Study of a Class of Two-Position DC Voltage Stabilizers
Abstract Generated abstract
This paper analyzes a class of two-position switching DC voltage stabilizers in which transistor switching is governed by the choke current reaching zero and a saturation-current threshold set by the output-voltage control circuit. Under simplifying assumptions, the stabilizer dynamics are formulated in dimensionless form and studied on a two-sheet phase plane using point transformations to determine stable self-oscillations. Parametric correspondence functions are derived for the transistor-open and transistor-closed intervals, reducing the calculation of oscillation parameters to finding a stable fixed point of the resulting transformation. An example calculation gives the self-oscillation frequency and ripple amplitude, and comparison over varying load resistance shows satisfactory agreement between computed and experimental results.
Full Text
STUDY OF A CLASS OF TWO-POSITION DC VOLTAGE STABILIZERS
O. S. VADUTOV, Yu. N. MALCHIK
(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)
A class of switching DC voltage stabilizers is known in which the transition of a power transistor, operating in switching mode, from the closed state to the open state and from the open state to the closed state occurs at the instants when the choke current reaches, respectively, zero and the value \(I_{\text{k.max}}\), equal to the transistor saturation current [1, 2]. A simplified circuit of such a stabilizer is shown in Fig. 1. The saturation current is determined by the base current; by varying it as a function of the output voltage, stabilization of the output voltage is effected. Then
\[ I_{\text{k.max}} = f(U_{\text{op}}, U_{\text{n}}). \tag{1} \]
Stabilizers of this type operate in a regime of stable self-oscillations with an amplitude and frequency that depend on the parameters of the filter, the load, and the supply voltage \(U_{\text{p}}\). Determination of the amplitude and frequency of these self-oscillations is an important problem from a practical point of view, but, as noted in [3], a rather complicated one. In the present work this problem is solved by the methods of the phase plane and point transformations, which have found wide application in the study of various nonlinear systems.
Fig. 1. Simplified circuit of a DC voltage stabilizer.
Equations of the stabilizer and the phase plane. To simplify the analysis, we shall adopt the following assumptions: the resistance of the open transistor, the diode in the forward direction, the internal resistance of the supply source, and the active resistance of the choke and filter capacitor are equal to zero, while those of the closed transistor and the diode in the reverse direction are infinity; the switching processes occur instantaneously.
Taking the adopted assumptions into account, the processes in the stabilizer can be described by the equations
\[ R_{\text{n}} C \frac{dU_{\text{n}}}{dt} + U_{\text{n}} = R_{\text{n}} i_1, \]
\[ L_1 \frac{di_1}{dt} = U_{\text{p}} - U_{\text{n}} \tag{2} \]
on the time intervals when the transistor is open, and by the equations
\[ R_{\mathrm{n}} C \frac{dU_{\mathrm{n}}}{dt} + U_{\mathrm{n}} = R_{\mathrm{n}} i_2, \]
\[ L_2 \frac{di_2}{dt} = - U_{\mathrm{n}} \tag{3} \]
on the time intervals when the transistor is closed; moreover, at the moment when the state of the transistor changes, the currents \(i_1\) and \(i_2\), which are respectively the currents in windings \(W_1\) and \(W_2\) of the filter choke, are related by
\[ n i_1 = i_2, \tag{4} \]
where \(n = \dfrac{W_1}{W_2}\).
For convenience, let us write the equations of the stabilizer, respectively for the open and closed states of the transistor, in dimensionless units:
\[ \frac{1}{2 \xi_1}\frac{dx}{d\tau} + x = y_1, \]
\[ 2 \xi_1 \frac{dy_1}{d\tau} = 1 - x \tag{5} \]
\[ \frac{1}{2 \xi_2}\frac{dx}{d\theta} + x = y_2, \]
\[ 2 \xi_2 \frac{dy_2}{d\theta} = -x, \tag{6} \]
where
\[ x = \frac{U_{\mathrm{n}}}{U_{\mathrm{p}}}; \qquad y_k = \frac{R_{\mathrm{n}} i_k}{U_{\mathrm{p}}}; \qquad \xi_k = \frac{\sqrt{L_k}}{2 R_{\mathrm{n}} \sqrt{C}} \quad (k = 1, 2); \]
\[ \tau = \frac{t}{\sqrt{L_1 C}}; \qquad \theta = \frac{t}{\sqrt{L_2 C}}. \]
The processes in the stabilizer can be investigated on the phase plane (Fig. 2), consisting of two sheets I and II (according to the number of transistor states), respectively with coordinates \(x, y_1\) and \(x, y_2\). Sheet I displays the processes in the stabilizer with the transistor open, and sheet II—with the transistor closed. The motion of the representative point on sheet I is governed by equations (5). The boundary of sheet I (line \(S\)) is determined from the condition for changing the open state of the transistor to the closed state, that is, by equation (1), represented in relative units as
Fig. 2. Phase plane
\[ y_1 = \varphi(a, bx). \tag{7} \]
Since we are interested only in the processes near steady-state regimes, equation (7) may be approximated by the straight line
\[ y_1 = a - bx. \tag{8} \]
Upon reaching the boundary of sheet I, the representative point passes to sheet II in accordance with relation (4), which in relative units has the form
\[ y_2 = n y_1 \tag{9} \]
On sheet II the motion of the representative point is governed by equations (6). The edge of sheet II (the line \(S_1\)), upon reaching which the representative point passes to sheet 1, is determined from the condition for the transistor to pass into the open state
\[ i_2=0 \]
or, in relative units,
\[ y_2=0. \]
Determination of the parameters of stable self-oscillations in the stabilizer reduces to finding a stable limit cycle on the phase plane (Fig. 2).
Point transformation. To determine the limit cycle, let us construct the point transformation \(\Pi\) of the line \(S\) into itself. This transformation can be represented as the product of two transformations: the transformation \(\Pi_1\) of the line \(S\) into the line \(S_1\), carried out along trajectories on sheet II, and the transformation \(\Pi_2\) of the line \(S_1\) into the line \(S\), which is carried out along trajectories of sheet 1.
The transformation \(\Pi_1\) consists of a “fast” and a “slow” part. The “fast” part of the transformation is caused by the stepwise change in the choke current at the instant when the transistor closes, according to expression (9)
\[ s' = ns. \]
The “slow” part of the transformation can be obtained if, in the solution of the system of equations (6),
\[ x=e^{-\xi_2\theta}\left(x_0\cos\omega_2\theta-\frac{2\xi_2y_{20}-\xi_2x_0}{\omega_2}\sin\omega_2\theta\right), \]
\[ y_2=e^{-\xi_2\theta}\left(y_{20}\cos\omega_2\theta-\frac{2\xi_2^2y_{20}-x_0}{2\xi_2\omega_2}\sin\omega_2\theta\right), \]
where
\[ \omega_2=+\sqrt{1-\xi_2^2}, \]
we substitute the coordinates of the point \(s'(x_0,s')\), from which the representative point begins its motion, and the coordinates of the point \(s_1(y_2=0,x=s_1)\), at which the representative point arrives after the time \(\theta=\theta_1/\omega_2\). Solving the equalities thereby obtained with respect to \(s\) and \(s_1\), we obtain the correspondence function \(s_1=f(s)\) of the transformation \(\Pi_1\), written in parametric form
\[ s=\frac{a}{b}\, \frac{\dfrac{1}{2\xi_2 n\omega_2}\sin\theta_1} {\cos\theta_1+\dfrac{2\xi_2^2bn+1}{2\xi_2bn\omega_2}\sin\theta_1}; \tag{10} \]
\[ s_1=\frac{a}{b}\, \frac{e^{-\frac{\xi_2}{\omega_2}\theta_1}} {\cos\theta_1+\dfrac{2\xi_2^2bn+1}{2\xi_2bn\omega_2}\sin\theta_1}, \]
where \(\theta_1\) is the parameter of the transformation.
The transformation \(\Pi_2\) consists only of the “slow” part. If we assume that the representative point at the instant \(\tau=0\) begins its motion from the point \(s_1(y_{10}=0,\ x=s_1)\), and after the time \(\tau=\tau_1/\omega_1\)
arrives at the point \(s_2\) \(\left(y_1=s_2,\ x=\dfrac{a-s_2}{b}\right)\), substitute the coordinates of the points \(s_1\) and \(s_2\) into the solution of equations (5)
\[ x=e^{-\xi_1\tau}\left[(x_0-1)\cos\omega_1\tau+ \frac{2\xi_1 y_{10}-(x_0+1)\xi_1}{\omega_1}\sin\omega_1\tau\right]+1, \]
\[ y_1=e^{-\xi_1\tau}\left[(y_{10}-1)\cos\omega_1\tau+ \frac{2\xi_1^{2}(y_{10}-1)-x_0+1}{2\xi_1\omega_1}\sin\omega_1\tau\right]+1, \]
where
\[ \omega_1=+\sqrt{1-\xi_1^2}, \]
and then solve the equalities thus obtained with respect to \(s_1\) and \(s_2\). We then obtain the correspondence function \(s_2=\varphi(s_1)\) of the transformation \(\Pi_2\), written in parametric form:
\[ s_2=1+ \frac{ \dfrac{b-a+1}{2\xi_1 b\omega_1}\sin\tau_1-e^{-\frac{\xi_1}{\omega_1}\tau_1} }{ \cos\tau_1-\dfrac{2\xi_1^2 b+1}{2\xi_1 b\omega_1}\sin\tau_1 }; \]
\[ s_1= \frac{ \dfrac{b+1}{b}\cos\tau_1+ \dfrac{2\xi_1^2(b+1)-1}{2\xi_1 b\omega_1}\sin\tau_1- \dfrac{b-a+1}{b}e^{\frac{\xi_1}{\omega_1}\tau} }{ \cos\tau_1-\dfrac{2\xi_1^2 b+1}{2\xi_1 b\omega_1}\sin\tau_1 }. \tag{11} \]
To determine the stable limit cycle, it is necessary to find the corresponding stable fixed point of the point transformation \(\Pi\), i.e., to solve the system of transcendental equations
\[ s_2(\tau_1)=s(\Theta_1), \]
\[ s_1(t_1)=s_1(\Theta_1). \tag{12} \]
Fig. 3. Schematic diagram of a dc voltage stabilizer
We find the solution of system (12) graphically. The two points of intersection of the curves \(s=f(s_1)\), \(s_2=\varphi(s_1)\) on the Lameray diagram (Fig. 3) are the fixed points of the point transformation \(\Pi\). The point with coordinates
\[ \left(s_1=\frac{a}{b},\ s=0\right) \]
corresponds to an unstable limit cycle, since all sequences taken in its neighborhood diverge, while the point with coordinates \((s_1=\bar{s}_1,\ s=\bar{s})\) determines the parameters of the stable limit cycle.
Parameters of self-oscillations. The frequency of the self-oscillations can be determined from the formula
\[ F=\left(\frac{\bar{\theta}_1}{\omega_2}+\frac{\bar{\tau}_1}{\omega_1}\right)^{-1}, \]
where \(\dfrac{\bar{\theta}_1}{\omega_2}\) and \(\dfrac{\bar{\tau}_1}{\omega_1}\) are, respectively, the travel times of the representative point
along the trajectory of the limit cycle from the line \(S\) to the line \(S_1\), and the time of the return run. In dimensional units the frequency is
\[ f=\left(\sqrt{L_2C}\,\frac{\theta_1}{\omega_2}+\sqrt{L_1C}\,\frac{\overline{\tau}_1}{\omega_1}\right)^{-1}. \tag{13} \]
The amplitude of the self-oscillations can be determined (Fig. 2) from the formula
\[ \frac{\Delta U}{2}=\frac{x_2-x_1}{2}\,U_{\text{p}}. \tag{14} \]
Example. Determine the frequency and amplitude of the self-oscillations in a DC-voltage stabilizer whose circuit is shown in Fig. 3.
Initial data: \(U_{\text{p}}=22\ \text{V}\); \(U_{\text{n}}=15\ \text{V}\); \(R_1=2.2\ \text{k}\Omega\); \(R_2=385\ \Omega\); \(R_3=615\ \Omega\); \(C_1=400\ \mu\text{F}\); \(L_{\text{dr}}=2.4\ \text{mH}\); reference voltage of the Stabilitron \(U_{\text{op}}=9.1\ \text{V}\); transformer transformation ratio \(k=1.5\); gain coefficients of transistors T1, T2, and T3, respectively, \(\beta_1=22\), \(\beta_2=29\), \(\beta_3=18\).
- We determine analytical expressions for the coefficients \(a\) and \(b\). We approximate the input characteristics of transistors T2 and T3 by the broken line
\[ I_{b i}=\frac{1}{r_{b i}}\left(U_{e b i}-U_{\text{op}}\right)\quad (i=2,3), \tag{15} \]
where \(r_{b i}\) is the dynamic resistance of transistors T2 and T3 at the operating point.
Taking equation (15) into account, we determine an expression for the base current \(I_{b2}\) of transistor T2:
\[ I_{b2}=\frac{k\left(U_{\text{p}}-U_{\text{n}}\right)-U_{01}}{R_1+r_{b3}} -\frac{R_1}{R_1+r_{b2}}\, \frac{\beta_3}{r_{b3}} \left[ \left(\frac{R_3}{R_2+R_3}\,U_{\text{n}}-U_{\text{op}}\right)-U_{03} \right] \tag{16} \]
and, substituting the expression obtained into the equation
\[ I_{k\max}=\beta_1(1+\beta_2)I_{b2}, \]
we obtain, in relative units,
\[ a=\frac{\beta_1(1+\beta_2)}{R_1+r_{b2}} \left[ k+\frac{R_1\beta_3}{r_{b3}}\cdot \frac{U_{\text{op}}+U_{03}}{U_{\text{p}}} -\frac{U_{02}}{U_{\text{p}}} \right]R_{\text{n}}, \]
\[ b=\frac{\beta_1(1+\beta_2)}{R_1+r_{b2}} \left[ k+\frac{R_1R_3\beta_3}{r_{b3}(R_2+R_3)} \right]R_{\text{n}}. \tag{17} \]
For a load resistance \(R_{\text{n}}=30\ \Omega\),
\[ a=380;\qquad b=555.1. \]
- Having constructed the iteration functions \(S=f(S_1)\) and \(S_2=\varphi(S_1)\) [from formulas (10) and (11)] on the Lameray diagram (Fig. 4), we find that the point with coordinates \(\overline{S}=1.5\), \(\overline{S}=0.679\) is a fixed point of the point transformation \(\Pi\). Here
\[ \frac{\theta_1}{\omega_2}=\frac{0.22}{0.998};\qquad \frac{\overline{\tau}_1}{\omega_1}=\frac{0.45}{0.993}. \]
Fig. 4. Lameray diagram
- The frequency of self-oscillations in the stabilizer is found from formula (13)
\[ f=\frac{1}{\sqrt{2.4\cdot 0.4\cdot 10^{-6}}}\cdot \frac{0.998}{0.22+0.45}=1500\ \text{Hz}. \]
- The amplitude of the self-oscillations is determined from formula [4]:
\[ \Delta U_{\sim}=\frac{U_{\text{n}}\cdot T\cdot t_{\text{off}}}{8LC}= \]
\[ =\frac{15\cdot 0.67\cdot 0.22}{8\cdot 2.4\cdot 0.4\cdot 10^{-6}}=0.21\ \text{V}. \]
Fig. 5. Dependence of the frequency and amplitude of pulsations; experimental data
The calculation is then repeated for various values of the load resistance. From the obtained calculation results, the dependences \(f=\varphi(I_{\text{n}})\) and \(\Delta U_{\sim}=\psi(I_{\text{n}})\) are plotted (respectively curves 1 and 2 in Fig. 5). The experimental data are also given here. The agreement between the calculated data and the experimental data may be considered satisfactory.
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