How to Derive LLC Converter AC Circuit

If you are asking how to derive LLC converter AC circuit, this article is perfect for you. The derivation is based on the first harmonic approximation (FHA). FHA is popular method in deriving AC circuit like those of an LLC which only considering the fundamental signal of a square wave. In this article I will give you some insights in the aim to help you understand faster. If this is what you are looking, this is perfectly for you.

Knowing the Square Wave Components

The square wave is a result of the sum of the fundamental and the odd harmonics of a sine wave. I am not an expert about this topic so I just based this to what I read. The first harmonic approximation then only focuses on the fundamental part of the waveform. The fundamental part of the the waveform shows the perfect sinusoidal shape.

 how to derive LLC converter AC circuit

How to Derive LLC Converter AC Circuit – Step by Step Procedure

Below is the step by step process on how to derive LLC converter AC circuit. AC circuit is very useful to easily simulate and understand the behavior of LLC resonant converter.

Step 1: Remove the Primary Switch and Derive the Equivalent Input Voltage Equation

There are few ways to make the LLC power section. Assuming your LLC circuit is like below diagram.

Remove the switches S1 and S2; they are the primary switches. After doing so, the voltage feed to the Lres and Cres is a continuous pulse wave. By the way, Lres is the resonant inductance while Cres is the resonant capacitance while Lmag is magnetizing inductance.

 how to derive LLC converter AC circuit - Remove the primary switches

The resulting circuit had changed. Both primary switches are gone. The input voltage to the tank circuit is now Vd. Tank circuit is often the term used to describe the network of series inductance, series capacitance and the magnetizing inductance or the transformer.

The fundamental equation of Vd is expressed below.

Step 2: Represent the Secondary Circuit with an AC Resistance

The next critical step on how to derive LLC converter AC circuit is to represent the secondary circuit with an AC resistance. This part is difficult. You must be a highly knowledgeable in electrical circuits and in mathematics. But just follow below derivations and sure you can get over this.

Resistance is always voltage over current as ohm’s law defined it. So, let us get the equivalent voltage and current in the secondary side. These voltage and current must be a fundamental value to satisfy FHA.

The fundamental value of the secondary load side voltage is

On the other hand, the fundamental of the secondary current is

Both the secondary voltage and current equations are known. The AC resistance of the secondary side is now easy to express as

Step 3: Transfer the AC Resistance on the Secondary to the Primary Side

We are getting close. Next step to perform is to transfer the AC resistance from the secondary side to the primary side. This could be done using the turns ratio and resistance equation.

Step 4: Finally, this is the answer to the question how to derive LLC converter AC circuit

There you go! The equivalent AC circuit of an LLC resonant converter is illustrated below.

 how to derive LLC converter AC circuit - The FHA circuit

Look how simple it is now compared to the first circuit above. Is this circuit model accurate? Since this is just an approximation (as the term FHA), the answer is not. However, as long as the LLC resonant converter operation is maintained at the resonant and not considering the effect of parasitic components, this circuit model is accurate. An LLC resonant converter could go away from the resonance depending on the circuit values.

Example on how to use the LLC resonant converter AC circuit in simulation

The main purpose in deriving the AC circuit is to use it in simulation and better understand the behavior of LLC resonant converter. The main parameters that could be extracted are the gain and the frequency relationship. It could be extracted from the simulation also how much design margin is present.

On the first example, we are going to plot the gain-frequency curve of the LLC with a fixed output voltage and current. Supposing the given are; Vo = 12V, Io = 10A and the turns ratio is 16.

The peak most of the curve is the boundary between capacitive and inductive region. For LLC, the desired operation is at the inductive region. If you are going to design an LLC, you should limit the circuit maximum gain below the peak gain to the right of the curve. Otherwise, failure will occur.

On the second example, we are sweeping the value of Rac from 50 to 250 ohms with 50 ohms apart. The result of the simulation shows the different gain-frequency curve per equivalent AC resistance.

How to Derive LLC Converter AC Circuit Summary

  • LLC resonant converter AC circuit derivation is based on FHA.
  • FHA stands for first harmonic approximation.
  • The equivalent AC circuit is only composed of Lres, Cres, Lmag and Rac
  • Lres is the resonant inductance, Cres is the resonant capacitance, Lmag is the magnetizing inductance while Rac is the ac resistance.
  • Rac is a function of turns ratio, output voltage and output current.
  • FHA is only accurate when the LLC resonant converter operates at resonance.

Should you have an easier way on how to derive LLC converter AC circuit, please share it. Let’s discuss.


    1. Thank you Want to Learn for a great question. When the resonant inductance will change, the LLC will not anymore operate at resonance. The LLC losses its ability to soft switch.

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