Problems of the Statics of a Pulse Automatic Control System Containing an Ionic Converter Operating into a Capacitance

V. A. Kochenurov, A. A. Tereshchenko

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

In most installations intended for obtaining pulsed magnetic fields of high intensity, the current source consists of batteries of electrical capacitors charged from rectifiers. Since the load resistance of the capacitors in the transient mode (during charging) varies from zero to infinity, ballast resistances are usually used to limit the charging current. However, this leads to a reduction in the efficiency of the installation. The economy of the charging process and the service life of the power equipment of a pulse current generator increase if the charging current is limited by a corresponding change in the control angle of the rectifier during the charging process, by introducing feedbacks with such a calculation that the charging current remains constant. From the experimental point of view, in most cases requirements are also imposed on the stability of the pre-discharge voltage on the capacitive storage unit. Solving the problem of creating the most advanced automatic control system for the charging current with stabilization of the pre-discharge voltage on the capacitive storage unit is associated with certain difficulties. First of all, it is necessary to determine the parameters of the system, control by which will ensure the minimum static control error for a given sensitivity of the measuring devices. In addition, for a given control accuracy it is necessary to establish the corresponding structure and transfer coefficients of the regulator. The block diagram of a pulse current generator in the general case may be represented in the form of sequentially connected elements: a grid-control system, a six-phase rectifier, a capacitive energy storage unit, and a forming device (Fig. 1). The system under consideration is nonlinear; therefore an analytical expression for determining the static accuracy of control in the general case can be obtained under the assumption that the disturbances acting in the system are small and that changes in the controlled quantities occur within a limited range.

The disturbances causing deviation of the current and voltage on the capacitors from the specified value are fluctuations of the supply-network voltage, the control voltage, changes in the residual voltage on the capacitors during operation, and other destabilizing factors. The initial equations of electrical equilibrium determining the operation of the system have the form:

\[ i_c Z + u_c(t) + 2E_b = \frac{3}{\pi}\sqrt{3}\,E_m \sin v \]

\[ v = (f u_{\mathrm{ctrl}}) \]

\[ u_{\mathrm{ctrl}} = f\{i_c,u_c(t),E_m,v,t\}; \tag{1} \]

Fig. 1.
Diagram labels: rectifier; tracking-control system; device forming the control signal; \(u_{\mathrm{ctrl}}\); \(\Delta \alpha\); \(C\); 1; 2.

Fig. 2.
Diagram labels: equivalent generator; \(E_e\); \(R\); \(L\); \(E_e=\dfrac{3}{\pi}\sqrt{3}E_m\sin v\); load; \(i_c\).

Let us write these equations in increments, which makes it possible to linearize the individual dependences. The statics equation of the converter equivalent circuit (Fig. 2) is written as

\[ i_{c0}Z+\Delta i_c Z+u_{c0}+\frac{i_{c0}\Delta t}{C}+\frac{\Delta i_c \Delta t}{C}+2E_b = \]

\[ = \frac{3}{\pi}\cdot\sqrt{3}\,(E_{m0}+\Delta E_m)\sin(v_0+\Delta v), \tag{2} \]

where

\[ i_c=i_{c0}+\Delta i_c; \]

\[ u_c=u_{c0}+\Delta u_c(t)=u_{c0}+\frac{i_c\Delta t}{C} =u_{c0}+\frac{i_{c0}\cdot\Delta t}{C}+\frac{\Delta i_c\cdot\Delta t}{C}; \]

\[ \Delta i_c \]
is the increment of the current in the charging circuit;

\[ \Delta u_c(t) \]
is the increment of the voltage on the capacitive storage element;

\[ \Delta E_m \]
is the increment of the supply-network voltage;

\[ \Delta v \]
is the increment of the firing angle of the ionic converter;

\[ Z \]
is the complex impedance taking into account all active and inductive resistances of the equivalent circuit.

Transform equation (2) and solve it with respect to the current increment:

\[ i_c Z+\Delta i_c Z+u_{c0}+\frac{i_{c0}\Delta t}{C}+\frac{\Delta i_c\Delta t}{C}+2E_b =\frac{3}{\pi}\sqrt{3}(E_{m0}+\Delta E_m)\cdot \]

\[ \cdot(\sin v_0\cdot\cos\Delta v+\cos v_0\sin\Delta v). \]

For small

\[ \Delta v;\qquad \cos\Delta v=1;\qquad \sin\Delta v=\Delta v; \]

\[ \Delta i_c= \frac{ \frac{3}{\pi}\sqrt{3}E_{m0}\sin v_0 \left[ \left(1+\frac{\Delta E_m}{E_{m0}}\right)\operatorname{ctg}v_0\Delta v+ \frac{\Delta E_m}{E_{m0}} \right] -\frac{i_{c0}\Delta t}{C} }{ Z+\frac{\Delta t}{C} }. \tag{3} \]

From equation (3) it is evident that the regulation must be programmed, since \(\Delta i_c\) is a function not only of \(\Delta E_m\) and \(\Delta v\), but also

and time. Let us consider the static accuracy of current regulation for three cases: control of the ionic converter is carried out only by the charging current, by the voltage across the capacitance, or jointly by current and voltage. The equation of the grid-control device, which will be the same for all three cases, is written as

\[ \Delta v=f(u_{\mathrm{ctrl}})=k_1u_{\mathrm{ctrl}}, \tag{4} \]

where \(k_1\) determines the slope of the characteristic of the grid-control system. Let us consider the indicated cases in order:

\[ u_{\mathrm{ctrl}}=k_y(u_{0i}-k_{gi}i_c) =k_y\left[u_{0i}-(i_{c0}+\Delta i_c)\cdot k_{gi}\right] =-k_y\cdot k_{gi}\cdot \Delta i_c, \tag{5} \]

where

\[ \begin{aligned} k_y&\text{ is the gain coefficient of the amplifier;}\\ k_{gi}&\text{ is the proportionality coefficient between the charging current and}\\ &\text{the voltage at the output of sensor 1 (Fig. 1);}\\ u_{0i}&\text{ is the reference voltage (set current).} \end{aligned} \]

The simultaneous solution of equations (3, 4, and 5) gives the static accuracy of current regulation for the first case:

\[ \Delta i_c= \frac{ \dfrac{3}{\pi}\sqrt{3}\,\mathrm{Sin}\,v_0\,\Delta E_m-\dfrac{i_{c0}\Delta t}{C} }{ Z+\dfrac{\Delta t}{C} +\dfrac{3}{\pi}\sqrt{3}E_{m0}\,\mathrm{Cos}\,v_0 \left(1+\dfrac{\Delta E_m}{E_{m0}}\right)k_1k_yk_{gi} }. \tag{6} \]

It follows from expression (6) that the static accuracy of current regulation depends on the time \(\Delta t\), as well as on the change in voltage across the capacitors,

\[ u_c(t)=\frac{i_{c0}\Delta t}{C}. \]

As the latter increases, the static error increases.

Since in expression (6) the magnitude of the static error is a linear function of time, the control signal may be represented in the form

\[ u_{\mathrm{ctrl}}=k_t\Delta t \tag{7} \]

Solving equations (7, 4, and 3) simultaneously, we obtain

\[ \Delta i_c= \frac{ \dfrac{3}{\pi}\sqrt{3}\,\mathrm{Sin}\,v_0\,\Delta E_m +\Delta t\left[ \dfrac{3}{\pi}\sqrt{3}E_{m0}\,\mathrm{Cos}\,v_0 \left(1+\dfrac{\Delta E_m}{E_{m0}}\right)k_1k_yk_t -\dfrac{i_{c0}}{C} \right] }{ Z+\dfrac{\Delta t}{C} }. \tag{8} \]

If

\[ \frac{3}{\pi}\sqrt{3}E_{m0}\cdot \mathrm{Cos}\,v_0 \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1k_yk_t -\frac{i_{c0}}{C}=0, \tag{9} \]

then

\[ \Delta i_c= \frac{ \dfrac{3}{\pi}\sqrt{3}\,\mathrm{Sin}\,v_0\,\Delta E_m }{ Z+\dfrac{\Delta t}{C} }, \tag{10} \]

i.e., introducing a linear function of time into the control law reduces the static error in proportion to the time \(\Delta t\).

\[ u_{\mathrm{ctrl}}=-k_{gi}\Delta i_c+k_t\Delta t. \tag{11} \]

Solving equations (11, 4, and 3) with respect to the current increment, we obtain

\[ \Delta i_c = \frac{ \frac{3}{\pi}\sqrt{3}\,\sin v_o\,\Delta E_m }{ Z+\frac{\Delta t}{C}+\frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1 k_y k_{gi} } + \frac{ \frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1 k_y k_{gi} -\frac{i_{c0}}{C} }{ Z+\frac{\Delta t}{C}+\frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1 k_y k_{gi} }. \tag{12} \]

If the condition

\[ \frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1 k_y k_t-\frac{i}{C} \tag{13} \]

is satisfied,

\[ \Delta i_c = \frac{ \frac{3}{\pi}\sqrt{3}\,\sin v_o\,\Delta E_m }{ Z+\frac{\Delta t}{C}+\frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right)k_1 k_y k_{gi} }. \tag{14} \]

The coefficients can be determined from condition (13):

\[ k_1 k_y k_t = \frac{i_{c0}}{ \frac{3}{\pi}\sqrt{3}\,E_{m0}\cos v_o \left(1+\frac{\Delta E_m}{E_{m0}}\right) }. \tag{15} \]

It follows from expression (14) that, when condition (13) is fulfilled, the static accuracy of regulation increases in proportion to the time \(\Delta t\). Since \(u_c(t)\) varies in proportion to time (under the condition \(i_c=\mathrm{const}\)), the relationship between \(\Delta t\) and \(u_c(t)\) can be expressed as

\[ \Delta t=\frac{t_z}{\Delta u_{cm0}}\cdot \Delta u_c(t). \tag{16} \]

Equation (11), taking (16) into account, is written as

\[ u_{\mathrm{ctrl}}=-k_{gi}\Delta i_c+\frac{k_t t_z}{\Delta u_{cm0}}\,\Delta u_c(t). \tag{17} \]

Thus, analysis of equations (6, 8, 12) shows that, in order to ensure the maximum static accuracy of regulation at a given sensitivity of the measuring devices, it is necessary to control the ionic converter by the charging current and by the voltage on the capacitive storage element. In this case, the transfer coefficients of the feedback circuit of the regulation system are determined from expression (15).

References

1. B. M. Gutkin. Ionic DC Drive. Energiya, 1965.