Chapter 20 Quasi-Resonant Converters

Fundamentals of Power Electronics 1 Chapter 20: Quasi-Resonant Converters

Chapter 20Quasi-Resonant Converters

Introduction

20.1 The zero-current-switching quasi-resonant switch cell20.1.1 Waveforms of the half-wave ZCS quasi-resonant switch cell20.1.2 The average terminal waveforms20.1.3 The full-wave ZCS quasi-resonant switch cell

20.2 Resonant switch topologies20.2.1 The zero-voltage-switching quasi-resonant switch20.2.2 The zero-voltage-switching multiresonant switch20.2.3 Quasi-square-wave resonant switches

20.3 Ac modeling of quasi-resonant converters

20.4 Summary of key points



The resonant switch concept

A quite general idea:1. PWM switch network is replaced by a resonant switch network2. This leads to a quasi-resonant version of the original PWM converter

Example: realization of the switch cell in the buck converter

+–

L

C R

+

v(t)

–

vg(t)

i(t)+

v2(t)

–

i1(t) i2(t)

Switchcell

+

v1(t)

–

+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

PWM switch cell


Two quasi-resonant switch cells

+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

Lr

Cr

Half-wave ZCS quasi-resonant switch cell

Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

Lr

Cr

Full-wave ZCS quasi-resonant switch cell

Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

+–

L

C R

+

v(t)

–

vg(t)

i(t)+

v2(t)

–

i1(t) i2(t)

Switchcell

+

v1(t)

–

Insert either of the above switch cells into the buck converter, to obtain a ZCS quasi-resonant version of the buck converter. Lr and Cr are small in value, and their resonant frequency f0 is greater than the switching frequency fs.


20.1 The zero-current-switchingquasi-resonant switch cell

+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

Lr

Cr


Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

+

v2(t)

–

i1(t) i2(t)+

v1(t)

–

Lr

Cr

Full-wave ZCS quasi-resonant switch cell

Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

Tank inductor Lr in series with transistor: transistor switches at zero crossings of inductor current waveform

Tank capacitor Cr in parallel with diode D2 : diode switches at zero crossings of capacitor voltage waveform

Two-quadrant switch is required:

Half-wave: Q1 and D1 in series, transistor turns off at first zero crossing of current waveform

Full-wave: Q1 and D1 in parallel, transistor turns off at second zero crossing of current waveform

Performances of half-wave and full-wave cells differ significantly.


Averaged switch modeling of ZCS cells

It is assumed that the converter filter elements are large, such that their switching ripples are small. Hence, we can make the small ripple approximation as usual, for these elements:

In steady state, we can further approximate these quantities by their dc values:

Modeling objective: find the average values of the terminal waveforms

v2(t) Ts and i1(t) Ts


The switch conversion ratio µ

+–

+

v2(t)

–

i1(t)

v1(t)Ts

Lr

Cr


Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

i2(t)Ts

In steady state:

A generalization of the duty cycle d(t)

The switch conversion ratio µ is the ratio of the average terminal voltages of the switch network. It can be applied to non-PWM switch networks. For the CCM PWM case, µ = d.

If V/Vg = M(d) for a PWM CCM converter, then V/Vg = M(µ) for the same converter with a switch network having conversion ratio µ.

Generalized switch averaging, and µ, are defined and discussed in Section 10.3.


20.1.1 Waveforms of the half-wave ZCSquasi-resonant switch cell

+–

+

v2(t)

–

i1(t)

v1(t)Ts

Lr

Cr


Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

i2(t)Ts

The half-wave ZCS quasi-resonant switch cell, driven by the terminal quantities v1(t)Ts and i2(t)Ts.

= 0t

i1(t)

I2

v2(t)

0Ts

Vc1

Subinterval: 1 2 3 4

Conductingdevices:

Q1

D2

D1

Q1D1

D2X

Waveforms:

Each switching period contains four subintervals


Subinterval 1

+–

+

v2(t)

–

i1(t)

V1

Lr

I2

Diode D2 is initially conducting the filter inductor current I2. Transistor Q1 turns on, and the tank inductor current i1 starts to increase. So all semiconductor devices conduct during this subinterval, and the circuit reduces to:

Circuit equations:

with i1(0) = 0

Solution:

where

This subinterval ends when diode D2 becomes reverse-biased. This occurs at time 0t = , when i1(t) = I2.



Subinterval 2

Diode D2 is off. Transistor Q1 conducts, and the tank inductor and tank capacitor ring sinusoidally. The circuit reduces to:

+–

+

v2(t)

–

i1(t)

V1

Lr

I2Cr

ic(t)

The circuit equations are

The solution is

The dc components of these waveforms are the dc solution of the circuit, while the sinusoidal components have magnitudes that depend on the initial conditions and on the characteristic impedance R0.


Subinterval 2continued

Peak inductor current:

This subinterval ends at the first zero crossing of i1(t). Define = angular length of subinterval 2. Then

= 0t

i1(t)

I2

v2(t)

0Ts

Vc1


Conductingdevices:

Q1

D2

D1

Q1D1

D2X

= 0t

i1(t)

I2

v2(t)

0Ts

Vc1


Conductingdevices:

Q1

D2

D1

Q1D1

D2X

Must use care to select the correct branch of the arcsine function. Note (from the i1(t) waveform) that .

Hence


Boundary of zero current switching

If the requirement

is violated, then the inductor current never reaches zero. In consequence, the transistor cannot switch off at zero current.

The resonant switch operates with zero current switching only for load currents less than the above value. The characteristic impedance must be sufficiently small, so that the ringing component of the current is greater than the dc load current.

Capacitor voltage at the end of subinterval 2 is


Subinterval 3

All semiconductor devices are off. The circuit reduces to:

The circuit equations are

The solution is

+

v2(t)

–

I2Cr

Subinterval 3 ends when the tank capacitor voltage reaches zero, and diode D2 becomes forward-biased. Define = angular length of subinterval 3. Then


Subinterval 4

Subinterval 4, of angular length , is identical to the diode conduction interval of the conventional PWM switch network.

Diode D2 conducts the filter inductor current I2

The tank capacitor voltage v2(t) is equal to zero.

Transistor Q1 is off, and the input current i1(t) is equal to zero.

The length of subinterval 4 can be used as a control variable. Increasing the length of this interval reduces the average output voltage.


Maximum switching frequency

The length of the fourth subinterval cannot be negative, and the switching period must be at least long enough for the tank current and voltage to return to zero by the end of the switching period.

The angular length of the switching period is

where the normalized switching frequency F is defined as

So the minimum switching period is

Substitute previous solutions for subinterval lengths:


20.1.2 The average terminal waveforms

+–

+

v2(t)

–

i1(t)

v1(t)Ts

Lr

Cr


Switch network

+

v1r(t)

–

i2r(t)D1

D2

Q1

i2(t)Ts

Averaged switch modeling: we need to determine the average values of i1(t) and v2(t). The average switch input current is given by

i1(t)

I2 i1(t)Ts

t

q2q1

q1 and q2 are the areas under the current waveform during subintervals 1 and 2. q1 is given by the triangle area formula:


Charge arguments: computation of q2

i1(t)

I2 i1(t)Ts

t

q2q1

+–

+

v2(t)

–

i1(t)

V1

Lr

I2Cr

ic(t)

Circuit during subinterval 2

Node equation for subinterval 2:

Substitute:

Second term is integral of constant I2:


Charge argumentscontinued

First term: integral of the capacitor current over subinterval 2. This can be related to the change in capacitor voltage :

= 0t

i1(t)

I2

v2(t)

0Ts

Vc1


Conductingdevices:

Q1

D2

D1

Q1D1

D2XSubstitute results for the two integrals:

Substitute into expression for average switch input current:


Switch conversion ratio µ

Eliminate , , Vc1 using previous results:

where


Analysis result: switch conversion ratio µ

Switch conversion ratio:

with

This is of the form

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1Js


Characteristics of the half-wave ZCS resonant switch

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Js

ZCS boundary

F = 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Js ≤ 1

Switch characteristics:

Mode boundary:


Buck converter containing half-wave ZCS quasi-resonant switch

Conversion ratio of the buck converter is (from inductor volt-second balance):

For the buck converter,

ZCS occurs when

Output voltage varies over the range

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Js

ZCS boundary

F = 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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Chapter 20 Quasi-Resonant Converters