AC modeling of converters containing resonant switches State-Space Averaging: see textbook section 7.3 Averaged Circuit Modeling and Circuit Averaging: see textbook section 7.4

Averaged Switch Modeling Separate switch elements from remainder of converter Remainder of converter consists of linear circuit The converter applies signals xT to the switch network The switch network generates output signals xs We have solved for how xs depends on xT

Block diagram of converter Switch network as a two-port circuit:

The linear time-invariant network

The circuit averaging step To model the low-frequency components of the converter waveforms, average the switch output waveforms (in xs(t)) over one switching period.

Relating the result to previously-derived PWM converter models: a buck is a buck, regardless of the switch We can do this if we can express the average xs(t) in the form

PWM switch: finding Xs1 and Xs2

Finding µ: ZCS example where, from previous slide,

Derivation of the averaged system equations of the resonant switch converter Equations of the linear network (previous Eq. 1): Substitute the averaged switch network equation: Result: Next: try to manipulate into same form as PWM state-space averaged result

Conventional state-space equations: PWM converter with switches in position 1 In the derivation of state-space averaging for subinterval 1: the converter equations can be written as a set of linear differential equations in the following standard form (Eq. 7.90): These equations must be equal: Solve for the relevant terms: But our Eq. 1 predicts that the circuit equations for this interval are:

Conventional state-space equations: PWM converter with switches in position 2 Same arguments yield the following result: and

Manipulation to standard state-space form Eliminate Xs1 and Xs2 from previous equations. Result is: Collect terms, and use the identity µ + µ’ = 1: —same as PWM result, but with d  µ

Perturbation and Linearization The switch conversion ratio µ is generally a fairly complex function. Must use multivariable Taylor series, evaluating slopes at the operating point:

Small signal model Substitute and eliminate nonlinear terms, to obtain: Same form of equations as PWM small signal model. Hence same model applies, including the canonical model of Section 7.5. The dependence of µ on converter signals constitutes built-in feedback.

Salient features of small-signal transfer functions, for basic converters

Parameters for various resonant switch networks

Example 1: full-wave ZCS Small-signal ac model

Low-frequency model

Example 2: Half-wave ZCS quasi-resonant buck

Small-signal modeling

Equivalent circuit model

Low frequency model: set tank elements to zero

Predicted small-signal transfer functions Half-wave ZCS buck