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Outline of discussion DIRECT MODELING APPROACH How small-signal variations in the switching frequency affect the spectrum of the switch network output voltage vs1(t) Passing the frequency-modulated voltage vs1(t) through the tank transfer function H(s) leads to amplitude modulation of the output voltage v(t) How to recover the envelope of the output voltage and determine the small-signal control-to-output-envelope transfer function Genv(s) PHASOR TRANSFORMATION APPROACH Equivalent circuit modeling via the phasor transform PSPICE simulation of Genv(s) using the phasor transform

Large signal phasor models of circuit elements Express all circuit signals in phasor form and substitute into v-i equations for the circuit elements. Freqeuncy modulation is represented by imaginary resistors in inductor and capacitor models.

Small-signal phasor models of circuit elements From perturbing and linearizing large signal equations/models. Split each large signal waveform into DC plus small signal and substitute into large signal equations for each circuit component:

Inductor equivalent circuit model

Inductor equivalent circuit model

Inductor equivalent circuit model

Modelling the switch network

One way

More general – model the switch network with an equivalent transformer

Steps

Steps

The generalized phasor transformer model

Power balance and current scaling for phasors defined using peak amplitudes

Power balance and current scaling for phasors defined using RMS amplitudes

Linearization of the phasor transformer model

Linearization of the phasor transformer model

Linearization of the phasor transformer model

Linearization of the transformer model

The linearized phasor transformer model Steady-state plus small-signal

Example: Full-bridge parallel resonant inverter 1 Example: Full-bridge parallel resonant inverter 1. Construct steady-state plus small-signal phasor model

Example: Full-bridge parallel resonant inverter 2 Example: Full-bridge parallel resonant inverter 2. Steady-state solution

Example: Full-bridge parallel resonant inverter 3 Example: Full-bridge parallel resonant inverter 3. Small-signal perturbation in output phasor

Example: Full-bridge parallel resonant inverter 3 Example: Full-bridge parallel resonant inverter 3. Small-signal perturbation in output phasor

Recovering the actual output envelope

Recovering the actual output envelope

Recovering the actual output envelope

Simulation using SPICE

SPICE inductor model

SPICE models

SPICE models

Example: Full-bridge parallel resonant inverter SPICE-compatible model

Recovering the actual output envelope in SPICE

Recovering the actual output envelope in SPICE