Chapter 6
Small signal analysis and control design of
LLC converter
6.1 Introduction
In previous chapters, the characteristic, design and advantages of LLC resonant converter were discusd. As demonstrated in chapter 3, LLC resonant converter has very low switching loss. Becau of low voltage stress on condary rectifier, low voltage rated diodes could be ud, conduction loss is also much reduced compared with PWM converter. With DC analysis and understanding of the operation of LLC resonant converter, power stage parameters could be designed to meet given specifications.
To u LLC resonant converter as front end DC/DC converter, still another important issue need to be investigated: small signal characteristic. Small signal characteristic is esntial for the feedback loop design. For front end DC/DC converter, feedback control is needed to provide a tight regulation of output voltage with load and input variation, which happens all the time for front end DC/DC converter.
In Figure 6.1, the whole converter with control circuit is shown. For LLC resonant converter, variable frequency control is ud. To achieve variable frequency control, instead of PWM comparator in PWM controller, a Voltage Controlled Oscillator (VCO) is ud to convert control
voltage Vc to the variable frequency square wave, which is ud to drive the switches. To design the compensator, we have to know the small signal characteristic of the converter. In this part, the small signal characteristic of LLC resonant converter with VCO will be investigated. Ba on the small signal characteristic of LLC resonant converter, the compensator design will be investigated later.
Figure 6.1 LLC resonant converter with feedback control
For PWM converter, state space average method has been widely ud. State space average method provides simple and accurate solution for up to half switching frequency. It has been verified and the theoretical system has been well established. With the small signal model derived from state space average method, small signal characteristic of PWM converter can be studied and control circuit can be designed accordingly.
Unfortunately, state space averaging method cannot be applied for frequency controlled resonant converter. This is becau of the totally different ways of energy processing methods for the two kinds of power converter. For PWM converter, the natural frequency of the linear network (output filter) is much lower than the switching frequency. The modulation of the converter is achieved through the low frequency content in the control signal. With this character, the average method can provide approximate linear solution of the nonlinear state equations. The derived model has a continuous form and is accurate up to half of switching frequency. However, for resonant converter, the switching frequency is clo to the natural frequency of the linear network (resonant tank). The states contain mainly switching frequency harmonics instead of low frequency content in PWM converter. The modulation of the resonant converter is achieved by the interaction between switchin
g frequency and resonant frequency. Since average method will eliminate the information of switching frequency, it cannot predict the dynamic performance of resonant converter [D-6][D-7].
In the past, veral methods were tried to solve this problem. Among the methods, some made too many simplifications that the results cannot match with test results. Some of them are very complex and difficult to u [D-8][D-9].
In this disrtation, two methods were ud. One is Extended Describing Function method developed by Dr. Eric X. Yang. This method is a simplified version of describing function method. A software package in Matlab is also
developed to realize this method. With the software package, small signal characteristic of a converter could be derived with short simulation time.
Another method ud in this disrtation is a simulation-bad method. This method us simulation tools to emulate the function of impedance analyzer to get the small signal respon of the converter. The method is bad on time domain switching model simulation, which is a necessary for every converter design. So no extra modeling effort is needed for this method. It could be ud to any periodical operating converter. It is a very effective method to deal with complex topology, which
is difficult to deal with conventional method. Also, the impact of parasitic could also be easily included into this method.
This chapter is organized in following way. First, two methods: extended describing function method and simulation-bad method, will be introduced. With the two methods, small signal characteristic of LLC resonant converter will be studied. Load impact, and resonant tank value impact will be studied with the tools. Finally, the results from the two methods will be compared with test results.
With the information of small signal characteristic of LLC resonant converter, the design of the compensator will be discusd.
6.2 Extended Describing Function analysis
Dr. Eric X. Yang published extended describing function method in [D-12]. This method is a simplified modeling method bad on the describing function method published by J. O. Groves [D-9]. With this method, the small signal model of a periodical operating converter could be derived with any order of harmonics of switching frequency taken into consideration. This method could be ud for PWM converter. With only DC components of state variables taken into consideration, it is same
as state space averaging method. For resonant converter, since switching frequency and its harmonics also play important roles in the power transfer process. State space averaging method could not be applied. With extended describing function, high order harmonics could be included so that an accurate model could be derived. The detail of extended describing function method and introduction of the software package could be found in [D-12]. The process of building the model for extended describing function is discusd in Appendix D. The model file of LLC resonant converter needed to perform the analysis are attached in Appendix D too.
In next part, the small signal characteristic of LLC resonant converter will be discusd using extended describing function method. The circuit parameters ud for this analysis is shown in Figure 6.2.
