On the Calculation of a Switching DC Voltage Stabilizer

O. S. VADUTOV

(Submitted by the scientific seminar
of the Department of Automation and Telemechanics)

A two-position DC voltage stabilizer (Fig. 1), consisting of a transistor \(T\) operating in switching mode, an \(LC\) filter, a load \(R\), a diode \(D\), a comparison circuit \(CC\), and a threshold modulator \(PM\) (Schmitt trigger), belongs to relay automatic-control systems and operates in a regime of stable self-oscillations.

In calculating a stabilizer of this type, the following problems arise:

1) determining the frequency and amplitude of the self-oscillations for known stabilizer parameters;

2) selecting some stabilizer parameter for prescribed values of the frequency and amplitude of the self-oscillations.

Fig. 1

To solve these problems, the method of harmonic linearization [1] and Ya. Z. Tsypkin’s frequency method [2] are used. However, the procedure proposed in [1, 2] for solving the second problem requires numerous graphical constructions.

In the present work, the final result of investigating a two-position stabilizer by the method of harmonic linearization is reduced to a nomogram, which makes it possible to solve both problems for any practically possible stabilizer parameters. For simplicity of the investigation, as in [1, 2], it is assumed that the switch (transistor \(T\)), the inductance \(L\), the diode \(D\), and the capacitor \(C\) are ideal elements.

The block diagram of the stabilizer is shown in Fig. 2,a. The transfer function of the linear part of the system has the form

\[ W_\ell(s)=\frac{2}{LCs^2+2\varepsilon\sqrt{LC}\,s+1}, \tag{1} \]

where \(\varepsilon=\dfrac{1}{2R}\sqrt{\dfrac{L}{C}}\), and the characteristic of the relay element is given in Fig. 2,b.

The type of DC voltage stabilizer under consideration operates in a regime of stable asymmetric self-oscillations. According to the method of harmonic linearization, the input signal of the relay element in the steady-state regime can be written in the form

\[ U_1 = U_0 - U_{\text{out}} = \Delta U_{\text{out}} + U_{\text{out}_m}\sin \omega t, \tag{2} \]

where \(\Delta U_{\text{out}}\) is the constant component, which may be called the static error of the stabilizer; \(U_{\text{out}_m}\) is the amplitude of the self-oscillations. The output quantity of the relay element in the steady-state regime is a sequence of voltage pulses of height \(U_{\text{in}}\), frequency \(\omega\), and duty ratio \(\gamma\).

Fig. 2

As is known from [3], the equivalent complex gain of a relay element with hysteresis is equal to

\[ W_{\text{n}}(U_{\text{out}_m}, \gamma) = \frac{2U_{\text{in}}}{\pi U_{\text{out}_m}} \sin \pi\gamma \cdot \exp\left( j \arcsin \frac{\Delta U}{U_{\text{out}_m}\sin \pi\gamma} \right), \tag{3} \]

where the constant component at the input of the relay element \(\Delta U_{\text{out}}\) is taken into account through the duty ratio of the output pulses \(\gamma\). The relation between \(\Delta U_{\text{out}}\) and \(\gamma\), according to [3], is determined by the expression

\[ \Delta U_{\text{out}_m} = \sqrt{ U_{\text{out}_m}^{2}\cos^{2}\pi\gamma - \Delta U^{2}\ctg^{2}\pi\gamma }. \tag{4} \]

The conditions for the existence of self-oscillations in the stabilizer are written in the form of two equations, respectively for the periodic and constant components:

\[ W_{\text{n}}(U_{\text{out}_m}, \gamma) W_{\text{l}}(j\omega) + 1 = 0; \tag{5} \]

\[ U_0 = U_{\text{out}_0} + \Delta U_{\text{out}} = \gamma U_{\text{in}} + \sqrt{ U_{\text{out}_m}^{2}\cos^{2}\pi\gamma - \Delta U^{2}\ctg^{2}\pi\gamma }. \tag{6} \]

To determine the steady-state values of the amplitude \(U_{\text{out}_m}\), frequency \(\omega\) of the self-oscillations, and the duty ratio of the pulses \(\gamma\), for known stabilizer parameters and a specified value of the output voltage \(U_0\), it is necessary to solve equations (5) and (6) jointly, which presents considerable difficulty. The problem is substantially simplified if \(\gamma\) is known. In a first approximation it may be assumed that \(\Delta U_{\text{out}} \approx 0\), and then

\[ \gamma \approx \frac{U_0}{U_{\text{in}}}. \tag{7} \]

We shall assume that \(U_{\text{out}_0}\) is the specified value of the output voltage, and then

\[ \gamma = \frac{U_{\text{out}_0}}{U_{\text{in}}}, \tag{8} \]

while the quantity \(U_0\) is unknown and must be determined.

Having solved equation (5) for a known \(\gamma\), one can determine the amplitude and frequency of the self-oscillations, and then determine the value \(U_0\) from equation (6). Equation (5) is usually solved graphically in the complex plane by constructing the characteristics \(W_\ell(j\omega)\) and \(-1/W_n\). However, when the parameters of the relay element and of the linear part are changed, these characteristics must be replotted. Taking into account the properties of expression (3) and writing equation (5) in the form of amplitude- and phase-balance conditions,

\[ \left|W_\ell(j\omega)\right|=-\frac{1}{W_n(U_{\text{out}m},\gamma)}; \tag{9} \]

\[ \arg W_\ell(j\omega)=\arg\left(-\frac{1}{W_n(U_{\text{out}m},\gamma)}\right), \tag{10} \]

one can propose, for determining the parameters of the self-oscillations, the nomogram shown in Fig. 3.

In the first quadrant, in the coordinates \(20\lg |W_\ell(j\omega)|\), \(\arg W_\ell(j\omega)\), frequency characteristics of the linear part of the system are plotted for different values of \(\varepsilon\). Lines of equal relative frequencies \(\Omega=\sqrt{LC}\omega\) are also drawn there. In the second quadrant, in the coordinates \(20\lg\left|-1/W_n\right|\), \(U_{\text{out}m}/U_{\text{out}}\), the dependence

\[ 20\lg\left|-\frac{1}{W_n}\right| = 20\lg\frac{\pi\gamma}{2\sin\pi\gamma}\cdot \frac{U_{\text{out}m}}{U_{\text{out}}}. \tag{11} \]

is plotted for different \(\gamma\).

In the fourth quadrant, for different \(\gamma\), the dependence of \(\arg\left|-1/W_n\right|\) on the quantity \(U_{\text{out}m}/U_{\text{out}}\) is plotted; it is determined by the expression

\[ \arg\left|-\frac{1}{W_n}\right| = \pi-\arcsin \frac{\Delta U}{U_{\text{out}m}\sin\pi\gamma}. \tag{12} \]

And in the third quadrant lines are drawn for equal values of the relative width of the hysteresis loop of the relay element, \(\Delta U/U_{\text{out}}\), which, for a known value of \(\Delta U/U_{\text{out}}\), make it possible to pass from the axis \(U_{\text{out}m}/U_{\text{out}}\) to the axis \(U_{\text{out}m}/\Delta U\).

The amplitude and relative frequency of the self-oscillations, for known values of \(\varepsilon\), \(\gamma\), and \(\Delta U/U_{\text{out}}\), are determined very simply by trial selection.

Figure 3 shows an example of determining the parameters of the self-oscillations for the following parameter values: \(\varepsilon=0.15\); \(\gamma=0.6\); \(\Delta U/U_{\text{out}}=0.0003\). In this case the following result was obtained: the relative amplitude of the self-oscillations \(U_{\text{out}m}/U_{\text{out}}=0.0103\) and the relative frequency \(\Omega=10\). It should be noted that the nomogram gives approximate results because of the approximate nature of the harmonic-linearization method underlying the nomogram.

Analysis of the obtained nomogram makes it possible to draw several conclusions useful in designing this type of stabilizer:

Fig. 3.

Visible labels and inscriptions in the graph:

• \(20\lg \dfrac{1}{|W_n|},\ 20\lg |W_n(j\Omega)|,\ \mathrm{dB}\)
• \(\arg W_n(j\Omega),\ \arg\left(-\dfrac{1}{W_n}\right),\ \mathrm{deg}\)
• \(\dfrac{U_{\mathrm{out}\,m}}{U_{\mathrm{out}}}\)
• \(\dfrac{\Delta U}{U_{\mathrm{out}}}\)
• \(\dfrac{U_{\mathrm{out}\,m}}{\Delta U}\)

Curve and scale markings visible in the plots:

• \(\gamma = 0.1,\ 0.3,\ 0.5,\ 0.6,\ 0.7,\ 0.75,\ 0.8,\ 0.9\)
• \(\Omega = 20,\ 18,\ 16,\ 14,\ 12,\ 10,\ 9,\ 8,\ 7,\ 6,\ 5,\ 4\)
• \(\chi = 0.1,\ 0.15,\ 0.2,\ 0.25,\ 0.3,\ 0.35,\ 0.4,\ 0.5,\ 0.6,\ 0.7,\ 0.8,\ 0.9,\ 1.0,\ 1.1,\ 1.2\)
• \(\chi = 0.5\)
• \(0.6(0.4)\)
• \(0.7(0.3)\)
• \(0.9(0.1)\)
• \(0.8(0.2)\)
• \(0.75(0.25)\)
• \(0.00004,\ 0.00008,\ 0.00012,\ 0.00016\)
• \(0.0001,\ 0.0002,\ 0.0003,\ 0.0004,\ 0.0005\)
• \(0.0006,\ 0.0007,\ 0.0008,\ 0.0009\)

a) the greater, for \(\varepsilon=\mathrm{const}\) and \(\gamma=\mathrm{const}\), the width of the hysteresis loop, the greater the amplitude and relative frequency of the established self-oscillations;

b) the greater \(\varepsilon\) for \(\gamma=\mathrm{const}\) and \(\Delta U=\mathrm{const}\), the smaller the amplitude and the greater the relative frequency of the self-oscillations;

c) the greater the width of the hysteresis loop for \(\gamma=\mathrm{const}\), the greater \(\varepsilon\) must be in order to obtain the same amplitude of self-oscillations.

In designing the stabilizer, using the nomogram one can determine the values of \(\Delta U\), \(\varepsilon\), and \(\Omega\) from the specified permissible amplitude and desired frequency of self-oscillations. The quantities \(L\) and \(C\) are determined from the expressions

\[ \varepsilon=\frac{1}{2R}\sqrt{\frac{L}{C}}; \tag{13} \]

\[ \Omega=\omega\sqrt{LC}, \tag{14} \]

where \(\omega\) is the desired switching frequency of the key, determined from additional conditions, in particular by the efficiency.

From expressions (13) and (14) we obtain

\[ L=\frac{2R\varepsilon\Omega}{\omega}; \tag{15} \]

\[ C=\frac{\Omega}{2R\varepsilon\omega}. \tag{16} \]

In conclusion, we note that the nomogram makes it possible to quickly analyze the influence of changes in the supply voltage \(U_{\mathrm{in}}\) and load \(R\) on the parameters of the self-oscillations, and to choose the stabilizer parameters so that the amplitude and frequency of the self-oscillations remain within permissible limits for any changes in \(U_{\mathrm{in}}\) and \(R\).

Conclusions

1) A nomogram is proposed that makes it possible to select the parameters of a two-position stabilizer according to a specified amplitude and frequency of self-oscillations.

2) With a corresponding readjustment of the frequency characteristics of the linear part, the nomogram can be used in the calculation of other relay systems with a similar characteristic of the relay element.

References

1. V. E. Kitaev, G. S. Plotnikov, V. S. Shevelev, A. A. Bokunyaev. Two-position DC voltage stabilizer. Proceedings of Institutes of Communications, L., issue 37, 1968.

2. V. E. Kitaev, A. A. Bokunyaev. Switching DC voltage stabilizer. Elektrosvyaz’, No. 7, 1968.

3. O. J. M. Smith. Automatic Control. Fizmatgiz, Moscow, 1962.