Harmonic Transfer Coefficient of the EMF Control Loop of Automatic-Control Systems for Electric Drives with Dependent Motor Excitation

V. A. Beinarovich, A. A. Zubok

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

When designing automatic-control systems (ACS) for electric drives with subordinate regulation of the field of the drive motor, owing to the dependence of the EMF on the product of speed and excitation flux, the use of well-developed frequency-domain calculation methods is complicated by the presence of nonlinearities of the “multiplication” type. In these cases, the method of harmonic linearization [1], which makes it possible to determine equivalent frequency characteristics of nonlinear systems [2], may prove very effective.

Below a derivation is given of the harmonic transfer coefficient of the EMF control loop of a drive motor in an ACS with dependent field regulation. The block diagram of the loop is shown in Fig. 1,a,

Fig. 1.

where, in relative units, the following Laplace images are denoted: the EMF \(e_0\), from which field weakening begins; the instantaneous values of the EMF \(e\), speed \(v\), and flux \(\varphi\); the voltage of the field regulator \(U_p\); and also the transfer functions of the EMF regulator \(G_p(S)\) and of the excitation-flux regulating device \(G_b(S)\). For \(e \ll e_0\) the EMF regulator oper—

operates in the limiting mode, providing a constant initial excitation flux \(\varphi_0\). When the speed exceeds the value \(v_0=\dfrac{e_0}{\varphi_0}\) by \(\Delta v\), field weakening \(\Delta\varphi\) begins,

\[ \Delta\varphi=-(v_0+\Delta v)(\varphi_0+\Delta\varphi)G_p(S)\cdot G_v(S), \tag{1} \]

which is reflected by the block diagram in Fig. 1,b. For a time-specified function of the speed deviation \(\Delta v(t)\), whose Laplace transform is \(U(S)\), the time function of the flux deviation \(\Delta\varphi(t)\) will have the transform \(\Phi(S)\), and the block diagram can be represented as in Fig. 1,c, where the low-pass filter is the element

\[ G(S)=\frac{G_p(S)G_v(S)}{1+v_0G_p(S)G_v(S)}. \tag{2} \]

The Laplace transform of the product of functions is represented as

\[ L[\Delta v(t)\cdot \Delta\varphi(t)] = \sum_{\kappa=1}^{q} \frac{A(S_\kappa)}{B'(S_\kappa)} \Phi(S-S_\kappa), \tag{3} \]

where \(U(S)=\dfrac{A(S)}{B'(S)}\) has \(q\) simple roots \(S_\kappa\), and \(\kappa\) is the number of the root.

Taking (3) into account, for Fig. 1,c one can write

\[ \Phi(S)=G(S)\left[\varphi_0\cdot U(S)- \sum_{\kappa=1}^{\infty} \frac{A(S_\kappa)}{B'(S_\kappa)} \Phi(S-S_\kappa)\right]. \tag{4} \]

Let us apply a harmonic input action (a cosine wave) to the input of the loop, with transform

\[ U(S)=|u|\frac{S}{S^2+\omega^2}. \tag{5} \]

Substituting (5) into (4) and considering the solution on the \(j\omega\) axis, we determine the transform of the output quantity in the form of a series sum

\[ \Phi(S)=\sum_{n=0}^{\infty}\Phi(jn\omega) = \varphi_0U(j\omega)G(j\omega)-- \]

\[ -\frac{|u|}{2} \sum_{n=0}^{\infty} G_n(jn\omega)\{\Phi_n[(n+1)j\omega]+\Phi_n[(n-1)j\omega]\}. \tag{6} \]

Equating terms of the same order \(n\) in (6), we obtain a system of recurrence equations

\[ \left. \begin{aligned} \Phi_0&=-\frac{|u|}{2}G_0(\Phi_1+\Phi_{-1});\\ \Phi_1&=G_1U_1\varphi_0-\frac{|u|}{2}G_1(\Phi_2+\Phi_0);\\ \Phi_2&=-\frac{|u|}{2}G_2(\Phi_3+\Phi_1);\\ &\quad\cdots\\ \Phi_n&=-\frac{|u|}{2}G_n(\Phi_{n+1}+\Phi_{n-1}), \end{aligned} \right\} \tag{7} \]

where \(\Phi_{-1}=\Phi(-j\omega)\); \(\Phi_0=\Phi(0)\); \(\Phi_1=\Phi(j\omega);\ldots \Phi_n=\Phi(jn\omega)\);

\[ G_0=G(0);\quad G_1=G(j\omega);\quad (G_2=G(j2\omega));\ldots G_n=G(jn\omega); \]

\[ U_1=U(j\omega)=|u|;\quad \Phi_1=|\Phi_1|e^{j\psi_1}. \]

Let us note that

\[ \Phi_1+\Phi_{-1}=\Phi_1(1+e^{-2j\psi_1})=\Phi_1\,2e^{-j\psi_1}\cos\psi_1, \tag{8} \]

where \(\psi_1\) is the phase shift of the output signal \(\Phi_1\) relative to the input \(U_1\).

Eliminating in (7) all \(\Phi_k\) except \(\Phi_1\), we find the harmonic transfer coefficient of the loop (Fig. 1, c)

\[ G_{\mathrm e}=\frac{\Phi_1}{U_1} = \frac{\varphi_0G_1} { 1-G_0G_1\frac{|u|^2}{2}e^{-j\psi_1}\cos\psi_1 - \cfrac{G_1G_2\frac{|u|^2}{4}} {1-\cfrac{G_2G_3\frac{|u|^2}{4}} {1-\cfrac{G_3G_4\frac{|u|^2}{4}} {1-\ldots}}} }. \tag{9} \]

For convenience of notation, let us denote

\[ M= \frac{\varphi_0G_1} { 1-\cfrac{G_1G_2\frac{|u|^2}{4}} {1-\cfrac{G_2G_3\frac{|u|^2}{4}} {1-\ldots}} } = \frac{\varphi_0} { \cfrac{1}{G_1} - \cfrac{\frac{|u|^2}{4}} {\cfrac{1}{G_2} - \cfrac{\frac{|u|^2}{4}} {\cfrac{1}{G_3}-\ldots}} }. \tag{10} \]

Then from (9) and (10) one can write

\[ \frac{1}{M}=\frac{1}{G_{\mathrm e}}+\frac{|u|^2G_0}{2\varphi_0}e^{-j\psi_1}\cos\psi_1; \tag{11} \]

but

\[ G_{\mathrm e}=\frac{\Phi_1}{U_1}=\frac{|\Phi_1|e^{j\psi_1}}{|u|}; \]

therefore

\[ M= \frac{|\Phi_1|e^{j\psi_1}} {|u|+\dfrac{|u|^2G_0}{2\varphi_0}|\Phi_1|\cos\psi_1} = |M|e^{j\psi_1} \tag{12} \]

and

\[ \psi_1=\operatorname{arctg}\frac{\operatorname{Im}\{M\}}{\operatorname{Re}\{M\}}. \tag{13} \]

Taking (13) into account, the expression for the harmonic coefficient from (11) is determined as

\[ G_{\mathrm{e}}= \frac{M}{1-\dfrac{|u|^{2}G_{o}}{2\varphi_{o}}\,|M|\cos\psi_{1}} = \frac{M}{1-\dfrac{|u|^{2}G_{o}}{2\varphi_{o}}\,Re\{M\}} . \tag{14} \]

Using the calculation formulas (10), (13), and (14), for the emf-control loop with linear part (2), under a harmonic deviation of the speed \(\Delta v\) with amplitude \(|u|\) in the vicinity of the initial values \(\varphi_{0}=v_{0}=1\), the amplitude and phase frequency characteristics of the harmonic transfer coefficient were calculated; these are shown in Fig. 2.

Figure labels: \(20\lg |G|,\ \mathrm{dB}\); \(\arg G\); \(|u|=1.0\); \(|u|=0.6\); \(|u|=0\); \(L(|G|)\); \(\psi\); \(\omega T_{\mathrm{b}}\).

Fig. 2.

In the calculation it was assumed that the emf regulator was adjusted according to the technical optimum [4]

\[ G_{p}(S)=\frac{1}{4T_{\mathrm{b}}S} \]

(where \(T_{\mathrm{b}}\) is the equivalent uncompensated time constant of the excitation loop), and the excitation-current control loop was approximated by a first-order equation.

Examination of the characteristics in Fig. 2 shows that the harmonic transfer coefficient of the EMF control loop depends substantially on the amplitude of the input harmonic action. At small amplitudes of the input action, the EMF control loop can be represented by an equivalent dynamic element that does not take account of the effect of multiplication and has invariant logarithmic frequency characteristics. Thus, for $|u| \leqslant 0.5$, the error in the logarithmic amplitude-frequency characteristics does not exceed 2 dB, and the error in the logarithmic phase-frequency characteristics is no more than 3–4°.

The relationships obtained make it possible to use the known frequency-domain methods [2] for the analysis and synthesis of nonlinear automatic-control systems containing nonlinearities of the “multiplication” type.

REFERENCES

1. The Method of Harmonic Linearization in the Design of Nonlinear Automatic-Control Systems. Collection edited by Yu. I. Topcheev, Moscow, “Mashinostroenie,” 1970.

2. A. A. Vavilov. Frequency-Domain Methods for the Calculation of Nonlinear Systems. Moscow, “Energiya,” 1970.

3. M. F. Gardner, J. L. Barnes. Transients in Linear Systems. Moscow, Fizmatgiz, 1961.

4. E. D. Lebedev, V. E. Neimark, M. Ya. Pistrak, O. V. Slezhanovskii. Control of DC Converter Electric Drives. Moscow, “Energiya,” 1970.

5. I. Miljan. Messen-Reglen-Steueren. Heft 4, 1969, pp. 145–149.