Class-D Resonant Converter — Phil's Workshop

1Overview

The Class-D half-bridge with a series-resonant tank is the workhorse topology for high-frequency AC loads such as Dielectric Barrier Discharge (DBD) plasma reactors. Conventional design relies on the First-Harmonic Approximation (FHA) — a linearization that treats the square-wave output as a sinusoid at the fundamental frequency. FHA gives closed-form design equations but breaks down away from resonance and entirely for nonlinear loads.

This work has two objectives: (1) validate a full nonlinear ODE model against the FHA baseline, and (2) characterize the ZVS–power trade-off using the ODE, identifying where conventional analytical design fails and motivating an RL-based approach.

ODE vs FHA at resonance: 0.18% error — ODE correctly implemented

ZVS onset: f_sw/f₀ ≈ 1.05

Power penalty at 1.1f₀: ~8%

FHA error at 0.5f₀: 46% — harmonics dominate

2Circuit Model

The half-bridge produces a square wave switching between 0 and V_dc at frequency f_sw. This drives a series resonant tank (L_r, C_r) into a load R. In the soft-switching model, a midpoint capacitance C_o (MOSFET output capacitance) is added in parallel with the lower switch.

  V_dc ─── S1 ──┬── L_r ── C_r ── R ──┐
                │                      │
               C_o  v_mid              │
                │                      │
   0V  ─── S2 ──┴──────────────────────┘

The two state variables during conduction are the inductor current i_L and capacitor voltage v_C:

di_L/dt = (v_in(t) − v_C − i_L·R) / L_r

dv_C/dt = i_L / C_r

During dead time, v_mid becomes a free variable and the system becomes 3rd order — C_o charges or discharges driven entirely by i_L.

Parameter	Value	Unit
V_dc	400	V
L_r	50	μH
C_r	100	nF
R	10	Ω
C_o	500	pF
f₀	71.18	kHz
Q = ω₀L_r/R	2.24	—

3FHA Baseline

The half-bridge square wave has a Fourier fundamental amplitude of V₁ = 2V_dc/π. The DC component is blocked by C_r. Retaining only the fundamental, the voltage transfer function is:

H(jω) = R / [R + j(ωL_r − 1/ωC_r)]

|H| = R / √[R² + (ωL_r − 1/ωC_r)²]

At resonance ω_sw = ω₀ = 1/√(L_rC_r), the reactive terms cancel (Z = R), giving peak current I_peak = 2V_dc/(πR) = 25.46 A and peak load voltage V_R = 254.65 V.

4ODE Validation

The ODE is integrated with scipy's Radau solver (max step T_sw/200) for 80 cycles to reach steady state. At resonance, the ODE matches the FHA to within 0.18% for both peak current and peak voltage — confirming correct implementation and sufficient solver accuracy.

ODE vs FHA steady-state waveforms — **Fig. 1** — Steady-state waveforms at resonance. Blue: ODE. Red dashed: FHA fundamental. Top: switching voltage v_in. Middle: inductor current i_L. Bottom: load voltage v_R.

Quantity	FHA	ODE	Error
I_peak	25.46 A	25.42 A	0.18%
V_R,peak	254.65 V	254.19 V	0.18%

5Bode Plot & Frequency Sweep

The FHA transfer function is compared against ODE-extracted gain and phase across a 4:1 frequency range. Near resonance (0.85 ≤ f_sw/f₀ ≤ 1.15) agreement is within a few percent. At low frequencies the error grows sharply — at 0.5f₀, the third harmonic lands near the resonant peak and is amplified, which the FHA cannot capture.

Bode plot FHA vs ODE — **Fig. 2** — Bode plot. Solid: analytical FHA. Red dots: ODE-extracted gain and phase. Dashed vertical: f₀. ODE diverges from FHA below ~0.85f₀ due to harmonic amplification.

f_sw/f₀	f_sw [kHz]	\|H\| FHA	\|H\| ODE	Error
0.50	35.59	0.286	0.418	46.3%
0.70	49.82	0.523	0.596	13.9%
0.85	60.50	0.808	0.868	7.4%
0.95	67.62	0.975	0.998	2.4%
1.00	71.18	1.000	0.998	0.2%
1.05	74.74	0.977	0.954	2.4%
1.15	81.85	0.847	0.809	4.5%
1.30	92.53	0.644	0.624	3.1%
1.50	106.76	0.473	0.488	3.3%
2.00	142.35	0.286	0.327	14.4%

Waveforms at different switching frequencies — **Fig. 3** — Waveforms at 0.7f₀, f₀, and 1.3f₀. Below resonance: tank is capacitive, current leads, waveforms non-sinusoidal. At resonance: ODE and FHA overlap. Above resonance: harmonics well-attenuated.

6ZVS & Dead-Time Design

ZVS requires i_L > 0 at the switching instant so that C_o can discharge from V_dc to zero before the incoming switch turns on. The FHA gives the current at the half-period and the minimum required dead time:

i_L(T/2) = −(V₁/|Z|) · sin φ_I

τ_min = C_o · V_dc / |i_L(T/2)|

At resonance φ_I = 0, so i_L(T/2) = 0 and τ_min → ∞ — ZVS is impossible without operating above resonance where the tank becomes inductive.

**Fig. 4** — FHA design curves vs. f_sw/f₀. Top: voltage gain |H|. Middle: current at the switching instant (positive = ZVS capable). Bottom: minimum dead time τ_min, which diverges at resonance.

	f_sw = f₀	f_sw = 1.1f₀
\|H\|	1.000	0.920
I_peak	25.46 A	23.42 A
φ_I	0°	−23.1°
i_L(T/2)	0 A	9.19 A
τ_min	∞	21.8 ns
P_load	3242 W	2743 W

ODE verification with dead time — **Fig. 5** — ODE simulation with C_o and dead time. Left: f_sw = f₀ — slow transition (low current). Center: f_sw = 1.1f₀ — fast ZVS transition. Right: zoomed dead-time detail. Orange shading marks the dead-time interval.

7Design Trade-off

A sweep of f_sw/f₀ from 0.9 to 1.3 quantifies the fundamental ZVS–power trade-off. The ODE measures actual v_mid at the end of dead time and actual delivered power, revealing where the FHA current estimate is optimistic vs. pessimistic.

ZVS margin vs power trade-off — **Fig. 6** — ZVS–power trade-off sweep. Top: residual v_mid at end of dead time — green = full ZVS achieved. Middle: switching-instant current, FHA vs ODE. Bottom: delivered power falls as f_sw moves away from f₀.

ZVS transition detail — **Fig. 7** — Zoomed dead-time transitions at three operating points. Only above ~1.05f₀ does v_mid fully reach zero before the next switch turns on.

ZVS onset: f_sw/f₀ ≈ 1.05

Power penalty: ~8% at 1.1f₀, ~20% at 1.2f₀

FHA accuracy: underestimates ZVS current below resonance (harmonics help); overestimates above — ODE verification required

8Conclusions

The conventional FHA-based design pipeline is accurate only near resonance with linear loads, and fails in two key ways:

Harmonic content: At f_sw < 0.85f₀, higher harmonics are amplified by the tank — FHA underestimates current by up to 46%. The ODE captures this correctly.
Nonlinear loads: For DBD loads where impedance alternates between capacitive and resistive within each cycle, FHA linearization fails entirely.

This validated ODE forms the simulation environment for an RL agent tasked with jointly optimizing (L_r, C_r, τ, f_sw) for a DBD load — replacing the iterative manual design loop with a single learned policy.

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