## Accurate Small-Signal Modeling for Resonant Converters

##### Abstract

In comparison with PWM converters, resonant converters are gaining increasing popularity for cases in which efficiency and power density are at a premium. However, the lack of an accurate small-signal model has become an impediment to performance optimization. Many modeling attempts have been made to date. Besides the discrete time-domain modeling, most continuous-time modeling approaches are based on fundamental approximation, and are thus unable to provide sufficient accuracy for practical use. An equivalent circuit model was proposed by Yang, which works well for series resonant converters (SRCs) with high Q (quality factor), but which is inadequate for LLC resonant converters. Furthermore, the model is rather complicated, with system orders that are as high as five and seven for the SRC and LLC converter, respectively.
The crux of the modeling difficulty is due to the underlying assumption based on the use of a band-pass filter for the resonant tank in conjunction with a low-pass output filter, which is not the case for most practical applications. The matter is further complicated by the presence of a rectifier, which is a nonlinearity that mixes and matches the original modulation frequency. Thus, the modulation signal becomes intractable when using a frequency-domain modeling approach.
This dissertation proposes an extended describing function modeling that is based on a Fourier analysis on the continuous-time-domain waveforms. Therefore, all important contributions from harmonics are taken into account. This modeling approach is demonstrated on the frequency-controlled SRC and LLC converters. The modeling is further extended to, with great accuracy, a charge-controlled LLC converter.
In the case of frequency control, a simple third-order equivalent circuit model is provided with high accuracy up to half of the switching frequency. The simplified low-frequency model consists of a double pole and a pair of right-half-plane (RHP) zeros. The double pole, when operated at a high switching frequency, manifests the property of a well-known beat frequency between the switching frequency and the resonant frequency. As the switching frequency approaches the resonant frequency of the tank, a new pair of poles is formed, representing the interaction of the resonant tank and the output filter. The pair of RHP zeros, which contributes to additional phase delay, was not recognized in earlier modeling attempts.
In the case of charge control, a simple second-order equivalent circuit model is provided. With capacitor voltage feedback, the order of the system is reduced. Consequently, the resonant tank behaves as an equivalent current source and the tank property is characterized by a single pole. The other low-frequency pole represents the output capacitor and the load. However, the capacitor voltage feedback cannot eliminate the high-frequency poles and the RHP zeros.
These RHP zeros may be an impediment for high-bandwidth design if not properly treated. Based on the proposed model, these unwanted RHP zeros can be mitigated by either changing the resonant tank design or by proper feedback compensation. The accurate model is essential for a high-performance high-bandwidth LLC converter.

##### General Audience Abstract

For high-frequency power conversion, resonant converters are increasingly popular. However, the lack of an accurate small-signal model has become an impediment to performance optimization. The existing equivalent circuit model and its simplified circuit were based on fundamental approximation, where the resonant tank was deemed a good band-pass filter. These models work well for series resonant converters (SRCs) with high Q (quality factor), but are inadequate for LLC resonant converters.
The crux of the modeling difficulty is due to the fact that the operation of this type of resonant converter is based on the use of a band-pass filter in conjunction with a low-pass filter. The matter is further complicated by the presence of a rectifier, which is a nonlinearity that mixes and matches the original modulation frequency. Thus, the modulation signal becomes intractable when using a frequency-domain modeling approach.
This dissertation proposes an extended describing function modeling that is based on a Fourier analysis on the continuous-time-domain waveforms. Therefore, all important contributions from harmonics are taken into account. This modeling approach is demonstrated on the frequency-controlled SRC, frequency-controlled LLC converter, and charge-controlled LLC converter, and the resulting models are proven to be accurate at all frequencies.
A simple equivalent circuit model is provided that targets the frequency range below the switching frequency. This simple, accurate model is able to predict the small-signal behaviors of the LLC converter with high accuracy at half of the switching frequency.
At high modulation frequencies, the resonant converter behaves like a non-minimum phase system, which was neither recognized nor characterized before. This property can be represented by RHP zeros, and these RHP zeros may be an impediment for high-bandwidth design if not properly treated. Based on the proposed model, these unwanted RHP zeros can be mitigated by either changing the resonant tank design or by proper feedback compensation. Accurate modeling is essential for a high-performance high-bandwidth LLC converter.

##### Collections

- Doctoral Dissertations [15532]

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