Problems of the General Theory of Electrical Machines as Applied to Brushless DC Electric-Machine Amplifiers (BEMAs)
Abstract Generated abstract
This paper applies the general theory of electrical machines to brushless DC electromechanical amplifiers and related fixed-frequency AC amplifier configurations. Using Lagrange-Maxwell equations, it treats the cascades of a combined machine as interacting stages and derives equations for electromotive forces, electric powers, electromechanical forces, and electromechanical powers for the relevant stator and rotor windings. The analysis specializes these relations to BEMA windings in steady-state operation, obtaining expressions for currents, induced electromotive forces, powers, and torques in the two amplifier stages. The paper argues that this framework provides the necessary quantities for analyzing operating modes of DC or AC BEMA machines, with feedback windings incorporated by the same method.
Full Text
Issues in the General Theory of Electric Machines as Applied to Brushless DC Electromechanical Amplifiers (BEMAs)
A. I. Skorospeshkin
(Recommended by the seminar of the Department of Electrical Machines and General Electrical Engineering)
The development of issues in the general theory of electric machines makes it possible to carry out a comparative assessment of various classes of electric machines and opens prospects for the development of new electric machines.
Combined electric machines of the converter and amplifier type constitute a specific class of machines, and their consideration from the standpoint of the general theory of electric machines is extremely necessary. A general approach to them, in our opinion, can be developed on the basis of Maxwell’s theory of electromagnetic phenomena and Lagrange’s equations [1, 2, 3]. In this case, the physical properties of the machines are taken directly into account.
When the conditions of combination are fulfilled, each cascade may be considered as a separate machine, with subsequent establishment of the connection between the individual cascades. The Lagrange–Maxwell equations make it possible to consider each of the windings separately and to write for it Lagrange equations of the second kind for electromechanical and electromotive forces. In general, in each cascade of a combined machine, for each of the windings, equations are written for electromotive forces and electric powers, and equations for electromechanical forces and electromechanical powers, taking into account the interaction of the stator and rotor windings.
In final form these equations are written as follows:
- Equations of e.m.f.
\[ e_k = i_k r_k + \frac{d(l_k i_k)}{dt} + \sum \frac{d(m_{ka} i_a)}{dt}. \tag{1} \]
- Equations of electric powers
\[ P_{\text{el}} = e_k i_k = i_k^2 r_k + i_k \frac{d(l_k i_k)}{dt} + i_k \sum \frac{d(m_{ka} i_a)}{dt}. \tag{2} \]
- Equations of electromechanical forces
\[ M_{\text{em}} = - \frac{1}{2} i_a i_v \frac{d m_{av}}{d\alpha}. \tag{3} \]
- Equations of electromechanical powers
\[ P_{\text{em}} = - \frac{1}{2} i_a i_v \frac{d m_{av}}{dt}. \tag{4} \]
Notation in the formulas:
\(e_k\) — generalized e.m.f.
\(M_{\text{em}}\) — generalized electromechanical force,
\(\alpha\) — geometric coordinate.
\(r_k, l_k, m_{ab}, i_a, i_b, i_k\) — active resistance, inductances, mutual inductances, and currents of the windings under consideration.
In nonsalient-pole machines the inductances and mutual inductances between phases may be regarded as constant, and only the mutual inductances between the stator and rotor windings vary.
From the system of equations (1)—(4) one can determine the currents, powers, and torques and thus establish the relationship between the individual stages.
Let us determine the necessary quantities as applied to a brushless DC electromechanical amplifier (BEMA) and to an AC BEMA with fixed frequency, whose circuits and operating principle are explained in Fig. 1.
In the figure the following notation is adopted:
Fig. 1. Circuit of a DC BEMA.
A — control winding. Located on the stator and supplied with direct current;
B, C — rotor windings of the first and second amplifier stages;
D — output winding. Located on the stator.
\(B_2\) and \(B_1\) — rectifier units in the rotor and on the stator. In the absence of \(B_2\), an alternating current of fixed frequency is obtained at the output.
The numbers 1, 2, 3 indicate the order of phase sequence in the three-row windings of the rotor and stator.
\(\omega_p\) — angular velocity of rotation of the rotor,
\(\omega_1; \omega_2\) — angular velocities of rotation of the fields in windings B and D.
The arrows indicate the directions of rotation of the rotor and of the fields in the I and II stages.
\(P_1\) and \(P_2\) — numbers of pole pairs in the I and II stages.
Let us write the emf balance equations for windings A, B, C, and D. In the most general form they are written by the following expressions.
For winding A,
\[ U_A = i_A r_A + \frac{d(L_A i_A)}{dt} + \frac{d(m_{1AB} i_{1B})}{dt} + \frac{d(m_{2AB} i_{2B})}{dt} + \frac{d(m_{3AB} i_{3B})}{dt}. \tag{5} \]
For the first phase of winding B
\[ 0=i_{1B}r_B+\frac{d(L_B i_{1B})}{dt} -\frac{d(M_B i_{2B})}{dt} -\frac{d(M_B i_{3B})}{dt} +\frac{d(m_{1AB}i_A)}{dt} = i_{1B}r_B+1.5L_B\frac{di_{1B}}{dt} +i_A\frac{dm_{1AB}}{dt} +m_{1AB}\frac{di_A}{dt}. \tag{6} \]
For winding C
\[ U_C=i_Cr_C+\frac{d(L_Ci_C)}{dt} +\frac{d(m_{C1D}i_{1D})}{dt} +\frac{d(m_{C2D}i_{2D})}{dt} +\frac{d(m_{C3D}i_{3D})}{dt}. \tag{7} \]
For winding D
\[ U_{1D}=i_{1D}r_D+\frac{d(L_Di_{1D})}{dt} -\frac{d(M_Di_{2D})}{dt} -\frac{d(M_Di_{3D})}{dt} +\frac{d(m_{1DC}i_C)}{dt} = i_{1D}r_D+1.5L_D\frac{di_{1D}}{dt} +i_C\frac{dm_{1CD}}{dt} +m_{1DC}\frac{di_C}{dt}. \tag{8} \]
Notation in the formulas:
\(i_A,i_B,i_C,i_{1D}\) — instantaneous values of the currents in windings A, B, C, and D.
\(r_A,r_B,r_C,r_D,L_A,L_B,L_C,L_D\), — active resistances and inductances of windings A, B, C, and D.
\(M_D, M_B\) — mutual inductances between phases of windings B and D.
The letter \(m\), with the corresponding subscripts, denotes variable mutual inductances between the stator and rotor windings.
\(U_A\) — the constant voltage applied to the control winding.
\(U_{C1}, U_D\) — instantaneous values of the voltages in winding C and in a phase of winding D.
According to [4], we take \(M_B=0.5L_B,\; M_D=0.5L_D\).
Equations (5)—(8) are written for instantaneous values and are valid for any steady-state or transient regime. For the time being we shall consider only the steady-state operating regime and shall take into account only the fundamental harmonics.
In this case, the variations of the currents and mutual inductances will be determined by the equations:
\[ \begin{aligned} i_{1B}&=I_{mB}\sin(\omega_1t+\alpha_1); & m_{1AB}&=M_{AB}\sin\omega_1t;\\ i_{2B}&=I_{mB}\sin\left(\omega_1t+\alpha_1+\frac{2\pi}{3}\right); & m_{2AB}&=M_{AB}\sin\left(\omega_1t+\frac{2\pi}{3}\right);\\ i_{3B}&=I_{mB}\sin\left(\omega_1t+\alpha_1-\frac{2\pi}{3}\right); & m_{3AB}&=M_{AB}\sin\left(\omega_1t-\frac{2\pi}{3}\right); \end{aligned} \tag{9} \]
\[ \begin{aligned} m_{DC1}&=M_{CD}\sin\omega_2t; & i_{1D}&=I_{mD}\sin(\omega_2t+\alpha_2);\\ m_{C2D}&=M_{CD}\sin\left(\omega_2t+\frac{2\pi}{3}\right); & i_{2D}&=I_{mD}\sin\left(\omega_2t+\alpha_2-\frac{2\pi}{3}\right);\\ m_{C3D}&=M_{CD}\sin\left(\omega_2t-\frac{2\pi}{3}\right); & i_{3D}&=I_{mD}\sin\left(\omega_2t+\alpha_2+\frac{2\pi}{3}\right); \end{aligned} \]
\[ \begin{aligned} m_{C1D}&=m_{1DC},\\ m_{C2D}&=m_{2DC},\\ m_{C3D}&=m_{3DC}. \end{aligned} \]
For the steady-state regime, (5) takes the form:
\[ U_A = I_A r_A, \qquad I_A = \frac{U_A}{r_A}. \tag{10} \]
Equation (6)
\[ 0 = I_{mB}\sin(\omega_1 t+\alpha_1)r_B +1.5L_B I_{mB}\omega_1\cos(\omega_1t+\alpha_1) + I_A\omega_1 M_{AB}\cos\omega_1 t . \tag{11} \]
Whence
\[ I_{mB}= \frac{-I_A\omega M_{AB}\cos\omega_1 t} {\sin(\omega_1t+\alpha_1)r_B+1.5L_B\omega_1\cos(\omega_1t+\alpha_1)} . \tag{12} \]
Equation (7)
\[ U_C = I_C r_C \quad \text{and} \quad I_C=\frac{U_C}{r_C}, \tag{13} \]
where \(U_C\) and \(I_C\) are the values of the rectified voltage and current.
Equation (8)
\[ U_{mD}\sin\omega_2t = I_{mD}\left[\sin(\omega_2t+\alpha_2)r_D +1.5L_D\omega_2\cos(\omega_2t+\alpha_2)\right] + I_C\omega_2 M_{CD}\cos\omega_2t, \tag{14} \]
whence
\[ I_{mD}= \frac{U_{mD}\sin\omega_2t} {\sin(\omega_2t+\alpha_2)r_D+1.5L_D\omega^2\cos(\omega_2t+\alpha_2)} - \frac{I_C\omega_2M_{CD}\cos\omega_2t} {\sin(\omega_2t+\alpha_2)r_D+1.5L_D\omega_2\cos(\omega_2t+\alpha_2)} . \tag{15} \]
When rectifier unit \(B_2\) is present, the rectified output current and voltage are taken into account.
The required e.m.f.’s for windings B and D are
\[ e_{1B}=I_A\omega_1M_{AB}\cos\omega_1t, \]
\[ e_{1D}=I_C\omega_2M_{CD}\cos\omega_2t, \tag{16} \]
The electrical powers of windings A and C and the powers falling per phase of windings B and D are
\[ P_{\text{el}}=I_A^2 r, \]
\[ P_{\text{el}\,1B}=i_{1B}e_{1B} =I_{mB}\sin(\omega_1t+\alpha_1)I_A\omega_1M_{AB}\cos\omega_1t. \tag{17} \]
\[ P_{\text{el}\,C}=I_C^2 r_C; \]
\[ P_{\text{el}\,1D}=i_{1D}e_{1D} =I_{mD}\sin(\omega_2t+\alpha_2)\cdot I_C\omega_2M_{Cd}\cos\omega_2t. \]
Electromechanical powers per phase for stages I and II are
\[ P_{\text{em}.1AB}= -\frac{1}{2}I_AI_{mB}M_{AB}\,\omega_1\sin(\omega_1t+\alpha_1)\cos\omega_1t; \]
\[ P_{\text{em}.C1D}= -\frac{1}{2}I_CI_{mD}M_{CD}\,\omega_2\sin(\omega_2t+\alpha_2)\cos\omega_2t;; \tag{18} \]
Rotating torques for stages I and II are
\[ M_{\text{em}.1AB}= -\frac{1}{2}I_AI_{mB}M_{AB}\sin(\omega_1t+\alpha_1)\cos\omega_1t, \]
\[ M_{\text{em}.C1D}= -\frac{1}{2}I_CI_{mD}M_{CD}\sin(\omega_2t+\alpha_2)\cos\omega_2t \tag{19} \]
Thus, on the basis of the general Lagrange–Maxwell equations, all the necessary data of a combined electrical machine of the BEMA type, of either direct or alternating current, are determined. From them the necessary analysis of all operating modes of the amplifier can be carried out.
In amplifiers, as a rule, feedback windings are provided to compensate the armature reaction. The currents, powers, and torques produced by them are taken into account in an analogous manner.
REFERENCES
-
A. G. Kaplyanskii. Introduction to the General Theory of Electrical Machines, GEI, 1941.
-
V. V. Bazilevich. On the question of the general theory of electrical machines, Elektrichestvo, No. 21, 1930.
-
I. M. Sadovskii. Electrodynamics of commutator electrical machines. Elektrichestvo, No. 4, 1949.
-
G. N. Petrov. Electrical Machines, GEI, 1963.
-
B. Atkins. General Theory of Electrical Machines, GEI, 1960.
-
M. I. Alyab’ev. General Theory of Shipboard Electrical Machines, Sudostroenie Publishing House, 1965.