A METHOD FOR ANALYZING BRIDGE \(m\)-PHASE VOLTAGE INVERTERS

V. I. Ivanchura, B. P. Soustin

(Presented by the Scientific-Technical Seminar of the Research Institute of Electrical Machine-Building)

If the \(m\)-phase load is symmetrical, then the system of linear equations reduces to a single vector equation, which, for zero initial conditions, is written as

\[ \overline{Y}(p)=\overline{U}(p)\cdot \overline{K}_{\mathrm{н}}(p). \tag{1} \]

Here \(\overline{Y}(p)\) is the operational image of the output quantity of the vector;
\(\overline{U}(p)\) is the image of the generalized \(m\)-phase voltage vector;
\(\overline{K}_{\mathrm{н}}(p)\) is the vector transfer function of the load.

If the form of the potential functions [1] does not depend on the load, then the \(m\)-phase bridge inverter can be replaced by a pulse element whose input receives a unit action of period \(T\), \(X(t)=1\). The pulse element forms the generalized vector of the \(m\)-phase voltage, defined over the interval \(T\). The transfer function of the pulse element under such an action coincides with its impulse characteristic,

\[ K_U(p)=\frac{L\{\overline{U}(t)\}}{L\{\delta(t)\}}=\overline{U}(p). \tag{2} \]

Therefore the output quantity of such an inverter is found as the response of an open-loop pulse system, to whose input a unit action of period \(T\) is applied. Then equation (1), when the discrete Laplace transform [2] is used, is written as

\[ \overline{Y}^{*}(q,\varepsilon)=\overline{K}^{*}(q,\varepsilon)\cdot \overline{X}^{*}(q,0), \tag{3} \]

where \(\overline{K}^{*}(q,\varepsilon)\) is the transfer function of the reduced continuous part, \(\overline{Y}^{*}(q,\varepsilon)\), \(\overline{X}^{*}(q,0)\) are the image of the output vector and the image of the input action.

The response \(\overline{Y}^{*}(q,\varepsilon)\) contains two components: the quasi-steady-state process \(\overline{Y}(\infty,\varepsilon)\) and the transient process \(\overline{Y}(n,\varepsilon)\).

\[ \overline{Y}(\infty,\varepsilon)=\overline{Y}(\varepsilon)= \lim_{q\to 0}\left[(e^{q}-1)\overline{Y}^{*}(q,\varepsilon)\right], \tag{4} \]

\[ \overline{Y}(n,\varepsilon)= \sum_{\nu}\operatorname{Res}_{e^{q_{\nu}}} \left[\overline{Y}^{*}(q,\varepsilon)e^{q(n-1)}\right]. \tag{5} \]

Here \(e^{q_s}\) are the poles of the function \(Y^*(q,\varepsilon)\), \(q_s \ne 0\).

Using the expression for the input action \(X(q,0)=\dfrac{e^q}{e^q-1}\), we find the component of the quasi-steady-state and transient processes:

\[ \overline{Y}(\varepsilon)=\lim_{q\to 0}\mathbf{D}\{K_{\mathrm{н}}(q)\cdot \widetilde{K}_{\mathrm{н}}(q)\}, \tag{6} \]

\[ \overline{Y}(n,\varepsilon)=\sum_s \operatorname{Res}_{e^{q_s}} \left\{ \frac{e^q}{e^{q_s}-1}\, \mathbf{D}\left[K_{\mathrm{н}}(q)e^{q(n-1)}\right] \right\}. \tag{7} \]

The generalized response vector \(\overline{Y}(\varepsilon)\) and \(\overline{Y}(n,\varepsilon)\) can be decomposed into phase components

\[ \overline{Y}_i(\varepsilon)=\operatorname{Re}\left[\overline{a}^{\,i-1}Y(\varepsilon)\right], \qquad \overline{Y}_i(n,\varepsilon)=\operatorname{Re}\left[\overline{a}^{\,i-1}Y(n,\varepsilon)\right]. \tag{8} \]

To determine the inverter response under pulse-width modulation, it is necessary to determine the image of the generalized voltage vector, which depends both on the form of the switching function and on the time characteristic of the supply source. If we assume that \(E_n(\varepsilon)=E_n=\mathrm{const}\), then, analogously to [3], we may write

\[ \overline{U}(q)=0.5E_n\sum_{i=1}^{m}\overline{a}^{\,i-1}\cdot F_i(q) = \frac{2E_n}{qm}\sum_{i=1}^{m}\sum_{k=0}^{N-1}\overline{a}^{\,i-1}e^{-\gamma_i(k)q}, \tag{9} \]

\[ F_i(q)=\frac{1}{T}F_i\left(\frac{q}{T}\right). \]

Expression (9) is valid for pulse-width modulation according to any law. Thus, if the phase voltage is modulated by rectangular pulses of equal width,

\[ \gamma_i(k)=\frac{\gamma}{2} \left\{ 1+\mu\,\operatorname{sign}\sin \left[ k\frac{2\pi}{N}+\xi_1+\Delta\xi+\frac{2\pi}{m}(i-1) \right] \right\} +k\gamma. \]

Here \(\mu\) is the modulation depth, and \(N\) is the number of pulses.

For pulse-width modulation according to a sinusoidal law,

\[ \gamma_i(k)=k\gamma+\frac{\gamma}{2} \left\{ 1+\mu\sin \left[ k\frac{2\pi}{N}+\xi_1+\frac{2\pi}{m}(i-1) \right] \right\}. \]

The angle \(\Delta\xi\) is introduced for definiteness of calculation on a digital computer; if \(\xi\ne 0\), \(\Delta\xi=0\).

It is obvious that for even \(N\), \(\mu=1\) and \(\xi=0\) will be a special case of \(180^\circ\) control.

Solving jointly (9) and (6), (7) for \(E_n=\mathrm{const}\),

\[ \overline{Y}(\varepsilon)=\lim_{q\to 0} \mathbf{D} \left\{ -\frac{2E_n}{qm} \sum_{k=0}^{N}\sum_{i=1}^{m} \overline{a}^{\,i-1}\cdot e^{-\gamma_i(k)q}\cdot \widetilde{K}_{\mathrm{н}}(q) \right\}, \tag{10} \]

\[ \overline{Y}(n,\varepsilon)= \sum_{\gamma}\operatorname{Res}_{e^{q_\gamma}} \left[ \frac{e^2}{e^q-1} \mathbf{D} \left\{ -\frac{2E_n}{qm}\cdot \sum_{k=0}^{N}\sum_{i=1}^{m} \overline{a}^{\,i-1}\cdot e^{-\gamma_i(k)q}\cdot \widetilde{K}_{\mathrm{н}}(q) \right\} e^{q(n-1)} \right]. \tag{11} \]

Practical inverter circuits contain switches made up of controlled elements connected in antiparallel (transistors, thyristors, etc.) and diodes. The currents of the elements are found from the following logical forms:

\[ \begin{aligned} &[(F_i \equiv +1)\wedge(\operatorname{sign} i_i \equiv +1)] \supset (i_{\mathrm{T}2i-1}\equiv +i_i),\\ &[(F_i \equiv +1)\wedge(\operatorname{sign} i_i \equiv -1)] \supset (i_{\mathrm{D}2i-1}\equiv -i_i),\\ &[(F_i \equiv -1)\wedge(\operatorname{sign} i_i \equiv -1)] \supset (i_{\mathrm{T}2i-1}\equiv -i_i),\\ &[(F_i \equiv -1)\wedge(\operatorname{sign} i_i \equiv +1)] \supset (i_{\mathrm{D}2i-1}\equiv +i_i). \end{aligned} \tag{12} \]

Here the switch current is decomposed into the currents of the elements in accordance with the logical function \(\operatorname{sign} i_i\).

The current of the power supply source \(i_n(t)\) is determined by

Fig. 1. Time characteristics of the inverter–asynchronous motor system: a) with \(180^\circ\) voltage; b) with PWM, equal pulse duration; c) PWM according to a sinusoidal law; d) flux linkages of the stator and rotor

\[ i_n(t)=\frac{1}{2}\sum_{i=1}^{i=m} F_i\cdot i_i. \]

Using the expressions given above, as an example, algorithms have been compiled for calculating the inverter currents, flux linkages, and torques of an asynchronous motor in a quasi-steady-state process. The solutions obtained are presented in Fig. 1.

Conclusions

A method is proposed for calculating \(m\)-phase voltage inverters for a linear load on the basis of the discrete Laplace transform using switching functions. The obtained expressions for the reaction are valid both for a static active-inductive load and for a motor load.

References

1. V. I. Ivanchura, B. P. Soustin. Determination of the output voltages of \(m\)-phase inverters through switching functions. (This collection).

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

3. V. I. Ivanchura, B. P. Soustin. Investigation of quasi-steady-state modes of an asynchronous motor when supplied from an inverter with nonsinusoidal voltage. Proceedings of TPI, vol. 243. TSU Publishing House, Tomsk, 1971.