right)\approx -1, \]

\[ \sin \Theta''=-\Theta''+\pi-\frac{k_i}{2}. \tag{18} \]

The graphical solution of this equation is given in Fig. 5. The value \(\Theta'_{\mathrm{gr}}\) in the first zone is found from the solution of two equations. The first equation is obtained from the condition of equality of the slopes of the approximating straight line and the tangent at the point \(\Theta_1\); the second equation characterizes the intersection of the tangent and the upper sinusoid. The solution of these equations gives an implicit expression, and its approximate value has the form:

\[ \Theta'_{\mathrm{gr}}=\Theta'+\operatorname{ctg}\varphi. \]

Similarly, \(\Theta''_{\mathrm{gr}}\) is determined in the second zone, which gives

\[ \Theta''_{\mathrm{gr}}=\pi-\frac{k_i}{2}-\operatorname{tg}\varphi. \]

From comparison of (16) and (19) it is seen that, with the first compensation method, the value of the reproducible frequency is somewhat higher than with the second method. However, it should be noted that at high output frequencies it is necessary to switch the inverter operation from the second mode to the first; i.e., in terms of the frequency range the inverter is the same both with compensation by a direct-current source and with combined compensation.

Fig. 5.

The maximum switching frequency at low frequencies is

\[ f^{*}_{k \max} \cong \frac{R}{L}\, \frac{\Theta_{\mathrm{gr}}}{\pi}\, \frac{2k_u - 1}{4k_i k_u} \cong \frac{R}{L}\, \frac{2k_u + 1}{4k_i k_u}. \]

A comparison of these frequencies,

\[ \frac{f_{k \max}}{f^{*}_{k \max}} = \frac{ \frac{R}{2L}\, \frac{4k_u^{2} - 1}{4k_i k_u} }{ \frac{R}{L}\, \frac{2k_u + 1}{4k_u k_i} } = K_u - 0.5, \]

shows an increase in the switching frequency at low output frequencies when the first compensation method is used.

Comparison of the results obtained from the formulas given above with the results of exact graphical constructions and experimental data gives an error in determining the switching frequency within 3–10%.

Conclusions

When forming a sinusoidal current, the switching frequency of power valves when compensating the reactive power of the load by a direct-current source is higher than with combined compensation. The first method is recommended at high output frequencies, and the second at low frequencies. If it is necessary to vary the switching frequency, additional controllable reactors must be inserted into the load circuit, or the supply-source voltage must be changed.

References

1. V. N. Brodovskii, E. S. Ivanov, Yu. P. Kuznetsov. Static converters for frequency-current control of AC machines. Proceedings of the Academy of Sciences of the USSR, Power Engineering and Transport, No. 2, 1969.

2. I. F. Kalinin, B. P. Soustin. Inverter with a closed system for forming the current waveform. In the collection: Power-Conversion Engineering Devices, issue 2, Naukova Dumka, Kiev, 1969.

3. V. M. Ordyncev. Mathematical Description of Automation Objects. Mashinostroenie, Moscow, 1965.