Electromagnetic Processes in a Multiphase Bridge Voltage Inverter

V. I. Ivanchura, B. P. Soustin, Yu. A. Shurygin

(Presented by the scientific-technical seminar of the Research Institute of Automation and Electromechanics)

In [1, 2, 3], switching functions \(F_i\) of the connection point of load line \(i\) were introduced, which made it possible to express the inverter response \(Y(n,\varepsilon)\) in generalized form using the discrete Laplace transform [4]. The discrete function \(\gamma_i(k)\)—the duration of the control pulse—completely determines the form of \(F_i\) and is its time representation.

If the load is an \(RL\) circuit, then the load transfer function with respect to current has two poles \(q=0\) and \(q_1=-\beta=-2\pi \operatorname{ctg}\varphi\), whose residues are \(C=1\), \(C_1=-1\). For any type of pulse-width modulation, the quasi-steady component of the inverter current is written in relative quantities as:

\[ \bar I_{kl}(\varepsilon) = -\frac{4}{m}a^l \frac{e^{-\beta[\varepsilon+1-\gamma_l(k)]}}{1-e^{-\beta}}, \qquad 0 \le \varepsilon \le \gamma_l(k), \]

\[ \bar I_{kl}(\varepsilon) = \frac{4}{m}a^l \left( 1- \frac{e^{-\beta[\varepsilon-\gamma_l(k)]}}{1-e^{-\beta}} \right), \qquad \gamma_l(k)\le \varepsilon < 1, \tag{1} \]

\[ \bar I(\varepsilon)= \sum_{k=0}^{N-1}\sum_{l=0}^{m-1}\bar I_{kl}(\varepsilon). \]

The current consumed from the power source is:

\[ i_p(\varepsilon) = \frac{1}{2}\sum_{i=1}^{m}F_i(\varepsilon)\cdot I_i(\varepsilon) = \frac{m}{4}\operatorname{Re}\,\bar E(\varepsilon)\cdot \bar I(\varepsilon). \tag{2} \]

The transient component of the current is described by a single expression

\[ \bar I_p(n,\varepsilon) = -\frac{4}{m}\sum_{k=0}^{m-1}a^l \frac{e^{-\beta[1+\varepsilon-\gamma_l(k)+n]}}{1-e^{-\beta}}. \tag{3} \]

For sinusoidal (SPWM) and rectangular (RPWM) types of modulation, programs were compiled for calculating quasi-steady processes for a BESM-4 computer. The printout included instantaneous average and maximum voltages, the magnitudes of the load current and power-source current for \(m=3\) and various \(N\), \(\mu\), \(\operatorname{tg}\varphi\).

From analysis of the calculated data it is seen that the conditions for exchange of the load reactive energy (LRE) between the source and the load are better in the case of SPWM; however, compensation of LRE by transfer from phase to phase is better in the case of RPWM.

With equality of the maximum returned REN for both types of modulation, there exists a limiting ratio of the carrier and output frequencies

\[ N_{\mathrm{gr}}= -\frac{6.92\,\sin^{2}\!\left(\dfrac{\psi}{2}-\dfrac{\pi}{12}\right)} {1-\sin\!\left(\dfrac{\pi}{6}+\varphi\right)} . \tag{4} \]

From (4) it is clear what \(N\) should be chosen so that ShIMS is better than ShIMP. If \(N>N_{\mathrm{gr}}\) is physically impossible to realize, then one should switch to ShIMP. In addition to the criterion given above, the current harmonic coefficient \(K_{\mathrm{tg}}\) is used; in this case there exists \(N'_{\mathrm{gr}}\), at which ShIMP has a better \(K_{\mathrm{tg}}\) in comparison with ShIMS, which is practically independent of the load and lies between 12 and 18.

For one and the same rms value of the first-harmonic current, the mean values of the inverter elements for both types of modulation are approximately the same. As an example, Fig. 1 gives all the characteristic quantities as functions of the modulation depth for \(\tg \varphi=0.1\).

Fig. 1.

The error in determining the current values and the maximum returned REN from the approximate expressions [5] and those obtained depends on \(\tg \varphi\), \(N\), \(\mu\), and increases significantly as \(\mu\), \(N\), and \(\tg \varphi\) decrease. Thus, for \(N=6\), \(\tg \varphi=0.5\), and \(\mu=0.1\), the error reaches 20%.

It can be shown that there exists a critical value of \(\tg \varphi\) up to which REN is not returned to the source. To determine \(\tg \varphi_{\mathrm{cr}}\), the superposition method was used.

The condition for the initial equation is equality to zero of the supply-source current at the origin of coordinates, \(i_{\mathrm{p}}(0)=0\). The fictitious current of the first phase is

\[ i_{\phi}'(\varepsilon)= \sum_{k=0}^{N_{1}-1} C_{1}(k)\cdot C_{0}\cdot \lambda(k) + \sum_{k=0}^{N_{1}-1}\sum_{\nu} \frac{C_{1}(k)\,C_{\nu}\,e^{q_{\nu}\left|\varepsilon+\gamma_{1}'(k)\right|}} {1-e^{q_{\nu}}}. \tag{5} \]

Taking into account that

\[ C=1, \qquad C_{\gamma}=1_{1}=-1, \qquad q_{\gamma}=-\beta \]

\[ C_{1}(0)=1 \qquad \lambda(0)=1; \qquad \gamma_{1}'(0)=0; \]

\[ C_{1}(1)=-2; \qquad \lambda(1)=0; \qquad \gamma_{1}'(1)=\frac{1}{2}; \]

\[ C_{1}(2)=1; \qquad \lambda(2)=0; \qquad \gamma_{1}'(2)=0, \]

the fictitious current on the interval is determined as

\[ i_{\phi}'(\varepsilon)=1-\frac{2}{1+a^{m}}e^{-\beta \varepsilon} =1+Ce^{-\beta \varepsilon}, \tag{6} \]

where

\[ a=e^{-\frac{\beta}{2m}} . \]

By symmetry,

\[ i_{\phi}'\left(\varepsilon+\frac{1}{2}\right)=-i_{\phi}'(\varepsilon). \]

Taking into account that, for an odd number of phases \(m\), the number of negative \(\Gamma\)-functions is \(0.5(m-1)\), and the number of positive ones is \(0.5(m+1)\), after the transformations, when the load is connected in a star, the supply-source current is determined as

\[ i_{\mathrm{p}}(\varepsilon)= \frac{m^{2}-1}{m} -\frac{2(m-1)}{m(1+a)}e^{-\beta \varepsilon} -\frac{4}{1+a^{m}} \sum_{l=1}^{\frac{m-1}{2}} a^{2l-1}e^{-\beta \varepsilon}. \tag{7} \]

From (7), after elimination of extraneous roots, one obtains the equation for determining \(a_{\mathrm{cr}}\):

\[ \sum_{l=0}^{\frac{m-1}{m}} \left\{ \left( \frac{m^{2}-1}{m} -\frac{m+1}{2}l \right) a_{\mathrm{cr}}^{\,m-2(l+1)} - \left[ \frac{(m-1)^{2}}{4} -\frac{m-1}{2}l \right] a_{\mathrm{cr}}^{\,2l} \right\}=0. \tag{8} \]

Analogous reasoning makes it possible to obtain an expression for an odd number of phases \(m\) and connection of the load in a polygon:

\[ a_{\mathrm{cr}}^{\,m-2} +a_{\mathrm{cr}}^{\,m}\left(1-a_{\mathrm{cr}}^{-1}\right) \sum_{i=1}^{\frac{m+1}{2}} a_{\mathrm{cr}}^{-2i}=0. \tag{9} \]

For an even number of phases and connection of the load in a star and in a polygon,

\[ \sum_{l=0}^{\frac{m}{2}-1} \left(\frac{m}{2}-2l\right) a_{\mathrm{cr}}^{\,\frac{m}{2}-(l+1)}=0, \tag{10} \]

\[ a_{\mathrm{cr}}^{\,m-2}=0. \tag{11} \]

From (8)--(10) it is evident that the best variant for using a multiphase inverter when operating into an active-inductive load, from the point of view of the reactive energy returned to the source, is an inverter with an odd number of phases when the load is connected in a star. If the number of load phases does not coincide with the required number, an intermediate phase converter can be used. If a multiphase asynchronous motor (a.m.) with a squirrel-cage rotor is taken as the load, then, in order to determine the electromagnetic processes, the a.m. equation should be reduced to an equivalent two-phase machine.

\[ \begin{gathered} \frac{d\overline{\psi}_1}{dt} = \overline{U}_1(t) - (\omega_0\alpha_s' + j\omega_k)\overline{\psi}_1 + \omega_0\alpha_s' k_r \overline{\psi}_2; \\[4pt] \frac{d\overline{\psi}_2}{dt} = - [\omega_0\alpha_r' + j(\omega_k-\omega)]\overline{\psi}_2 + \omega_0\alpha_r' k_s \overline{\psi}_1; \\[4pt] M=\frac{m}{2} Z_p \omega_0 \frac{k_r}{x_s\sigma}\operatorname{Im}(\overline{\psi}_1\cdot \dot{\overline{\psi}}_2); \\[4pt] \frac{d\omega}{dt}=\frac{Z_p}{J}(M-M_c). \end{gathered} \tag{12} \]

Here \(\omega_0\), \(\omega\), \(\omega_k\) are the angular velocities of rotation of the stator field, the rotor, and the coordinate system; \(\overline{\Psi}_1\), \(\overline{\Psi}_2\) are the flux-linkage vectors of the stator and rotor of the idealized induction motor; \(M\), \(M_c\), \(J\) are the motor torque, load torque, and inertia; \(Z_p\) is the number of pole pairs of the motor; \(\alpha_s'\), \(\alpha_r'\), \(k_r\), \(k_s\), \(x_s\), \(\sigma\) are quantities expressed through the motor parameters.

System (12) is essentially nonlinear; however, for the case \(\omega_k=\mathrm{const}\), with zero initial conditions,

\[ k_i\cdot a\cdot \pi\cdot \overline{F}(q) = (q+2\pi a\alpha_s')\overline{\psi}_1^{\,0}(q) - 2\pi a\alpha_s' k_r \overline{\psi}_2^{\,0}(q) \tag{13} \]

\[ a = -2\pi a\alpha_r' k_s \overline{\psi}_1^{\,0}(q) + (q+2\pi a\alpha_r' - j2\pi b)\overline{\psi}_2^{\,0}(q); \]

whence the ordinary vector transfer functions are found:

\[ \widetilde{K}_1(q) = \frac{\overline{\psi}_1^{\,0}(q)}{\overline{F}(q)} = k_i\cdot \pi\cdot a\, \frac{q+2\pi a\left(\alpha_r' - j\frac{b}{a}\right)} {(q-q_1)(q-q_2)}; \tag{14} \]

\[ \widetilde{K}_2(q) = \frac{\overline{\psi}_2^{\,0}(q)}{\overline{F}(q)} = k_i\cdot \pi\cdot a\, \frac{2\pi\cdot a\cdot \alpha_r' k_s} {(q-q_1)(q-q_2)}. \]

Here

\[ 2\pi\cdot a=\omega_0 T; \qquad 2\pi b=\omega T; \]

\[ q_{1,2} = -\pi a\left(\alpha_r' + \alpha_s' - j\frac{b}{a}\right) \pm \frac{k_i=\dfrac{E_n}{U_n}} {\sqrt{ \left(\alpha_r' + \alpha_s' - j\frac{b}{a}\right)^2 + 4\alpha_r'\alpha_s'(1-b) }}. \]

The solution for the quasi-steady-state flux-linkage vectors of an equivalent two-phase machine under PWM according to any law and with coincidence of the modulation periods is

\[ \overline{\psi}_{1kl}^{\,0} = -\frac{4}{m} \left[ C_0^{(1)}\lambda(k) + \sum_{\nu} \frac{C_{k\nu}^{(1)} e^{q_\nu[\varepsilon+\varepsilon'(k)]}} {1-e^{q_\nu}} \right]; \qquad \overline{\psi}_1^{\,0} = \sum_{k=0}^{N-1}\sum_{l=0}^{m-1}\overline{\psi}_{1kl}^{\,0}; \]

\[ \overline{\psi}_{2kl}^{\,0} = -\frac{4}{m} \left[ C_0^{(2)}\lambda(k) + \sum_{\nu} \frac{C_{k\nu}^{(2)} e^{q_\nu[\varepsilon+\varepsilon'(k)]}} {1-e^{q_\nu}} \right]; \qquad \overline{\psi}_2^{\,0} = \sum_{k=0}^{N-1}\sum_{l=0}^{m-1}\overline{\psi}_{2kl}^{\,0}, \tag{15} \]

where

\[ C_\nu^{1(2)} = \operatorname{Res}_{q_\nu} \left\{ \frac{1}{q}K_{1(2)\nu}(q) \right\}. \]

Using (15), an algorithm and a calculation program for a BESM-4-type digital computer were compiled. The calculation results show that torque pulsations

with PWM have period \(\frac{1}{2m}\), with PWM \(S\)—\(\frac{1}{m}\), \(m\) being odd.

In this case there also exists \(N''_{\mathrm{gr}}\), up to which PWMП is better than PWM \(S\) in terms of the magnitude of the torque pulsations, which lies within the limits 12 and 18.

In the case where in (15) the \(F\)-function has \(\mu=1\) for PWMП and \(N=2km\), the determination of the flux-linkage vectors is better carried out by the superposition method.

The current of any phase of the stator or rotor of the motor is obtained from the expression, in relative units,

\[ i_{si}^{\partial}(\varepsilon)= \operatorname{Re}\left\{ \frac{\alpha_s'}{r_1^{\partial}} \left[\bar{\psi}_1^{\partial}(\varepsilon)-k_s\bar{\psi}_2^{\partial}(\varepsilon)\right] \right\}. \tag{16} \]

\[ i_{ri}^{\partial}(\varepsilon)= \operatorname{Re}\left\{ \frac{\alpha_r'}{r_2^{\partial}} \left[\bar{\psi}_2^{\partial}\varepsilon-k_s\bar{\psi}_1^{\partial}(\varepsilon)\right] \right\}. \]

The determination of the maximum, average, and rms values of the currents of the inverter elements is performed in accordance with (16) and [1] and has a dependence similar to Fig. 1.

Conclusions

1. The use of switching functions makes it possible to carry out a generalized approach to all bridge voltage inverters for any type of modulation of the output voltage.

2. Compensation of the RPN by phase-to-phase transfer is most favorable with PWMП and depends on the number and parity of the phases and on the load connection. An odd number of phases and a star-connected load are preferable.

3. The energy processes for an odd number of phases (source current, motor torque at constant rotational speed) have a repetition period \(\frac{1}{2m}\) with PWMП and \(\frac{1}{m}\) with PWM \(S\).

4. The field form of the machine improves as the number of phases increases; at the same time the amplitude of the torque and speed pulsations decreases.

References

1. V. I. Ivanchura, B. P. Soustin. Investigation of \(m\)-phase bridge voltage inverters. In: Improving the Efficiency of Converter-Engineering Devices, vol. 2, Kiev, “Naukova Dumka,” 1972.

2. V. I. Ivanchura, B. P. Soustin. Determination of the output voltages of \(m\)-phase inverters through switching functions, Izvestiya TPI, vol. 262, Tomsk, Tomsk State University Press, 1972.

3. V. I. Ivanchura, B. P. Soustin. A method for analyzing bridge \(m\)-phase voltage inverters, Izvestiya TPI, vol. 262, Tomsk, Tomsk State University Press, 1972.

4. Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Moscow, Fizmatgiz, 1963.

5. A. S. Sandler, R. S. Sarbatov. Frequency Converters for Controlling Asynchronous Motors. Moscow, “Energiya,” 1966.