Static Characteristics of a Pulse-Width DC Voltage Converter

A. I. Zaitsev, B. A. Lapin

(Presented by the scientific seminar of the Electromechanical Faculty)

Power amplifiers that use the principle of pulse-width modulation of the voltage of the primary source and subsequent demodulation by means of a smoothing filter (Fig. 1) are widely used in converter technology, in particular as power supplies for radio-electronic equipment. The switching mode of the regulating element and the absence of an alternating-current link account for the high technical-and-economic indices of pulse converters. Despite the large number of publications devoted to the development of the circuit principles of such devices [1, 2, 3], their theory has been developed only poorly [2]. In the present article the characteristics of the converter are investigated in the mode of continuous and discontinuous current of the smoothing choke when operating from a source with zero internal resistance.

Fig. 1.

External Characteristics of the Converter in the Region of Continuous Current

To derive the equation of the external characteristic, we shall use the method, known in the theory of pulse systems [4], for determining the mean value of the steady-state process through the transfer function of the equivalent continuous part:

\[ u_0=\lim \frac{1-e^{-\gamma q}}{q}\, \frac{\beta_1\beta_2 r_{\mathrm{н}}} {q^2+(\beta_1+\beta_2)q+(1+r_{\mathrm{н}})\beta_1\beta_2} \tag{1} \]

\[ i_0=\frac{u_0}{r_{\mathrm{н}}} =\gamma\,\frac{1}{1+r_{\mathrm{н}}} \tag{2} \]

Here

\[ u_0=\frac{U_0}{E_{\mathrm{s}}}; \qquad i_0=\frac{I_0}{E_{\mathrm{s}}}R_L; \qquad r_{\mathrm{n}}=\frac{R_{\mathrm{n}}}{R_L} \]

are the voltage, current, and resistance in relative units,

\(q\) is the dimensionless parameter of the Laplace transformation.

From (1) and (2) one can obtain the equation of the converter external characteristic, valid in the region of continuous choke current (Fig. 2):

Fig. 2.

\[ u_0=\gamma-i_0 \tag{3} \]

Multiplying (3) by \(E_{\mathrm{s}}\), we obtain, in absolute units:

\[ U_0=\gamma E_{\mathrm{s}}-I_0R_L. \tag{4} \]

If the voltage drop across the controlled \((\Delta U_1)\) and uncontrolled \((\Delta U_2)\) valves of the converter is taken into account, equation (4) is transformed into the form:

\[ U_0=\gamma(E_{\mathrm{s}}-\Delta U_1) -(1-\gamma)\Delta U_2-I_0R_L. \tag{5} \]

Analysis of expressions (3)—(5) shows that in the region of continuous current it is possible to smoothly regulate the output voltage over the range from 0 to the supply-source voltage, while the stiffness of the characteristics over the entire regulation range is unchanged and equal to

\[ G=-\frac{di}{du}=-1 \]

A radical means of increasing the stiffness is the use of voltage and current feedback. In the first case the equation of the characteristic has the form:

\[ U_0 \simeq \frac{1}{K_u} \left( U_y-I_0\frac{R_L}{k_u\cdot k_{\mathrm{sh}}\cdot E_{\mathrm{s}}} \right). \tag{6} \]

Here \(U_y\) is the control voltage,

\(k_u\) is the voltage-feedback coefficient,

\(k_y\) is the amplifier gain coefficient,

\(k_{\mathrm{sh}}\) is the conversion coefficient of the width modulator.

It follows from (6) that in a closed-loop system the stiffness increases substantially, with the theoretical limit being absolutely stiff characteristics at

\[ k_y\cdot k_{\mathrm{sh}}\to\infty \]

The characteristics of a converter with positive current feedback have the form:

\[ U_0=U_y\cdot k_{\mathrm{sh}}\cdot E_{\mathrm{s}} -I_0\left(R_L-k_I\cdot k_{\mathrm{sh}}\cdot E_{\mathrm{s}}\right). \tag{7} \]

Here \(k_I\) is the current-feedback coefficient.

It is obvious that by choosing \(k_I\) one can achieve absolute stiffness and even obtain rising characteristics.

Current and Voltage Ripple

The investigation of the ripple of the output voltage and of the choke current in exact discrete form for the given system is laborious and unjustified because of the unwieldiness of the final result. This problem can be substantially simplified if the influence of the voltage ripple on the magnitude and shape of the choke current is neglected. The error arising in this case will be estimated later with the aid of the second approximation. In order to simplify the presentation, it is expedient to operate only with the variable components of the current and voltage, separating out their mean values, determined by (1), (2). Taking what has been said into account, we have:

\[ \frac{d i_{\mathrm{ch}.\sim}}{d\bar t}+\beta_1 i_{\mathrm{ch}.\sim} = \beta_1(1-\gamma-u_{\sim}) \simeq \beta_1(1-\gamma), \qquad 0\leq \bar t \leq \gamma \tag{8} \]

\[ \frac{d i_{\mathrm{ch}.\sim}}{d\bar t}+\beta_1 i_{\mathrm{ch}.\sim} = -\beta_1(\gamma+u_{\sim}) \simeq -\beta_1\gamma, \qquad \gamma\leq \bar t \leq 1 \tag{8a} \]

Here \(\bar t\) is relative time.

The solutions of (8) and (8a) and their linear approximations will be:

\[ i_{\mathrm{ch}.\sim} = (1-\gamma) - \frac{1-e^{-\beta_1(1-\gamma)}}{1-e^{-\beta_1}} e^{-\beta_1\bar t} \simeq (1-\gamma)\beta_1\bar t - \frac{\beta_1\gamma(1-\gamma)}{2}, \qquad 0\leq \bar t \leq \gamma \tag{9} \]

\[ i_{\mathrm{ch}.\sim} = -\gamma + \frac{1-e^{-\beta_1\gamma}}{1-e^{-\beta_1}} e^{-\beta_1(\bar t-\gamma)} \simeq -\beta_1\gamma(\bar t-\gamma) + \frac{\beta_1\gamma(1-\gamma)}{2}, \qquad \gamma\leq \bar t \leq 1 \tag{9a} \]

From (9), (9a) the amplitude of the current ripple is found:

\[ \Delta i_{\mathrm{ch}.} = \frac{ \left(1-e^{-\beta_1\gamma}\right) \left[1-e^{-\beta_1(1-\gamma)}\right] }{ \left[1-e^{-\beta_1}\right] } \simeq \beta_1\gamma(1-\gamma) \tag{10} \]

and the ripple coefficient:

\[ \chi_i = \frac{i_{\mathrm{ch}\,m}}{i_0} = \frac{1}{2}\beta_1(1-\gamma)(1+r_H) \tag{11} \]

Simple and sufficiently accurate expressions for the output voltage \(u_{\sim}\) can be obtained if the conductance of the load for the variable component of the current is neglected. This assumption is generally accepted in the theory of smoothing filters and is fulfilled with sufficient accuracy. Taking this into account, the initial equations are written in the form:

\[ \frac{d u_{\sim}}{d\bar t} = \beta_1\beta_2 r_H(1-\gamma)\bar t - \frac{1}{2}\beta_1\beta_2 r_H\gamma(1-\gamma), \qquad 0\leq \bar t \leq \gamma \tag{12} \]

\[ \frac{d u_{\sim}}{d\bar t} = -\beta_1\beta_2 r_H\gamma(\bar t-\gamma) + \frac{1}{2}\beta_1\beta_2 r_H\gamma(1-\gamma), \qquad \gamma\leq \bar t \leq 1 \tag{12a} \]

The solutions of (12), (12a), describing the forced process in the system, are the expressions:

\[ u_{\sim} = \frac{1}{2}\beta_1\beta_2 r_H(1-\gamma) \left[ \bar t^{\,2} - \gamma\bar t + \frac{1}{6}\gamma(2\gamma-1) \right], \qquad 0\leq \bar t \leq \gamma \tag{13} \]

\[ u_{\sim} = -\frac{1}{2}\beta_1\beta_2 r_H\gamma \left[ (\bar t-\gamma)^2 - (1-\gamma)(\bar t-\gamma) - \frac{1}{6}(1-\gamma)(2\gamma-1) \right], \qquad \gamma\leq \bar t \leq 1 \tag{13a} \]

The voltage curve attains its amplitude values at \(\bar t=\dfrac{1}{2}\gamma\) and \(\bar t=\dfrac{1+\gamma}{2}\):

\[ u_{m\sim}^{(-)}=-\frac{1}{24}\,\beta_1\beta_2 r_{\mathrm{n}}\,(1-\gamma)(2-\gamma) \tag{14} \]

\[ u_{m\sim}^{(+)}=\frac{1}{24}\,\beta_1\beta_2 r_{\mathrm{n}}\,\gamma(1-\gamma)(1+\gamma) \tag{14a} \]

Fig. 3.

It follows from (14), (14a) that the positive and negative amplitudes \(u_{\sim}\) are not equal to one another and change when the pulse width changes (Fig. 3). The negative amplitude has the greatest value at

\[ \gamma=1-\frac{1}{\sqrt{3}}: \]

\[ u_{m\sim\max}^{(-)} = -\frac{\sqrt{3}}{108}\,\beta_1\beta_2 r_{\mathrm{n}} = -\frac{\sqrt{3}}{108}\cdot \frac{1}{LC\cdot f^{2}}. \tag{15} \]

The positive amplitude \(u_{\sim}\) reaches its maximum at \(\gamma=\dfrac{1}{\sqrt{3}}\):

\[ u_{m\sim\max}^{(+)} = \frac{\sqrt{3}}{108}\,\beta_1\beta_2 r_{\mathrm{n}} = \frac{\sqrt{3}}{108}\cdot \frac{1}{LC\cdot f^{2}}. \tag{15a} \]

In addition to the average value \(u_0\) and the ripple amplitude, the output voltage is characterized by the ripple swing

\[ \Delta u = u_{m\sim}^{(+)}-u_{m\sim}^{(-)} = \frac{1}{2}\,\beta_1\beta_2 r_{\mathrm{n}}\,\gamma(1-\gamma) = \frac{0.125}{LC\cdot f^{2}}\,\gamma(1-\gamma) \tag{16} \]

and by the ripple coefficients:

\[ \chi_u^{(-)} = \frac{\left|u_{m\sim}^{(-)}\right|}{u_0} = \frac{1}{24}\,\beta_1\beta_2(1+r_{\mathrm{n}})(1-\gamma)(2-\gamma) \simeq \]

\[ \simeq \frac{1}{24}\, \frac{(1-\gamma)(2-\gamma)}{LC\cdot f^{2}} \tag{17} \]

\[ \chi_u^{(+)} = \frac{\left|u_{m\sim}^{(+)}\right|}{u_0} = \frac{1}{24}\,\beta_1\beta_2(1+r_{\mathrm{n}})(1-\gamma)(1+\gamma) \simeq \]

\[ \simeq \frac{1}{24}\, \frac{(1-\gamma)(1+\gamma)}{LC\cdot f^{2}} \tag{17a} \]

As is evident from Fig. 3, both coefficients increase monotonically as \(\gamma\) changes from 1 to 0, with \(\chi_u^{(-)}>\chi_u^{(+)}\) for \(\gamma<0.5\); \(\chi_u^{(-)}<\chi_u^{(+)}\) for \(\gamma>0.5\).

Taking into account that

\[ \beta_1\beta_2 r_{\mathrm{n}}=\frac{1}{LC\cdot f^{2}}, \]

on the basis of (13)—(17a), it may be concluded that the ripple of the output voltage does not depend on the average value of the load current.

The error in the solution of equations (8), (8a), arising when \(u_{\sim}\) is neglected, is estimated with the aid of (13), (13a), (18). It is possible to—

show that it does not exceed \(-\frac{1}{2}x_{i\,\mathrm{allow}}\), where \(x_{i\,\mathrm{allow}}\) is the ripple factor admissible under the technical specifications, usually not exceeding 1–2%.

Converter characteristics in the discontinuous-current region

The discontinuous-current mode of the choke is characterized by the fact that by the end of the pause the choke current decreases to zero. On this basis, the equation of the boundary between the two regions is found from the condition

\[ i_0+i_{\mathrm{ch}.\sim}(\bar t=1)=0 . \]

Using (9a), we obtain:

\[ i_{0\,\mathrm{bnd}}=-\frac{1}{2}\beta_1\gamma(1-\gamma) \tag{18} \]

The greatest boundary current occurs at \(\gamma=0.5\) (Fig. 2).

An approximate expression for the external characteristic in the discontinuous-current region has the form:

\[ u_0=1-\frac{2i_0}{\beta_1^2+2i_0} \tag{19} \]

The stiffness of the characteristics is determined by the expression

\[ G=-\frac{(\beta_1\gamma^2-2i_0)^2}{2\beta_1\gamma^2} \tag{20} \]

Analysis of (19), (20) shows that, upon transition into the discontinuous-current region, the stiffness of the characteristics decreases considerably. Within the region the stiffness is maximal at the boundary:

\[ G_{\max}=-\frac{1}{\beta_1} \]

and minimal at the point of ideal no-load operation:

\[ G_{\min}=-\frac{1}{2}\beta\gamma^2 \]

The poor regulating properties of the converter in the discontinuous-current region indicate the advisability of operation in the continuous-current region. In order that the operating point not go beyond its limits when the load current decreases, the parameter \(\beta_1\) must be chosen according to the condition:

\[ \beta_1<8i_{n\,\min}. \tag{21} \]

If the minimum load current is so small that satisfying condition (21) is difficult or impractical, a ballast resistance may be connected to the converter output, satisfying the condition:

\[ r_B\leq \frac{2-\beta_1(1-u_0)_{\min}} {\beta_1(1-u_{0\,\min})}. \tag{22} \]

Here \(u_{0\,\min}\) is the lower regulation limit.

The range of output-voltage ripple in the discontinuous-current region is approximately determined by the expression

\[ \Delta u= \frac{\beta_2 i_n i_0}{4\beta_1^2 z^2} \left[\beta_1(2-\gamma)-2i_0\right]^2 . \]

The value \(\Delta u\) attains its greatest value at the boundary of the modes

\[ \Delta u_{\max}=\frac{1}{2}\,\beta I_{1,2}^{0} r_{\mathrm{н}}\gamma(1-\gamma) \]

and becomes zero at the point of ideal no-load operation.

Comparison of (17), (23), and (24) shows that the amplitude of the pulsations within the region of discontinuous currents is smaller than in the region of continuous current.

REFERENCES

1. O. A. Kossov. Power Amplifiers on Transistors in Switching Mode, “Energiya,” 1964.
2. V. M. Porsov. Key stabilized rectifier using transistors, “Elektrichestvo,” No. 5, 1965.
3. V. I. Aleksandrin. Pulse voltage stabilizers using transistors. “Semiconductor Devices and Their Application,” issue 7, 1961.
4. Ya. E. Tsypkin. Theory of Linear Pulse Systems, 1963.