Pulse control of current in an active-inductive load
Abstract Generated abstract
This paper analyzes pulse-width current regulation in active-inductive loads using semiconductor switching devices and an uncontrolled shunting valve. It derives transient and steady-state current expressions for three practical circuit arrangements: a reversible bridge circuit, a nonreversible switching circuit, and a circuit with periodic shunting of an added resistance. The authors obtain formulas for maximum and minimum current, ripple amplitude, ripple coefficient, and smooth current components, then use these results to discuss switching-frequency selection under permissible ripple and semiconductor heating constraints. The work indicates that regulation at a constant ripple factor can prevent discontinuous-current operation, but requires pulse-width modulators capable of varying both duty cycle and pulse repetition frequency.
Full Text
PULSE CONTROL OF CURRENT IN AN ACTIVE-INDUCTIVE LOAD
A. I. Zaitsev, A. P. Zaitsev
(Presented by the scientific-technical seminar of the EPP Department of the Electromechanical Faculty)
Introduction
Recently, current regulators operating on the principle of pulse-width modulation have become widely used. In the case of an active-inductive load shunted by an uncontrolled valve, it becomes possible to maintain the load current continuous over a wide control range while the supply voltage has a discrete character. Such regulators employ semiconductor amplifiers operating in a switching mode and controlled by pulse-width modulators. In this case, the utilization factor of the semiconductor devices, which is the ratio of the power in the load to the power loss in the device, increases hundreds of times in comparison with the linear amplification mode. In view of the promise and the increasing spread of the pulse method of current regulation in an active-inductive load (field windings of electrical machines, current stabilizers, etc.), questions of calculating the transient processes associated with establishing the current, and of selecting the switching frequency, are of undoubted interest.
In practice, three methods are used for regulating current in an active-inductive load:
1) the load is connected in a bridge circuit to a source with constant voltage; the bridge circuit provides a periodic reversal of the voltage polarity on the load, and current regulation is carried out by changing the duration of the voltage pulses of both one polarity and the other (reversible circuit);
2) the load is connected to a source with constant voltage through a periodically opening and closing semiconductor switch (non-reversible circuit);
3) the load is connected to a source with constant voltage through an additional resistance; current regulation is carried out by periodic shunting, by a semiconductor switch, of the additional resistance (circuit with shunting of the additional resistance).
In analyzing the processes in the circuits we shall assume that:
1) the uncontrolled valve shunting the load is ideal,
2) the inductance of the load does not depend on the current.
Reversing circuit
Let us denote
\(T\) — the repetition period of the load-voltage pulses,
\(\gamma\) — the relative duration of switching on,
\(n\) — a discrete argument,
\(U_0\) — the amplitude of the pulsed load voltage,
\(i\) — the load current,
\(L_{\text{н}}\) — the load inductance,
\(R_{\text{н}}\) — the active resistance of the load,
\[ T_{\text{н}}=\frac{L_{\text{н}}}{R_{\text{н}}}, \qquad \beta=\frac{T}{T_{\text{н}}}, \qquad I=\frac{U_0}{R_{\text{н}}}. \]
In the time intervals \(nT \leq t \leq T(n+\gamma)\), the circuit equation has the form
\[ L_{\text{н}}\frac{di}{dt}+iR_{\text{н}}=U_0. \tag{1} \]
In the time intervals \(T(n+\gamma)\leq t\leq T(n+1)\), the circuit equation has the form
\[ L_{\text{н}}\frac{di}{dt}+iR_{\text{н}}=-U_0. \tag{2} \]
Introduce a new dependent variable—the relative time
\[ \bar t=\frac{t}{T}. \]
Then (1) and (2), taking into account the above notation, are transformed as
\[ \frac{1}{\beta}\frac{di}{d\bar t}+i=I, \qquad n\leq \bar t\leq n+\gamma, \tag{3} \]
\[ \frac{1}{\beta}\frac{di}{d\bar t}+i=-I, \qquad n+\gamma\leq \bar t\leq n+1. \tag{4} \]
The solution of (3) will be
\[ i(\bar t)=I+C_1e^{-\beta(\bar t-n)}, \]
\[ n\leq \bar t\leq n+\gamma. \tag{5} \]
We determine the integration constant from the initial conditions. For \(\bar t=n\), \(i(\bar t)=i(n)\). Substituting these values into (5), we obtain
\[ C_1=i(n)-I. \tag{6} \]
Substituting (6) into (5), we obtain
\[ i(\bar t)=I+[i(n)-I]e^{-\beta(\bar t-n)}, \tag{7} \]
\[ n\leq \bar t\leq n+\gamma. \]
The solution of (4) will be
\[ i(\bar t)=-I+C_2e^{-\beta(\bar t-n-\gamma)}, \]
\[ n+\gamma\leq \bar t\leq n+1. \tag{8} \]
For \(\bar t=n+\gamma\), from (7) we determine
\[ i(n+\gamma)=I+[i(n)-I]e^{-\beta\gamma}. \tag{9} \]
Minimum steady-state current value \((n=\infty,\ \varepsilon=1)\)
\[ i_{\min}=\frac{I(1-e^{-\beta\gamma})e^{-\beta(1-\gamma)}}{1-e^{-\beta}}. \tag{33} \]
The full swing of the current pulsations in the steady-state regime is
\[ \Delta i=I\frac{(1-e^{-\beta\gamma})[1-e^{-\beta(1-\gamma)}]}{1-e^{-\beta}}. \tag{34} \]
Current pulsation coefficient
\[ \varkappa=e^{\beta(1-\gamma)}. \tag{35} \]
Equation of the smooth component of the current
\[ i_{\mathrm{g}}(t)=I\gamma\left[1-\frac{(1-e^{-\beta\gamma})e^{-\beta(1.5-\gamma)}}{\beta\gamma}\,e^{-\frac{t}{T_{\mathrm{n}}}}\right]\approx \tag{36} \]
\[ \approx I\gamma\left(1-e^{-\frac{t}{T_{\mathrm{n}}}}\right). \]
Circuit with shunting of the added resistance
Initial equations
\[ L_{\mathrm{n}}\frac{di}{dt}+R_0i=U_0,\quad nT\leq t\leq(n+\gamma)T; \tag{37} \]
\[ L_{\mathrm{n}}\frac{di}{dt}+R_{\mathrm{n}}i=U_0,\quad (n+\gamma)T\leq t\leq(n+1)T, \tag{38} \]
where
\[ R_0=R_1+R_{\mathrm{n}}. \]
\(R_1\) is the added resistance.
Expressions for the current in terms of shifted lattice functions
\[ i[n,\varepsilon]=\frac{U_0}{R_0}+\left[R\frac{1-e^{-\beta_{\mathrm{e}}n}}{1-e^{-\beta_{\mathrm{e}}}}-\frac{U_0}{R_0}\right]e^{-\beta'\varepsilon}, \tag{39} \]
\[ 0\leq\varepsilon\leq\gamma; \]
\[ i[n,\varepsilon]=\frac{U_0}{R_{\mathrm{n}}} +\left\{\left[R\frac{1-e^{-\beta_{\mathrm{e}}n}}{1-e^{-\beta_{\mathrm{e}}}}-\frac{U_0}{R_0}\right]e^{-\beta'\gamma} -\frac{U_0R_1}{R_0R_{\mathrm{n}}}\right\}e^{-\beta''(\varepsilon-\gamma)}, \tag{40} \]
\[ \gamma\leq\varepsilon\leq1. \]
Here
\[ \beta'=\frac{T\cdot R_0}{L_{\mathrm{n}}}, \]
\[ \beta''=\frac{TR_{\mathrm{n}}}{L_{\mathrm{n}}}, \]
\[ \beta_{\mathrm{e}}=\beta'\gamma+\beta''(1-\gamma), \]
\[ R=\frac{U_0}{R_0R_{\mathrm{n}}}\left[R_0-R_{\mathrm{n}}e^{-\beta_{\mathrm{e}}}-R_1e^{-\beta''(1-\gamma)}\right]. \]
the minimum steady-state value of the current \((n=\infty,\ \varepsilon=1)\)
\[ i_{\min}=-\frac{I\left[1-2e^{-\beta(1-\gamma)}+e^{-\beta}\right]}{1-e^{-\beta}}; \tag{22} \]
the total amplitude of the current pulsations
\[ \Delta i=i_{\max}-i_{\min}= \frac{2I\left[1-e^{-\beta\gamma}-e^{-\beta(1-\gamma)}+e^{-\beta}\right]}{1-e^{-\beta}}; \tag{23} \]
and the current pulsation coefficient
\[ \varkappa=\frac{i_{\max}}{i_{\min}}= -\frac{1-2e^{-\beta\gamma}+e^{-\beta}} {1-2e^{-\beta(1-\gamma)}+e^{-\beta}}. \tag{24} \]
It is very often necessary to determine the general character of the transient process, which may be represented by the smooth component according to the expression
\[ i_{\mathrm{g}}(n)=\int_0^\gamma i[n,\varepsilon]\,d\varepsilon+ \int_\gamma^1 i[n,\varepsilon]\,d\varepsilon. \tag{25} \]
After evaluating the definite integrals and passing to real time, the smooth component of the current has the form
\[ i_{\mathrm{g}}(t)=I(2\gamma-1)\left\{1- \frac{\left[2e^{-\beta(1-\gamma)}-e^{-\beta}-1\right]e^{-0.5\beta}} {\beta(2\gamma-1)} e^{-\frac{t}{T_{\mathrm{н}}}}\right\}. \tag{26} \]
In the case where \(T_{\mathrm{н}}\gg T\), applying the expansion formulas for exponential functions, we obtain
\[ i_{\mathrm{g}}(t)=I(2\gamma-1)\left(1-e^{-\frac{t}{T_{\mathrm{н}}}}\right). \tag{27} \]
Nonreversing circuit
The initial differential equations for the current over the different time intervals are
\[ L_{\mathrm{н}}\frac{di}{dt}+R_{\mathrm{н}}i=U_0, \qquad nT\leq t\leq T(n+\gamma); \tag{28} \]
\[ L_{\mathrm{н}}\frac{di}{dt}+R_{\mathrm{н}}i=0, \qquad T(n+\gamma)\leq t\leq T(n+1). \tag{29} \]
Using the above method of constructing the difference equation, it is easy to obtain expressions for the load current in terms of shifted lattice functions
\[ i[n,\varepsilon]=I+ \left[ \frac{Ie^{-\beta(1-\gamma)}(1-e^{-\beta\gamma})(1-e^{-\beta n})}{1-e^{-\beta}} -I \right]e^{-\beta\varepsilon}; \tag{30} \]
\[ 0\leq\varepsilon\leq\gamma; \]
\[ i[n,\varepsilon]= \left\{ I+ \left[ \frac{Ie^{-\beta(1-\gamma)}(1-e^{-\beta\gamma})(1-e^{-\beta n})}{1-e^{-\beta}} -I \right]e^{-\beta\gamma} \right\}e^{-\beta(\varepsilon-\gamma)}; \tag{31} \]
\[ \gamma\leq\varepsilon\leq1. \]
The maximum steady-state value of the current \((n=\infty,\ \varepsilon=\gamma)\)
\[ i_{\max}=\frac{I(1-e^{-\beta\gamma})}{1-e^{-\beta}}. \tag{32} \]
is determined from (44) after a number of transformations and neglect of quantities of the second order of smallness.
\[ T \approx \frac{x - 1}{\gamma(1-\gamma)} T'. \tag{47} \]
Here
\[ T'=\frac{L_{\mathrm{н}}}{xR_0 - R_{\mathrm{н}}}\approx \frac{L_{\mathrm{н}}}{R_1}. \]
The power losses in the semiconductor triode increase with the growth of the absolute value of the current through the triode and of the switching frequency. Therefore, as the absolute value of the current increases, it is expedient to reduce the switching frequency so that the temperature of the collector junction remains permissible over the entire range of current regulation. This condition is satisfied by current regulation at a constant ripple factor. With current regulation by this method, the regime of discontinuous currents is impossible in principle. The law of variation \(T=f(\gamma)\) for \(x=\mathrm{const}\) is constructed on the basis of (45), (46), (47).
For current regulation at \(x=\mathrm{const}\), pulse-width modulators are required that make it possible to vary both the relative duration of switching on and the repetition frequency of the pulse-width-modulated pulses.
REFERENCES
- Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Fizmatgiz, 1963.
Maximum steady-state value of the current \((n=\infty,\ \varepsilon=1)\)
\[ i_{\max}= \frac{U_0\left[R_0-R_1 e^{-\beta''(1-\gamma)}-R_{\mathrm n}e^{-\beta_3}\right]} {R_0R_{\mathrm n}\left(1-e^{-\beta_3}\right)} . \tag{41} \]
Minimum steady-state value of the current \((n=\infty,\ \varepsilon=\gamma)\)
\[ i_{\min}= \frac{U_0\left(R_{\mathrm n}+R_1e^{-\beta'\gamma}-R_0e^{-\beta_3}\right)} {R_0R_{\mathrm n}\left(1-e^{-\beta_3}\right)} . \tag{42} \]
The full amplitude of the current ripple in the steady-state regime is
\[ \Delta i= \frac{U_0R_1}{R_0R_{\mathrm n}} \left(1-e^{-\beta'\gamma}\right) \left[1-e^{-\beta''(1-\gamma)}\right]. \tag{43} \]
The current ripple coefficient is
\[ \chi= \frac{ R_{\mathrm n}\left(1-e^{-\beta_3}\right)+ R_1\left[1-e^{-\beta''(1-\gamma)}\right] }{ R_{\mathrm n}\left(1-e^{-\beta_3}\right)+ R_1\left[1-e^{-\beta''(1-\gamma)}\right]e^{-\beta'\gamma} }. \tag{44} \]
An exact derivation of the equation for the smooth component of the current leads to a very complicated expression. Approximately, the smooth component may be determined according to the formula
\[ i_{\mathrm g}(n)\approx \frac{i[n,\gamma]+i[n,1]}{2}. \]
Selection of the switching frequency
The basic requirement imposed in most cases on systems of pulse control of current in an active-inductive load is to obtain, in the steady-state regime, such a waveform of the current curve as would approach the smooth component as closely as possible. From this point of view, it is advantageous to choose the switching frequency as high as possible. However, the switching frequency can be increased only up to a certain limit, at which the effects of switching delay in semiconductor devices and switching losses become appreciable, leading to device failure and degrading the regulating qualities of the systems. From the point of view of heating of semiconductor devices and reliability of system operation, it is advantageous to choose the switching frequency as low as possible under the condition of the permissible ripple of the load current. When selecting the minimum possible switching frequency, the main criterion must be the specified permissible current ripple coefficient \(\chi\) at the lower limit of regulation (for a given \(\gamma\)). Then the period of the switching frequency for the reversible circuit
\[ T=\left(1-\frac{1}{\chi}\right)\left(1-\frac{1}{4\gamma-1}\right)T_{\mathrm n} \tag{45} \]
is determined from (24) by expanding the exponential functions in powers and neglecting quantities of third order of smallness.
For the nonreversible circuit, \(T\) is determined from (35):
\[ T=\frac{\ln\chi}{1-\gamma}T_{\mathrm n}. \tag{46} \]
For the circuit with shunting by an additional resistance, the exact expression for \(T\) is obtained as rather complicated. An approximate expres-