Issues in the Theory of a Brushless AC Electric-Machine Amplifier
Abstract Generated abstract
This paper develops a transient theory for a brushless AC electromechanical amplifier consisting of an inverted synchronous generator and an asynchronous frequency converter in a common magnetic core. Using Park, Gorev idealization, symmetry assumptions, and transformations from three-phase variables to two-phase rotor-axis and complex coordinates, it derives linear equations for symmetrical operating regimes and applies them to a no-load step change of the control voltage. The analysis shows that, neglecting a rapidly decaying transformer component, the amplifier behaves with respect to the control signal as an aperiodic element whose time constant is determined by the field circuit parameters and mutual coupling to the rotor circuit. Experimental results for a 300 W, 50 Hz sample are reported to agree with calculated characteristics and to confirm the predicted time constant and response behavior under load.
Full Text
PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV
Volume 190 1968
QUESTIONS IN THE THEORY OF A CONTACTLESS AC ELECTROMECHANICAL AMPLIFIER
M. L. KOSTYREV, A. I. SKOROSPEШKIN
(Recommended by the seminar of the departments of electrical machines
and general electrical engineering)
With the development of automatic-control systems operating on alternating current, a need has arisen for power amplifier-converters. As one such amplifier, a contactless electromechanical amplifier (BEMU) with an output at elevated frequency has comparatively recently been proposed [1, 2]. The amplifier (Fig. 1) consists of an inverted, non-salient-pole synchronous generator and an asynchronous frequency converter, combined in a single magnetic core and rotated by a drive motor. To obtain high gain, fast response, and overload capacity, a compensating winding $W_k$, fed from a phase-compounding circuit, is used in the BEMU [2].
Fig. 1. BEMU circuit
The amplifier usually operates in dynamic regimes with a varying load and control signals. In this connection it is necessary to develop a theory of BEMU operation in transient regimes. The present article is devoted to these questions.
In the analysis we consider the machine idealized according to Park–Gorev. In addition, we assume that the rotor speed is constant, there is no magnetic coupling between the stages, and the machine load is ...
symmetric. We consider the energetically most expedient mode corresponding to the coordinated rotation of the field and the rotor of the second cascade.
Initial equations
Starting from the general theory of an EMVC [3], for instantaneous values of currents and voltages in matrix form, one may write the following equations in the intrinsic axes of the windings:
\[ \begin{gathered} \left\| U^{S}_{fkabc}{}^{R1} \right\| = \left\| R^{S}_{fkabc}{}^{R1} \right\| \cdot \left\| i^{S}_{fkabc}{}^{R1} \right\| + p\left\| L^{S}_{fkabc}{}^{R1} \right\| \cdot \left\| i^{S}_{fkabc}{}^{R1} \right\| , \\ \left\| U^{s}_{abc\,abc}{}^{R2} \right\| = \left\| R^{s}_{abc\,abc}{}^{R2} \right\| \cdot \left\| i^{s}_{abc\,abc}{}^{R2} \right\| + p\left\| L^{s}_{abc\,abc}{}^{R2} \right\| \cdot \left\| i^{s}_{abc\,abc}{}^{R2} \right\| , \\ \left\| U^{R1}_{abc} \right\| = - \left\| U^{R2}_{acb} \right\| , \qquad \left\| i^{R1}_{abc} \right\| = \left\| i^{R2}_{acb} \right\| . \end{gathered} \tag{1} \]
Voltage and current matrices
\[ \left\| U^{S}_{fkabc}{}^{R1} \right\| = \left\| \begin{array}{c} U^{S}_{f} \\ U^{S}_{k} \\ U^{R1}_{a} \\ U^{R1}_{b} \\ U^{R1}_{c} \end{array} \right\|, \qquad \left\| U^{s}_{abc\,abc}{}^{R2} \right\| = \left\| \begin{array}{c} U^{s}_{a} \\ U^{s}_{b} \\ U^{s}_{c} \\ U^{R2}_{a} \\ U^{R2}_{b} \\ U^{R2}_{c} \end{array} \right\| \]
\[ \left\| i^{s}_{fk\,abc}{}^{R1} \right\| = \left\| \begin{array}{c} i^{s}_{f} \\ i^{s}_{k} \\ i^{R1}_{a} \\ i^{R1}_{b} \\ i^{R1}_{c} \end{array} \right\|, \qquad \left\| i^{s}_{abc\,abc}{}^{R2} \right\| = \left\| \begin{array}{c} i^{s}_{a} \\ i^{s}_{b} \\ i^{s}_{c} \\ i^{R2}_{a} \\ i^{R2}_{b} \\ i^{R2}_{c} \end{array} \right\| \]
Active-resistance matrices
\[ \left\| R^{S}_{fk\,abc}{}^{R1} \right\| = \left\| \begin{array}{cc} \begin{array}{cc} R_f & 0 \\ 0 & R_k \end{array} & 0 \\ 0 & R_{R1}\cdot \left\| E \right\| \end{array} \right\| \]
\[ \left\| R^{s}_{abc\,abc}{}^{R2} \right\| = \left\| \begin{array}{cc} R_s\left\| E \right\| & 0 \\ 0 & R_{R2}\left\| E \right\| \end{array} \right\| \]
\[ \left\| E \right\| \text{ is the identity matrix,} \]
\(R_f, R_k, R_s, R_{R1}, R_{R2}\) are the active resistances of the field, control, compensation, and phase windings of the stator and rotor.
Matrices of inductive reactances
\[ \left\| L^{s}_{fk\,abc}{}^{R1}\right\|= \begin{vmatrix} \begin{array}{c|c||c} L_f & M_{fk} & \left\|1^{s\,R1}_{fk\,abc}\right\|\\ \hline M_{fk} & L_k & \\ \hline \left\|1^{R1}_{abc}{}^{s}_{fk}\right\| & & \left\|1^{R1}_{abc}\right\| \end{array} \end{vmatrix} \]
\[ \left\| L^{s}_{abc}{}^{R2}_{abc}\right\|= \begin{vmatrix} \left\|1^{s}_{abc}\right\| & \left\|1^{s\,R2}_{abc}\right\|\\ \hline \left\|1^{R2}_{abc}{}^{s}\right\| & \left\|1^{R2}_{abc}\right\| \end{vmatrix} \]
Submatrices of inductive reactances
\[ \left\|1^{R1}_{abc}\right\|=L^{R1}_{a}\left\|A_1\right\|,\qquad \left\|1^{R2}_{abc}\right\|=L^{R2}_{a}\left\|A_1\right\|, \]
\[ \left\|1^{s}_{abc}\right\|=L^{s}_{a}\left\|A_1\right\|,\qquad \left\|1^{R1}_{abc}{}^{s}_{fk}\right\|=\left\|1^{s\,R1}_{fk\,abc}\right\|^{t}, \]
\[ \left\|1^{R2}_{abc}{}^{s}\right\|=\left\|1^{s\,R2}_{abc}\right\|^{t}, \]
\[ \left\|A_1\right\|= \begin{vmatrix} 1 & \cos\dfrac{2\pi}{3} & \cos\dfrac{4\pi}{3}\\ \hline \cos\dfrac{2\pi}{3} & 1 & \cos\dfrac{2\pi}{3}\\ \hline \cos\dfrac{4\pi}{3} & \cos\dfrac{2\pi}{3} & 1 \end{vmatrix} \]
\[ \left\|1^{s\,R1}_{fk\,abc}\right\|= \begin{vmatrix} M^{sR1}_{fa}\cos\theta_1 & M^{sR1}_{fa}\cos\left(\theta_1+\dfrac{2\pi}{3}\right) & M^{sR1}_{fa}\cos\left(\theta_1+\dfrac{4\pi}{3}\right) \\ \hline M^{sR1}_{ka}\cos\theta_1 & M^{sR1}_{ka}\cos\left(\theta_1+\dfrac{2\pi}{3}\right) & M^{sR1}_{ka}\cos\left(\theta_1+\dfrac{4\pi}{3}\right) \end{vmatrix} \]
\[ \left\|1^{sR2}_{abc}\right\|= M^{sR2}_{a}\cdot \begin{vmatrix} \cos\theta_2 & \cos\left(\theta_2-\dfrac{2\pi}{3}\right) & \cos\left(\theta_2+\dfrac{4\pi}{3}\right) \\ \hline \cos\left(\theta_2-\dfrac{2\pi}{3}\right) & \cos\theta_2 & \cos\left(\theta_2+\dfrac{2\pi}{3}\right) \\ \hline \cos\left(\theta_2-\dfrac{4\pi}{3}\right) & \cos\left(\theta_2-\dfrac{2\pi}{3}\right) & \cos\theta_2 \end{vmatrix} \]
\(L_f,\ L_k,\ L^{R1}_{a},\ L^{R2}_{a},\ L^{s}_{a}\) are the total inductances, respectively, of the windings \(W_f,\ W_k\) and of the phases of windings \(W_{R1},\ W_{R2},\ W_s\).
\(M_{fk},\ M^{sR1}_{fa},\ M^{sR1}_{ka},\ M^{sR2}_{a}\) are the mutual inductances of the phases of the corresponding windings when their axes coincide.
Angles between the axes of the windings
\[ \theta_1 = p_1 \Omega t + \theta_{01} = \omega t + \theta_{01}, \]
\[ \theta_2 = p_2 \Omega t + \theta_{02} = k \omega t + \theta_{02}, \]
where \(P_1, P_2\) are the numbers of pole pairs of the stages,
\[ k = \frac{p_2}{p_1}. \]
\(\Omega\) is the mechanical speed of the rotor,
\(\omega = p_1 \Omega\) is the current frequency in the rotor of the BEMU,
\(\Theta_{01}, \Theta_{02}\) are the initial angles.
Coordinate Transformation
The differential equations (1) contain periodic coefficients. To solve these equations, we choose new coordinate axes so as to obtain a system of linear equations. For this purpose we transform the machine to an equivalent two-phase one (variables \(O, \alpha, \beta\)) and choose the coordinate axes \(d—q\) on the rotor (Fig. 2). The transformations are carried out according to the well-known formulas:
\[ \|U\|=\|A\|\cdot\|U'\| \qquad \|i\|=\|A\|\cdot\|i'\| \]
\[ \|Z'\|=\|A\|^{-1}\cdot\|Z\|\cdot\|A\|, \tag{1} \]
where \(\|U'\|\), \(\|i'\|\), \(\|Z'\|\) are the matrices of currents, voltages, and resistances in the new coordinate system,
\[ \|A\| \]
is the transformation matrix.
Fig. 2. Coordinate axes of the BEMU
In passing from a three-phase machine to an equivalent two-phase one, we use the transformation matrix
\[ \|A_2\|=\sqrt{\frac{2}{3}}\cdot \begin{Vmatrix} \dfrac{\sqrt{2}}{2} & 1 & 0 \\[6pt] \dfrac{\sqrt{2}}{2} & -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \\[6pt] \dfrac{\sqrt{2}}{2} & -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{Vmatrix} \]
The factor \(\sqrt{\dfrac{2}{3}}\), used in the transformation, ensures the invariance of the transformation power and the reversibility of the mutual inductions between the stator and rotor windings [5]. Since zero-sequence currents
sequences in the windings of the circuit under investigation do not flow, then the corresponding equations are omitted.
The coordinate axis \(d\) (Fig. 2) coincides with the axis of phase \(a\) of the rotor winding, and also with the axis \(\alpha\) of the equivalent two-phase winding. The axis \(q\) leads the axis \(d\) by 90 electrical degrees and coincides with the axis \(\beta\).
With this choice of axes, the matrices for transforming the variables \(\alpha,\beta\) into the variables \(d,q\) for the rotor windings are
\[ \left\| A_3 \right\|= \begin{array}{c|cc} & d & q\\ \hline \alpha & 1 & 0\\ \beta & 0 & 1 \end{array} \]
for the stator windings
\[ \left\| A_4 \right\|= \begin{array}{c|cc} & d & q\\ \hline \alpha & \cos\theta & -\sin\theta\\ \beta & \sin\theta & \cos\theta \end{array} \]
From the variables \(d\)—\(q\) we pass to complex quantities, using the relations
\[ \dot U = U_d + jU_q \qquad \dot I = I_d + jI_q \]
\[ \dot U^{*} = U_d - jU_q \qquad \dot I^{*} = I_d - jI_q \]
After the transformations we obtain the equations of the EMM in the rotor axes:
\[ \begin{gathered} \begin{bmatrix} \dot U_f\\ \dot U_k\\ 0\\ \dot U_s^{*} \end{bmatrix} = \begin{bmatrix} R_f+L_f(p+j\omega) & M_{fk}(p+j\omega)\\ M_{fk}(p+j\omega) & R_k+L_k(p+j\omega)\\ pM_{fR} & pM_{kR}\\ 0 & 0 \end{bmatrix} \;\longrightarrow \\[1em] \longleftarrow\; \begin{bmatrix} M_{fR}(p+j\omega) & 0\\ M_{kR}(p+j\omega) & 0\\ R_R+pL_R & pM_{sR}\\ M_{sR}(p-jk\omega) & R_s+L_s(p-jk\omega) \end{bmatrix} \cdot \begin{bmatrix} \dot i_f\\ \dot i_k\\ I_R\\ \dot I_s^{*} \end{bmatrix} \end{gathered} \tag{3} \]
Mutual inductances of the transformed windings
\[ M_{fR}=\sqrt{\frac{3}{2}}\,M_f^{s}{}_{a}^{r} \]
\[ M_{kR}=\sqrt{\frac{3}{2}}\,M_k^{s}{}_{a}^{k} \]
\[ M_{sR}=\frac{3}{2}\,M_a^{sR} \]
Self-inductances of the transformed windings
\[ L_R=\frac{3}{2}\left(L_a^{R1}+L_a^{R2}\right), \qquad L_s=\frac{3}{2}L_a^{s} \]
Active resistance of the rotor circuit
\[ R_R = R_{R1} + R_{R2}. \]
Let us pass to a system of relative units, referring the windings to the rotor circuit and choosing as base current, voltage, resistance, and frequency, respectively, the nominal values of the reduced output winding \(W_S\) and the frequency of the current in the rotor.
In the system of relative units
\[ p\theta_1=\omega=1, \]
\[ p\theta_2=k\omega=S-1. \]
The slip of the second cascade is
\[ S=\frac{p_1+p_2}{p_1}. \]
The equations of the CEM in the system of relative units:
\[ \left| \begin{array}{c} \dot U_f\\ \dot U_k\\ 0\\ \dot U_s^{*} \end{array} \right| = \left| \begin{array}{cc} \dfrac{r_f+x_f(p+j)}{x_{m1}(p+j)} & \dfrac{x_{m1}(p+j)}{r_k+x_k(p+j)}\\[1.0em] \dfrac{p x_{m1}}{} & \dfrac{p x_{m1}}{}\\[1.0em] 0 & 0 \end{array} \right| \to \]
\[ \leftarrow \left| \begin{array}{cc} x_{m1}(p+j) & 0\\[0.7em] \dfrac{x_{m1}(p+j)}{r_f+p x_f} & 0\\[1.0em] x_{m2}\,[p+j(1-S)] & \dfrac{p x_{m2}}{r_s+x_s[p+j(1-S)]} \end{array} \right| \cdot \left| \begin{array}{c} \dot i_f\\ \dot i_k\\ \dot i_R\\ \dot i_s^{*} \end{array} \right| \tag{4} \]
The linear equations (4) describe the CEM in any symmetrical regime.
Solution of the CEM equations
In the steady-state regime the frequencies of the transformed circuits of the CEM are
\[ \omega_f=\omega_k=0-p_1\Omega=-p_1\Omega \]
\[ \omega_2=-p_1\Omega=-p_1\Omega-0 \]
\[ \omega_s=(p_1+p_2)\Omega-p_2\Omega=p_1\Omega. \]
Therefore, in the system of relative units for the steady-state regime
\[ p\dot i_f(t)=p\dot i_f e^{-jt}=-j\dot i_f \qquad p\dot i_R(t)=p\dot i_R e^{-jt}=-j\dot i_R \]
\[ p\dot i_k(t)=p\dot i_k e^{-jt}=-j\dot i_k \qquad p\dot i_s^{*}(t)=p\dot i_s^{*}e^{-jt}=-j\dot i_s^{*} \]
Thus, in order to pass to the CEM equations in the steady-state regime it is sufficient in equations (4) to put \(P=-j\).
Let us consider the simplest transient process—the stepwise increase of the control voltage of the CEM without feedback in the no-load regime with zero initial conditions:
\[ U_f=0 \quad (t=0), \qquad \theta_{01}=0, \qquad \theta_{02}=0. \]
Let us find the roots of the corresponding characteristic equation
\[ [r_f+x_f(p+j)](r_f+p x_f)-p(p+j)x_{m1}^{2}=0. \]
Introducing the notation
\[ T_f=\frac{x_f}{r_f}, \qquad T_r=\frac{x_r}{r_z}, \qquad T=\left(x_f-\frac{x_{m1}^{2}}{x_r}\right)\frac{1}{r_f}, \]
\[ a_1=\frac{T_f}{T\cdot T_r}, \qquad a_2=\frac{1}{T}, \]
we obtain the equation in the form
\[ p^2+p(a_1+a_2+j)+ja_1+\frac{1}{T_r\cdot T_f}=0. \]
Since, in the adopted system of units for real BEMUs,
\[ a_1<1, \qquad a_2<1, \qquad \frac{1}{T_r\cdot T_f}<1, \]
the roots of the characteristic equation are
\[ p_1\approx -a_1, \qquad p_2\approx -a_2-j. \]
For a step action, the Carson transform of the control-voltage phasor is
\[ U_f(t)=U_f\cdot e^{-jt}\doteq U_f\frac{p}{p+j}. \]
From equation (4), the voltage at the BEMU output is
\[ \overset{*}{U}_s(p)=x_{m2}\bigl[p+j(1-S)\bigr]\overset{*}{I}_r(p) =-\frac{U_s\cdot a_2p^2\bigl[p+j(1-S)\bigr]} {(p+a_1)(p+j)(p+a_2+j)}, \]
where
\[ U_s=\frac{U_f x_{m1}x_{m2}}{r_f x_r}. \]
Passing from the transform to the original and neglecting quantities of second order of smallness, we obtain
\[ \overset{*}{U}_s(t)=U_s(A\cos t+B\sin t)+jU_s(B\cos t-A\sin t), \]
where
\[ A=a_2(S+1)e^{-a_2t}+S\cdot a_1, \]
\[ B=S(1-e^{-a_2t}). \]
The inverse transformations are carried out on the basis of the following expressions:
\[ \overset{*}{U}_s(t)=U_{ds}-jU_{qs}, \]
\[ U_{sq}=U_{sa}=U_{ds}\cos(S-1)t-U_{qs}\sin(S-1)t. \]
As a result of the solution, we obtain the voltage in the phase of the real machine when the control current is switched on:
\[ U_{sa}=U_s\bigl[a_2(S+1)e^{-a_2t}+Sa_1\bigr]\cos St +U_sS(1-e^{-a_2t})\sin St. \]
If the transformer component of the voltage at the output is neglected because of the smallness of its amplitude and its rapid decay, then
\[ U_{sa}\approx U_s\cdot S(1-e^{-t/T})\sin St. \]
Thus, in the no-load mode, the BEMU, with respect to the control signal, behaves as an aperiodic element with time constant
\[ T=\left(x_f-\frac{x_{m1}^{2}}{x_r}\right)\frac{1}{r_f}\cdot \ [\text{electr. rad}], \tag{5} \]
or
\[ T \pm \left( L_f - \frac{M^2_{Rf}}{L_R} \right)\frac{1}{R_f}\quad [\text{sec}], \]
The obtained equations were verified on an experimental sample of a contactless electromechanical amplifier (BEMU) with a power of 300 W and a frequency of 50 Hz, with the following parameters:
\[ R_f=68\ \Omega,\quad L_f=3.4\ \text{H},\quad L_R=0.54\ \text{H},\quad M_{fR}=1\ \text{H}. \]
The external and control characteristics of the BEMU, calculated with allowance for saturation, proved to be close to the experimental ones [6].
The calculated value \(T=0.023\) sec. was confirmed experimentally \((0.02—0.025\ \text{sec})\). The experiments showed that the speed of response of a BEMU without feedback under nominal active and active-inductive load, determined by the time constant of the voltage rise at the output, practically does not differ from the speed of response in the no-load mode.
Conclusions
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Linear equations of a BEMU have been obtained for symmetrical transient processes.
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The time constant of a BEMU (5) determines its speed of response and is recommended for use in design.
Literature
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E. Mishkin. Poly-field Alternating-Current Induction Machines. U.S. Patent No. 2966623. 27, 12, 1960.
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M. L. Kostyrev, A. I. Skorospeshkin. Phase compound excitation of an AC BEMU. Izvestiya TPI, vol. 172, 1967.
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A. I. Skorospeshkin. Questions of the general theory of electrical machines as applied to contactless alternating-current electromechanical amplifiers. Izvestiya TPI, vol. 145, 1966.
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D. Wyatt, G. Judson. Electromechanical Energy Conversion. “Energiya,” 1964.
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M. L. Kostyrev, A. I. Skorospeshkin. Calculation of the characteristics of an alternating-current EMU. Izvestiya TPI, vol. 160, 1966.