1

AC modeling of quasi-resonant convertersExtension of State-Space Averaging to model non-PWM switches

Use averaged switch modeling technique: apply averaged PWM model, with d replaced by µ

Buck example with full-wave ZCS quasi-resonant cell:

+–

L

C R

+

v(t)

–

vg(t)

i(t)+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

Lr

Cr

Full-wave ZCS quasi-resonant switch cell

+

v1r(t)

–

i2r(t)D1

D2

Q1

Frequencymodulator

Gatedriver

vc(t)

µ = F

2

3

4

Equilibrium (dc) state-space averaged model

Provided that the natural frequencies of the converter, as well as the frequencies of variations of the converter inputs, are much slower than the switching frequency, then the state-space averaged model that describes the converter in equilibrium is

where the averaged matrices are

and the equilibrium dc components are

5

Small-signal ac state-space averaged model

where

So if we can write the converter state equations during subintervals 1 and 2, then we can always find the averaged dc and small-signal ac models

6

Relevant background

State-Space Averaging: see textbook section 7.3Averaged Switch Modeling and Circuit Averaging: see textbook

section 7.4

7

Circuit averaging and averaged switch modeling

• Separate switch elements from remainder of converter

• Choose the independent input signals xT to the switch network

• The switch network generates dependent output signals xs

• Average switch waveforms

• Solve for how <xs> depends on <xT>

8

9

Basic switch networks and their PWM CCM large-signal, nonlinear, averaged switch models

for non-isolated converters

10

Basic switch networks and their PWM CCM dc + small-signal averaged switch models

for non-isolated converters

11

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>

• Replace switch network with its averaged switch model

12

Block diagram of converter

Switch network as a two-port circuit:

13

The linear time-invariant network

14

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.

15

Relating the result to previously-derived PWM converter models

We can do this if we can express the average xs(t) in the form

16

PWM switch: finding Xs1 and Xs2

17

Finding µ: ZCS example

where, from previous slide,

18

Derivation of the averaged system equationsof the resonant switch converter

Equations of the linear network(previous Eq. 1):

Substitute the averaged switch network equation:

Result:

Next: try to manipulate into sameform as PWM state-space averaged result

19

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 lineardifferential equations in the following standard form (Eq. 7.90):

But our Eq. 1 predicts that the circuit equations for this interval are:

These equations must be equal:

Solve for the relevant terms:

20

Conventional state-space equations: PWM converter with switches in position 2

Same arguments yield the following result:

and

21

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

22

Perturbation and Linearization

The switch conversion ratio µ is generally a fairlycomplex function. Must use multivariable Taylor series,evaluating slopes at the operating point:

23

24

25

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.

26

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

27

Parameters for various resonant switch networks

28

Example 1: full-wave ZCSSmall-signal ac model

29

Low-frequency model

30

Example 2: Half-wave ZCS quasi-resonant buck

31

Small-signal modeling

32

Equivalent circuit model

33

Low frequency model: set tank elements to zero

34

Predicted small-signal transfer functionsHalf-wave ZCS buck