# Control-to-output Transfer Function of the Open-loop Step-up Converter in CCM Operation

• Wang, Faqiang (State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi'an Jiaotong University) ;
• Ma, Xikui (State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi'an Jiaotong University)
• Received : 2013.10.06
• Accepted : 2014.03.01
• Published : 2014.09.01
• 136 14

#### Abstract

Based on the average method and the geometrical technique to calculate the average value, the average model of the open-loop step-up converter in CCM operation is established. The DC equilibrium point and corresponding small signal model is derived. The control-to-output transfer function is presented and analyzed. The theoretical analysis and PSIM simulations shows that the control-to-output transfer function includes not only the DC input voltage and the DC duty cycle, but also the two inductors, the two energy-transferring capacitors, the switching frequency and the load. Finally, the hardware circuit is designed, and the circuit experimental results are given to confirm the effectiveness of theoretical derivations and analysis.

# 1. Introduction

As one of the important research area in the filed of DC-DC power supply, exploring a new topology of DC-DC converter to obtain the good performance in practical engineering is very meaningful and valuable work and many good results have been presented [1-12]. Among of them, a novel step-up converter, which combining the KY converter and the traditional Buck-Boost converter, has been proposed by K. I. Hwu et al in 2011 to boost the low input voltage to the high output voltage [5]. Comparing to the existing DC-DC boosting converter [7-16], this novel step-up converter has some outstanding advantages [5]. For example, by comparing to the traditional Boost converter [7], this novel step-up converter has no pulsating current through the output capacitor since there is an output inductor, thereby causing the output voltage ripple to be small [5].

By comparing to the KY converter whose voltage conversion ratio up to two [8], the voltage converter ratio of this novel step-up converter can be up to infinite. By comparing to the DC-DC boosting converters that include the coupled inductors with turn ratio [9-10], there is no leakage inductance should be considered in this novel step-up converter. Also, by comparing to the DC-DC boosting converters whose power switch is floating and need the isolated driving signal [11-12], it is easy to drive this novel step-up converter since no isolated gate driver is needed instead of one half-bridge gate driver. Accordingly, it can be believed that this novel step-up converter will be widely accepted and used in the area of boosting the low input voltage to the high output voltage to obtain enough output power and voltage amplitude, such as uninterruptible power supplies, solar cell powering systems, battery powering systems, and so on, to instead of the traditional Boost converter and the voltage boosting converter which uses the coupling or needs the isolated gate driver to obtain the high output voltage in the future. Therefore, it is necessary to establish the accurate model for this novel step-up converter to prepare for consequent designing.

However, up to now, there are only a few studies to do this point. In [5], under the assumption that the two energy-transferring capacitors were large enough to keep the voltage across themselves at some values, the formulas for DC output voltage which only includes the DC duty cycle and the DC input voltage has been derived. But, the absence of the average model and corresponding small signal model of the open-loop step-up converter in [5] leads to hard to design the controller to obtain the good performance to satisfy the requirements in practical engineering. In other words, establishing the average model and small signal model of the open-loop step-up converter is the key and important step for consequent designing and analysis.

The rest of the paper is organized as follows. In section 2, the circuit operation, the mathematical model and some PSIM simulations of the open-loop step-up converter in CCM operation are given. Then, the average model is established. The DC equilibrium point and the control-to-output transfer function are derived and analyzed. In section 4, the circuit experiments are given for confirmation. Finally, some concluding remarks are given in section 5.

# 2. Circuit Operation, Mathematical Model and PSIM Simulations

The circuit schematic of this step-up converter is shown in Fig. 1. It can be seen that this step-up converter consists of the input voltage vin, two inductors: L1 and L2, two energy-transferring capacitors: C1 and C2, two power switches: S1 and S2, one diode D1, one output capacitor C0 and one load R.

Fig. 1.The circuit schematic of step-up converter

The switches S1 and S2 are driven by one half-bridge gate driver with the switching frequency being f and the DC duty cycle being D. The voltage across C0, C1 and C2 are defined as v0, v1 and v2, respectively. The current through the inductor L1 and L2 are defined as iL1 and iL2, respectively. The current through C1 and C2 are defined as iC1 and iC2, respectively. Note that, here, only the continuous conduction mode (CCM) operation is concerned, i.e., there are only two operation modes in this novel step-up converter, which are shown in Figs. 2(a) and (b), respectively. Additionally, the circuit parameters here are chosen as vin=6V, C1=2μF, C2=4.7μF, L1=3mH, L2=1mH, C0 = 40μF, R = 80Ω, G = 1/R, f= 25kHz, T = 1/f, and D = 0.5.

Fig. 2.The power flows of the two operation modes of step-up converter: (a) mode 1; (b) mode 2.

Mode 1: the power switch S1 is turned on whereas S2 is turned off and the diode D1 is not conducted for its inverse-biased voltage. The power flows for this mode are shown in Fig. 2(a). According to the circuit theory, the equations for describing this mode can be derived as follows

Mode 2: the power switch S1 is turned off whereas S2 is turned on and the diode D1 is conducted for its forward-biased voltage. The power flows for this mode are shown in Fig. 2(b). According to the circuit theory, the equations for describing this mode can also be derived as follows

Based on the PSIM software which is widely used to simulate the motor and the power electronics [17, 18], the PSIM simulations for the voltage v1+vin, v2 and the PWM signal vd are shown in Fig. 3.

Fig. 3.PSIM simulations: (a) the voltage v1+vin and the PWM signal vd; (b) the voltage v2 and the PWM signal vd.

From the time-domain waveform of the voltage v1+vin in Fig. 3(a) and the voltage v2 in Fig. 3(b), it is easily obtained that the ripple of the voltage v1+vin is within (9.9V, 13.8V) and the voltage v2 is within (9.1V, 10.5V). Thus, both the ripples of the voltage v1+vin and v2 can not be considered as being equal to zero when calculating their average values since their ripples are not small enough. In other words, in [5], the assumption that the two energy-transferring capacitors are large enough to keep the voltage across themselves being constant at some values is only an extreme case, i.e., the assumption in [5] can not be satisfied in some cases in practical engineering.

Also, from the characteristics of PSIM software [19-20], i.e., bode diagram of switching power converter can be directly obtained by using its switch mode form, bode diagram of the control-to-output transfer function (Gvd (s)) of this novel step-up converter can be obtained, which is shown in Fig. 4. These PSIM simulations will be used to preliminary confirm the effectiveness of the theoretical derivations and analysis in the following sections.

Fig. 4.Bode diagram of Gvd (s) from PSIM simulations

# 3. Theoretical Derivations and Analysis

Based on the average method [21] and the Eqs. (1) and (2), the average model of this open-loop step-up converter can be easily obtained as follows

where 〈iL1〉, 〈iL2〉, 〈iC1〉, 〈v0〉, 〈v1〉, 〈v2〉 and 〈vin〉 are the corresponding average value of iL1, iL2, iC1, v0, v1, v2 and vin, respectively, and d is the duty cycle.

Obviously, in order to derive the average model of this open-loop step-up converter completely, the expressions for 〈v1〉 and 〈iC1〉 must be derived. Here, for convenience to derive the expression for 〈v1〉, the time-domain waveforms of the voltage v1+vin, v2 and the PWM signal vd are replotted in Fig. 5.

Fig. 5.The re-plotted of the voltage v1+vin (dashed line), v2 (solid line) and the PWM signal vd

In Fig. 5, it is clearly seen that the voltage v1+vin is the same as the voltage v2 within ((N+d)T, (N+1)T). However, they are different from each other within (NT, (N+d)T). Also, one can see that there are jumps on the voltage across the two energy-transferring capacitors since they are abruptly changing at (N+d)T, respectively. Therefore, this novel step-up converter is very different from the traditional DC-DC converters which have no jumps on the voltage across the capacitors, such as Cuk or Sepic converters. Thus, unlike establishing average model and small signal model for Cuk or Sepic converters which using the average method directly, the jumps on the voltage across the two energy-transferring capacitors in this novel step-up converter must be firstly considered and calculated by using the geometrical technique.

Here, the voltage v1+vin and v2 are all denoted as VN0 after abruptly changing at (N+d)T and VN+1 at (N+1)T, respectively. Also, they are denoted as VN1 and VN2 before abruptly changing at (N+d)T, respectively, and VN at NT. Note that VN =VN+1. Moreover, assuming that the voltage v1+vin decreases linearly within ((N+d)T, (N+1)T) and increases linearly within (NT, (N+d)T), and the voltage v2 decreases linearly within (NT, (N+d)T) and ((N+d)T, (N+1)T). Based on the geometrical technique to calculate the average value in the field of power converter and Fig. 5, the following formulas can be obtained

where

Thus, the expression for 〈v1〉 can be derived by making (4) minus (5) and the result is shown as follows

where α = (C1-C2)T / (2C1C2) and β =T / (2C1).

In addition, based on the ampere-second balance, 〈iC1〉 can be expressed as a function of 〈iL1〉 and 〈iL2〉, given by

Therefore, the average model of the open-loop step-up converter can be obtained by taking (8) and (9) into (3), and its expression is

The DC equilibrium point and the small signal model of (10) can be derived by using the perturbation and linearization of (10).

Assuming that IL1, IL2, V0, V2, Vin and D are the DC values of 〈iL1〉, 〈iL2〉, 〈v0〉, 〈v2〉, 〈vin〉 and d, respectively, and and are their small ac values. Additionally, the following equations are assumed

Thus, the DC equilibrium point of the open-loop step-up converter can be obtained by substituting (11) into (10) and then separating the DC values out. The result is

where G =1/R. From the formula for DC output voltage, i.e., V0, it is easily seen that its expression includes not only the DC input voltage Vin and the DC duty cycle D, but also the two energy-transferring capacitors C1 and C2, the switching frequency f and the load R.

By calculating the voltage V0 over the voltage Vin, the voltage conversion ratio can also be derived, which is shown as follows

It is obvious that the voltage conversion ratio here is very different from the result in [5] which only includes the DC duty cycle D.

Moreover, from (1), (2), (12) and Fig. 5, the current ripples (ΔiL1 and ΔiL2) of two inductors (L1 and L2) and the voltage ripples (Δv1, Δv2 and Δv0) of three capacitors (C1, C2 and C0) can be derived, which are shown as follows. In other words, if these current ripples and voltage ripples, the DC duty cycle D, the DC input voltage Vin, the load R (G=1/R) and the switching period T have already been given, the values of L1, L2, C1, C2 and C0 can be calculated and these are very helpful for designing these parameters to satisfy the requirements in practical engineering.

Furthermore, the small signal model of the open-loop step-up converter can be derived by substituting (11) into (10) and then separating the small ac variations out and omitting the second and higher order terms since their values are very small. The result is

Therefore, the control-to-output transfer function (Gvd (s)) can be derived by using the Laplace transform on (15) and its definition, and the expression is

where

From (16), it is obvious that the derived control-to-output transfer function Gvd(s) includes not only the DC input voltage Vin and the DC duty cycle D, but also the two inductors: L1 and L2, the two energy-transferring capacitors: C1 and C2, the switching frequency f and the load R.

Fig. 6 shows bode diagram of Gvd(s) from the theoretical calculations. At the same time, for confirming the effectiveness of the derived Gvd(s) preliminary, the PSIM simulations about bode diagram of Gvd(s) are also replotted in Fig. 6. Obviously, the theoretical calculations are in good agreement with the PSIM simulations. Thus, the derived control-to-output transfer function Gvd(s) here is an effective model for describing the small signal dynamical behaviors of the open-loop step-up converter.

Fig. 6.Comparisons about bode diagram of Gvd(s) between theoretical calculations and PSIM simulations

# 4. Circuit Experiments

In order to confirm the effectiveness of the theoretical derivations and analysis further, the hardware circuit of the step-up converter is implemented by using MOSFET IRFP460 for realizing the switch S1 and S2, and MUR1560 for realizing the diode D1. The digital oscilloscope GDS 3254 is employed to capture the measured time-domain waveforms in the probes and the Agilent E5061B LF-RF network analyzer is employed to capture the measured gain and phase in the probes. Taking the circuit parameters as in section 2, the voltage v1+vin, v2 and the PWM signal vd from the circuit experiments are shown in Fig. 7. Comparing Fig. 7 with Fig. 3, it can be seen that they are in good agreement with each other. Therefore, it is confirmed that the voltage v1+vin is really not equal to the voltage v2 within the whole switching period.

Fig. 7.Time-domain waveforms from the circuit experiments with time scale: 20μs/div: (a) v1+vin (upper: 5V/div) and vd (lower: 10V/div); (b) v2 (upper: 5V/div) and vd (lower: 10V/div).

Moreover, by using the Agilent E5061B LF-RF network analyzer, bode diagram of Gvd (s) from the circuit experiments is directly obtained, as shown in Fig. 8. Comparing Fig. 8 with the Fig. 4 and Fig. 6, one can see that the circuit experimental results are in good agreement with the PSIM simulations and the theoretical calculations. Therefore, it is confirmed further that the derived control-to-output transfer function Gvd(s) is an effective model for the open-loop step-up converter. In other words, the derived average model and small signal model here are effective for describing the small signal dynamical behaviors of the step-up converter.

Fig. 8.Bode diagram of Gvd(s) from the circuit experiments

# 5. Conclusion

By using the average method and the geometric technique to calculate the average value, the average model and the corresponding small signal model of the open-loop step-up converter are established. The derived DC equilibrium point shows that the DC output voltage includes not only the DC input voltage and the DC duty cycle, but also the two energy-transferring capacitors, the switching frequency and the load. Moreover, the control-to-output transfer function is also derived and its theoretical calculations are in good agreement with the PSIM simulations and the circuit experiments. Therefore, the derived average model, the small signal model and the control-to-output transfer function are effective for the open-loop step-up converter and these will be helpful for designing the step-up converter in practical engineering.

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