ASYNCHRONOUS GENERATOR WITH A CONTROLLED SEMICONDUCTOR SWITCH

B. E. TROFIMENKO, A. I. SKOROSPËSHKIN

(Recommended by the seminar of the departments of electrical machines and general electrical engineering)

In [1], on the basis of an asynchronous machine with excitation from a direct-current source through a controlled semiconductor switch (CSS), an electric-machine amplifier of regulated frequency (EMA-RF) was proposed.

Figure 1 shows the circuit of the EMA-RF, consisting of an asynchronous generator (windings \(W_r, W_c\)) and an excitation winding (\(W_{\mathrm{v}}\)). The excitation winding, together with the output winding, is placed in the stator slots and constitutes a six-phase star with a neutral conductor. The supply of the phases is single-half-period. The CSS circuit provides regular sequential connection and disconnection of the phases of the excitation winding, owing to which a rotating excitation m.m.f. is created in the machine.

The rotor of the EMA-RF is rotated by a prime mover (in principle, any kind) at variable or constant speed. At any speed of the drive shaft, the asynchronous generator, operating with negative slip, delivers active power to the load and to the excitation source.

At small loads of the amplifier, a mode of its self-excitation is possible, i.e., loss of control over the output voltage. To ensure normal operation of the EMA-RF over a wide range of loads, it is provided with a biasing winding \(W_{\mathrm{p}}\), wound around the stator back and connected into the excitation circuit \(U_{\mathrm{v}}\) (Fig. 1).

The purpose of the present work is a theoretical investigation of the proposed amplifier variant. In view of the discrete switching of the phases of the excitation winding by means of the CSS, it is impossible to study the EMA-RF as an ordinary asynchronous machine.

Consideration of both transient and steady-state regimes is possible over intercommutation time intervals, during which the number of acting e.m.f.’s and of circuits participating in the process remains unchanged.

The regularities of the flow of currents and voltages in separate time intervals can be described by a system of differential equations written with respect to instantaneous phase variables.

For the formulation of the system of equations and the subsequent analytical investigation, in addition to the assumptions generally known in the theory of electrical machines—the magnetic circuit is unsaturated, iron losses and current displacement are absent, etc.—a number of additional assumptions are adopted: the rotor speed is constant; the load of the EMA-RF and its windings are symmetrical; the transistor switches of the CSS are ideal, i.e., the commutation angle of the phases of winding \(W_{\mathrm{v}}\) is equal to zero.

Let us write, for the interval when the CSS switches K1, K2, K3 are open, the equations of the phase variables for the windings of the EMA-RF:

\[ |U_{\mathrm{v}}| = r_{\mathrm{v}} |i^{\mathrm{v}}_{m}| + \frac{d}{dt} |\psi^{\mathrm{v}}_{m}| ; \]

Fig. 1. Equivalent circuit of the EMU-RCh.

\[ 0 = r'_p \left| i^p_n \right| + \frac{d}{dt}\left| \psi^p_n \right|, \tag{1} \]

\[ 0 = r'_c \left| i^c_q \right| + \frac{d}{dt}\left| \psi^c_q \right|, \]

where \(\left| U_{\mathrm{v}} \right|\), \(\left| i^{\mathrm{v}}_m \right|\), \(\left| i^p_n \right|\), \(\left| i^c_q \right|\), \(\left| \psi^{\mathrm{v}}_m \right|\), \(\left| \psi^p_n \right|\), \(\left| \psi^c_q \right|\) are column matrices of the voltages, currents, and flux linkages of the phases of the amplifier windings.

The total flux linkages of the phases relative to the chosen orthogonal coordinate axes \(\alpha_c, j\beta_c\) on the stator and \(\alpha_p, j\beta_p\) on the rotor are written as

\[ \psi_m^{b} = L_{\sigma b}\, i_m^{b} + \sum_{t=1}^{m} M^{bb} i_t^{b}\cos(\alpha_m^{b}-\alpha_t^{b}) + \sum_{k=1}^{n} M^{bp} i_k^{p}\cos(\alpha_k^{p}+\Theta_p-\alpha_m^{b}) + \]

\[ + \sum_{s=1}^{q} M^{bc} i_s^{c}\cos(\alpha_q^{c}-\alpha_m^{b}), \]

\[ \psi_q^{c} = L'_{\sigma c}\, i_q^{c} + \sum_{s=1}^{q} M^{cc} i_s^{c}\cos(\alpha_q^{c}-\alpha_s^{c}) + \sum_{t=1}^{m} M^{cb} i_t^{b}\cos(\alpha_m^{b}-\alpha_q^{c}) + \]

\[ + \sum_{k=1}^{n} M^{cp} i_k^{p}\cos(\alpha_k^{p}+\Theta_p-\alpha_q^{c}), \]

\[ \psi_n^{p} = L_{\sigma p}\, i_n^{p} + \sum_{k=1}^{n} M^{pp} i_k^{p}\cos(\alpha_n^{p}-\alpha_k^{p}) + \sum_{t=1}^{m} M^{pb} i_t^{b}\cos\!\left[\alpha_t^{b}-(\alpha_n^{p}+\theta_p)\right] + \]

\[ + \sum_{s=1}^{q} M^{pc} i_s^{c}\cos\!\left[\alpha_s^{c}-(\alpha_n^{p}+\theta_p)\right], \]

where

\[ \begin{aligned} m &= 1, 2, \ldots m_b, & t &= 1, 2, \ldots m_b,\\ n &= 1, 2, \ldots n_p, & K &= 1, 2, \ldots n_p,\\ q &= 1, 2, \ldots q_c, & s &= 1, 2, \ldots q_c, \end{aligned} \]

\(m_b\) is the number of phases of the excitation winding;
\(n_p\) is the number of phases of the rotor winding;
\(q_c\) is the number of phases of the output winding;
\(M^{bb}, M^{pp}, M^{cc}, M^{bp}=M^{pb}, M^{bc}=M^{cb}, M^{cp}, M^{pc}\) are the mutual inductances of the winding phases when their magnetic axes coincide;
\(r_b, r_c', r_p'\) are the active resistances in the phases of the windings \(W_b, W_c, W_p\);
\(L_{\sigma b}, L_{\sigma c}, L_{\sigma p}\) are the leakage inductances of the phases of the EMF–RC windings;
\(\alpha_m^{b}, \alpha_n^{p}, \alpha_q^{c}\) are the displacement angles of the magnetic axes of the winding phases relative to the selected stator and rotor axes at the initial instant \(t=0\);
\(\theta_p=\omega_p t\) is the angle of the longitudinal axis of the rotor, depending on its rotation.

The system of equations (1) is a system of differential equations with periodic coefficients.

To obtain a system of linear equations, a transition is made from the axes of the winding phases to the selected axes on the stator and rotor. In this case the transformation of variables is performed by multiplying equations (1) by the system (2)

\[ \left|A\right|^{b} = \left|e^{j\alpha_1^{b}}, e^{j\alpha_2^{b}}, \ldots e^{j\alpha_m^{b}}\right|. \]

\[ \left|A\right|^{c} = \left|e^{j\alpha_1^{c}}, e^{j\alpha_2^{c}}, \ldots e^{j\alpha_q^{c}}\right|, \tag{2} \]

\[ \left|A\right|^{p} = e^{j\theta_p}\left|e^{j\alpha_1^{p}}, e^{j\alpha_2^{p}}, \ldots e^{j\alpha_n^{p}}\right|. \]

After multiplying (1) by (2), respectively, equations with new complex variables are obtained:

\[ \overline{U}_b = r_b\,\overline{I}_b + \frac{d}{dt}\,\overline{\psi}_b, \]

\[ 0 = r_c'\overline{I}_c + \frac{d}{dt}\,\overline{\psi}_c, \tag{3} \]

\[ 0 = r_p'\overline{I} + \frac{d}{dt}\,\overline{\psi}_p - j\omega_p\overline{\psi}_p, \]

where

\[ \overline{U}_{\mathrm{v}}=\sum_{t=1}^{m_{\mathrm{v}}} u_t^{\mathrm{v}} e^{j\alpha_t^{\mathrm{v}}}, \qquad \overline{e}_{\mathrm{v}}=\sum_{t=1}^{m_{\mathrm{v}}} i_t^{\mathrm{v}}\cdot e^{j\alpha_t^{\mathrm{v}}}, \]

\[ \overline{I}_{\mathrm{c}}=\sum_{s=1}^{q_{\mathrm{c}}} i_s^{\mathrm{c}}\cdot e^{j\alpha_s^{\mathrm{c}}}, \qquad \overline{I}_{\mathrm{p}}=e^{j\theta_{\mathrm{p}}}\cdot \sum_{k=1}^{n_{\mathrm{p}}} i_k^{\mathrm{p}}\cdot e^{j\alpha_k^{\mathrm{p}}} \]

are the resultant vectors of currents and voltages, represented in the complex plane and characterizing their instantaneous values; the flux linkages are written as

\[ \left| \begin{array}{c} \overline{\psi}_{\mathrm{v}}\\ \overline{\psi}_{\mathrm{c}}\\ \overline{\psi}_{\mathrm{p}} \end{array} \right| = \left| \begin{array}{ccc} L_{\sigma\mathrm{v}}+\dfrac{m_{\mathrm{v}}}{2}M^{\mathrm{vv}} & \dfrac{m_{\mathrm{v}}}{2}M^{\mathrm{vc}} & \dfrac{m_{\mathrm{v}}}{2}M^{\mathrm{vp}} \\[6pt] \dfrac{q_{\mathrm{c}}}{2}M^{\mathrm{cv}} & L'_{\sigma\mathrm{c}}+\dfrac{q_{\mathrm{c}}}{2}M^{\mathrm{cc}} & \dfrac{q_{\mathrm{c}}}{2}M^{\mathrm{cp}} \\[6pt] \dfrac{n}{2}M^{\mathrm{pv}} & \dfrac{n}{2}M^{\mathrm{pc}} & L_{\sigma\mathrm{p}}+\dfrac{n}{2}M^{\mathrm{pp}} \end{array} \right| \cdot \left| \begin{array}{c} \overline{I}_{\mathrm{v}}\\ \overline{I}_{\mathrm{c}}\\ \overline{I}_{\mathrm{p}} \end{array} \right|. \]

Thus, the transformation made it possible to obtain the linear system of equations (3) with constant coefficients and new complex variables.

Equations (3) for complex relative quantities in operational form, after reducing the windings to the rotor one and taking the initial conditions into account, are written in the image domain as:

\[ \overline{U}_{\mathrm{v}}+p x_{\mathrm{v}}\overline{I}_{0\mathrm{v}} +p x_m\overline{I}_{0\mathrm{c}} +p x_m\overline{I}_{0\mathrm{p}} = (r_{\mathrm{v}}+p x_{\mathrm{v}})\overline{I}_{\mathrm{v}} +p x_m\overline{I}_{\mathrm{c}} +p x_m\overline{I}_{\mathrm{p}}, \tag{4} \]

\[ p x_m\overline{I}_{0\mathrm{v}} +p x_{\mathrm{c}}\overline{I}_{0\mathrm{c}} +p x_m\overline{I}_{0\mathrm{p}} = p x_m\overline{I}_{\mathrm{v}} +(r'_{\mathrm{c}}+p x_{\mathrm{c}})\overline{I}_{\mathrm{c}} +p x_m\overline{I}_{\mathrm{p}}, \]

\[ p x_m\overline{I}_{0\mathrm{v}} +p x_m\overline{I}_{0\mathrm{c}} +p x_{\mathrm{p}}\overline{I}_{0\mathrm{p}} = (p-j)x_m\overline{I}_{\mathrm{v}} +(p-j)x_m\overline{I}_{\mathrm{c}} +\bigl[r'_{\mathrm{p}}+x_{\mathrm{p}}(p-j)\bigr]\overline{I}_{\mathrm{p}}, \]

where

\[ x_{\mathrm{v}}=x_m+x_{\sigma\mathrm{v}}, \]

\[ x_{\mathrm{p}}=x_m+x_{\sigma\mathrm{p}}, \]

\[ x_{\mathrm{c}}=x_m+x'_{\sigma\mathrm{c}}=x_m+x_{\sigma\mathrm{c}}+x_{\mathrm{n}}, \qquad \omega=1. \]

To simplify the solution of system (4), on the basis of the fact that the character of the transient processes in the machine is determined mainly by the total inductance of the windings, it is assumed that \(x_{\sigma\mathrm{v}}=0\), \(x_{\sigma\mathrm{p}}=0\). The assumption is valid for the operating frequency range, since at large slips \(r'_{\mathrm{p}}\gg x_{\sigma\mathrm{p}}\).

As a result of the solution, the resultant current vectors of the windings of the EMU-RC were found:

\[ \overline{I}_{\mathrm{v}}(t)= \frac{\overline{U}_{\mathrm{v}}}{r_{\mathrm{v}}} + \frac{ r'_{\mathrm{p}}x_m\left(r'_{\mathrm{c}}\overline{I}_{0\mathrm{c}}+\overline{U}_{\mathrm{v}}\right) }{ r_{\mathrm{v}}x'_{\sigma\mathrm{c}}(r'_{\mathrm{p}}-j x_m)-x_m r'_{\mathrm{c}}(r_{\mathrm{v}}+r'_{\mathrm{p}}) } \cdot e^{a\tau} + \]

\[ + \frac{ r_{\mathrm{v}}^{2}\Sigma\overline{I}_0\,(x_m x'_{\sigma\mathrm{c}}+r'_{\mathrm{p}}r'_{\mathrm{c}}-j r'_{\mathrm{c}}x_m) + \bigl[(r_{\mathrm{v}}-r'_{\mathrm{c}})(r'_{\mathrm{c}}+r'_{\mathrm{p}})-j r'_{\mathrm{c}}x_{\mathrm{c}}\bigr]\, r'_{\mathrm{p}}\overline{U}_{\mathrm{v}} }{ r_{\mathrm{v}}x'_{\sigma\mathrm{c}}(r_{\mathrm{v}}+r'_{\mathrm{p}})^2 } \, e^{(a+j\beta)\tau}, \]

\[ \overline{I}_{\mathrm{p}}(t)= \frac{ r_{\mathrm{v}}r'_{\mathrm{c}}x_m\overline{I}_{0\mathrm{v}} + j x_m r_{\mathrm{v}}\left(x_m\Sigma\overline{I}_0+x'_{\sigma\mathrm{c}}\overline{I}_{0\mathrm{c}}\right) }{ r_{\mathrm{v}}x'_{\sigma\mathrm{c}}(r'_{\mathrm{p}}-j x_m)-x_m r'_{\mathrm{c}}(r_{\mathrm{v}}+r'_{\mathrm{p}}) } e^{a\tau} + \]

\[ + \frac{ r_{\mathrm{v}}x_m\Sigma\overline{I}_0 \left[(r_{\mathrm{v}}x_m+r_{\mathrm{v}}x'_{\sigma\mathrm{c}}+r'_{\mathrm{c}}x_m) +j(x_m x'_{\sigma\mathrm{c}}-r_{\mathrm{v}}r'_{\mathrm{c}})\right] + j r'_{\mathrm{p}}\overline{U}_{\mathrm{v}}(r_{\mathrm{v}}x'_{\sigma\mathrm{c}}-r'_{\mathrm{c}}x_m) }{ x_m(r_{\mathrm{v}}+r'_{\mathrm{p}})\,[r_{\mathrm{v}}x'_{\sigma\mathrm{c}}-j(r'_{\mathrm{c}})^2] } \times \]

\[ \times e^{(a+j\beta)\tau}, \tag{5} \]

\[ \overline{I}_{\mathrm{c}}(t)= \]

\[ = \frac{ r'_{\mathrm{p}}x_m(r_{\mathrm{v}}\Sigma\overline{I}_0-\overline{U}_{\mathrm{v}}) + \overline{I}_{0\mathrm{c}}(r_{\mathrm{v}}r'_{\mathrm{p}}x'_{\sigma\mathrm{c}} -r_{\mathrm{v}}r'_{\mathrm{c}}x_m-r'_{\mathrm{p}}r'_{\mathrm{c}}x_m) - j x_m r_{\mathrm{v}}(x_m\Sigma\overline{I}_0+x'_{\sigma\mathrm{c}}\overline{I}_{0\mathrm{c}}) }{ r_{\mathrm{v}}x'_{\sigma\mathrm{c}}(r'_{\mathrm{p}}-j x_m)-r'_{\mathrm{c}}x_m(r_{\mathrm{v}}+r'_{\mathrm{p}}) } \times \]

\[ \times e^{a\tau} + \frac{ r'_{\mathrm{p}}x_m(r_{\mathrm{v}}\Sigma\overline{I}_0-\overline{U}_{\mathrm{v}}) -j r_{\mathrm{v}}x_m^{2}\Sigma\overline{I}_0 }{ r'_{\mathrm{c}}x_m(r'_{\mathrm{p}}+r'_{\mathrm{c}}) -r_{\mathrm{v}}x'_{\sigma\mathrm{c}}(r'_{\mathrm{p}}-j x_m) } e^{(a+j\beta)\tau}, \]

where

\[ \Sigma \overline{I}_{o}=\overline{I}_{ov}+\overline{I}_{oc}+\overline{I}_{or}, \]

\[ a=-\frac{r_v \cdot r'_r}{x_m\cdot (r_v+r'_r)}, \qquad b=\frac{r_v}{r_v+r'_r}, \]

\[ x'_{\sigma c}=x_{\sigma c}+x_n, \qquad x_n,\ R_n \text{ are the load parameters of the EMI–RTs,} \]

\[ r'_c=r_c+R_n, \]

\[ r'_r=r_r+R_r, \]

\(R_r\) is the adjustable additional resistance in the rotor circuit for creating a critical-slip mode at any output frequencies of the EMI–RT.

Fig. 2. Change in the magnitude and position of the resultant vector of the excitation winding when \(U_v\) is switched on.

Equations (5) make it possible to calculate transient and steady-state processes in the EMI–RT, taking into account the operation of the controlled transistor switch. The calculation is performed over time intervals.

Given the complex currents \(\overline{I}_v\), \(\overline{I}_c\), \(\overline{I}_r\), the instantaneous values of the currents in the phases of the amplifier windings are found from the inverse-transformation expressions.

The instantaneous current values are equal to:

\[ |i_m^{v}|=Re\ [[\overline{I}_v\cdot |T|^{v}], \]

\[ |i_q^{c}|=Re\ [[\overline{I}_c\cdot |T|^{c}], \]

\[ |i_n^{r}|=Re\ [[\overline{I}_r\cdot |T|^{r}], \]

where \(|T|^{v}\), \(|T|^{c}\), \(|T|^{r}\) are column matrices of the inverse transformation.

In the case of three phases, the matrices are written as:

\[ |T|^{v,c}= \frac{1}{\Delta_{v,c,p}} \begin{vmatrix} (\sin \alpha^{v,c}_{3}-\sin \alpha^{v,c}_{2})+j(\cos \alpha^{v,c}_{3}-\cos \alpha^{v,c}_{2})\\ (\sin \alpha^{v,c}_{1}-\sin \alpha^{v,c}_{3})+j(\cos \alpha^{v,c}_{1}-\cos \alpha^{v,c}_{3})\\ (\sin \alpha^{v,c}_{2}-\sin \alpha^{v,c}_{1})+j(\cos \alpha^{v,c}_{2}-\cos \alpha^{v,c}_{1}) \end{vmatrix}, \]

\[ |T|^{p}= \frac{e^{-j\theta_p}}{\Delta_{v,c,p}} \begin{vmatrix} (\sin \alpha^{p}_{3}-\sin \alpha^{p}_{2})+j(\cos \alpha^{p}_{3}-\cos \alpha^{p}_{2})\\ (\sin \alpha^{p}_{1}-\sin \alpha^{p}_{3})+j(\cos \alpha^{p}_{1}-\cos \alpha^{p}_{3})\\ (\sin \alpha^{p}_{2}-\sin \alpha^{p}_{1})+j(\cos \alpha^{p}_{2}-\cos \alpha^{p}_{1}) \end{vmatrix}, \]

where

\[ \Delta_{v,c,p}= \sin(\alpha^{v,c,p}_{1}-\alpha^{v,c,p}_{2}) +\sin(\alpha^{v,c,p}_{2}-\alpha^{v,c,p}_{3}) +\sin(\alpha^{v,c,p}_{3}-\alpha^{v,c,p}_{1}). \]

In the matrix \( |T|^{v} \), the angles \(\alpha^{v}_{1}, \alpha^{v}_{2}, \alpha^{v}_{3}\) are the phase angles of the winding \(W_v\), which in the time interval under consideration are connected to the source \(U_v\).

In expressions (5), the influence of the bias winding on the operation of the EMU-RCh has not been taken into account. Since the winding \(W_p\) changes the machine’s \(x_m\) during operation, taking its influence into account, with known control current and the magnetization curve of the machine iron, presents no difficulty.

Figure 2 gives the calculated settling of the resultant vector of the excitation-winding current when the source \(U_v\) is switched on. It is evident from the figure that the process of settling of the current vector is oscillatory in character. From the hodograph of the settling of the resultant vector, the time constant and speed of response of the amplifier can be determined. In principle, the hodograph can be constructed for any EMU-RCh winding and any of its modes.

Thus, as a result of the investigation carried out on the EMU-RCh, expressions have been obtained for the currents of its windings, making it possible to calculate established transient processes as well as the principal characteristics.

References

1. Sh. S. Roiz, B. E. Trofimenko, A. I. Skorospeshkin, M. L. Kostyrev, E. F. Obergauz. “An Electromachine Amplifier with a Controlled Semiconductor Commutator,” Izv. TPI, vol. 212.

2. Sh. I. Gutidze. Fundamentals of the Theory of Electric Machines with a Controlled Semiconductor Commutator. Moscow: Nauka, 1968.