Theoretical Study of a Variable-Frequency Electric Machine Amplifier with Allowance for a Controlled Semiconductor Commutator (CSC)
Abstract Generated abstract
This paper develops a theoretical model of a variable-frequency electric-machine amplifier that explicitly includes the controlled semiconductor commutator and its interaction with the control winding. Treating the commutator and control winding as an m-phase inverter, the authors derive voltage, current, and slip relations, construct equivalent circuits for contactless and contact versions of the amplifier, and examine the effects of feedback diodes and auxiliary cascades on operating behavior. The analysis yields expressions for external characteristics and amplification factor as functions of winding parameters, load, rotor parameters, and slip, with reported experimental comparisons showing satisfactory agreement with calculated input-frequency and external characteristics.
Full Text
THEORETICAL STUDY OF A VARIABLE-FREQUENCY ELECTRIC MACHINE AMPLIFIER TAKING INTO ACCOUNT A CONTROLLED SEMICONDUCTOR SWITCH (CSS)
Sh. S. Roiz, A. I. Skorospeshkin
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
A variable-frequency electric machine amplifier is an electric-machine–semiconductor complex consisting of a CSS and an electric-machine part. A theoretical study of the amplifier must be carried out taking into account all the links of the amplifier, considering the specific features of the operation of the CSS and of the electric-machine part. The CSS together with the control winding may be regarded as an \(m\)-phase inverter. This makes it possible to consider all the processes occurring in the amplifier as a whole. Moreover, such a representation of the control winding and the CSS makes it possible to obtain the fundamental equation and to construct the equivalent circuit required for practical calculations.
Fig. 1. Circuit of a variable-frequency electric machine amplifier in a contactless design
Figure 1 shows the circuit of an electric machine amplifier with semiconductor switches in the control-winding circuit. The designations in the figure are: \(T_1, T_2, \ldots, T_6\) are semiconductor switches; \(D_1, D_2, \ldots, D_6\) are return diodes; \(1, 2, \ldots, 6\) are the phases of the control winding; \(W_1; W_2\) is the output winding of the amplifier; \(W_3\) and \(W_4\) are the rotor windings of the main and auxiliary stages; \(W_5\) is the stator winding of the auxiliary stage.
The purpose of the feedback diodes in the CSS circuit is that, when the amplifier operates in the no-load or partial-load mode, the resultant magnetic flux induces an emf in the control winding. The alternating current flowing through the control winding is rectified by the feedback diodes and enters the dc network. The presence of feedback diodes in the CSS circuit also has a favorable effect on the operation of the CSS, reducing the magnitude of overvoltages on the power elements of the commutator. Thus, the feedback diodes have a positive influence on the operation of the CSS and also make it possible to transfer the rotor power in the no-load and partial-load modes of the amplifier to the dc network, which reduces the control power and increases the gain of the amplifier.
The dc voltage applied to each phase of the control winding under a symmetrical commutation law consists of rectangular pulses (Fig. 2) whose duration is equal to one half of the period of variation of the control frequency.
Fig. 2
The rms value of the first harmonic of the equivalent ac voltage applied to the control winding is equal to
\[ U_{y\sim}=\frac{4}{\pi}\cdot\frac{1}{\sqrt{2}}\,U_y, \tag{1} \]
where
\[ \frac{4}{\pi}U_y \]
is the amplitude of the equivalent ac voltage,
\(U_y\) is the voltage of the dc network.
In the absence of feedback diodes, the relation between the direct current and the equivalent alternating current is determined by the expression
\[ I_y = K_I I_{y\sim}, \tag{2} \]
where
\(I_y\) — equivalent alternating current flowing through each phase of the control winding;
\(I_y\) — direct current consumed from the dc network;
\(K_p\) — current coefficient taking into account the CSS circuit.
In the presence of feedback diodes in the CSS circuit, the control current flowing in the control circuit is equal to
\[ I_y = I_y' - I_y'', \tag{3} \]
where
\(I_y\) — resultant current of the control circuit,
\(I_y'\) — current consumed from the dc network,
\(I_y''\) — direct current due to the active component of the rotor current.
Fig. 3. Input frequency characteristic \(J_y=f(f_y)\) at \(U_y=\mathrm{const}=5.5\ \mathrm{V}\)
The experimental and calculated characteristics \(I_y=f(f_y)\) at \(U_y=\mathrm{const}=5.5\ \mathrm{V}\) and with the load and rotor circuits open (Fig. 3) show sufficient agreement, which makes it possible to conclude that the expressions obtained, (1), (2), (3), are valid and may be used in calculation.
Taking expressions (1), (2), (3) into account, the circuit of the electric-machine amplifier, whose control winding is supplied from an ac source, may be considered.
Let us write the emf equilibrium equations for one of the phases of the windings \(W_1, W_2, W_3, W_4, W_5\):
\[ \dot U_1=\dot I_1(r_1+j\omega_y P_1 L_1)+j\dot I_2\omega_y M_{12}P_1+j\dot I_3 M_{13}P_1, \tag{4} \]
\[ \dot U_2=-\dot I_2(r_2+j\omega_y P_1 L_1)-j\dot I_3\omega_y M_{23}P_1+j\dot I_1\omega_y M_{21}P_1 \tag{5} \]
\[ 0=\dot I_3\{(r_3+r_4)+j[(\omega_y-\omega)P_1L_3+j(\omega_y-\omega)P_1L_4]\}+ \]
\[ +j\dot I_1\omega_y P_1M_{31}+j\dot I_2\omega_yM_{32}+j\dot I_5M_{45}(\omega_y-\omega)P_1, \tag{6} \]
\[ 0=\dot I_5\{r_5+j[\omega_yP_1-\omega(P_1\pm P_2)]L_5\}+j\dot I_3M_{54}\{\omega_yP_1-\omega(P_1\pm P_2)\}, \tag{7} \]
where
\(L_1, L_2, L_3, L_4, L_5\) are the self-inductances of windings \(W_1, W_2, W_3, W_4, W_5\);
\(M_1=M_{31}=M_{21}=M_{32}=M_{12}=M_{13}=M_{23}\) are the mutual inductances in the main cascade;
\(M_2=M_{45}=M_{54}\) are the mutual inductances in the supplementary cascade;
\(\omega_y\) and \(\omega\) are the angular velocities of the control field and of the rotor;
\(P_1\) and \(P_2\) are the numbers of pole pairs of the main and supplementary cascades.
The signs \(\pm\) in (6) denote the concordant and opposite directions of rotation of the field of winding \(W_4\) and of the rotor.
Transform equation (6), dividing its right- and left-hand sides by \(S_1\),
\[
0=\dot I_3\left[\frac{r_3+r_4}{S_1}+j\omega_y P_1(L_3+L_4)\right]+j(\dot I_1+\dot I_2)M_1\omega_y P_1+
\]
\[
+j(\dot I_3+\dot I_5)M_2\omega_y P_1,
\tag{8}
\]
where
\[ S_1=\frac{\omega_y-\omega}{\omega_y} \]
is the slip of the main cascade.
We carry out analogous transformations with equation (7), dividing its right- and left-hand sides by \(S_1\) and \(S_2\),
\[ 0=\dot I_5\left(\frac{r_5}{S_1S_2}+jL_5\omega_y P_1\right)+j\dot I_3M_2\omega_y P_1, \tag{9} \]
where
\[ S_2=\frac{\omega_yP_1-\omega(P_1\pm P_2)}{(\omega_y-\omega)P_1} \]
is the slip of the supplementary cascade.
Assuming that the mutual inductances and currents vary according to a sinusoidal law, equations (4), (5), (8), (9), after a number of transformations, take the form:
\[ \dot U_1=\dot I_1(r_1+jx_{\sigma1})+\dot I_{01}Z_{01}, \]
\[ \dot U_2=-\dot I_2(r_2+jx_{\sigma2})-\dot I_{01}Z_{01}, \]
\[ 0=\dot I_3\left[\frac{r_3+r_4}{S_1}+j(X_{\sigma3}+X_{\sigma4})\right]+\dot I_{01}Z_{01}+\dot I_{02}Z_{02}, \tag{10} \]
\[ 0=\dot I_5\left(\frac{r_5}{S_1S_2}+jX_{\sigma5}\right)+\dot I_{02}Z_{02}, \]
where
\[ \dot I_{01}=\dot I_1+\dot I_2+\dot I_3 \]
is the magnetizing current of the main cascade,
\[ \dot I_{02}=\dot I_3+\dot I_5 \]
is the magnetizing current of the supplementary cascade,
\(Z_{01}\) is the impedance of the magnetizing circuit of the main cascade,
\(Z_{02}\) is the impedance of the magnetizing circuit of the supplementary cascade.
Using the system of equations (10), we construct the equivalent circuit of the amplifier (Fig. 4). The obtained system of equations (10) and the constructed equivalent circuit make it possible to calculate the possible operating modes of the amplifier.
For convenience of analysis we transform the equivalent circuit (Fig. 4) by combining the magnetizing and secondary circuits of the supplementary cascade (Fig. 5).
The values of the parameters \(R\) and \(X\) in the equivalent circuit are determined by the expressions
Fig. 4. Equivalent circuit of the amplifier in the contactless design.
Fig. 5. Transformed equivalent circuit of the amplifier in the contactless design
\[ R = \frac{ \dfrac{r_5}{S_1 S_2}\cdot X_{02}^{\,2} }{ \left(\dfrac{r_5}{S_1 S_2}\right)^2 + (X_{02}+X_{65})^2 }, \tag{11} \]
\[ X = \frac{ X_{02}\left[\left(\dfrac{r_5}{S_1 S_2}\right)^2 + (X_{02}+X_{52})X_{55}\right] }{ \left(\dfrac{r_5}{S_1 S_2}\right)^2 + (X_{02}+X_{65})^2 }, \tag{12} \]
Let us analyze the parameters \(R\) and \(X\) as functions of the operating mode of the amplifier.
The slip value \(S_1\) is always assumed to be negative, proceeding from the operating principle of the main cascade of the amplifier (asynchronous-generator mode).
The slip values \(S_2\) may be either positive or negative, depending on the operating mode of the auxiliary cascade.
The parameter \(X\) in the equivalent circuit is always taken as positive, independently of the slip values \(S_1\) and \(S_2\), which indicates that an additional inductive load is present in the rotor circuit of the main cascade, caused by the presence of the auxiliary cascade. The latter causes an increase in the reactive component of the control-winding current, which increases the calculated power of the FEMC.
The parameter \(R\) in the equivalent circuit may take both positive—
both positive and negative values depending on the values of the slip \(S_2\). The slip \(S_2\) has positive values in the mode of an asynchronous frequency converter (the field of winding \(W_4\) and the rotor rotate in opposite directions). The parameter \(R\) takes negative values, which indicates that the losses in the amplifier in this mode increase because of the additional stage.
The parameter \(R\) also takes negative values in the case when \(S_2>0\) (motor mode). This mode is valid when \(\omega_y P_1 < \omega(P_1-P_2)\).
The parameter \(R\) takes positive values in the case when \(S_2<0\) (generator mode). This mode occurs when \(\omega_y P_1 > \omega(P_1-P_2)\).
The external characteristic is one of the principal characteristics of the amplifier. The expression for it can be obtained in the form:
\[ \dot U_2 = \dot E_{01} - \dot I_2 Z_2, \tag{13} \]
where
\[ E_{01} = \frac{\dfrac{4}{\pi\sqrt{2}}\, U_y K_{u1}} {1+\dfrac{Z_{\mathrm{э}1}(Z_{01}+Z_3)}{Z_{01}+Z_3}} \quad \text{— the no-load emf of the amplifier,} \]
\[ Z_3 = \frac{r_3+r_4}{S_1} + j(X_{33}+X_{34}) + R + jX \]
\[ Z_{\mathrm{э}1} = r_1 + jX_{\mathrm{э}1} \]
\(K_{u1}\) is the voltage coefficient.
The experimental and calculated external characteristics are shown in Fig. 6. As is evident from the figure, there is sufficiently good agreement between calculation and experiment.
Fig. 6. External characteristic of the amplifier.
The additional stage operates in motor mode.
The amplification factor is one of the important indicators of the amplifier. The amplification factor is equal to the ratio of the output and input powers.
The output power, expressed through the load parameters, is
\[ P_2=\left(\frac{E_{01}}{Z_{c2}+Z_{\mathrm{H}}}\right)^2 Z_{\mathrm{H}}\cos\varphi_{\mathrm{H}}\,m_2 . \tag{14} \]
The control power required to create the magnetic flux and to compensate the reactive components of the rotor and the load is determined by the formula
\[ P_{\mathrm{y}}= \frac{ K_1\frac{4}{\pi\sqrt{2}}U_{\mathrm{y}}^2K_{\mathrm{u}1}(Z_{01}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+Z_{01}Z_3 }{ Z_1\left[(Z_{01}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+X_{01}X_3\right]+Z_{01}Z_3(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}}) }. \tag{15} \]
The amplification factor, taking (14) and (15) into account, is equal to
\[ K_{\mathrm{y}}= \frac{ m_2K_{\mathrm{u}1}Z_{\mathrm{H}}\cos\varphi_{\mathrm{H}} \left\{Z_{c1}\left[(Z_{01}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+Z_{03}Z_3\right]+\right. }{ K_1(Z_{c2}+Z_{\mathrm{H}})^2\left[(Z_{c2}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+Z_{01}Z_3\right. } \rightarrow \]
\[ \rightarrow \frac{ \left.+Z_{01}Z_3(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})\right\} }{ 1 }. \tag{16} \]
Analysis of expression (16) shows that the amplification factor is determined by the parameters of the amplifier windings, as well as by the nature and magnitude of the load.
The amplification factor can be represented in another form:
\[ K_{\mathrm{y}}= \frac{ m_3K_{\mathrm{u}1}R_3\left[Z_{01}\left\{(Z_{01}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+Z_{01}Z_3(Z_2+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})\right\}\right] }{ S_1K_1\left[(Z_{01}+Z_3)(Z_{c2}+Z_{\mathrm{H}}\sin\varphi_{\mathrm{H}})+Z_{01}Z_3\right]Z_3 }. \tag{17} \]
Expression (17) is obtained from the condition that the output power of the amplifier is equal to the rotor power:
\[ P_{\mathrm{out}}=P_3=\left(\frac{E_{01}}{Z_3}\right)^2\frac{R_3m_3}{S_1}. \tag{18} \]
As is evident from expression (17), the amplification factor also depends on the rotor parameters and on the slip \(S_1\).
A variable-frequency amplifier can also be made in a contact version (Fig. 7).
In the contact design of the amplifier, the equivalent circuit is simplified (Fig. 8), since \(Z_{c2}=\infty\), \(Z_{34}=0\). The system of equations describing the possible operating modes of the amplifier in the contact design takes the form:
\[ \dot U_1=\dot I_1(r_1+jX_{c1})+\dot I_{01}Z_{01}, \]
\[ \dot U_2=-\dot I_2(r_2+jX_{c2})-\dot I_{01}Z_{01}, \tag{19} \]
\[ 0=\dot I_3\left(\frac{r_3}{S_1}+jX_{c3}\right)+\dot I_{01}Z_{01}. \]
The expressions for the external characteristic (13) and the amplification factor (16) remain the same as in the noncontact design of the amplifier, with \(R=X=0\).
It should be noted that an electromechanical amplifier in the contact design can operate in motor mode (there is no torque on the shaft of the driving motor and winding \(W_2\) is open) and in re-
Fig. 7. Circuit diagram of a variable-frequency electric-machine amplifier in the contact version
Fig. 8. Equivalent circuit of the amplifier in the contact version
... mode of the converter (winding \(W_3\) is open-circuited). The system of equations for the motor mode takes the form:
\[ \dot U_1 = \dot I_1 (r_1 + jX_{\sigma 1}) + \dot I_{01} Z_{01}; \]
\[ 0 = \dot I_3 \left(-\frac{r_3}{S_1} + jX_{\sigma 3}\right) + \dot I_{01} Z_{01}. \tag{20} \]
For the converter mode, the following equations hold:
\[ \dot U_1 = \dot I_1 (r_1 + jX_{\sigma 1}) + \dot I_{01} Z_{01}, \]
\[ \dot U_2 = -\dot I_2 (r_2 + jX_{\sigma 2}) - \dot I_{01} Z_{01}. \tag{21} \]
Overall, the results of the work may be formulated in the following conclusions:
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In an amplifier with adjustable frequency, the CSS and the control winding can be represented as an \(m\)-phase inverter.
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Under these conditions, the basic equations have been obtained which make it possible, for contactless and contact versions, to construct equivalent circuits and to analyze all operating modes of the amplifier.
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Equations have been obtained for the external characteristic and the amplification coefficient, making it possible to construct and analyze their dependence on various parameters.