**Non-linear Time Domain Simulation**Full analytical solutions to the "complete" non-linear LLC circuit in the time-domain seem quite involved and I haven't seen any complete analysis of all 6 operational modes in a CLLC or LLC system though there are some papers with partial analysis. So likewise I might do some analysis of narrow non-linear time-domain cases (e.g. ZVS conditions) but otherwise I'm going to rely on LTspice simulation.

**Square Wave Input - Rectified Output**Before getting dug into detailed analysis of time-domain waveforms and behaviours of a spice circuit modelling "real" components including input inverter switching behaviour and output rectifiers I wanted to have a look at the simplified non-linear model used as the basis for the linearised FHA (first/fundamental harmonic analysis) and see how the linearised behaviour compares to the non-linear model.

Here is the LTspice circuit. Almost exactly the simplified model CLLC introduced earlier but I've put in voltage source left at 0V which I can use to help simulate a battery later on. The input square wave is parametrised to allow easy .step simulation over the frequency range. There are also a bunch of .meas directives to obtain various operating point characteristics. I've attached the .asc file of the simulation.

A quick look at waveforms first as a sanity check to make sure the circuit is behaving as it should and there's no major modelling errors (again will do a detailed analysis of waveforms later).

Some initial instability as the output capacitor charges then a steady state.

A smaller time scale look at the waveforms over a steady state cycle, things look as expected at resonance as per waveforms shown in literature (also checked above and below resonance but leaving out screenshots to reduce clutter).

Actually checking the frequency response of the system using the .step directive now.

The screenshot shows the circuit, the stacked Vout waveforms of each simulated frequency and the .meas plots. Note duration of simulation and measurement directives have been increased from above to make sure all iterations can reach steady state.

*Nonlinear time-domain sim vs AC sim: Rload = 18, Rac = 14.6**Nonlinear time-domain sim vs AC sim: Rload = 2.5, Rac = 2.02**Nonlinear time-domain sim vs AC sim: Rload = 1.33, Rac = 1.08*The nonlinear time domain simulation frequency response matches the AC simulation frequency response quite well when well above the the primary + transformer resonance. In particular, the Vout matches very closely apart from light load around primary + transformer resonance. Current values also match pretty closely in terms of peak values apart from light load around primary + transformer resonance again. The maximum peak capacitor voltage doesn't agree with the AC simulation even accounting for \$\sqrt{2}\$ factor to convert RMS to sinusoidal peak.

R_load (Ω) | R_ac (Ω) | AC RMS (V) | AC Peak (V) | Time-Doman Peak (V) |

1.33 | 1.08 | 357.8 | 506.0 | 480 |

2.5 | 2.02 | 219.2 | 310.0 | 335 |

18 | 14.6 | 259.1 | 366.4 | 275 |

The observed discrepencies as well as agreements between the nonlinear time domain simulation and the AC simulation could be explained due to the way the models are setup. The FHA is setup with an RMS equivalent input and output which works well enough since the input and output in the "actual" time domain model are effectively "DC" constant magnitude voltage. The intermediate network values are sinusoidal near primary tank resonance and roughly sinusoidal above resonance but below resonance they become significantly nonsinusoidal and all the extra harmonic content likely causes behaviour to start to deviate significantly from the FHA.

*Waveforms at primary + transformer resonance showing non-sinusoidal form*Additionally, there is a small bump in the resonant capacitor voltage and transformer magnetising current frequency response around ~180kHz and currently I have no clue what it is due to.

Phase of the secondary current (important to implementing syncronus rectififcation) is similarly fair close between the time-domain and AC sims near primary resonance but starts to deviate at lower frequencies and is completely different by primary + transformer resonance. There are also some "glitches" in the phase plot due to aliasing type effects in the way .meas was setup to measure zero crossing time and derive phase. Perhaps there is a better way of doing it?