What is the meaning of the Quality factor of LLC resonant converter? For LLC resonant converter, the quality factor Q is used to describe the operating range of the LLC. Operating range means the region in the gain-frequency curve wherein the LLC can be operated without damage. When the quality factor is high, the circuit is a narrow band. On the other hand, if the quality factor is low, the circuit is wide band.
Quality factor (Q) is the ratio of the characteristic impedance to the ac resistance.
Q = [ sqrt (Lr / Cr) ] / Rac
where
Lr – is the resonant inductor
Cr – is the resonant capacitor
sqrt (Lr/Cr) – is the characteristic impedance of an LC circuit
Rac – is the equivalent AC resistance
Above parameters are shown in the AC circuit of LLC resonant converter below.
If you are interested to know how the AC circuit is derived, read How to Derive LLC Converter AC Circuit.
Quality Factor of LLC Resonant Converter and Frequency-Gain Curve
In order to understand further the importance of the Quality factor to the LLC resonant converter, consider the graph below. It is called a Frequency-gain curve or gain-frequency curve by some. The x-axis is the frequency while the y-axis is the gain.
There are 5 plots or curves in the graph as you can see in different colors. The red vertical line is the resonant frequency. All the plots or curves are taken from the same AC circuit and by just varying the AC resistance to generate 5 different plots. The peak of each plot or curve that is highlighted by a big dot is the so called peak gain.
An LLC resonant converter can operate without damage starting from the peak gain going to the right portion of the curve. This is the operating range of an LLC resonant converter.
Why only limit the operation to the right of the peak gain? It is because the right side of the peak gain is inductive region. In the inductive region, soft switching occurs which mean there is no overlap between voltage and current.
If you remember the basic electrical circuit theory, in an inductive circuit, the current lags the voltage. If we will convert this to a switching MOSFET, the current will only flow when the drain to source voltage VDS is already zero. Thus, there is no power loss ideally.
On the other hand, the left side of the peak gain is a capacitive region. When the circuit is capacitive, the current leads the voltage. This will result to an overlapping voltage and current to a switching MOSFET. The end effect is very huge power loss that could immediately break the MOSFET.
In general, if the Quality factor of LLC resonant converter a high, the LLC converter is a narrow band while if the Quality factor of LLC resonant converter is low, the LLC converter is a wide band.
Let us go back to the frequency-gain curve to explain this. The yellow green plot or curve is the narrowest band of all the curves. It is because it has the lowest Rac which give the highest quality factor Q.
On the other hand, the widest band of all is the pink plot or curve and it has the highest Rac that yields the lowest quality factor Q. Lowest Rac means the LLC is heavily loaded while the highest Rac means otherwise.