PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
named after S. M. Kirov

Vol. 262　　　　　　　　　　　　　　　　　　　　　　　　　　　1973

EQUATIONS AND TRANSFER FUNCTIONS OF A RELAY AUTOMATIC CONTROL SYSTEM FOR THE STARTING CURRENT OF A SERIES-EXCITATION TRACTION MOTOR

A. P. ZAITSEV, Yu. F. MIKHEEV

(Presented by the Scientific and Technical Seminar of the Research Institute of AEM)

The article considers the derivation of equations and transfer functions of a relay automatic control system (ACS) for the direct starting of a series-excitation motor, taking into account armature reaction, saturation of the magnetic circuit, leakage of the main poles, and eddy currents arising in the massive parts of the magnetic core.

A simplified diagram of the ACS is presented in Fig. 1,a, where the following designations are used:

RE — relay element,
TC — thyristor converter,
ACS — armature-current sensor.

Control is effected by maintaining the starting armature current at a certain level corresponding to the reference voltage \(V_\mathrm{e}\). Monitoring of the current value is performed by the ACS. If the ACS signal \(V_\mathrm{d} < V_\mathrm{e}\), then the thyristor converter is switched on, the full voltage of the power source is applied to the motor, and the armature current increases. When \(V_\mathrm{d} > V_\mathrm{e}\), the thyristor converter is switched off. Consequently, the motor current will be maintained at a certain level, pulsating about the mean value.

A characteristic feature of the ACS under consideration in this operating mode is the stepwise change in the voltage applied to the motor at each switching of the TC. In addition, immediately before the next switching, the ACS has nonzero values of the output quantity and its derivatives. In this case, when formulating the differential equations it is expedient to use generalized functions [1].

In formulating the equations and transfer functions we shall adopt the following assumptions:

1) the commutation period of the thyristor converter at the minimum control frequency in the system is much smaller than the electromechanical time constant of the drive;

2) the armature-reaction flux is proportional to the armature current for small deviations of it;

3) the rotational speed in the quasi-steady-state mode remains constant;

4) the power source is of infinite capacity.

Taking the adopted assumptions into account, the circuit of Fig. 1,a is described by the following system of differential equations:

\[ \left. \begin{gathered} (L_{\mathrm{в}}+L_{\mathrm{я}}+L_{\mathrm{р}})\frac{di}{dt}+(r_{\mathrm{в}}+r_{\mathrm{я}})i +M\frac{di_{\mathrm{к}}}{dt}+C_e\omega\Phi_e=V_c,\\[4pt] L_{\mathrm{к}}\frac{di_{\mathrm{к}}}{dt}+i_{\mathrm{к}}r_{\mathrm{к}}+M\frac{di}{dt}=0,\\[4pt] \Phi_e=\Phi_\mu-K_\rho\cdot i,\\[4pt] V_c=f(V_y)= \begin{cases} \max, & \text{when } \dfrac{di}{dt}>0,\\[4pt] 0, & \text{when } \dfrac{di}{dt}<0, \end{cases}\\[4pt] V_y=V_{\mathrm{д}}-V_{\mathrm{э}},\\ V_{\mathrm{д}}=K_{\mathrm{п}}\cdot i. \end{gathered} \right\} \tag{1} \]

Fig. 1

where

\(L_{\mathrm{в}}\) — inductance of the field winding,
\(L_{\mathrm{я}}\) — inductance of the armature winding,
\(L_{\mathrm{р}}\) — leakage inductance of the main poles,
\(L_{\mathrm{к}}\) — inductance of the fictitious eddy-current winding,
\(M\) — mutual inductance of the field winding and the fictitious eddy-current winding,

$r_{\mathrm{a}}$ — armature-winding resistance,
$r_{\mathrm{v}}$ — excitation-winding resistance,
$r_{\mathrm{k}}$ — eddy-current winding resistance,
$C_{\mathrm{e}}$ — design constant of the motor,
$\omega$ — rotational speed of the motor,
$\Phi_{\mathrm{e}}$ — useful magnetic flux,
$\Phi_{\mu}$ — magnetizing flux,
$k_{\mathrm{p}}$ — armature-reaction coefficient,
$i$ — armature current,
$i_{\mathrm{k}}$ — current in the eddy-current circuit,
$V_{\mathrm{y}}$ — control voltage of the RE,
$k_{\mathrm{p}}$ — current-sensor transfer coefficient,
$V_{\mathrm{c}}$ — line voltage.

Since the fictitious winding characterizing the eddy-current circuit has no leakage fluxes, whereas the excitation winding has leakage fluxes, the circuit of Fig. 1,a may be represented in the form shown in Fig. 1,b, where the following are denoted:

$L_{\mu}$ — inductance of the magnetizing circuit,
$i_{\mu}$ — magnetizing current.

The remaining notation is clear from Fig. 1,b. For the on state of the thyristor converter, the equivalent circuit is described by the equations:

\[ \left. \begin{aligned} &(L_{\mathrm{a}}+L_{\mathrm{p}})\frac{di}{dt}+(r_{\mathrm{a}}+r_{\mathrm{v}})i+L_{\mu}\frac{di_{\mu}}{dt}+C_{\mathrm{e}}\omega\Phi_{\mathrm{e}}=V_{\mathrm{c}}\\ &L_{\mu}\frac{di_{\mu}}{dt}=i_{\mathrm{k}}r_{\mathrm{k}}.\\ &i=i_{\mu}+i_{\mathrm{k}}. \end{aligned} \right\} \tag{2} \]

This system of equations, written in full coordinates, is nonlinear. In studying quasi-steady-state processes, when the deviations of the output coordinate (armature current) are small, it may be assumed that the ACS is described by linear differential equations. Under this assumption, the values of the parameters entering equations (2) may be determined for the mean value of the magnetizing current. In this case the motor flux $\Phi_{\mu}$ can be determined directly from the no-load characteristic, and the increment of the flux $\Delta \Phi_{\mu}$, for a change of current by $\Delta i_{\mu}$, can be determined as

\[ \Delta \Phi_{\mu}=k_{\mu}\cdot \Delta i_{\mu}, \tag{3} \]

where $k_{\mu}$ is the tangent of the angle of inclination of the tangent to the characteristic at the point of conditional equilibrium, determined by the mean value of the current.

The system of equations (2) for coordinate increments, taking into account expression (3), may be written, after first replacing $\dfrac{d}{dt}=D$, as

\[ \left. \begin{aligned} &(L_{\mathrm{a}}+L_{\mathrm{p}})Di+i(r_{\mathrm{a}}+r_{\mathrm{v}})+L_{\mu}Di_{\mu}+C_{\mathrm{e}}\omega(k_{\mu}i_{\mu}-k_{\mathrm{p}}i)=V_{\mathrm{c}}\\ &L_{\mu}Di_{\mu}=i_{\mathrm{k}}r_{\mathrm{k}}\\ &i=i_{\mu}+i_{\mathrm{k}}. \end{aligned} \right\} \tag{4} \]

The increment sign $\Delta$ has been omitted in the system of equations (4). Carrying out simple transformations, the system of equations (4) can be reduced to system (5):

\[ \begin{cases} (T_sD+1)i+(T_{\mu}D+\kappa)i_{\mu}=\kappa_c V_c,\\ (T_kD+1)i_{\mu}=i, \end{cases} \tag{5} \]

where

\[ T_s=\frac{L_{\mathrm{a}}+L_p}{r};\qquad T_{\mu}=\frac{L_{\mu}}{r};\qquad \kappa=\frac{C_e\omega\kappa_{\mu}}{r};\qquad \kappa_c=\frac{1}{r} \]

\[ r=r_{\mathrm{a}}+r_{\mathrm{v}}-C_e\omega\kappa_p;\qquad T_k=\frac{L_{\mu}}{r_k}. \]

Solving equations (5) jointly and carrying out the Laplace transformations, we obtain:

\[ \begin{aligned} &\bigl[T_sT_kp^2+(T_{\mu}+T_k+T_s)p+1+\kappa\bigr]I(p) =\kappa_cV_c(p)[T_kp+1]-\\ &\qquad{}-\kappa_cV_0T_k+\bigl[T_sT_kp+(T_{\mu}+T_k+T_s)\bigr]i_0+T_sT_ki_0', \end{aligned} \tag{6} \]

where \(i\) and \(i'\) are the initial values of the output coordinate.

From expression (6) we determine the transfer function with respect to the main input \(W_f(p)\) and with respect to the \(\delta\)-input \(W_{\delta}(p)\)

\[ W_f(p)=\frac{\kappa_c(T_kp+1)} {T_sT_kp^2+(T_{\mu}+T_k+T_s)p+1+\kappa}, \tag{7} \]

\[ W_{\delta}(p)= \frac{T_sT_ki_0'+\bigl[T_sT_kp+(T_{\mu}+T_k+T_s)\bigr]i_0-\kappa_cV_0T_k} {T_sT_kp^2+(T_{\mu}+T_k+T_s)p+1+\kappa}. \tag{8} \]

The image of the output coordinate is determined by the relation

\[ I(p)=W_f(p)\cdot V_c(p)+W_{\delta}(p)\delta(p). \tag{9} \]

In the considered automatic control system, the disturbing action—the voltage of the source—has a stepwise character, i.e.,

\[ V_c(t)=h(t)=\mathrm{const};\qquad V_c(p)=\frac{h}{p}. \tag{10} \]

Taking into account that

\[ L\{\delta(t)\}=\delta(p)=1, \tag{11} \]

we write, with allowance for (10) and (11):

\[ I(p)=\bigl[W_f(p)\cdot h+pW_{\delta}(p)\bigr]\frac{1}{p} =W_{\phi}(p)\cdot\frac{1}{p}, \tag{12} \]

where

\[ W_{\phi}(p)=hW_f(p)+pW_{\delta}(p). \]

Expression (12) shows that the current can be considered as the output coordinate of a certain fictitious system, to which, under zero initial conditions, an external action of the unit-step type is applied.

The transfer function of the fictitious system \(W_{\phi}(p)\) can be represented in the general form as

\[ W_{\phi}(p)= \frac{\beta_2T^2p^2+\beta_1Tp+\beta_0} {T^2p^2+2\varepsilon Tp+1}, \tag{13} \]

where

\[ T^{2}=\frac{T_s\cdot T_{\mathrm{k}}}{1+\kappa},\qquad 2\varepsilon T=\frac{T_s+T_{\mathrm{k}}+T_{\mu}}{1+\kappa};\qquad \beta_1=2\varepsilon i_0+T i_0' + \frac{T_{\mathrm{k}}\cdot \kappa_c(h-V_0)}{T(1+\kappa)}; \]

\[ \beta_0=\frac{h\cdot \kappa_c}{1+\kappa};\qquad \beta_2=i_0. \]

After the motor reaches its natural characteristic, regulation of the starting current is carried out by weakening the field. A feature of this operating mode is the stepwise change of the system parameters before each switching operation. The equivalent circuit for this operating mode is presented in Fig. 1, c, and, taking into account the adopted assumptions, may be described for one of the states of the TP by the following system of equations:

\[ \left. \begin{aligned} &L_{\mathrm{ya}}\frac{di}{dt}+r_{\mathrm{ya}}i+L_{\mu}\frac{di_{\mu}}{dt}+i_{\mathrm{v}}r_{\mathrm{v}}+L_s\frac{di_{\mathrm{v}}}{dt} +C_e\omega(\kappa_{\mu}i_{\mu}-\kappa_{\rho}i)=V_c,\\ &i_{\mathrm{sh}}r_{\mathrm{sh}}=L_{\mu}\frac{di_{\mu}}{dt}+i_{\mathrm{v}}r_{\mathrm{v}}+L_s\frac{di_{\mathrm{v}}}{dt},\\ &L_{\mu}\frac{di_{\mu}}{dt}=i_{\mathrm{k}}r_{\mathrm{k}},\\ &i=i_{\mathrm{sh}}+i_{\mathrm{v}},\\ &i_{\mathrm{v}}=i_{\mu}+i_{\mathrm{k}}. \end{aligned} \right\} \tag{14} \]

After transformations analogous to those carried out above, we obtain the transfer functions of the automatic control system with respect to the main input, relative to the image of the network voltage,

\[ W_f(p)= \frac{ \kappa_c\left[T_sT_{\mathrm{k}}p^2+(T_{\mu}+T_s+T_{\mathrm{k}})p+1\right]\times }{ T_{\mathrm{ya}}T_sT_{\mathrm{k}}p^3+ \left[T_{\mu}T_{\mathrm{ya}}+T_sT_{\mathrm{ya}}+T_{\mathrm{ya}}T_{\mathrm{k}}+T_sT_{\mathrm{k}}(1+\kappa_1)\right]p^2+ } \]

\[ \frac{ \times 1 }{ +\left[T_{\mathrm{ya}}+(1+\kappa_1\kappa_2)T_{\mathrm{k}}+(1+\kappa_1)(T_s+T_{\mu})\right]p+ \left[1+\kappa_1\kappa_2+\kappa_1\kappa_2\kappa_3\right] }; \tag{15} \]

and with respect to the \(\delta\)-input,

\[ W_{\delta}(p)= \frac{ \left\{T_{\mathrm{ya}}T_sT_{\mathrm{k}}p^2+ \left[T_{\mu}T_{\mathrm{ya}}+T_sT_{\mathrm{ya}}+T_{\mathrm{ya}}T_{\mathrm{k}}+T_sT_{\mathrm{k}}(1+\kappa_1)\right]p+ \left[T_{\mathrm{ya}}+(1+\kappa_1\kappa_2)T_{\mathrm{k}}+\right.\right. }{ \left\{T_{\mathrm{ya}}T_sT_{\mathrm{k}}p^3+ \left[T_{\mu}T_{\mathrm{ya}}+T_sT_{\mathrm{ya}}+T_{\mathrm{ya}}T_{\mathrm{k}}+T_sT_{\mathrm{k}}(1+\kappa_1)\right]p^2+ \right. } \]

\[ \frac{ \left.\left. +(1+\kappa_1)(T_s+T_{\mu})\right]\right\}i_0+ \left[T_{\mathrm{ya}}T_sT_{\mathrm{k}}p+T_{\mu}T_{\mathrm{ya}}+T_sT_{\mathrm{ya}}+T_{\mathrm{ya}}T_{\mathrm{k}}+T_sT_{\mathrm{k}}(1+\kappa_1)\right]i_0' }{ +\left[T_{\mathrm{ya}}+(1+\kappa_1\kappa_2)T_{\mathrm{k}}+(1+\kappa_1)(T_s+T_{\mu})\right]p+ \left[1+\kappa_1\kappa_2+\kappa_1\kappa_2\kappa_3\right]\right\}\times } \]

\[ \frac{ +T_{\mathrm{ya}}T_sT_{\mathrm{k}}i_0'' -\kappa_c\left[T_sT_{\mathrm{k}}V_0+(T_{\mu}+T_s+T_{\mathrm{k}})V_0+T_sT_{\mathrm{k}}V_0'\right] }{ \times 1 }, \tag{16} \]

where

\[ T_{\mathrm{ya}}=\frac{L_{\mathrm{ya}}}{r};\qquad T_{\mu}=\frac{L_{\mu}}{r_{\mathrm{sh}}+r_{\mathrm{v}}};\qquad T_s=\frac{L_s}{r_{\mathrm{sh}}+r_{\mathrm{v}}};\qquad T_{\mathrm{k}}=\frac{L_{\mu}}{r_{\mathrm{k}}}; \]

\[ \kappa_1=\frac{r_{\mathrm{sh}}}{r};\qquad \kappa_2=\frac{r_{\mathrm{v}}}{r_{\mathrm{sh}}+r_{\mathrm{v}}};\qquad \kappa_3=\frac{C_e\omega\kappa_{\mu}}{r_{\mathrm{v}}};\qquad \kappa_c=\frac{1}{r}, \]

\[ r=r_{\mathrm{ya}}-C_e\omega\kappa_{\rho}. \]

\(i_0,\ i_0',\ i_0''\) are the initial values of the output coordinate,

\(V_0,\ V_0'\) are the initial values of the disturbing action.

For a step change in the supply-network voltage, \(\Delta V_c(t)=h(t)=\mathrm{const}\) and \(\Delta V_c(p)=\dfrac{h}{p}\), the image of the output coordinate is written as

\[ I(p)=\bigl[W_f(p)\cdot h+pW_\delta(p)\bigr]\frac{1}{p} = W_\phi(p)\cdot \frac{1}{p}, \tag{17} \]

where

\[ W_\phi(p)=hW_f(p)+pW_\delta(p). \tag{18} \]

\(W_\phi(p)\)—the fictitious transfer function of the system—after certain transformations is reduced to the form

\[ W_\phi(p)= \frac{\beta_3T^3p^3+\beta_2T^2p^2+\beta_1Tp+\beta_0} {T^3p^3+T^2p^2+2\varepsilon Tp+1}, \tag{19} \]

where

\[ T^3=\frac{T_{\mathrm{я}}T_sT_k}{1+K_1K_2+K_1K_2K_3}; \qquad T^2=\frac{T_\mu T_{\mathrm{я}}+T_sT_{\mathrm{я}}+T_{\mathrm{я}}T_k+T_sT_k(1+K_1)} {1+K_1K_2+K_1K_2K_3}; \]

\[ 2\varepsilon T= \frac{T_{\mathrm{я}}+T_k(1+K_1K_2)+(1+K_1)\cdot(T_s+T_\mu)} {1+K_1K_2+K_1K_2K_3}; \]

\[ \beta_0= \frac{K_ch}{1+K_1K_2+K_1K_2K_3}; \]

\[ \beta_2T^2=T^3i_0''+T^2i_0' +\frac{T_sT_kK_c(h-V_0)} {1+K_1K_2+K_1K_2K_3}; \]

\[ \beta_2=Ti_0'+i_0+ \frac{T_sT_kK_c(h-V_0)} {T^2(1+K_1K_2+K_1K_2K_3)}; \]

\[ \beta_1T=T^3i_0''+T^2i_0'+2\varepsilon Ti_0 +\frac{(T_\mu+T_s+T_k)K_c(h-V_0)} {1+K_1K_2+K_1K_2K_3} - \frac{T_sT_kK_cV_0'} {1+K_1K_2+K_1K_2K_3}; \]

\[ \beta_1=T^2i_0''+Ti_0'+2\varepsilon i_0 +\frac{(T_\mu+T_s+T_k)K_c(h-V_0)} {T(1+K_1K_2+K_1K_2K_3)} - \frac{T_sT_kK_cV_0'} {T(1+K_1K_2+K_1K_2K_3)}; \]

\[ \beta_3=i_0. \]

When using typical phase trajectories to calculate transients under step changes of the supply-network voltage, for the automatic control system under consideration it is necessary, over intervals of constant parameters, to represent the fictitious transfer function (19) as a parallel connection of first- and second-order links. This is possible if the multiplicity of the roots of the system’s characteristic polynomial does not exceed two [2].

The transfer function may be represented as a sum of fractions

\[ W_\phi(p)= \frac{\beta_{10}}{T_1p+1} + \frac{\beta_{22}T_2^2p^2+\beta_{21}T_2p+\beta_{20}} {T_2^2p^2+2\varepsilon_2T_2p+1}, \tag{20} \]

where

\[ \beta_{10}= \frac{\beta_0 T_1^{3}-\beta_1 T T_1^{2}+\beta_2 T^{2}T_1-\beta_3 T^{3}} {T_1T_2^{2}+T^{2}(T_1-2\varepsilon_2T_2)}; \]

\[ \beta_{22}=\beta_3-i_0; \]

\[ \beta_{21}= \frac{\beta_2T^{2}T_1(T_1-2\varepsilon_2T_2)-\beta_0T_1^{2}T_2^{2} +\beta_1T\cdot T_1T_2^{2}-\beta_3T^{3}(T_1-2\varepsilon_2T_2)} {\left[T^{3}+\dfrac{T_1T^{3}}{T_2^{2}}(T_1-2\varepsilon_2T_2^{2})\right]T_2}; \]

\[ \beta_{20}= \frac{\beta_1TT_1^{2}-\beta_2T^{2}T_1+\beta_3T^{3}-\beta_0T_1^{2}\cdot 2\varepsilon T_2+\beta_0T_1T_2^{2}} {T^{3}+\dfrac{T_1T^{3}}{T_2^{2}}(T_1-2\varepsilon_2T_2^{2})}; \]

\[ T_1=\frac{1}{p_1}; \qquad T_2^{2}=\frac{1}{p_2p_3}; \qquad 2\varepsilon_2T_2=\left(\frac{1}{p_2}+\frac{1}{p_3}\right). \]

\(P_1, P_2, P_3\) are the roots of the characteristic equation.

It follows from expression (20) that the fictitious transfer function for the field-weakening mode can be represented as consisting of a parallel-connected inertial element and a second-order element.

References

1. A. S. Rozenfeld, B. I. Yakhinson. Transient Processes and Generalized Functions. Moscow, “Nauka,” 1966.

2. L. P. Smolnikov. Calculation of Nonlinear Electromechanical Systems. Leningrad, “Energiya,” 1968.