Some Issues in the Design of a Variable-Frequency Electric Machine Amplifier
Abstract Generated abstract
This paper examines design problems for the electric-machine part of a contactless variable-frequency electric-machine amplifier with combined main and auxiliary stages in a single magnetic core. It derives relationships for distributing magnetic induction and linear loading between the stages, analyzes how this distribution affects output power, and obtains an optimum condition for the loading coefficients, indicating that the pole-pair ratio of the auxiliary and main stages should be minimized where possible. The paper also formulates calculations for the control winding, control power, and amplification factor, showing their dependence on rotor, load, and control-winding linear-load parameters.
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SOME ISSUES IN THE DESIGN OF A VARIABLE-FREQUENCY ELECTRIC MACHINE AMPLIFIER
Sh. S. Roiz, A. I. Skorospelkin
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
A contactless variable-frequency electric-machine amplifier is a combination of a controlled semiconductor commutator and an electric-machine part [1]. When designing the electric-machine part, it is necessary to take into account the circuit of the CSC, which is directly connected with the control winding.
The electric-machine part of a contactless variable-frequency amplifier is a two-stage electric machine. To reduce the weight and dimensions of the amplifier, its stages are made combined in a single magnetic core.
In the present article, issues related to the choice of electromagnetic loadings and the calculation of the control winding are considered.
The distribution of electromagnetic loadings between the combined stages is characterized by the coefficients
\[ C_{\mathrm{B}}=\frac{B_{\delta 1}}{\sum B_{\delta}}, \qquad C_{\mathrm{A}}=\frac{A_1}{\sum A}, \tag{1} \]
where
\(B_{\delta}\) and \(A_1\) — values of the magnetic induction and the linear loading of the main stage,
\(\sum B_{\delta}\) and \(\sum A\) — values of the total magnetic induction and total linear loading of the amplifier.
Studies [2, 3] have established that the value of the total linear loading is selected within the limits corresponding to asynchronous motors of the same power and construction.
The magnetic induction in the air gap of the amplifier is selected lower, taking into account forcing of the output voltage.
Proceeding from the equality of the number of phases and currents of the rotor windings, the following relation follows:
\[ \frac{A_3}{A_4}=\frac{W_3}{W_4}, \tag{2} \]
and the ratio of the e.m.f. of these same windings is written in the form:
\[ \frac{E_3}{E_4}=K_E=\frac{W_3 K_{w3} P_1 B_{\delta 1}}{W_4 K_{w4} P_2 B_{\delta 2}} . \tag{3} \]
Solving (1), (2), (3), we obtain
\[ C_A=\frac{1-C_B}{1-C_B(1-K_1)}, \tag{4} \]
where
\[ K_1=\frac{K_E\cdot K_{w1}\cdot P_2}{K_{w3}\cdot P_1}. \tag{4a} \]
\(P_1\) and \(P_2\) are the numbers of poles of the main and auxiliary stages. Using (1) and (4), and adopting definite values of \(\sum A\), \(\sum B_\delta\), and \(C_B\), one can determine the electromagnetic loadings of the amplifier stages.
The choice of the coefficient \(C_B\) is not arbitrary. Let us consider the influence of the coefficient \(C_B\) on one of the principal parameters—the output power of the amplifier. For known dimensions of the amplifier, the output power is equal to
\[ P_{\text{out}}=m_2U_2I_2\cos\varphi_n. \tag{5} \]
The reactive component of the load current is supplied by the control winding. When the amplifier operates in the nominal mode, we assume that all the rotor power is transmitted to the amplifier output. Using the vector diagram (Fig. 1), one can write that the active components of the rotor and load currents are equal:
\[ I_2\cos\varphi_n=I_3'\cos\varphi_p=I_3K_i\cos\varphi_p. \tag{6} \]
Fig. 1. Vector diagram of the amplifier
Expressing the value of the current \(I_2\) through the linear load \(A_3\), we obtain
\[ I_2=\frac{A_3D}{2m_2W_2}\left(\frac{\cos\varphi_p}{\cos\varphi_n}\right). \tag{7} \]
Taking (5), (6), and (7) into account, the expression for the output power takes the form
\[ P_{\text{out}}= \frac{4.44 f_1 \alpha \tau^2 D^2 l \sum A \sum B_\delta C_A C_B K_{w2}} {4P_1K_E}\cos\varphi_p. \tag{8} \]
For analysis of the output power as a function of the coefficient \(C_{\mathrm{B}}\), let us take the derivative \(\dfrac{dP_{\text{out}}}{dC_{\mathrm{B}}}\) and, setting it equal to zero, obtain
\[ C_{A0}=C_{B0}=\frac{1-K_1}{1-\sqrt{K_1}} . \tag{9} \]
Note: We assume that the parameters \(\Sigma A\), \(\Sigma B\delta\), \(D\), \(l\), \(i_1\) remain unchanged.
Thus, the optimal values of the coefficients \(C_{B0}\) and \(C_{A0}\) are obtained from expression (9). As is evident from (9), different values of the coefficient \(K_1\) correspond to their own optimal values of \(C_{A0}\) and \(C_{B0}\).
Taking into account that \(K_1\) is determined by expression (4, a), one may conclude that, when combining the stages of a variable-frequency amplifier in a single magnetic core, the ratio of the numbers of pole pairs \(\dfrac{P_2}{P_1}\) must be the smallest possible.
Figure 2 presents the curves \(P_{\text{out}}=f(C_{\mathrm{B}})\) and \(C_A=f(C_{\mathrm{B}})\). It follows from the figure that the optimal value of the coefficient \(C_{B0}\) corresponds to the maximum output power. These curves are plotted for the values \(K_1=0.27\), \(P_1=4\), \(P_2=1\). The value of \(P_{\text{out}}\) is taken in relative units. In design, great importance is attached to the calculation of the control winding, which is directly associated with the power elements of the VFA.
In the graph: vertical axis \(P_{\text{out}},\, C_A\); horizontal axis \(C_{\mathrm{B}}\); curves \(P_{\text{out}}=f(C_{\mathrm{B}})\) and \(C_A=f(C_{\mathrm{B}})\); marked values \(C_{A0}\) and \(C_{B0}\).
Fig. 2. Dependence \(P_{\text{out}}=f(C_{\mathrm{B}})\) and
\(C_A=f(C_{\mathrm{B}})\)
The magnetizing force of the control winding according to the vector diagram is equal to
\[ \bar F_y=\bar F_{01}+\bar F_2\sin\varphi_1+\bar F_3\sin\varphi_p, \tag{10} \]
where \(\bar F_{01}\) is the m.m.f. of the magnetizing circuit of the main stage, and \(\bar F_2\) and \(\bar F_3\) are the m.m.f. of windings \(W_2\) and \(W_3\).
The equivalent alternating current flowing through each phase of the control winding can be determined from the formula
\[ I_{y\sim}=\frac{F_y}{0.9m_1W_1K_{w1}} . \tag{11} \]
The current drawn from the dc network is equal to
\[ I_y=K_I I_{y\sim}, \tag{12} \]
where \(K_I\) is the current coefficient taking into account the inversion circuit.
The amplification factor is one of the most important indicators of any amplifier. The amplification factor is equal to the ratio of output power to input power.
The control power is equal to
\[ P_y=I_y\cdot U_y . \tag{13} \]
The voltage \(U_y\) can be determined from the formula
\[ U_y=\frac{4}{\pi\sqrt{2}}\cdot 4.44f_1\cdot B_{\delta 1}al\frac{\tau D}{2p_1}W_1K_{w1}. \tag{14} \]
The value of the control current through the linear load \(A_y\) is equal to:
\[ I_{y\sim}=K_I\frac{A_y\tau D}{2m_1W_1}, \tag{15} \]
where \(A_y=A_{01}+A_2\sin\varphi_n+A_3\sin\varphi_p\) is the linear load of the control winding.
Taking (14) and (15) into account, the control power takes the form:
\[ P_y=K_I\,4.44f_1aB_\delta lK_{w1} \frac{A_y\tau^2D^2}{\pi\sqrt{2}\,m_1P_1}. \tag{16} \]
Taking (8) and (16) into account, the amplification factor is determined by the expression:
\[ K_y= \frac{A_3}{A_{01}+A_2\sin\varphi_n+A_3\sin\varphi_p} \cdot \frac{K_IK_{w2}}{K_eK_{w1}\dfrac{4}{\pi\sqrt{2}}} \cdot \cos\varphi_p . \tag{17} \]
Analysis of expression (17) shows that the amplification factor is determined by the parameters of the rotor, the load, and also by the linear loads of the control windings.
As a result of the work performed, the following conclusions may be formulated:
- Recommendations are given for the selection of the electromagnetic loadings of the amplifier stages.
- Expressions have been obtained for calculating the control winding and the amplification factor.
REFERENCES
- Sh. S. Roiz, A. I. Skorospeshkin. Theoretical study of a variable-frequency electric-machine amplifier taking into account a controlled semiconductor commutator. Izvestiya TPI, in the present collection.
- V. S. Novokshonov. Study of a brushless frequency converter. Dissertation, Tomsk, 1960.
- M. L. Kostyrev, A. I. Skorospeshkin. On the design of a contactless ac EMA. Izvestiya TPI, vol. 190, 1968.