Electromagnetic Processes in a DC Electric Machine under Pulse Excitation

A. I. Zaitsev, A. P. Zaitsev, A. D. Mitaenko

(Presented by the scientific and technical seminar of the Scientific Research Institute of AEM)

Pulse excitation of DC electric machines is finding ever broader application in controlled electric drives and in converter electric-machine units. One of the main indicators of the quality of the process of pulse control of an electric machine from the excitation-winding side in quasi-static regimes is the magnitude of the pulsations of the e.m.f. and of the armature current. The exact analytical determination of the ripple amplitude of the e.m.f. and of the armature current under pulse-width control of the excitation current is a complex problem, mainly because of the complicated nature of the electromagnetic processes occurring in the eddy-current circuits in the massive parts of the magnetic circuit and exerting a substantial influence on the pulsations of the excitation flux.

Of the numerous methods for taking into account the action of eddy currents \[1, 2\], the most convenient mathematically is the method of replacing the action of numerous eddy-current circuits by the action of a single equivalent \(LR\)-circuit coupled without magnetic leakage to the main magnetizing circuit.

Below are given the results of an analysis of quasi-static electromagnetic processes under pulse excitation of a DC electric machine with allowance for eddy currents by the above-mentioned method. In doing so, the problem of simplifying the analysis is considered on the basis of taking into account the action only of the fundamental harmonic of the variable component of the magnetizing current.

Fig. 1. Equivalent circuit of an electric machine in generator mode.

The equivalent circuit of the electrical machine in generator mode is shown in Fig. 1. Regulation of the excitation current \(i_{\mathrm{v}}\) is carried out by a controlled pulse-width method with switch \(K\), with switching period \(T\) and relative on-time \(\gamma\). The equivalent circuit of the eddy currents in the massive sections of the magnetic circuit is accounted for by the resistance \(R_{\mathrm{e}}\), through which the equivalent eddy current \(i_{\mathrm{e}}\) flows. The magnetizing circuit includes the inductance \(L_{\mu}\), through which the magnetizing current \(i_{\mu}\) flows. The scattering of the main poles is accounted for by the inductance \(L_{\sigma}\).

For the pulse interval \(\gamma T\), the following system of equations corresponds to the equivalent circuit:

\[ \left. \begin{aligned} i_{\mathrm{v}}R_{\mathrm{v}} + L_{\sigma}\frac{di_{\mathrm{v}}}{dt} + L_{\mu}\frac{di_{\mu}}{dt} &= U;\\ L_{\mu}\frac{di_{\mu}}{dt} &= i_{\mathrm{e}}R_{\mathrm{e}};\\ i_{\mathrm{v}} - i_{\mu} - i_{\mathrm{e}} &= 0. \end{aligned} \right\} \tag{1} \]

From (1) the differential equations in operational form are determined with respect to the currents \(i_{\mathrm{v}}\) and \(i_{\mu}\):

\[ T_{\sigma}T_{\mathrm{e}}p^{2}i_{\mathrm{v}} + (T_{\mu}+T_{\sigma}+T_{\mathrm{e}})pi_{\mathrm{v}} + i_{\mathrm{v}} = kT_{\mathrm{e}}pU + kU; \tag{2} \]

\[ T_{\sigma}T_{\mathrm{e}}p^{2}i_{\mu} + (T_{\mu}+T_{\sigma}+T_{\mathrm{e}})pi_{\mu} + i_{\mu} = kU, \tag{3} \]

where

\[ T_{\mu}=\frac{L_{\mu}}{R_{\mathrm{v}}},\qquad T_{\sigma}=\frac{L_{\sigma}}{R_{\mathrm{v}}},\qquad T_{\mathrm{e}}=\frac{L_{\mu}}{R_{\mathrm{e}}},\qquad k=\frac{1}{R_{\mathrm{v}}}. \]

For the pause interval in (2) and (3), it is necessary to set \(U=0\). The solutions of equations (2) and (3) are represented in the form:

\[ i_{\mathrm{v}1}=I_{y}+C_{1}(1+p_{1}T_{\mathrm{e}})e^{p_{1}t} +C_{2}(1+p_{2}T_{\mathrm{e}})e^{p_{2}t}; \tag{4} \]

\[ i_{\mu 1}=I_{y}+C_{1}e^{p_{1}t}+C_{2}e^{p_{2}t}, \tag{5} \]

where \(p_{1}, p_{2}\) are real, negative, and distinct roots of the characteristic equation,

\(C_{1}, C_{2}\) are integration constants,

\[ I_{y}=\frac{U}{R_{\mathrm{v}}}. \]

For the pause interval,

\[ i_{\mathrm{v}2}=C_{3}(1+p_{1}T_{\mathrm{e}})e^{p_{1}t} +C_{4}(1+p_{2}T_{\mathrm{e}})e^{p_{2}t}; \tag{6} \]

\[ i_{\mu 2}=C_{3}e^{p_{1}t}+C_{4}e^{p_{2}t}. \tag{7} \]

The integration constants are determined from the conditions \(i_{\mathrm{v}1}=i_{\mathrm{v}2}\) and \(i_{\mu 1}=i_{\mu 2}\) at the instants of switch commutation:

\[ \left. \begin{aligned} C_{1} &= q_{2}I_{y}\frac{1-e^{p_{1}(1-\gamma)T}}{1-e^{p_{1}T}}; & C_{2} &= -q_{1}I_{y}\frac{1-e^{p_{2}(1-\gamma)T}}{1-e^{p_{2}T}}; \\[6pt] C_{3} &= -q_{2}I_{y}\frac{1-e^{p_{1}\gamma T}}{1-e^{p_{1}T}}; & C_{4} &= q_{1}I_{y}\frac{1-e^{p_{2}\gamma T}}{1-e^{p_{2}T}}. \end{aligned} \right\} \tag{8} \]

In the expressions for \(C_{1}\)--\(C_{4}\), the following notation has been adopted:

\[ q_{1}=\frac{p_{1}}{p_{1}-p_{2}};\qquad q_{2}=\frac{p_{2}}{p_{1}-p_{2}}. \]

In (4)—(7), the time \(t\) is counted from the instants of switching of the key; therefore, for the pulse interval \(0 \leq t \leq \gamma T\), and for the pause interval \(0 \leq t \leq (1-\gamma)T\).

The extreme values of the current \(i_{\mathrm{в}}\) can be determined from (4) or (6) at the boundary values of \(t\).

Investigation of dependences (5) and (7) for an extremum makes it possible to determine the times of displacement of the extrema of the currents \(i_{\mathrm{в}}\) and \(i_{\mu}\) (Fig. 2):

Fig. 2. Curves of the currents \(i_{\mathrm{в}}\) and \(i_{\mu}\)

\[ t_1=-\frac{1}{p_1-p_2}\ln\left|\frac{C_4p_2}{C_3p_1}\right|; \qquad t_2=\frac{1}{p_1-p_2}\ln\left|\frac{C_2p_2}{C_1p_1}\right|, \tag{9} \]

and then, from the found extreme values of the current \(i_{\mu}\), calculate the ripple amplitude

\[ \Delta i_{\mu} = I_y \left\{ \frac{(1-e^{p_1T})^{q_2}}{(1-e^{p_2T})^{q_1}} \left[ \frac{(1-e^{p_2\gamma T})^{q_1}}{(1-e^{p_1\gamma T})^{q_2}} + \frac{(1-e^{p_2(1-\gamma)T})^{q_1}}{(1-e^{p_1(1-\gamma)T})^{q_2}} \right] -1 \right\}. \tag{10} \]

When the condition \(p_2 \gg p_1\) is satisfied, the maximum ripple amplitude will be at \(\gamma=0.5\)

\[ \Delta i_{\mu} = I_y \left[ \frac{4\left(1+e^{\frac{p_2T}{2}}\right)^{-q_1}}{4+p_1T} -1 \right]. \tag{11} \]

Equation (2) differs from (3) by the presence, in the right-hand side, of the term \(kT_{\mathrm{э}}pU\), containing the first derivative of the stepwise input function and existing only at the instants of switching of the key. At the instants of jumps of the input function, \(kT_{\mathrm{э}}pU=\infty\), which causes jumps of a definite magnitude in the first derivative of the current \(i_{\mathrm{в}}\) and determines the difference in the initial conditions in the solutions of equations (2) and (3). The curve of the current \(i_{\mu}\), which determines the pulsations of the useful flux and of the armature current, has an explicitly expressed smoothed character and differs substantially from the curve of the current \(i_{\mathrm{в}}\) both in ripple amplitude and in harmonic composition.

For harmonic analysis of the curve of the current \(i_{\mu}\), we separate out its variable component by excluding from (5) and (7) the constant component \(\gamma I_y\), and write the equations on the pulse and pause intervals, taking into account a single time reference from the beginning of the period \(T\):

\[ \sim i_{\mu1} = I_y(1-\gamma)+C_1e^{p_1t}+C_2e^{p_2t}, \qquad 0\leq t\leq \gamma T; \tag{12} \]

\[ \sim i_{\mu2} = C_3e^{p_1(t-\gamma T)}+C_4e^{p_2(t-\gamma T)}-\gamma I_y, \qquad \gamma T\leq t\leq T. \tag{13} \]

On the basis of (12) and (13) we obtain analytical dependences for determining the coefficients of the Fourier series:

\[ a_k=\frac{I_y p_1p_2T^2\{[p_1p_2T^2-(2\pi k)^2]\sin\beta+2\pi kT(p_1+p_2)(1-\cos\beta)\}} {\pi k[(p_1T)^2+(2\pi k)^2][(p_2T)^2+(2\pi k)^2]}, \tag{14} \]

\[ b_k=\frac{I_y p_1p_2T^2\{(1-\cos\beta)[p_1p_2T^2-(2\pi k)^2]-2\pi kT(p_1+p_2)\sin\beta\}} {\pi k[(p_1T)^2+(2\pi k)^2][(p_2T)^2+(2\pi k)^2]}, \tag{15} \]

where \(\beta=2\pi k\gamma\),
\(k\) is the harmonic number.

Represent the nonlinear function of the variable component \(\sim i_\mu\) by a Fourier series

\[ \sim i_\mu=\sum_{k=1}^{n} A_k \sin(k\omega t+\varphi_k), \tag{16} \]

where

\[ A_k=\sqrt{a_k^2+b_k^2} =\frac{0.45 I_y p_1p_2T^2}{k}\times \]

\[ \times\sqrt{ \frac{1-\cos\beta} {[(p_1T)^2+(2\pi k)^2][(p_2T)^2+(2\pi k)^2]} }, \]

\[ \tg \varphi_k=\frac{a_k}{b_k},\qquad \omega=\frac{2\pi}{T}. \]

Harmonic analysis of the expression for the variable component \(\sim i_\mu\) shows that already for electrical machines with a power exceeding 8–10 kW the amplitudes of the higher harmonics are substantially smaller than the amplitude of the first harmonic at control frequencies above 200 Hz. Therefore, without great error, in subsequent calculations one may take into account only the first harmonic of the current \(i_\mu\), and determine the pulsation range \(\Delta i_\mu\) from the doubled amplitude of the first harmonic. In this case the smallest error occurs for \(\gamma=0.4\text{--}0.6\).

The use in calculations of only the fundamental harmonic of the current \(i_\mu\) makes it possible to simplify substantially the analysis of electromagnetic processes in the armature circuit. With an active-inductive character of the resistance of the armature circuit, the differential equation with respect to the armature current is of third order, and, with respect to the input action, the electrical machine has sufficiently good filtering properties. The analytical solution of the third-order differential equation with respect to the armature current \(i_a\) is laborious. The use of the results of harmonic analysis, however, considerably simplifies the solution of this problem with satisfactory accuracy.

When only the first harmonic \((k=1)\) is taken into account, the variable component of the current \(i_\mu\), with (16) taken into account, is determined by the expression

\[ \sim i_\mu=A_1\sin(\omega t+\varphi_1), \tag{17} \]

and the amplitude of the variable component of the useful flux, without allowing for the armature reaction, is

\[ \Phi_m=k_\mu A_1, \tag{18} \]

where

\[ k_\mu \]

is the angular coefficient of the tangent to the magnetization curve at the operating point determined by the constant component of the current \(i_\mu\).

Accordingly, the amplitudes of the variable components of the e.m.f. and the armature current in both motor and generator modes may be represented in the form

\[ E_m = c_e \omega_v k_{\mu} A_1, \tag{19} \]

\[ I_{ya\,m}=\frac{E_m}{z}, \tag{20} \]

where

\[ \omega_v \]
is the angular velocity of rotation of the armature,

\[ z \]
is the complex impedance of the armature circuit.

The determination of the constant components of the armature emf and current is carried out in the usual manner, taking into account the average value of the magnetizing current \(I_{\mu\,avg}=i_{v\,avg}=\gamma I_y\).

References

1. E. A. Musin. Comparison of various methods for taking into account the influence of eddy currents on the process of flux establishment in DC machines. Proceedings of LETI, issue XLII, Leningrad, 1960.

2. O. V. Slezhanovskii. Reversible DC Electric Drive. Moscow, Metallurgiya, 1967.