USING THE RESULTS OF HARMONIC ANALYSIS TO CALCULATE CURRENT RIPPLE IN PULSE CONTROL OF ELECTRIC MACHINE EXCITATION

A. P. ZAITSEV, A. D. MITAENKO, Yu. V. KOLOKOLOV

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

In [1] an equivalent circuit and the basic equations are given that characterize the quasi-static regime of pulse-width control of the excitation of DC electric machines, taking into account the effect of eddy currents on the course of electromagnetic processes in the excitation-winding and armature circuits. The total current $i_{\mathrm{v}}$ in the excitation winding, consumed from the power source, is represented as two components: the magnetizing current $i_{\mu}$ and the equivalent current of the eddy-current circuit $i_{\mathrm{e}}$. The average over the commutation period $T$ value of the current $i_{\mathrm{e}}$ is zero, and thus the eddy-current circuit does not affect the average value of the magnetizing current $i_{\mu}$, but only has a damping effect on it. The magnetizing current, by determining the useful flux, to a considerable extent determines the character of the variation of the variable component of the current in the armature circuit. It is therefore important to know the character of the variation of the magnetizing-current curve in order to assess the influence of pulse control of excitation on the electromagnetic processes in the armature circuit. In the presence of a zero valve shunting the excitation winding, the expressions for the variable component of the magnetizing current, respectively for the pulse and pause intervals, have the form:

\[ i_{\mu\sim\mathrm{I}}(t)=I_y(1-\gamma)+C_1 e^{p_1 t}+C_2 e^{p_2 t} \tag{1} \]

for $0 \leq t \leq \gamma T$;

\[ i_{\mu\sim\mathrm{II}}(t)=C_3 e^{p_1(t-T)}+C_4 e^{p_2(t-T)}-\gamma I_y \tag{2} \]

for $\gamma T \leq t < T$.

In these equations

$T$ is the commutation period of the current in the excitation winding;

$\gamma$ is the relative duration of the switched-on state of the excitation winding connected to the power source with voltage $U$;

\[ I_y=\frac{U}{r_{\mathrm{v}}} \]

is the maximum current in the excitation winding corresponding to its continuous connection to the voltage $U$;

$r_{\mathrm{v}}$ is the total resistance of the excitation-winding circuit;

$C_1 \div C_4$ and $p_{1,2}$ are integration constants and roots of the characteristic equation, determined from [1].

Taking into account the damping effect of eddy currents, let us consider the possibility of replacing the actual curve of the variable component of the magnetizing current by a sinusoidal curve.

of the magnetizing current \(i_{\mu\sim}\) by a harmonic signal. Let us represent the variable component of the magnetizing current, in accordance with equations (1) and (2), by a trigonometric Fourier series:

\[ i_{\mu\sim}(t)=\frac{a_0}{2}+\sum_{k=1}^{n}\left(a_k\cos k\omega t+b_k\sin k\omega t\right), \tag{3} \]

where \(\omega=\dfrac{2\pi}{T}\).

Using equalities (1) and (2), we determine analytically the expressions for the coefficients of series (3), valid on the interval \(0\le t\le T\):

\[ \frac{a_0}{2}=\frac{1}{T}\left[\int_{0}^{\gamma T} i_{\mu\sim I}(t)\,dt+\int_{\gamma T}^{T} i_{\mu\sim II}(t)\,dt\right]; \tag{4} \]

\[ a_k=\frac{2}{T}\left[\int_{0}^{\gamma T} i_{\mu\sim I}(t)\cos k\omega t\,dt+\int_{\gamma T}^{T} i_{\mu\sim II}(t)\cos k\omega t\,dt\right]; \tag{5} \]

\[ b_k=\frac{2}{T}\left[\int_{0}^{\gamma T} i_{\mu\sim I}(t)\sin k\omega t\,dt+\int_{\gamma T}^{T} i_{\mu\sim II}(t)\sin k\omega t\,dt\right]. \tag{6} \]

After evaluating the integrals, taking into account the expressions for the constants of integration \(C_1\div C_4\) from [1], the following expressions were obtained for the coefficients of the Fourier series:

\[ \frac{a_0}{2}=0, \tag{7} \]

which corresponds to the adopted zero mean value of the variable component of the magnetizing current;

\[ a_k= \frac{ I_y p_1p_2T^2\left\{[p_1p_2T^2-(k2\pi)^2]\sin\beta+k2\pi T(p_1+p_2)(1-\cos\beta)\right\} }{ k\pi\left[(p_1T)^2+(k2\pi)^2\right]\left[(p_2T)^2+(k2\pi)^2\right] }; \tag{8} \]

\[ b_k= \frac{ I_y p_1p_2T^2\left\{(1-\cos\beta)[p_1p_2T^2-(k2\pi)^2]-k2\pi T(p_1+p_2)\sin\beta\right\} }{ k\pi\left[(p_1T)^2+(k2\pi)^2\right]\left[(p_2T)^2+(k2\pi)^2\right] }. \tag{9} \]

In these equalities \(\beta=k2\pi\gamma\).

Analysis of expressions (3), (8), and (9) shows that the harmonic composition of the curve of the variable component of the magnetizing current is determined both by the parameters of the excitation circuit (the values of the roots \(P_1\) and \(P_2\)), and by the switching frequency \(\dfrac{1}{T}\) and the mean value of the current in the excitation-winding circuit (the quantity \(\gamma\)).

For the case under consideration, series (3), taking (7) into account, may be written as follows:

\[ i_{\mu\sim}(t)=\sum_{k=1}^{n} A_k\sin(k\omega t+\varphi_k), \tag{10} \]

where \(A_k=\sqrt{a_k^2+b_k^2}\) is the amplitude of the \(k\)-th harmonic;

\[ \varphi_k=\operatorname{arctg}\frac{a_k}{b_k} \]

is the phase of the \(k\)-th harmonic.

Using equalities (8) and (9), after simple transformations we obtain:

\[ A_k=\frac{0.45 I_y p_1p_2T^2}{k} \sqrt{\frac{1-\cos\beta}{\left[(p_1T)^2+(k2\pi)^2\right]\left[(p_2T)^2+(k2\pi)^2\right]}} ; \tag{11} \]

\[ \operatorname{tg}\varphi_k= \frac{\sin\beta\left[p_1p_2T^2-(k2\pi)^2\right]+k2\pi T(p_1+p_2)(1-\cos\beta)} {(1-\cos\beta)\left[p_1p_2T^2-(k2\pi)^2\right]-k2\pi T(p_1+p_2)\sin\beta}. \tag{12} \]

Harmonic analysis of the expression for the alternating component of the magnetizing current \(i_{\mu\sim}\) under pulse-width control of the excitation circuits shows that the amplitude of the higher harmonics is considerably smaller than the amplitude of the fundamental harmonic. This is confirmed by the graphs calculated from equation (11) for a P52 motor with a power of

Fig. 1.

8 kW at \(k=1\div 5\). In this calculation, the dependence of the parameters of the excitation circuit, and consequently of the roots \(p_{1,2}\), on the magnitude of the average current in the excitation winding was taken into account.

Because of the damping action of eddy currents, for preliminary calculations the pulsations of the alternating component of the magnetizing current \(i_{\mu\sim}\) may be taken as sinusoidal, with a frequency determined by the commutation frequency of the current in the excitation winding,

\[ i_{\mu\sim}=A_1\sin(\omega t+\varphi_1). \tag{13} \]

Then the ripple range $\Delta i_\mu$ is determined by twice the amplitude $A_1$ of the fundamental harmonic. In this case the smallest error when neglecting the higher harmonics will be for values of $\gamma$ close to 0.5. In reality, taking into account the influence of eddy currents, the equation for the armature current is of third order. To calculate the maximum range of the armature-current ripple, it is necessary to determine the time of this maximum by solving a third-order equation. In the general case this problem cannot be solved analytically. In connection with this, using the results of harmonic analysis, one can simplify the calculation of armature-current ripples caused by pulsations of the magnetic flux.

REFERENCES

1. A. P. Zaitsev, A. D. Mitashenko. Analysis of electromagnetic processes in the excitation-winding circuit under pulse control. Reports of the VII Scientific and Technical Conference “Static Converters in Automation and Electric Drive.” Tomsk, 1971.