Small Signal Model of LCC Resonant Converter using Charge Mode Control with Capacitor as Output Filter

Søren Kjærulff Christensen*

*Bang & Olufsen A/S, Struer, 7600 DK (Tel: 0045-9684-4973; e-mail: skn@ bang-olufsen.dk)

Abstract: In this paper, a small signal model of a LCC resonant converter with charge mode control is described. Based on measurements, it is shown that the small signal loop gain for the LCC resonant converter with charge mode control has the same gain and phase curves as normal frequency controlled LCC converter. The frequency controlled LCC resonant converter has load dependent gain opposite to the converter with charge mode control, which has a gain independent of load. It is the aim with this paper to show that charge mode control off LCC resonant converter will give a simple small signal gain model, which is independent of load. This independence of load gives a very simple model, which is obvious to use in later calculations and simulations.

1. INTRODUCTION

In hunt of smaller and more compact power supplies, resonant converters with three and four storage elements have been investigated since the 1980th (Steigerwald, 1988). It has been shown (Batarseh, 1994) that from three storage elements there are 36 variants or combination possibilities. Among them, the two most preferred types are LLC and LCC converters. A LCC converter is shown in Fig. 1. The name comes from the arrangement of inductive and capacitive components. In Fig. 1, we have a series parallel resonant converter with capacitive load. Ls and Cs are series resonant component in series with the load and Cp is parallel to the load. Therefor the name LCC.

Resonant converters have higher efficiency compared to pulse with modulation (PWM) topologies, less switching loss and can go to higher switching frequency. This give more compact designs with less electromagnetic interference. The resonant converters are more difficult to model due to many stats and non-linear control characteristic. In this paper, we look at a LCC series parallel resonant converter, which is capacitive loaded as shown in Fig. 1.

Steady state calculations are well documented for the LCC resonant converter with capacitive load (Bhat, 1991) but when it comes to small signal analysis, only a few models have been made by (Fabiana & Kolar, 2005) and (Martin-Ramos, Dias, Pernia, Lopera, & Nuno, 2007). Unfortunately they are complicated to use. Moreever, in (Fabiana & Kolar, 2005) the small signal gain depends on load when frequency control is used. Variation in small signal gain with load will lower the total open loop gain of a feedback system.

To overcome the small signal gain variation with load an idea to regulate on charge from cycle to cycle instead of frequency a patent (Nielsen & Christensen, 2005) has been made which eliminates the gain dependence of the load. In section 3, it

will be shown that charge mode control only affects the gain without moving or introducing new poles or zeros into the system, and small signal gain will not change with load. This is achieved by introducing a new inner loop on the primary side of the LCC converter equivalent to current mode control in PWM control.

In 1994, small signal models were made of different resonant converters. Extended describing function were used to make mathematic DC equations. When introducing small perturbation on the DC equations, the small signal gain can be found (Agarwal & Bhat, 1994). However, this is done on a LCC converter with inductors as output load. The first model seen with a LCC converter with capacitive load is found in 2005 (Fabiana & Kolar, 2005) again here DC steady state solution is found and with this equation small perturbation is added to find small signal gain. No model is made but small signal gain is calculated with use of a MATLAB program.

The aim of this paper is to make a simple model, which is easy to use. It will be shown in section 3 that there will be no gain variation of the small signal gain with respect to load, if charge mode control is used.

LCC

Transformer.

Vin

Fig. 1. LCC resonant converter. Series-parallel resonant converter. Cs and Ls are in series with the load and Cp are parallel to the load. Output are capacitive loaded.

Section 2 shows how the small signal gain is measured. Section 3 gives the results of the measurements from which the model is made. In section 4, a small signal gain model for LCC converter with charge mode control is made and finally in section 5, conclusions are made.

2. METHODE

First, a short description on how the small signal gain is measured on a circuit with and without charge mode control. Fig. 2 shows a LCC converter with an input voltage that will have no variation. Only DC input. We like to find the small signal gain (1) from the control voltage input to output voltage.

controlout vvgain ~/~= (1)

In a system with very high gain, this cannot just be done by putting in a DC control voltage with a small perturbation on top of it and then measure the perturbation signal on the output voltage as in Fig. 2.

LCCConverter.

Fig. 2. A control vcontrol regulate the output. The DC part will give the DC working point and small perturbation give the small signal gain in this DC working point.

To control the output voltage and keep it stable, a normal feedback system is made as in Fig. 3 and a measuring signal is introduced in the feedback loop. This is further described in (R.D.Middlebrook, 1975). The small signal gain is measured between two points with low output impedance and high input impedance, which gives the voltage gain.

H

G+-

vin

vout olA =⇒

vcontrol vout

voutvin

vref

powerA =⇒ voutvcontrol

Fig. 3. Small signal power gain is measured in a closed loop setup. A signal is introduced in the loop and the voltage gain are measured from Vcontrol to Vout.

3. RESULTS

To demonstrate the difference in small signal loop gain for the power stage in Fig. 1. with frequency control and charge

mode control. The small signal gain and phase are measured with the following component values.

Ls = 330 uH Cs=22 nF Cp = 5 nF

N = 37/16 Vin = 400 V Vout = 70 V

Cout = 2 x 100 uF

With gain phase meter HP4194 and a test setup like Fig 3, small signal gain and phase were measured from Vcontrol to Vout. Measured small signal loop gains with use of the method in Fig. 3. is shown in Fig. 4.

10 100 1 103× 1 10

4× 1 105×

40−30−20−10−0

102030405060

Rload = 350 OhmRload = 70 OhmRload = 35 OhmRload = 28 Ohm

Measured gain. Frequency mode control.

frequency (Hz)

gain

(dB

)

10 100 1 103× 1 10

4× 1 105×

40−30−20−10−0

102030405060

Rload = 350 OhmRload = 70 OhmRload = 35 OhmRload = 28 Ohm

Measured gain. Charge mode control.

frequency (Hz)

gain

(dB

)

Fig. 4. Measured small signal loop gain for LCC converter with frequency control and with charge mode control for different load conditions.

4. DISCUSSION

The curves in Fig. 4 show the difference in small signal gain between frequency control and charge mode control of a LCC converter. With frequency control, the gain will change with different load, contrarily nearly no change is seen with charge mode control.

Above 10 kHz gain drops with lower load resistance or higher converter output power. In (Martin-Ramos, Dias, Pernia, Lopera, & Nuno, 2007) similar drop in gain in the

theoretical model for the frequency-controlled converter is seen. The double pole which comes from this gain drop is dependent on how close the switching frequency at the given load is to the series resonance of Ls, Cs in Fig. 1. When Rload decreases, the switching frequency is lowered and comes closer to the series resonance of Ls and Cs which gives a bigger influence on small signal gain from the double pole. Series resonance for the given values is 59 kHz and with 28 Ω load, the switching frequency is 96 kHz.

From measurements on small signal gain with charge mode control seen in Fig. 4, we can make a model as shown in Fig. 5.

Cout

Resr

Rload Vout

Ipower

Fig. 5. Small signal ac gain model of LCC converter with charge mode control.

The model in Fig. 5 have Cout with an equivalent series resistance. Rload is the load of the LCC converter.

( )( ) ( )

2

_

_

+∗=

fsP

PKI

outswPole

outswPolepower ω

ω (2)

Current generator Ipower has double pole as shown in (2). The model in Fig. 5. can be used in small signal gain calculation and simulations for LCC converters with charge mode control. K is found from DC calculations or from the measurements in Fig. 4. If frequency mode control is used, a similar model is used but then K will depend on output load.

10 100 1 103× 1 10

4× 1 105×

40−30−20−10−0

102030405060

Rload = 350 OhmRload = 70 OhmRload = 35 OhmRload = 28 Ohm

Model gain. Charge mode control.

frequency (kHz)

gai

n (d

B)

Fig. 6. Gain curve over LCC model with charge mode control.

If the model in Fig. 5 is used for the given values for the LCC converter, we get the curves in Fig. 6, which are quite similar

to the curves in Fig. 4, except for the blue 70 Ω load curve. The reason for this difference could be measurement errors.

5. CONCLUSION

Measurements on small signal gain for a frequency and charge mode controlled LCC converter were made and a model for the charge mode controlled converter was made. This model is simple in use and shows the advantage of charge mode control compared to frequency control, particular that the small signal gain does not change with load.

Next step is to make a theoretical model for the LCC converter with charge mode control. That can support the model.

ACKNOWLEDGEMENTS

Runo Nielsen has invented charge mode control. This work will not have been possible without his contribution.

REFERENCES

Agarwal, V., & Bhat, A. K. (1994). Small signal analysis of the LCC-type parallel resonant converter using discrete time domain modeling. Power Electronics Specialists Conference. 2, pp. 805 - 813. Taipei: IEEE. doi:10.1109/PESC.1994.373771

Batarseh, I. (1994, Jan). Resonant Converter Topologies with Three and Four Energy Storage Elements. IEEE Transactions on Power Electronics, 9(1), 64-73. doi:10.1109/63.285495

Bhat, A. K. (1991, MAY). Analysis and design of a series-parallel resonant converter with capacitive output filter. IEEE Transactions on Industry Applications, 523-530. doi:10.1109/28.81837

Fabiana, C. d., & Kolar, J. W. (2005). Small-Signal Model of a 5kW High-Output Voltage Capacitive-Loaded Series-Parallel. IEEE Power Electronics Specialists Conference, (pp. 1271-1277). doi:10.1109/PESC.2005.1581793

Martin-Ramos, J. A., Dias, J., Pernia, A. M., Lopera, J. M., & Nuno, F. (2007, AUG). Dynamic and Steady-State Models for the PRC-LCC Resonant Topology With a Capacitor as Output Filter. IEEE Transactions on Industrial Electronics, 54(4), 2262 - 2275. doi:10.1109/TIE.2007.894763

R.D.Middlebrook. (1975). Measurement of loop gain in feedback systems. International Journal of Electronics, 38(4), 485-512.

Runo Nielsen, Søren Kjærulff Christensen (2005, MAY). Charge Mode Control of a Series Resonance Converter. World Intellectual Property Organization, International Publication Number WO 2005/046037 A1.

Steigerwald, R. L. (1988, APR). A Comparison of Half-Bridge Resonant. IEEE Transactions on Power Electronics, 3(2), 174-182. doi:10.1109/63.4347