Study of the Steady-State Mode in a DC Boost Converter with Direct Coupling

A. I. Zaitsev, B. A. Lapin, A. T. Mikhalev

(Presented by the scientific-technical seminar of the Research Institute of AEM)

To obtain a regulated DC voltage exceeding the voltage of the primary power source, converters are generally used that contain an AC link and a voltage regulator located on the DC or AC side. In cases where galvanic isolation between the output and the input is not required, it is more expedient to make boost converters according to the direct-coupling circuit shown in Fig. 1. In this circuit, the excess of the output voltage over the voltage of the primary source is achieved by means of periodically repeated processes of charging the inductance from the power source during the intervals when switch \(K\) is closed, and discharging it through diode \(D\) into capacitor \(C\) during the intervals when this switch is open. The magnitude of the converter output voltage is regulated by changing the relative duration of the on-state of switch \(K\).

Fig. 1

Unlike ordinary converters with direct coupling, whose output voltage does not exceed the voltage of the power source, boost converters with direct coupling have been studied comparatively little.

In the present work, the quasi-steady-state mode in the converter circuit of Fig. 1 is investigated under the assumption that diode \(D\) and switch \(K\) have ideal properties.

Below, the most typical case is considered, in which the AC component of the output voltage is negligibly small in comparison with the average voltage. The equations of the electrical equilibrium of the circuit for both intervals of the period have the form:

\[ L\frac{dI_{\sim}}{dt}+(I_0+I_{\sim})R=E, \qquad 0\leq t\leq \gamma T; \tag{1} \]

\[ L\frac{dI_{\sim}}{dt}+(I_0+I_{\sim})R=E-(U_0+U_{\sim}), \qquad \gamma T\leq t\leq T; \tag{2} \]

\[ C\frac{dU_{\sim}}{dt}+\frac{U_0+U_{\sim}}{R_{\mathrm{n}}}=0, \qquad 0\leq t\leq \gamma T; \tag{3} \]

\[ C\frac{dU_{\sim}}{dt}+\frac{U_0+U_{\sim}}{R_{\mathrm{n}}}=I_0+I_{\sim}, \qquad \gamma T\leq t\leq T, \tag{4} \]

where

\(T\) is the repetition period;

\(\gamma\) is the relative duration of the on-state of switch \(K\);

\(I_0, U_0, I_{\sim}, U_{\sim}\) are, respectively, the constant and variable components of the choke current and of the output voltage. The components \(I_{\sim}, U_{\sim}\) satisfy the condition

\[ \int_0^T I_{\sim}\,dt=\int_0^T U_{\sim}\,dt=0. \tag{5} \]

The component \(I_0\) is found from the average voltage across the choke by Ohm’s law:

\[ I_0=\frac{1}{RT}\left\{\int_0^T E\,dt+\int_{\gamma T}^{T}\left[E-(U_0+U_{\sim})\right]dt\right\} \simeq \frac{E\gamma+(E-U_0)(1-\gamma)}{R}. \tag{6} \]

In finding the expression for the variable component of the choke current, it is expedient to neglect the variable component of the output voltage.

Under this condition, the equations for determining \(I_{\sim}\), after substituting (6) into (1) and (2), take the form

\[ L\frac{dI_{\sim}}{dt}+I_{\sim}R=U_0(1-\gamma), \qquad 0\leq t\leq \gamma T; \tag{7} \]

\[ L\frac{dI_{\sim}}{dt}+I_{\sim}R=-U_0\gamma, \qquad \gamma T\leq t\leq T. \tag{8} \]

The approximate solutions of equations (7), (8), under condition (5), corresponding to the steady-state operating mode, have the form

\[ I_{\sim}(t)=\frac{U_0}{L}(1-\gamma)t-\frac{1}{2}\frac{U_0}{L}\gamma(1-\gamma)T, \qquad 0\leq t\leq \gamma T, \tag{9} \]

\[ I_{\sim}(t)=-\frac{U_0}{L}\gamma(t-\gamma T)+\frac{1}{2}\frac{U_0}{L}\gamma(1-\gamma)T, \qquad \gamma T\leq t\leq T. \tag{10} \]

The waveform of the choke current is shown in Fig. 2.

Knowing the expression for the total choke current \((I_0+I_{\sim})\), it is not difficult to determine the component \(U_0\) of the output voltage.

In the steady-state operating mode, the charge received by the capacitor over a period is equal to zero. Therefore, taking into account that the capacitor is discharged by the current \(\dfrac{U_0+U_{\sim}}{R_{\mathrm{n}}}\) over the entire period and is charged by the current \(I_0+I_{\sim}\) only in the interval of the period \(\gamma T\leq t\leq T\), one may write the following equality:

\[ \int_{\gamma T}^{T} (I_0 - I_{\sim})\,dt - \int_{0}^{T} \frac{U_0 + U_{\sim}}{R_{\mathrm{n}}}\,dt = 0, \tag{11} \]

where \(I_{\sim}\) is determined by expression (10).

Solving (11) with respect to \(U_0\), taking into account relations (5), (6), and (10), we obtain the equation of the converter control characteristic

\[ u_0 = \frac{1-\gamma}{\dfrac{1}{r_{\mathrm{n}}} + (1-\gamma)^2}, \tag{12} \]

where

\[ u_0=\frac{U_0}{E};\quad r_{\mathrm{n}}=\frac{R_{\mathrm{n}}}{R} \]

are the relative values of the voltage and resistance.

Fig. 2. Current in the choke circuit

Fig. 3. Control characteristics of the converter

Figure 3 shows a family of control characteristics constructed according to (12) for three values of the parameter \(r_{\mathrm{n}}\).

Investigation of (12) shows that, for \(\gamma = 1 - \sqrt{\dfrac{1}{r_{\mathrm{n}}}}\), the output voltage is maximal and equal to

\[ u_{0\max}=\frac{1}{2}\sqrt{r_{\mathrm{n}}}. \tag{13} \]

The equation of the converter external characteristic is obtained from equation (12) by substituting \(r_{\mathrm{n}}=\dfrac{u_0}{i_{\mathrm{n}}}\).

\[ u_0=\frac{1}{1-\gamma} - \frac{i_{\mathrm{n}}}{(1-\gamma)^2}, \tag{14} \]

where \(i_{\mathrm{n}}=\dfrac{I_{\mathrm{n}}}{E}R\) is the relative current.

A family of external characteristics of the converter for a number of values of \(\gamma\), constructed according to (14), is presented in Fig. 4.

Relations (6) and (12) make it possible to obtain an approximate expression for the converter efficiency, not taking into account losses in the choke, switch K, and diode D:

\[ \eta=\frac{1}{1+\frac{1}{(1-\gamma)^2 r_{\text{н}}}} . \tag{15} \]

The equation of the variable component of the output voltage \(U_{\sim}\) can be found by integrating the original equations (3), (4). However, of greater interest are expressions for the ripple amplitude of the output voltage. The voltage ripple amplitude \(\Delta U\) is equal to the decrease in voltage over the interval of the period \(0\leq t\leq \gamma T\), when the capacitor is discharged by the current \(\dfrac{U_0}{R_{\text{н}}}\). On the basis of what has been said,

\[ \Delta U=\frac{1}{C}\int_0^{\gamma T}\frac{U_0}{R_{\text{н}}}\,dt= \frac{\gamma(1-\gamma)E}{[R+(1-\gamma)^2R_{\text{н}}]C}, \tag{16} \]

Fig. 4. External characteristics of the converter

the relative magnitude of the output-voltage ripple of the converter is equal to

\[ \Delta u=\frac{\Delta U}{U_0}=\gamma\frac{T}{R_{\text{н}}C}. \tag{17} \]

Testing on a prototype of the converter confirmed the correctness of the theoretical results obtained.

Conclusions

1. The converter according to the circuit of Fig. 1 makes it possible to regulate the output voltage upward from the voltage of the primary source; moreover, the regulation range is proportional to the load resistance;

2. The external characteristics of the converter are linear, and their slope increases with increasing output voltage.

3. The amplitude of the output-voltage ripple does not depend on the inductance of the choke and is inversely proportional to the time constant of the converter output circuit and to the switching frequency.

References

1. Sh. M. Razmadze. Converter Circuits and Systems. “Vysshaya shkola,” 1967.

2. B. Bedford, R. Hoft. Theory of Autonomous Inverters. “Energiya,” 1969.

3. O. I. Khasayev. Transistor Voltage and Frequency Converters. “Nauka,” 1966.