Modeling of non-ideal DC converters with peak current control

1 Introduction
The high frequency, soft switching technology and new control technology of switching converters have brought many urgent problems to the modeling of switching converters. Some ideal assumptions of previous converters will produce serious deviations. Especially in high current situations, the parasitic parameters of switching devices must be considered when modeling, such as the on-resistance of the power switch, the forward resistance and forward voltage of the diode. Peak current
control has the function of current limiting protection, which improves reliability and can prevent the core saturation of transformers in push-pull and bridge circuits. It has been widely used. Since its PWM signal modulation method is different from the general generation method, it will inevitably affect the dynamic performance. In order to study its dynamic performance, it is necessary to establish a dynamic model for it.
This paper draws on the basic ideas of the three-terminal switching device model method, the time-averaged equivalent circuit method and the energy conservation method. Taking the Buck-Boost converter based on peak current control as an example, the circuit average modeling method of the non-ideal converter in the peak current mode is studied, and the accurate model of the current control loop is derived.

2 Non-ideal Buck - Boost Small Signal Model Establishment

The Buck-Boost converter is shown in Figure 1(a). The switching period of the power switch S is T, the conduction time is , the duty cycle is D, and the disturbance of D is , the instantaneous value . Figure 1(b) is the equivalent circuit of the converter considering parasitic parameters. The power switch is equivalent to the series connection of an ideal switch and an on-resistance , the diode is equivalent to the series connection of an ideal switch, a forward voltage drop , and a forward resistance , is the equivalent series resistance of the inductor, and is the equivalent series resistance of the filter capacitor.

(a) Ideal Buck-boost circuit

(b) Equivalent circuit considering parasitic parameters

Figure 1 Buck-boost converter

[page]When S is turned on, , the effective value of the current of S is:

(1)

The turn-on loss of MOSFET is:

(2)

Therefore, the equivalent average resistance of the resistor in the switch S branch is .

When S is turned off and diode D is turned on, the current flowing through D is the effective value of the diode current: (3)

The power loss in the forward resistance of the diode is:

(4)

(5)

So the equivalent average resistance of the diode branch is . The equivalent average voltage of the diode branch is still .

Figure 2 Non-ideal Buck-Boost Large Signal Model

The ideal switch shown by the dotted line in FIG1(b) is represented by the

The controlled current source and controlled voltage source are substituted for ^[1] to obtain the large signal average model of the non-ideal Buck-Boost in continuous operation mode. The values ​​of the controlled current source and controlled voltage source are the average values ​​within the cycle.

According to the energy conservation principle, that is, the power loss of the switching device remains unchanged, the average parasitic components can be converted to the inductor branch, simplifying the large signal model. From equations (2), (4), and (5), the equivalent average resistance of the MOSFET converted to the equivalent resistance in the inductor branch is , the equivalent average resistance of the diode D is , and the equivalent average voltage is . The total resistance in the inductor branch is expressed by the equivalent average resistance : (6)

The simplified large signal averaging model is shown in Figure 3.

Figure 3 Simplified Buck-Boost large signal average model

In order to further simplify the model, the controlled current source and the controlled voltage source are replaced by an ideal transformer, and the large signal average model is obtained as shown in Figure 4.

Figure 4 Large signal average model equivalent to an ideal transformer

Decompose the average variables, , , . Then:

(7) (8)

When performing dynamic signal analysis, the steady-state components and are taken as zero, and the small signal product terms and are ignored , and the small signal model is obtained as shown in Figure 5.

Figure 5 Non-ideal Buck-boost small signal model

[page]3 Accurate model of peak current control loop

In order to design the voltage outer loop of the peak current model, the AC small signal model of the current inner loop must be established first, that is, the model of the new power stage including the current loop. The first-order model usually established ignores the inductor current ripple and the slope compensation current, so it is suitable for occasions where the inductor current ripple is small and the slope compensation current ripple is small.

This paper studies the circuit model when the inductor current ripple is large and there is slope compensation. It has higher accuracy.

Figure 6 Relationship between inductor current and control current

Considering the slope compensation of the peak current, the inductor current and the control current have the relationship shown in Figure 6. In Figure 6, the slope of the slope compensation is ma; the rising rate of the inductor current is m1, and the falling rate is m2; the control current is ic, then the difference between the peak value of the inductor current and the control current is equal to . Between [0, dTs] and [dTs, Ts], the difference between the peak value of the inductor current and its average value is m1dTs/2, m2(1-d)Ts/2 respectively.

Therefore, the average value of the inductor current is:

(9)

Decompose the mean variable: ; ;

; ; , Substituting into (9) and ignoring the small signal product term, we get:

(10)

Considering that in the Buck-Boost circuit, ; . At the same time, the steady-state relationship is , simplifying (10),

get:

Find: (12)

The small signal AC average model of the peak current control mode can be obtained from the small signal model of Buck-Boost as shown in Figure 7.

Figure 7 Small signal average model of peak current mode

In the figure ,

.

From Figure 7, we can write the relationship between the inductor current average value disturbance and the output voltage disturbance and :

(13)

(14)

Substitute (13) into the frequency domain expression of (12), and then substitute into (14), we can get:

(15)

The circuit consisting of the current control loop and the load is defined as the equivalent power level. It is the basis for designing the voltage outer loop. The transfer function of the equivalent power level is:

(16)

[page]From the non-ideal Buck-Boost small signal model in Figure 5, we can find:

(17)

(18)

At this point, the transfer function from equivalent power level control to output is obtained. The equivalent power level can be regarded as the control object of the voltage outer loop and is the basis for designing the voltage outer loop.

4 Conclusion:

Based on the small signal model of the non-ideal converter, this paper establishes an accurate model of the peak current control loop, taking into account the influence of the inductor current ripple and artificial slope compensation, which has higher accuracy than the first-order model. The derived transfer function from the control to the output of the equivalent power stage provides a basis for the stability analysis of the loop. The modeling method is also applicable to other converters.

References

[1] Xu jianping, Modeling of switching DC-DC converters by time averaging equivalent circuit approach. Part 1. Continuous conduction mode, International Journal of Electronics, 1993, Vol.74, No.3, 465~475.

[2]Unitrode Application Note. Modeling Analysis and Compensation of the Current-model converter, handbook 1993~1994, Unitrode.

[3] Ouyang Changlian, Modeling Analysis and Research of DC-DC Switching Converter, PhD Dissertation of Nanjing University of Aeronautics and Astronautics, 2004, 35~39.

[4] Zhang Weiping, Modeling and Control of Switching Converters, Beijing, China Electric Power Press, 2006.