ON THE CALCULATION OF A THREE-PHASE BRIDGE INVERTER CIRCUIT WITH PARALLEL CAPACITIVE COMPENSATION

Engineer M. B. KONOVALOV

In certain cases of practical application of static transistor AC converters, when it is possible to dispense with a power transformer, it is advantageous to use a bridge circuit. However, bridge circuits have substantial shortcomings associated with the occurrence of very large commutation overlap currents. In addition to direct short circuits of the DC source at the moment of transistor switching, short circuits of the parallel capacitances also arise during the intervals of overlap between the transistors of either the collector or the emitter groups.

To eliminate short-circuit currents it is advisable, first, to trigger the power transistor switches with pulses for which:

\[ 0.5 < \frac{\tau_o}{\tau_z} < 1.0 \tag{1} \]

(Here: \(\tau_o\) is the duration of the turn-on pulse, \(\tau_z\) is the duration of the turn-off pulse);

second, to separate the compensating capacitors from the inverter by special inductances. These inductances \(L\) (Fig. 1) also play the role of filters, suppressing the higher harmonic voltage components of the 5th, 7th, and 11th harmonics.

The choice of large inductances \(L\), although it improves the waveform of the output voltage, worsens the external characteristic of the converter by increasing the voltage loss on the internal resistances. Conversely, the choice of small inductances leads, first, to an excessive increase in the compensating capacitances and, second, does not sufficiently limit the commutation overlap currents.

The proposed article contains a description of considerations and methods for choosing optimal ratios for \(L\) and \(C\), which ensure the minimum weight of the filtering elements and create conditions for the maximum loading of the transistor switches for given load parameters \(L_n\) and \(R_n\), as well as for a known internal resistance of the DC source \(r_o\).

Equivalent Circuit of the Converter

The power inverter circuit shown in Fig. 1, when its triggering is carried out by pulses satisfying relation (1), operates with large overlaps between the switches of adjacent phases, which short-circuit the capacitors. To limit the discharge currents of the capacitors in this case, it is necessary to increase excessively the inductances of the filter choke. When the pulse ratio is decreased,

\[ \frac{\tau_0}{\tau_3} < 0.5 \]

the transistors are poorly utilized with respect to current because of the considerable reduction in the time of the on state. It is therefore clear that reducing \(\frac{\tau_0}{\tau_3}\) below 0.5 is inadvisable. We shall confine ourselves to considering the case when \(\frac{\tau_0}{\tau_3}=0.5\). Figure 2 gives the time diagram.

currents and voltages for an inverter without compensating capacitors, loaded by an active load, when fired by pulses with

\[ \frac{\tau_0}{\tau_3}=0.5. \]

The form of the phase and line voltages is that of stepped pulses containing no harmonic components that are multiples of three.

Fig. 2. Time diagram of currents and voltages at the transformer collectors and in the load for \(\cos\varphi=1\).

In accordance with the method for calculating steady-state processes in inverter circuits, set forth in [L1], the bridge circuit of transistor switches together with the direct-current source may be replaced by a three-phase system of symmetrical equivalent step-shaped emfs.

Figure 3 shows the equivalent replacement circuit of the inverter. In this circuit \(E_{a\mathrm{e}}\), \(E_{b\mathrm{e}}\), and \(E_{c\mathrm{e}}\) are equivalent emfs whose images are expressed by the relations [L2]

\[ \left. \begin{aligned} E_{a\mathrm{e}}&=U\frac{1-e^{-\frac{1}{3}p}}{1+e^{-\frac{1}{2}p}};\\[6pt] E_{b\mathrm{e}}&=U\frac{e^{-\frac{1}{3}p}-e^{-\frac{2}{3}Tp}}{1+e^{-\frac{1}{2}p}};\\[6pt] E_{c\mathrm{e}}&=U\frac{e^{-\frac{1}{3}p}-1}{1+e^{-\frac{1}{2}p}}. \end{aligned} \right\} \tag{2} \]

$L$ and $C$ are the filtering inductance and compensating capacitance. The resistance $R$ consists of the active resistance of the choke $r$, of the resistance of the saturated transistor $R_0$, and of the internal resistance of the direct-current source reduced to the three-phase circuit, $R'$. Since the source is traversed by a direct current, the reactive elements of the direct-current circuit may be disregarded. The influence of these elements can be substantial only in the presence of large commutation currents that vary sharply with time. For the Larionov circuit $R'=\frac{1}{2}r_0$, and therefore

Fig. 3. Equivalent circuit of the inverter.

\[ R=r+R_0+\frac{1}{2}r_0 \tag{3} \]

Calculation of the Steady-State Regime

For the loops indicated in Fig. 3 by dashed arrows, six equations may be written: for the steady-state regime with a symmetrical load:

\[ \left. \begin{aligned} E_{a\mathrm{e}}-E_{b\mathrm{e}}&=3J_{ab}R+3J_{ab}L_{\mathrm{p}}+J_{ab}\frac{1}{Cp}-I_{ab}\frac{1}{Cp},\\ E_{b\mathrm{e}}-E_{c\mathrm{e}}&=3J_{bc}R+3J_{bc}L_{\mathrm{p}}+J_{bc}\frac{1}{Cp}-I_{bc}\frac{1}{Cp},\\ E_{c\mathrm{e}}-E_{a\mathrm{e}}&=3J_{ca}R+3J_{ca}L_{\mathrm{p}}+J_{ca}\frac{1}{Cp}-I_{ca}\frac{1}{Cp},\\ 0&=3I_{ab}R+3I_{ab}L_{\mathrm{p}}-I_{ab}\frac{1}{Cp}-J_{ab}\frac{1}{Cp},\\ 0&=3I_{bc}R+3I_{bc}L_{\mathrm{p}}-I_{bc}\frac{1}{Cp}-J_{bc}\frac{1}{Cp},\\ 0&=3I_{ca}R+3I_{ca}L_{\mathrm{p}}-I_{ca}\frac{1}{Cp}-J_{ca}\frac{1}{Cp}. \end{aligned} \right\} \tag{4} \]

The last equations are written in operational form. Each equation essentially expresses Kirchhoff’s law for the equivalent emfs and voltage drops along the corresponding loop for all harmonic components of the spectrum of equivalent emfs. In view of the absence in the spectrum of the equivalent emfs of harmonic components that are multiples of three and two, all currents contain neither inverse, introducing asymmetric distortions, nor a zero phase sequence, which makes it possible to use the symmetry conditions of the system.

\[ I_{ab}+I_{bc}+I_{ca}=0; \]

Solving equations (4) makes it possible to find the transform for the currents and voltages at the load.

\[ \left. \begin{aligned} I_a&=\frac{2E_{a\mathrm{e}}-E_{b\mathrm{e}}-E_{c\mathrm{e}}}{M(p)},\\ I_b&=\frac{2E_{b\mathrm{e}}-E_{a\mathrm{e}}-E_{c\mathrm{e}}}{M(p)},\\ I_c&=\frac{2E_{c\mathrm{e}}-E_{a\mathrm{e}}-E_{b\mathrm{e}}}{M(p)},\\ U_{ab}&=(E_{a\mathrm{e}}-E_{b\mathrm{e}})\frac{R_{\mathrm{n}}+pL_{\mathrm{n}}}{M(p)},\\ U_{bc}&=(E_{b\mathrm{e}}-E_{c\mathrm{e}})\frac{R_{\mathrm{n}}+pL_{\mathrm{n}}}{M(p)},\\ U_{ca}&=(E_{c\mathrm{e}}-E_{a\mathrm{e}})\frac{R_{\mathrm{n}}+pL_{\mathrm{n}}}{M(p)}. \end{aligned} \right\} \tag{5} \]

In expressions (5), \(M(p)\) is a third-degree polynomial

\[ M(p)=3ap^3+3bp^2+3cp+3d \tag{6} \]

with coefficients

\[ \left. \begin{aligned} a&=3L_{\mathrm{n}}LC,\\ b&=3R_{\mathrm{n}}LC,\\ c&=3R_{\mathrm{n}}RC+L+L_{\mathrm{n}},\\ d&=R_{\mathrm{n}}+R. \end{aligned} \right\} \tag{7} \]

Finally, the steady-state line voltage at the load can be calculated by the formula:

\[ U_{ab}=\frac{3U}{a\pi n}\sum_{n=-\infty}^{n=\infty} e^{jn\omega t}\, \frac{R_{\mathrm{н}}+jn\omega L_{\mathrm{н}}} {n^{3}\omega^{3}-jn^{2}\omega^{2}\frac{b}{a}-n\omega\frac{c}{a}+j\frac{d}{a}} \tag{8} \]

in the form of a sum of harmonic components not divisible by two and three. Formula (8) is obtained from the image (5), taking into account the general inversion formula [Л2].

Resonance conditions in the inverter

In order to determine the resonance conditions in the circuit of an inverter operating into a specified load, it is necessary to consider the transient process. In the present case, only the character of variation of the transient quantities is of interest; therefore there is no need to connect the solutions with the initial conditions.

From equations (5) it can be established that the transient process is characterized by the equation

\[ M(p)=0 \]

In the resonance mode, the free components of the currents or voltages at the load must have periodic terms with natural frequency \(\omega\), equal to the angular switching frequency of the transistor switches. In other words, the equation

\[ ap^{3}+bp^{2}+cp+d=0 \]

must have a pair of conjugate roots:

\[ p_1;\qquad p_2=-\delta+j\omega;\qquad p_3=-\delta-j\omega; \]

In this article the consideration is limited to cases of practical interest, when \(|\delta|\ll\omega\). Owing to the relatively small damping decrement, these cases correspond to a good waveform of the output voltage.

The equation \(M(p)=0\) can be represented, through the dimensionless coefficients (9),

\[ \begin{aligned} \rho&=\frac{R_{\mathrm{н}}}{\omega L_{\mathrm{н}}},\\[4pt] k_c&=\rho\,\frac{R}{\omega L}+\left(1+\frac{L}{L_{\mathrm{н}}}\right)\frac{1}{3LC\omega^{2}},\\[4pt] k_d&=\left(1+\frac{R}{R_{\mathrm{н}}}\right)\frac{1}{3LC\omega^{2}}, \end{aligned} \tag{9} \]

as follows:

\[ p^{3}+\rho\omega p^{2}-k_c\omega^{2}p+\rho\omega^{3}k_d=0 \tag{10} \]

Certain relations [Л3] must hold between the coefficients and the roots of equation (10); after simplification, with terms proportional to \(\delta^{2}\) neglected, they take the form:

\[ \begin{aligned} p_1&=-k_d\rho\omega,\\ 2\delta&=-(1-k_d)\rho\omega,\\ k_c-1&=(1-k_d)k_d\rho^{2}. \end{aligned} \tag{11} \]

The first two relations determine the roots of equation (10). The last relation determines the relationship between the coefficients (9) under which the inverter operates in the resonance mode.

Choice of the filtering inductance \(L\) and the compensating capacitance \(C\)

Investigating formula (8), which expresses the value of the steady-state line voltage at the load, one can establish that the maximum of its first harmonic component corresponds to the resonance mode, when \(k_c\) and \(k_d\) have values close to unity, \(k_c > 1\), \(k_d < 1\). Indeed, the steady-state voltage at the load is proportional to the factor:

\[ \frac{1}{M(jn\omega)} = \frac{1}{-j\omega^3 n^3 + j n\omega k_c \omega^2 - n^2 \omega^2 \rho \omega + \rho \omega^3 k_d} \]

which, for \(n=1\), is equal to:

\[ \frac{1}{M(j\omega)} = \frac{1}{j\omega^3(k_c-1)-\rho\omega^3(k_d-1)} \tag{12} \]

Relation (12) shows that, at resonance, the line voltage increases greatly. At the same time, all higher harmonic components decrease sharply.

On the basis of the foregoing, the parameters \(L\) and \(C\) must be selected in such a way that a resonance mode is ensured in the circuit. In addition to the resonance conditions, the selection of the parameters of the inverter circuit must be based on: first, the condition of ensuring the necessary minimum overlap currents; second, the condition of minimum weight.

Since the specific weight of copper and steel is considerably greater than the specific weight of the dielectric, in practice the weight of the filtering elements will almost always be smaller when the reactor weight is smaller. Therefore, it is first of all necessary to strive to reduce the inductance \(L\).

The limiting small value can be determined from the condition of limiting the collector current in the transistor switches at the instant of their overlap.

Let \(\tau\) be the overlap time [14], and \(I_{k\,\mathrm{dop.}}\) the permissible current in the collector at which, first, power is dissipated in the transistor that does not cause overheating of the junctions above the permissible limits; and, second, the triode does not leave the saturated state at the corresponding base current. Then the necessary minimum inductance \(L\) can be calculated by formula (13), assuming that the voltage on the capacitor at the switching instant is close to the maximum value and is equal to \(U\):

\[ L=\frac{U\tau}{I_{k\,\mathrm{dop.}}} \tag{13} \]

Solving equations (9) and the last equation (11) jointly, one can obtain relations for the coefficients \(k_c\) and \(k_d\), as well as for the constant

\[ \frac{1}{3LC\omega^2}. \]

For this purpose, the constant

\[ \frac{1}{3LC\omega^2} \]

is eliminated from (9), and, taking equation (11) into account, an equation for \(k_d\) is obtained:

\[ k_d^2+2\alpha k_d-\beta=0 \tag{14} \]

here

\[ \alpha=\frac{1}{2\rho^{2}}\frac{1+\dfrac{L}{L_{\mathrm{н}}}}{1+\dfrac{R}{R_{\mathrm{н}}}}-\frac{1}{2}; \]

\[ \beta=\frac{R}{R_{\mathrm{н}}}\cdot\frac{L_{\mathrm{н}}}{L}-\frac{1}{\rho^{2}} \]

The coefficient \(k_d\) from (14) is equal to:

\[ k_d=-\alpha+\sqrt{\alpha^{2}-\beta} \tag{15} \]

Formula (15), for a considerable \(R\), when \(\beta\) becomes greater than \(\alpha^{2}\), loses its meaning, since \(k_d\) becomes an imaginary number. However, as was indicated earlier, these formulas are valid only for insignificant damping decrements, to which small values of \(R\) must correspond.

For \(R=0\) and \(\rho=1\)

\[ \left. \begin{aligned} \alpha&=\frac{L}{2L_{\mathrm{н}}}; \qquad \beta=-1;\\ k_d&=-\frac{L}{2L_{\mathrm{н}}}+\sqrt{\frac{L}{4L_{\mathrm{н}}^{2}}+1} \end{aligned} \right\} \tag{16} \]

The last case is very often encountered in practice. Indeed, \(\rho=1\) corresponds to \(\cos \varphi\) of the load equal to 0.7; and \(R\simeq0\) is quite probable, since the resistance of a saturated triode is negligibly small and the internal resistance of the storage battery is also very small. Since

\[ \frac{1}{3LC\omega^{2}}=\frac{k_d}{1+\dfrac{R}{R_{\mathrm{н}}}} \tag{17} \]

then, knowing the values of \(k_d\) and \(L\), the value of the compensating capacitance \(C\) can easily be determined.

Naturally, the calculation method set forth for \(L\) and \(C\) is suitable for a stable load \(L_{\mathrm{н}}\) and \(R_{\mathrm{н}}\). However, the resonance condition provided by the selected parameters \(L\) and \(C\) can also be maintained under certain changes in \(L_{\mathrm{н}}\). Indeed, for small changes in \(L_{\mathrm{н}}\) (for example, when it is increased), the coefficient \(k_c\) changes very little (decreases), while the coefficient \(k_d\) remains unchanged. But small changes (decreases) in \(k_c\) can cause significant changes (decreases) in the difference \(k_c-1\), if \(k_c\simeq1\) is chosen, so that equality (11) will be preserved, since \(\rho^{2}\) will also change (decrease) when \(L_{\mathrm{н}}\) changes (increases).

This latter investigation of the resonance condition (11) indicates the possibility of regulating the voltage within certain small limits by changing the inductive reactance of the included

connected in series with the load, while preserving the most advantageous resonance mode.

The realization of this possibility will require satisfaction of the additional condition

\[ k_c > 1, \tag{18} \]

under which \(k_c\) is greater than unity by a small and definite amount, so that for any changes in \(L_{\mathrm{н}}\) that ensure the voltage-regulation requirements, the coefficient \(k_{\mathrm{н}}\) never becomes less than unity. The latter condition (18) can be ensured only at considerably larger values of \(L\), and therefore it conflicts with the condition of minimum weight of the filtering elements \(L\) and \(C\).

CONCLUSIONS

The choice of the filtering elements \(L\) and \(C\) is in fact reduced to ensuring the resonance conditions and to limiting the commutation overlap currents.

It is advisable to determine the minimum possible value of the inductance \(L\) from the permissible values of the short-time collector currents for the transistors used in the converter. This same minimum value of \(L\) also determines the minimum weight of the filtering elements selected for resonance operation for specified load parameters \(R_{\mathrm{н}}\) and \(L_{\mathrm{н}}\). With a certain method of selecting the parameters, the resonance mode in the inverter circuit proves to be stable with respect to certain changes in the load inductance \(L_{\mathrm{н}}\), which makes it possible to use a special choke for regulating the output voltage within small limits.

REFERENCES

1. Shevchenko G. I., Operating modes of a full frequency converter. Proceedings of MEI, issue XIII, Moscow, 1953.

2. Lurie A. I., Operational Calculus. GITTL, Moscow, 1951.

3. Zaguskin V. L., Handbook of Numerical Methods for Solving Algebraic and Transcendental Equations. Fizmatgiz, Moscow—Leningrad, 1960.

4. Sandler A. S., Sarbatov R. S., Inertial properties of planar triodes when operating in switching mode into an active load. Vestnik elektropromyshlennosti, No. 7, 1961.