Investigation of Quasi-Steady-State Modes of an Induction Motor Supplied by an Inverter with Nonsinusoidal Voltage

V. I. Ivanchura, B. P. Soustin

(Submitted by the Department of Electrical Equipment)

When three-phase AC motors are supplied from static semiconductor converters, it is very difficult, and sometimes impossible, to obtain a sinusoidal voltage. The nonsinusoidal nature of the stator voltage leads to distortion of the currents and flux linkages, which causes pulsations of the motor torque. At constant rotor speed and with decayed electromagnetic transients, the currents, flux linkages, and torque can be expressed through the generalized voltage vector:

\[ \bar U = \frac{2}{3}\left(U_A+\bar a U_B+\bar a^2 U_C\right), \]

assuming that the commutation functions of the static converter are known. To obtain expressions for the currents, flux linkages, and torques, the discrete Laplace transform is used.

The output voltage of an inverter without a neutral conductor does not depend on the load if the condition

\[ \tau_{1A}+\tau_{2A}=\tau_{1B}+\tau_{2B}=\tau_{1C}+\tau_{2C}=\tau=\frac{T}{N}, \tag{1} \]

is satisfied, where

\[ \begin{aligned} \tau_1,\ \tau_2 &\text{ are the times during which the elements of the upper and lower inverter groups are switched on, respectively, for phases } A,\ B,\text{ and } C;\\ \tau &\text{ is the modulation period, if modulation exists;}\\ T &\text{ is the period of the output frequency;}\\ N &\text{ is the number of modulation pulses per period of the output frequency.} \end{aligned} \]

In this case, the output voltage is uniquely expressed through the control commutation functions \(F_A, F_B, F_C\), where

\[ F=1,\quad 0<t<\tau_1 \]

\[ F=-1,\quad \tau_1<t<\tau. \]

The line voltages are

\[ U_{AB}=\frac{1}{2}E_n(F_A-F_B), \]

\[ U_{BC}=\frac{1}{2}E_n(F_B-F_C), \tag{2} \]

\[ U_{CA}=\frac{1}{2}E_n(F_C-F_A). \]

The phase voltages are determined from the matrix equation

\[ \begin{bmatrix} U_A\\ U_B\\ U_C \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 2&1\\ -1&1\\ -1&-2 \end{bmatrix} \begin{bmatrix} U_{AB}\\ U_{BC} \end{bmatrix} \]

and have the form

\[ \begin{aligned} U_A&=\frac{1}{2}E_{\text{p}}\frac{2F_A-F_B-F_C}{3},\\ U_B&=\frac{1}{2}E_{\text{p}}\frac{2F_B-F_C-F_A}{3},\\ U_C&=-\frac{1}{2}E_{\text{p}}\frac{2F_C-F_A-F_B}{3}. \end{aligned} \tag{3} \]

Then the generalized vector of the three-phase voltage is

\[ \overline{U} = \frac{1}{2}E_{\text{p}}\frac{2}{3} \left(F_A+\overline{a}F_B+\overline{a}^{\,2}F_C\right) = \frac{1}{2}E_{\text{p}}\overline{F}, \tag{4} \]

where

\[ \overline{F} \]
— the generalized commutation vector;

\[ E_{\text{p}} \]
— the supply-source voltage at the inverter input, which is assumed constant over the time interval under consideration.

The commutation functions are potential functions of points \(A, B, C\) and are taken as bipolar.

In a circuit with a neutral conductor, the phase voltage corresponds to the form of the commutation function.

It follows from (3) that the first harmonics of the phase voltages are equal to the first harmonics of the corresponding commutation functions, multiplied by \(0.5E_{\text{p}}\).

The commutation functions form a symmetrical three-phase system. In order that the higher harmonics that are multiples of the carrier frequency should disappear from the phase voltage, the modulation periods [1] must coincide for all phases. Implementation of this condition in practice leads to simplification of the inverter control circuit [2]. In the particular case, if \(N\) is an integer, then \(N/3\) must also be an integer.

In Fig. 1 (see p. 186) the commutation functions of the inverter phases are given:

1, a — for inverters with \(180^\circ\) control;

1, b — with pulse-width modulation (PWM) and equal widths of the voltage pulses, \(N=6\);

1, c — PWM according to a sinusoidal law, \(N=6\).

The flux-linkage vectors of an induction motor \(\overline{\Psi}_1\) and \(\overline{\Psi}'_2\) can be represented as the output of pulsed open-loop systems to whose input a unit action of period \(T\) is applied (Fig. 2), \(X(n)=1\), where \(n\) is the current value

\[ \overline{\Psi}^{\,*}_{1(2)}(q,\varepsilon) = X^{*}(q,o)K^{*}_{1(2)}(q,\varepsilon). \tag{4} \]

Here \(\overline{\Psi}^{\,*}_{1(2)}\) is the image of the flux-linkage vectors when using the discrete Laplace transform;

\[ K^{*}_{1(2)}(q,\varepsilon) \]
— the transfer functions of the flux linkages;

\[ X^{*}(q,o) \]
— the image of the input action;

\[ q,\varepsilon \]
— the transform parameter and a real parameter.

The transfer functions \(K^{*}_{1}(q,\varepsilon)\) and \(K^{*}_{2}(q,\varepsilon)\) are the discrete pre-

Fig. 1.

Fig. 2.

the formation of the corresponding impulse responses \(K_1[n,\varepsilon]\) and \(K_2[n,\varepsilon]\).

\[ K^{*}_{1(2)}(q,\varepsilon)=D\{K_{1(2)}[n,\varepsilon]\}. \tag{5} \]

In turn, the impulse responses are expressed as the inverse transforms of the transfer functions of the reduced continuous part in dimensionless quantities:

\[ K_{1(2)}[n,\varepsilon]=L^{-1}\{K_{1(2)}(q)\}. \tag{6} \]

In the present case

\[ K_{1(2)}(q)=K_u(q)\,\widetilde K_{g1(2)}(q). \tag{7} \]

Here \(K_u(q)\) is the transfer function of the device forming the voltage vector in dimensionless quantities,

\[ K_u(q)=\frac{E_{\mathrm{p}}}{2}\,\overline{F}(q), \]

\(\widetilde{K}_{g1(2)}(q)\) are the transfer functions of the motor, relating the voltage vector and the flux-linkage vectors.

By jointly solving (4), (5), (6), (7), the flux linkages are found:

\[ \overline{\Psi}^{*}_{1(2)}(q,\varepsilon) = X^{*}(q,o)\cdot D\left\{ L^{-1}\left[ \frac{E_{\mathrm{p}}}{2}\, \overline{F}(q)\widetilde{K}_{g1(2)}(q) \right] \right\}. \tag{8} \]

The flux-linkage vectors can be decomposed into two components: the component of the quasi-steady-state process \(\overline{\Psi}_{1(2)}(\infty,\varepsilon)\) and the transient component \(\overline{\Psi}_{1(2)n}(n,\varepsilon)\).

\[ \overline{\Psi}_{1(2)}(\infty,\varepsilon) = \lim_{q\to 0} \left[(e^{q}-1)\overline{\Psi}^{*}_{1}(q,\varepsilon)\right], \]

\[ \overline{\Psi}_{1(2)(n,\varepsilon)} = \sum_{\nu} \operatorname{Res}_{e^{q_\nu}} \left[\overline{\Psi}^{*}_{1(2)}(q,\varepsilon)\right] e^{q(n-1)} \tag{9} \]

\[ q_\nu\ne 0 \]

For pulse-width modulation of the inverter output voltage, when \(N\) is an integer multiple of three, it is assumed that the law of variation of the switching functions is specified:

\[ \tau_{1A}(k),\quad \tau_{1B}(k),\quad \tau_{1C}(k). \]

\[ F_A(q) = \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{1A}(kq)}}{q}\cdot e^{-k\gamma q} - \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{2A}(k)q}}{q}\cdot e^{-[k\gamma+\gamma_{1A}(k)]q}, \]

\[ F_B(q) = \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{1B}(kq)}}{q}\cdot e^{-k\gamma q} - \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{2B}(k)q}}{q}\cdot e^{-[k\gamma+\gamma_{1B}(k)]q}, \]

\[ F_C(q) = \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{1C}(k)q}}{q}\cdot e^{-k\gamma q} - \sum_{k=0}^{N-1} \frac{1-e^{-\gamma_{2C}(k)q}}{q}\cdot e^{-[k\gamma+\gamma_{1C}(k)]q}. \tag{11} \]

After transformations,

\[ \overline{F}(q) = -\frac{4}{3q} \sum_{k=0}^{N-1} e^{-\gamma_A(k)q} + \overline{a}\,e^{-\gamma_B(k)q} + \overline{a}^{\,2}e^{-\gamma_C(k)q}. \tag{12} \]

Here

\[ \gamma_{1A}=\frac{\tau_{1A}(k)}{T}; \qquad \gamma_{1B}(k)=\frac{\tau_{1B}(k)}{T}; \qquad \gamma_{1C}(k)=\frac{\tau_{1C}(k)}{T}; \]

\[ \gamma=\frac{\tau}{T}; \qquad \gamma_A(k)=\gamma k+\gamma_{1A}(k); \qquad \gamma_B(k)=\gamma k+\gamma_{1B}(k); \qquad \gamma_C(k)=\gamma k+\gamma_{1C}(k), \]

\(k\) is the ordinal number of the switching interval.

Then the transfer function of the voltage-vector-forming element will have the form

\[ K_u(q) = -\frac{E_{\mathrm{p}}\cdot 2}{3q} \sum_{k=0}^{N-1} e^{-\gamma_A(k)q} + \overline{a}\,e^{-\gamma_B(k)q} + \overline{a}^{\,2}e^{-\gamma_C(k)q}. \tag{13} \]

Expression (13) is valid for PWM according to any law. The most commonly used PWM laws are:

1. For equal widths of the phase pulses

\[ \gamma_A(k)=k\gamma+\frac{\gamma}{2}(1+\mu),\quad 0\leq k\leq \frac{N}{2}-1; \]

\[ \gamma_A(k)=k\gamma+\frac{\gamma}{2}(1-\mu),\quad \frac{N}{2}\leq k\leq N-1. \]

1. When the pulse width varies according to a sinusoidal law

\[ \gamma_A(k)=k\gamma+\frac{\gamma}{2}\left(1+\mu\sin k\cdot\frac{2\pi}{N}\right). \]

Here \(\mu\) is the modulation depth, \(0\leq\mu\leq 1\).

The transfer functions of the motor \(\widetilde K_{1(2)}(q)\) are obtained from the vector equations [4] for a constant rotor speed and a fixed coordinate system

\[ \begin{aligned} \overline U_1(p)&=(p+\omega_0\alpha'_s)\overline\Psi_1(p)-\omega_0\alpha_s k_r\overline\Psi_2(p),\\ 0&=-\omega_0\alpha'_r k_s\overline\Psi_1(p)+(p+\omega_0\alpha'_r-j\omega)\overline\Psi_2(p), \end{aligned} \tag{14} \]

where

\[ \alpha'_s=\frac{\alpha_s}{\sigma};\quad \alpha_s=\frac{r_1}{x_s};\quad k_s=\frac{x_0}{x_s};\quad \sigma=1-k_r\cdot k_s; \]

\[ \alpha'_r=\frac{\alpha_r}{\sigma};\quad \alpha_r=\frac{r_2}{x_r};\quad k_r=\frac{x_0}{x_r}; \]

\[ \begin{aligned} r_1(r_2)&\text{ — active resistance of the stator (rotor);}\\ \omega_0&\text{ — base rated frequency;}\\ x_s(x_r)&\text{ — synchronous reactance of the stator (rotor) winding;}\\ \omega&\text{ — angular velocity of rotor rotation in electrical rad/sec.} \end{aligned} \]

With zero initial conditions, solving (14) with the substitution of \(P\) by \(\dfrac{q}{T}\), we obtain:

\[ \widetilde K_{g1}(q)=T\cdot \frac{q+2\pi\alpha'_r a-j2\pi b}{(q-q_1)(q-q_2)} =K_{g1}(q)T, \]

\[ \widetilde K_{g2}(q)=T\, \frac{2\pi\alpha'_r a k_s}{(q-q_1)(q-q_2)} =K_{g2}(q)T. \tag{15} \]

\[ q_1=p_1T;\quad q_2=p_2T;\quad a=\frac{\omega_0T}{2\pi};\quad b=\frac{\omega T}{2\pi}. \]

\(p_1\) and \(p_2\), the roots of the characteristic equation of system (14), are found by the method described in [4].

The image of the input action is

\[ X^*(q,o)=D\{1[n]\}=\frac{e^q}{e^q-1}. \tag{16} \]

From expressions (10), using (13), (15), and (16), the quasi-steady-state process under PWM according to any law is found:

\[ \overline\Psi_{1(2)}(\varepsilon)=\frac{E_n}{2}\cdot T\cdot \lim_{q\to 0}D\left\{L^{-1}\left[\overline F(q)\cdot K_{g1(2)}(q)\right]\right\}. \tag{17} \]

The discrete transform of the expression in braces depends on \(\varepsilon\), since there are shifting links

\[ \sum_{k=0}^{N-1} e^{-\gamma_A(k)q} +\overline a\,e^{-\gamma_B(k)q} +\overline a^{2}e^{-\gamma_C(k)q}. \]

\(\overline{\Psi}_{1(2)}(\varepsilon)\) is not difficult to obtain for any \(\varepsilon\), for example, when \(\varepsilon=m\gamma\),

\[ \begin{aligned} \overline{\Psi}_{1(2)}(m\gamma) &= \frac{E_n}{2}T \sum_{\nu=1}^{2} C_{\nu}^{1(2)} \Bigg\{ \frac{e^{q_\nu}}{1-e^{q_\nu}} \sum_{k=0}^{N-1} \left( e^{q_\nu[m\gamma-\gamma_A(k)]} +a e^{q_\nu[m\gamma-\gamma_B(k)]} +a^{-2} e^{q_\nu[m\gamma-\gamma_C(k)]} \right) \\ &\qquad\qquad\qquad\qquad +\sum_{k=0}^{m-1} \left( e^{q_\nu[m\gamma-\gamma_A(k)]} +\overline{a}e^{q_\nu[m\gamma-\gamma_A(k)]} +a^{-2}e^{q_\nu[m\gamma-\gamma_A(k)]} \right) \Bigg\}. \end{aligned} \tag{18} \]

Here

\[ C_{\nu}^{1(2)}=\lim_{q\to q_\nu}\left[(q-q_\nu)\cdot -\frac{4}{3q}K_{g1(2)}(q)\right]. \]

The torque of the machine in the quasi-steady-state mode is

\[ M(\varepsilon)=\frac{3}{2}z_p\frac{\omega_0 k_r}{x_s' \sigma}\operatorname{Im} \left[\overline{\Psi}_1(\varepsilon)\cdot\overline{\Psi}_2(\varepsilon)\right], \tag{19} \]

where \(z_p\) is the number of pole pairs.

The stator current of an induction machine is

\[ i_A(\varepsilon)=\frac{\omega_0 d_s'}{r_1} \left[\Psi_{a1}(\varepsilon)-\Psi_{a2}(\varepsilon)\right]=i_{a1}, \tag{20} \]

where

\[ \Psi_{a1}=\operatorname{Re}\overline{\Psi}_1;\qquad \Psi_{a2}=\operatorname{Re}\overline{\Psi}_2. \]

Expressions (17), (18), (19), and (20) make it possible to analyze, on an electronic digital computer, the currents, fluxes, and torque of a particular machine as functions of the law, depth, and frequency of modulation in the quasi-steady-state mode.

Conclusions

1. It has been shown that the generalized voltage vector, phase voltages, and line voltages are simply expressed through the switching functions in inverters with independent control.

2. Expressions have been obtained for quasi-steady-state processes using the discrete Laplace transform.

3. The quasi-steady-state mode when an induction motor is supplied from an inverter with nonsinusoidal output voltage should expediently be investigated with the aid of an electronic digital computer.

References

1. Schönung A., Stemmler H. Seregelter Drehstrom. Umkehrantrieb mit gesteurten Umrichter nach dem Unterschwingungs verfahren, BBC Mitt., 1964, No. 8/9, pp. 555–577.

2. A. S. Sandler, Yu. M. Gusyatskii. Thyristor Inverters with Pulse-Width Modulation. “Energiya,” 1968.

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

4. M. M. Sokolov, L. P. Petrov, L. B. Masandilov. Electromagnetic Transient Processes in an Induction Electric Drive. Moscow, “Energiya,” 1967.