# LLC Power Conversion Explained, Part 5: Explaining the Gain Equation

After a four-part introduction, the LLC gain equation is finally introduced. This portion was delayed due to the need to explain that the LLC is a switched mode power supply (SMPS), yet the gain has nothing to do with inductor ‘ON’ time like a continuous mode buck or boost converter. Similarly, the discontinuous mode flyback gain is calculated from the output energy. The LLC on the other hand gets its gain from the impedance divider equation as shown in the following figure. Image courtesy of On Semiconductor

Further on down in the ON semi application note, Figure 11 shows how the resonant tank circuit inductors have two different frequencies, fr1 and fr2, that affect each other. Image courtesy of On Semiconductor

The two resonant frequencies are based on the leakage inductance Lr (fr1) and magnetizing inductance Lm (fr2) as follows in the following excerpt from the ON Semi application note: Image courtesy of On Semiconductor

With the many modes of operation of the LLC, along with the various circumstances for creating a resonant current as well as a magnetizing current, this design is rather confusing. To add to the confusion, many of the references base a variable on a variable. Therefore there are uses of the variable Q, a common quality factor variable, an M, the gain equation. Instead of going off introducing more variables, it is best to jump right to the gain plot that shows the two resonant points versus frequency. This particular plot is from the Texas Instruments LLC evaluation board in Reference 8.

It is important to note that the leakage inductance resonant frequency (fr1) is at 93 kHz. This is the operating point for a gain of one for the resonant tank circuit. The controller will keep the switching frequency at this point while producing the square wave that is turned into a sine wave as outlined in LLC Power Conversion Explained, Part 2: Sine Wave from a Square Wave. Note that not all LLC power based circuits operate at a gain of one. The transformer turns ratio is used to adjust the voltage gain. For the TI design8 , the voltage is stepped down from a 400V PFC output to 42 volts for a Lithium ion battery. Image courtesy of Texas Instruments LLC evaluation board

I attempted to reproduce this gain in Excel and got the following results: It turns out that this is not a straight forward calculation. Even the Bo Yang dissertation and subsequent appendix1 really don’t give you the exact formula. My formula just used the basic impedance equation that is repeated here for convenience. Image courtesy of On Semiconductor

The gain is found using the impedance of a voltage divider in the figure above where

Vo = (Vin x X2) / (X1 + X2)

(Note that this is a correction from part 2 which erroneously left out the division sign.)

Where X2 = XLm||Rac and X1 = XCr + XLr when combining the impedances of the resonant tank circuit components in the last figure.

Note that I stumbled a bit when recreating the gain equation plot due to not taking the absolute value of the results as shown in the following figure. As a result, an important (and yet another) operating mode of the LLC converter became apparent. The negative spike is actually due to the capacitor having an imaginary number in the denominator, thus the absolute value of the gain equation accounts for the negative spike. This capacitive region defines the ZCS operating point of the LLC rectifier diodes. Rectifier diodes benefit from ZCS as it offsets the reverse recovery of the rectifier. The capacitive region is shown in the following excerpt from the Infineon application note5 . Image courtesy of Infineon in Reference 5

The ON Semi application note 2 goes a step further by introducing regions that have both inductor and capacitor based impact as shown below. These modes are crucial to ZVS and ZCS operation as detailed in the ON Semi application note 2 . Image courtesy of ON Semi in Reference 2

The LLC operation and gain have been presented in a series of five blogs. The next step will be to identify the waveforms associated with operating in and out of tune; which means in the region of fr1, the confluence point that all load lines converge through in the graphs of the gain presented in this blog.

References

• “Design Considerations for an LLC Resonant Converter” Fairchild Semiconductor Power Seminar 2007 Appendix A: White Papers; couldn’t get a website URL; suggest you Google the text in brackets [“Design Considerations for an LLC Resonant Converter” Fairchild Semiconductor Power Seminar 2007 Appendix A: White Papers]
• Square Wave Signals”, Chapter 7 – Mixed-Frequency AC Signals, All About Circuits website

## 2 comments on “LLC Power Conversion Explained, Part 5: Explaining the Gain Equation”

1. traneus
September 24, 2017

Scott, thanks for this series on LLC power converters. If one uses the AC output directly, this might be useful as an AM radio transmitter.

2. tecmaestro
September 27, 2017

Thanks

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