Transfer Functions of a DC Motor under Two-Channel Control
Abstract Generated abstract
This paper derives transfer functions for DC motors under two-channel control while accounting for eddy currents in massive magnetic-circuit parts, leakage fluxes, and armature inductance. Using equivalent electromagnetic circuits for the field winding and armature, the authors formulate linearized operator equations and transform the resulting block diagrams to obtain transfer functions for field-circuit control, armature-circuit control, common-source supply of both circuits, and load-torque disturbances. The resulting expressions are intended for analysis of fast electric-drive systems, including drives supplied by pulsating-voltage sources such as thyristor converters, where eddy currents can significantly alter electromagnetic transients by making excitation current change faster than flux and motor emf.
Full Text
TRANSFER FUNCTIONS OF A DC MOTOR UNDER TWO-CHANNEL CONTROL
V. A. Beinarovich, A. I. Sapozhnikov, V. K. Smirnov,
V. M. Stepanov
(Presented by the Scientific-Technical Seminar of the Research Institute of AEM)
A large number of works have been devoted to the study of automatic-control systems with DC motors under two-channel control, for example [2, 3, 4]. However, these works do not take into account the damping influence of eddy currents in the massive parts of the magnetic circuits of electric motors, which have a significant effect on the course of transient processes in fast-acting drives with motors of medium and high power. Some authors take the action of eddy currents into account by introducing the electromagnetic time constant of the eddy-current circuit \(T_k\), determined by the Rüdenberg formula [3], which gives considerably overestimated values of \(T_k\). A more accurate dynamic calculation of the system is obtained when the action of the eddy-current circuit is taken into account by the Zhitz method [5]. However, when the method [5] is used, the expression for the motor transfer function becomes significantly more complicated, which makes calculations difficult.
In the present work, the transfer function of a motor is derived for control through the field-winding circuit. The equations of the field winding are formulated on the basis of the equivalent circuit (Fig. 1) of the electromagnetic circuit of the field winding, taking into account eddy currents and leakage fluxes [4]. In this circuit, the leakage fluxes of the field winding are taken into account by the inductance \(L_{sv}\), while the equivalent eddy-current circuit, referred to the field winding, is taken into account by the resistance \(r_{kv}\), which should expediently be calculated by the method [1].
Fig. 1. Electromagnetic circuit of the field winding.
When formulating the motor equations with allowance for eddy currents in the massive parts of the magnetic circuit, leakage fluxes, and the inductance of the armature circuit, the following assumptions were adopted:
the characteristics are linearized in the region of small deviations from the steady operating state;
the inductances of the armature and field windings do not change;
the demagnetizing action of the armature reaction is compensated by the flux of the additional poles;
the leakage fluxes are proportional to the excitation current;
the demagnetizing action of the eddy currents is proportional to the rate of change of the main flux;
the mutual inductance of the excitation windings, additional poles, and armature is not taken into account.
In this case the equations of the electromagnetic circuit of the excitation winding can be written in the following form (Fig. 1):
\[ U_v = i_v (r_v + pL_{sv}) + r_{kv}\cdot i_{kv}, \tag{1} \]
\[ pL_{\mu v}\cdot i_{\mu v} = r_{kv}\cdot i_{kv}; \tag{2} \]
\[ i_v = i_{kv} + i_{\mu v}, \tag{3} \]
where
\[ \begin{aligned} &U_v,\ i_v,\ r_v &&\text{are the voltage, current, and resistance of the excitation circuit;}\\ &i_{\mu v} &&\text{is the magnetizing current of the excitation circuit;}\\ &i_{kv} &&\text{is the current of the equivalent eddy-current circuit, referred to the excitation winding;}\\ &L_{\mu v},\ L_{sv} &&\text{are the magnetizing inductance and the leakage inductance of the excitation circuit.} \end{aligned} \]
According to equations (1), (2), and (3), a block diagram of the excitation-winding circuit is constructed; it is shown in Fig. 3a. The transfer function for the excitation circuit of the motor is written as
\[ W_1(p)=\frac{\Delta i_{\mu v}(p)}{\Delta U_v(p)} = \frac{1/r_v}{1+pT_{sv}+pT_{\mu v}+pT_{kv}+p^2T_{sv}\cdot T_{kv}}, \tag{4} \]
where
\[ T_{sv}=\frac{L_{sv}}{r_v};\qquad T_{\mu v}=\frac{L_{\mu v}}{r_v};\qquad T_{kv}=\frac{L_{\mu v}}{r_{kv}}. \]
The equivalent circuit of the armature circuit of a compensated direct-current motor with parallel excitation, taking account of eddy currents and leakage fluxes, is shown in Fig. 2. In this circuit the leakage fluxes are taken into account by the inductance \(L_{sa}\), while the equivalent eddy-current circuit is taken into account by a resistance, determined analogously to the resistance \(r_{kv}\). The equivalent circuit of the armature circuit is described by the following equations:
Fig. 2. Electromagnetic circuit of the motor armature.
\[ U_a = i_a\left[r_{a\Sigma}+p\left(L_{sa}+L_a\right)\right]+i_{kd p}\cdot r_{kd p}-E; \tag{5} \]
\[ pL_{\mu d p}\cdot i_{\mu d p}=i_{kd p}\cdot r_{kd p}; \tag{6} \]
\[ i_a=i_{kd p}+i_{\mu d p}, \tag{7} \]
where
\[ \begin{aligned} &U_a,\ i_a,\ r_{a\Sigma} &&\text{are the supply voltage, current, and total resistance of the armature circuit;}\\ &i_{\mu d p} &&\text{is the magnetizing current of the additional poles;} \end{aligned} \]
$i_{\text{kdp}}$ is the current of the equivalent eddy-current circuit;
$L_{\mu\text{dp}},\ L_{\text{a}}$ are the magnetizing inductance of the additional poles and the inductance of the motor armature;
$E$ is the emf of rotation.
Fig. 3. Structural diagrams of a separately excited electric motor under two-channel control.
According to equations (5), (6), (7), the structural diagram of the armature circuit of the electric motor is constructed (Fig. 3, a). After transformation of the structural diagram (Fig. 3, a) to a single-loop form, the transfer function of the motor armature circuit is written as:
\[ W_2(p)=\frac{\Delta i_{\text{a}}(p)}{\Delta U_{\text{a}}(p)} = \frac{1+pT_{\text{kdp}}} {r_{\text{a}\Sigma}\left[1+p\left(T_{S\text{a}}+T_{\text{a}}\right)\right]\left(1+pT_{\text{kdp}}\right)+pT_{\text{kdp}}\cdot r_{\text{kdp}}}, \tag{8} \]
where
\[ T_{S\text{a}}=\frac{L_{S\text{a}}}{r_{\text{a}\Sigma}};\qquad T_{\text{a}}=\frac{L_{\text{a}}}{r_{\text{a}\Sigma}};\qquad T_{\text{kdp}}=\frac{L_{\mu\text{dp}}}{r_{\text{kdp}}}. \]
Let us write the operator equations in increments for the flux $\Delta\Phi$, the emf of rotation $\Delta E$, the electromagnetic torque $\Delta M$, and the speed $\Delta\omega$ of the electric motor [2].
\[ \Delta\Phi(p)=S\Phi_1\cdot\Delta i_{\text{f.v}}(p); \tag{9} \]
\[ \Delta E(p)=E_0\left[\frac{\Delta\Phi(p)}{\Phi_0}+\frac{\Delta\omega(p)}{\omega_0}\right]; \tag{10} \]
\[ \Delta M(p)=M_0\left[\frac{\Delta\Phi(p)}{\Phi_0}+\frac{\Delta I_{\text{a}}(p)}{I_0}\right]; \tag{11} \]
\[ \Delta\omega(p)=\frac{1}{Jp}\left[\Delta M(p)-\Delta M_{\text{n}}(p)\right]. \tag{12} \]
Using formulas (1)—(3), (5)—(7), and (9)—(11), Fig. 3, a shows a structural diagram of a separately excited electric motor under
two-channel control. Having transformed the block diagram into a single-loop one, the transfer function of the motor is written as follows:
for the control action through the field circuit
\[
W_{dv}(p)=\frac{\Delta \omega(p)}{\Delta U_v(p)}
=\frac{M_0}{I\cdot p}\cdot \frac{W_6(p)}{1+W_7(p)} ;
\tag{13}
\]
for the control action through the armature circuit
\[
W_{dya}(p)=\frac{\Delta \omega(p)}{\Delta U_y(p)}
=\frac{M_0}{I_c\cdot I_p}\cdot \frac{W_2(p)}{1+W_7(p)} ;
\tag{14}
\]
for the disturbing action
\[
W_M(p)=\frac{\Delta \omega(p)}{\Delta M(p)}
=-\frac{1}{I\cdot p}\cdot \frac{1}{1+W_7(p)} ,
\tag{15}
\]
where
\[
W_7(p)=\frac{1}{I\cdot p}-\frac{E_0}{\omega_0}\frac{1}{I_0}\cdot W_2(p)\cdot M_0 ;
\tag{16}
\]
\[
W_6(p)=W_3(p)\cdot W_5(p) ;
\tag{17}
\]
\[
W_5(p)=\frac{1}{I_0}\cdot W_4(p)+1 ;
\tag{18}
\]
\[
W_4(p)=-E_0\cdot W_2(p) ;
\tag{19}
\]
\[
W_3(p)=W_{1p}\frac{S\Phi_1}{\Phi_0}.
\tag{20}
\]
Such transfer functions of a DC motor are expedient to use for studying automatic-control systems in which the armature circuit and the field circuit (or one of them) are supplied from a source with pulsating voltage, for example, from a thyristor converter. In this case, eddy currents in the massive parts of the motor magnetic circuit have a substantial influence on electromagnetic transient processes. The influence of eddy currents changes the nature of the course of transient processes. The current in the field-winding circuit, under a step change of voltage at the beginning of the process, changes more intensively, while the magnetic flux and the motor emf change more slowly. In machines with laminated magnetic circuits, eddy currents are almost absent and the excitation flux changes identically with the excitation current [6].
When the motor field-winding circuit and armature circuit are supplied from a common source with voltage \(U\), the block diagram of the DC motor (Fig. 3, a) can be reduced to the form shown in Fig. 4. In this case the motor transfer functions are written as follows:
Fig. 4. Block diagram of a parallel-excitation electric motor when the field-winding circuit and the armature circuit are supplied from a common source.
with respect to the control action
\[ W_y(p)=\frac{\Delta\omega(p)}{\Delta U(p)} =\frac{W_8(p)\cdot M^0}{I\cdot p[1-W_{10}(p)]}; \tag{21} \]
with respect to the disturbing action
\[ W_M(p)=\frac{\Delta\omega(p)}{\Delta M(p)} =\frac{1}{I\cdot p}\cdot \frac{1}{1-W_{10}(p)}, \tag{22} \]
where
\[ W_{10}(p)=\frac{M_0}{I\cdot p}\cdot W_9(p); \tag{23} \]
\[ W_9(p)=\frac{1}{I_0}\cdot W_2(p)-\frac{E_0}{\omega_0}; \tag{24} \]
\[ W_8(p)=\frac{1}{I_0}\cdot W_2(p)+W_4(p). \tag{25} \]
The transfer functions presented make it possible to investigate electric-drive systems with direct-current motors, taking into account eddy currents and leakage fluxes, when control is effected through the field circuit, through the armature circuit, and under control with joint supply of the field and armature circuits from a common source, and also to investigate the influence of disturbances in the magnitude of the load torque.
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