ZCS notes

Power Electronics ZCS-QRC HK PolyU

Zero-current Switching Quasi-resonant Converters

1 Introduction Conventional switching mode power supplies operate in hard switching. Every time when the transistor is switched on or off, the overlapping of the voltage and current waveforms indicates the switching loss during the switching. This is illustrated in Fig. 1. Because there are one turn-on and one turn-off during each period, the switching loss therefore increases with switching frequency. One popular method is to use resonant techniques to force the switching devices to turn-on and turn-off at zero current or zero voltage, hence the switching loss becomes low.

Fig. 1 Illustration of the switching loss 2 Classification of resonant converters


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There are many ways to classify the resonant converters. They can be summarized in Table 1 In short, the Quasi-resonant converter only resonates for a portion of a operation cycle. For the load-resonant converter [1], the resonance occurs for the whole operation cycle. Also the load is part of the resonant circuit which is to make the circuit become an R-L-C resonance. The resonant transition converter only resonates during the switching transition [2]. It will return to the conventional switched mode afterwards.

Table 1 Summary of the various resonant converters Name Quasi-resonant converter Load-resonant converter Resonant Transition Comment Not complete resonance

cycle Load becomes part of the resonant circuit

High frequency on the switching transition and back to normal condition

Zero current switching Series resonant Phase-shifted converter Zero voltage switching Parallel resonant Extended-period quasi-

resonant

Types

Multi-resonant Series-parallel resonant 3 Resonant switches Fig. 2 shows three configurations of switches. This is the zero-current switching configuration and was developed many decades ago and it was promoted in mid 1980s by Lee [3]. Fig. 2a is a general topology. Fig. 2b is a half-wave with the current through the switch being one-directional and Fig. 1c is a full-wave switch in which the current through the switch is bi-directional. Lr and Cr are resonant components.

(a) General (b) Half-wave (c) Full-wave

Fig. 2 Resonant switch structure

4 Switching trajectory In the conventional inductive load switching, the switching loss is high. If the switch’s current ISW versus switch's voltage VSW is plotted, paths Hon and Hoff for the turn-on and turn-off switching trajectories will be seen respectively. The area enclosed by the curve with the coordinate axes represents the switching loss. The bigger the enclosed area, the higher the switching loss is. If the converter is operated under resonant switching, the path S may be obtained and the switching loss is low.


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Fig. 3 Load line trajectory (Hon-Hoff) Hard-switching (S) Resonant/soft switching

5 Zero current switching Quasi-resonant (ZCS QR) Buck converter 5.1 The circuit Fig. 4a shows a conventional Buck converter whereas Fig. 4b shows a ZCS-QR Buck converter. Observe the differences between the two circuits:

a) Classical (b) Resonant

Fig. 4 Buck converter

In the circuit, resonant components Lr and Cr have been added around the switch SW for the conversion into the zero-current switching topology. The other components are exactly the same. The bulk energy storage components CF and LF are usually large and their associated voltage across CF and current through LF are assumed to be constant in the analysis. For the circuit analysis, LF can be replaced by a current sink and the CF can be replaced by a voltage source. 5.2 Principle of operation The circuit can be redrawn into four equivalent circuits as shown in Fig. 5. There are four states of operation and the time domain waveform is shown in Figs 6a and 6b for half-wave and full-wave respectively. Again, LF is large, the current iLF is assumed to be equal to constant Io.


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Fig. 5 Equivalent circuits of the Buck Quasi-resonant converter during a switching cycle. A) Linear stage [t0-t1] (Fig. 5a): When Switch SW is turned on at t0, freewheeling diode

DF is still conducting the load current Io through LF in the previous stage The voltage across Lr is therefore equal to Vin. Input current iLr rises linearly and is governed by the state equations:

(a) Half-wave


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(b) Full-wave

Fig. 6:Waveforms of the Quasi-resonant ZCS Buck converter

inLr

r Vdt

diL = (1)

Solution: iLr = Vin (t-to) /Lr (2)

The way that the current rises linearly from zero makes the turn-on loss small and is considered as zero-current switching. During this state, the sum of iDF and iLr is equal to Io. This state finishes when iLr increases until iLr is greater than Io. iDF decreases to zero, DF is not required conduction any more. Io can then be supported by iLr which is shown in the next state. The duration of this state Td1 is: Lr Io / Vin (3) Boundary condition: iLr= Io (4)

B) Resonant state [t1-t2] (Fig. 5b) : Lr and Cr resonate and DF is off. The state equations are:

crin

rr vV

dtdiL −=

(5)


6

oLr

Crr Ii

dtdv

C −= (6)

The solution is:

(7)

))(cos1(

)(sin

1

1

ttVv

ttZ

VIi

oincr

oin

oLr

−−=

−+=

ω

ω

(8) where

(9)

r

r

rro

CLZ

CL

=

=1ω

(10)

The duration of this state Td2 is:

wavefull22/3

wavehalf2/3

sinwhere 12

παπ

παπ

αωα

<<

<<

−== −

in

o

od V

ZIT

For the half-wave mode, because there is a diode in series with the transistor, the current cannot be reversed. Therefore when the resonant current starts to enter the negative value, the diode stops conduction as shown in Fig 6a. The resonant state ends. However, for the full-wave mode, a diode is connected in parallel with the transistor. When the resonant current iLr changes to negative, the diode conducts. During this time, the transistor can be turned off so that when the resonant current cannot be conducted again when iLr returns from negative back to positive as shown in Fig. 6b. Equations (12) and (13) give the conduction angle α for the half-wave and full-wave respectively. Simply look at the resonant current waveform iLr, and it can be seen that α lies within [π, 3π/2] and [3π/2, 2π]. Boundary condition: iLr =0 (14)

At T2, )cos1( α−= incr Vv (15)

C) Recovering stage [t2, t3] (Fig. 5c): Resonant stops, Cr begins to be discharged through LF

with a discharging current equal to Io.

(11)

(12)

(13)


7

ocr

r Idt

dvC −= (16)

Solution: r

oinCr C

ttIVv

)()cos1( 2−−−= α (17)

Cr is discharged until its voltage reaches zero and DF then becomes forward bias. The duration of this state can be solved by equating equation (17) to zero. Duration: Td3= Cr Vin (1-cos α)/Io (18) Boundary condition: vCr=0 (19)

D) Free-wheeling stage [t3, t4] (Fig. 5d): Output current freewheels through the diode DF. Duration: Td4= Ts-Td1-Td2-Td3 (20) Ts is the duration of the switchinjg cycle. At t4, the converter will be turned on again and the same cycle repeats again, i.e. t4 is the same as t0 in the next cycle.

5.3 Condition for zero-current switching (ZCS) The condition for ZCS is that the resonant current must reach zero so that the switch can be turned off during this time. Therefore the condition for this is:

oin IZ

V≥ (21)

This condition is the same as solving equation (7) in order to obtain α using the arcsin. In order to make the arcsin function valid, equation (21) must be fulfilled. The timing for turning off the transistor is also an important issue. For the half-wave mode, the reverse voltage of the switch SW is annotated with D and T in Fig. 6a between t2 and t4 to indicate which device is standing for the voltage. Right after t2, the diode D inside the switch is in reverse bias and the reverse voltage appears across D. During this interval, the transistor T can be turned off under ZCS. After vsw becomes positive, the reverse voltage of the SW appears across T because D is in forward bias. Therefore T must be turned off before vsw is positive. For full-wave mode, T must be turned off when iLr is negative so that its anti-parallel diode is conducting all the current through SW, T can be turned off under zero-current switching. 5.4 DC Voltage conversion ratio


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The output voltage Vo can be solved by equating the input energy Ein and output energy Eo. Because the input current is the same as the iLr, based on Fig. 6, the resonant inductor Lr only conducts between t0 and t2. Hence, output energy Eo:

sooo

t

t

t

tLrLrinin

TIVE

dtidtiVE

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+= ∫ ∫

1

0

2

1

Hence, the voltage conversion ratio M is

⎟⎠⎞

⎜⎝⎛ −

Φ++

Φ= )cos1(1

22αα

π o

S

in

of

fVV

(24)

where in

oVZI

=Φ (25)

and fo=ωo/(2π) . Equation (25) can be rewritten as:

n

in

o

RM

ZR

VV

==Φ (26)

The characteristics of voltage conversion ratios are shown in Fig. 6. It is a recursive equation and numerical method is needed to calculate the characteristics. The ratio depends on the load resistance R for half-wave mode and relatively independent of R for full-wave mode. The above function in the parentheses, say F(Φ)

⎟⎠⎞

⎜⎝⎛ −

Φ++

Φ=Φ )cos1(1

2)( ααF (27)

has been plotted in Fig. 8 for full-wave operation. This function is very close to 2π for the range 0<Φ<1. However, the function for half-wave is not.

The output voltage can be alternatively calculated by the average value of the resonant capacitor voltage vcr which is illustrated in Fig. 9. It can be seen that for full-wave mode, the area A1 is about the same as A2, and A3 is about the same as A4. The average value of vCr between t1 and t3 is approximately equal to Vin. Therefore the average output voltage is approximately equal to Vin(Td2+Td3)/Ts.

The Buck ZCS-QR converter is one of the famous resonant converters for many power conversion applications. It also has a forward converter counterpart which is the transformer isolated version. The converter is easy to use and usually very stable in operation.

(22)

(23)


9

Fig. 7a DC voltage conversion of the Half-wave Buck version

Fig. 7b DC voltage conversion of the Full-wave Buck version


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Fig. 8. Approximation of the F(Φ)

Fig. 9 Alternative method to calculate output voltage by averaging vCr 6 Boost ZCS-QR converters

6.1 The circuit Fig. 10a shows a conventional Boost converter whereas Fig. 10b shows a ZCS-QR Boost converter. Observe the difference between the two circuits.


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(a) Classical (b) Resonant

Fig. 10 Boost converter

Again, it can be seen that the difference is the addition of resonant components Lr and Cr around the switch. This converter is also useful in many circuit designs as it can be used for power factor correction [4]. 6.2 Principle of operation The circuit can be redrawn into four equivalent circuits as shown in Fig. 11. There are four states of operation and the time domain waveform is shown in Fig. 12a and 4.12b for half-wave and full-wave respectively. LF is large so that its current iLF is assumed to be constant and equal to Iin.

Fig. 11 Equivalent circuits of the Boost Quasi-resonant converter during a switching cycle.

A) Linear stage [t0-t1] (Fig. 11a): When switch SW is turned on at t0, freewheeling diode

DF is still conducting the source current Iin through LF. The current iDF is also equal to Iin at t0. The voltage across vCr is therefore equal to Vo. Resonant inductor current iLr rises linearly and is governed by the state equations:

oLr

r Vdt

diL = (28)

Solution: iLr = Vo (t-t0) /Lr (29)


12

When t>t0, iDF decreases such that it is equal to Iin-iLr. This state finishes when iLr increases until iLr is greater than Iin. iDF decreases to zero. DF is not required for conduction any more. Iin can then be supported by iLr which is shown in next state. The duration of this state Td1 is: Lr Iin / Vo (30) Boundary condition: iLr=Iin (31)

B) Resonant state [t1-t2] (Fig. 11b): Lr and Cr resonate and DF is off. The state equations are:

cr

rr v

dtdiL =

(32)

Lrin

Crr iI

dtdv

C −= (33)

The solution is: (34)

)(cos

)(sin

1

1

ttVv

ttZ

VIi

oocr

oo

inLr

−=

−+=

ω

ω

(35)The constants ωo and Z are defined the same as the Buck version shown in equation (9-10). The duration of this state Td2 is:

(36)

(37)

wavefull22/3

wavehalf2/3

sinwhere 12

παπ

παπ

αωα

<<

<<

== −

o

in

od V

ZIT

(38)The consideration of the half- and full-wave operations is the same as in the Buck converter. Boundary condition: iLr =0 (39)

At t2, αcosocr Vv = (40)

C) Recovering stage [t2, t3] (Fig. 10c): Resonant stops, Cr begins to be charged by the input

current Iin through LF.

in

crr I

dtdv

C = (41)

Solution: r

inoCr C

ttIVv )(cos 2−+= α (42)


13

Cr is discharged until its voltage reaches zero and DF then becomes forward bias. The duration of this state can be solved by equating equation (42) to zero. Duration: Td3= Cr Vo (1-cos α)/Iin (43) Boundary condition: vCr=Vo (44)

D) Freewheeling stage [t3, t4] (Fig. 10d): Output current freewheels through DF. This stage finishes when the control gate voltage of SW is turned on again at t4. Duration: Td4= Ts-Td1-Td2-Td3 (45)

(a) Half-wave


14

(b) Full-wave

Fig. 12: Waveforms of the Quasi-resonant ZCS Boost converter 6.3 Condition for zero-current switching (ZCS) The condition for ZCS is that the resonant current must reach zero so that the switch can be turned off during this time. Therefore the condition for this is:

ino I

ZV

≥ (46)

This condition is the same as solving equation (36) in order to obtain α using the arcsin. In order to make the arcsin function valid, equation (46) must be fulfilled. The timing for turning off the transistor is also an important issue. For the half-wave mode, the reverse voltage of the switch S is annotated with D and T in Fig. 12a between t2 and t4 for which device is accounted for the voltage drop. Right after t2, the diode inside the switch is in reverse bias and the reverse voltage appears across the D. During this interval, the transistor T can be turned off under ZCS. After vsw becomes positive, the reverse voltage of the SW appears across T because D is in forward bias. Therefore T must be turned off before vsw is positive. For full-wave mode, T must be turned off when iLr is negative so that its anti-parallel diode is conducting all the current through SW. T can be turned off under zero-current switching.


15

6.4 DC Voltage conversion ratio The output voltage Vo can be solved by equating the input energy Ein and the output energy Eo. The input current is the same as the iLF which is a constant. Based on Fig. 11, the output current is the current of the diode DF, iDF. iDF only conducts in the linear state and freewheeling state. Hence, output energy Eo:

sininin

t

t

t

tinLrinoo

TIVE

dtIdtiIVE

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−= ∫ ∫

1

0

4

3

)(

Hence, the voltage conversion ratio M is

⎟⎠⎞

⎜⎝⎛ −

Φ++

Φ−

=)cos1(1

221

1

ααπ o

Sin

o

ffV

V

(49)

where o

inV

ZI=Φ

(50) and fo=ωo/(2π) . Equation (50) can be rewritten as:

n

o

in

RM

ZRII

==Φ

(51)

where Io is the output current, i.e. Vo/R. It is also equal to the average of iDF. The characteristics of voltage conversion ratios are shown in Fig. 13. It can be seen that the ratio depends on the load resistance R for half mode and independent of R for full load. The voltage conversion ratio cannot be higher than Rn otherwise the converter does not operate under zero-current switching. Therefore the characteristic curve does not show value for M>Rn. It can also seen that in equation (49), the function in the parenthese, is the same format as F(Φ) as shown in equation (27).

(47) (48)


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Fig. 13a DC voltage conversion of the Half-wave Boost version

Fig. 13b DC voltage conversion of the Full-wave Boost version

7 Buck-boost converters

(a) Classical (b) Resonant

Fig. 14 Buck-Boost converter


17

Fig. 15 Equivalent circuits of the Buck-Boost Quasi-resonant converter during a switching cycle. The DC voltage conversion ratios of the Buck-boost converter are shown in Fig 16a and 16b. It can be seen that the voltage conversion ratio of half–wave depends on the load whereas the full-wave b is independent of load.

Fig. 16a DC voltage conversion of the Half-wave Buck-Boost version


18

Fig. 16b DC voltage conversion of the Full-wave Buck-Boost version

8 Flyback resonant converter A Flyback resonant converter is shown in Fig 17a. This topology is popular because the components used are fewer than the other circuit. The resonant inductor is part of the leakage inductance of the transformer and magnetizing inductance is the couple inductor (LF, or L) of the flyback (Fig 17b and 17c).

Fig 17 Flyback converter

(a) Basic circuit (b) Equivalent circuit with transformer replaced by a T-model (c) Simplified equivalent circuit.

A practical implementation of the circuit is shown in Fig 18. It can be seen that a diode D2 is used to disable the internal body diode of the MOSFET Q1. The D1 is the external antiparallel diode for full-wave operation. C2, L2 and C3 are filter components. C1 and the leakage inductance of the transformer are resonant components.


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Fig 18 Practical implementation of a 1MHz 75W flyback ZCS quasi-resonant converter 9 Comparison and discussion of different topologies The above four converters have a lot of similarities especially in the voltage conversion ratio, M because they are can be expressed in term of F(Φ). During the derivation of M, it has also been found out that the duration of each stage can be reduced to a simple form. It has been summarized in Table 2.

Table 2: Duration of each stage of all four topologies Stage Duration Td1

oωΦ

Td2

oωα

Td3 Φ

−

oωαcos1

Td4 Ts-Td1-Td2-Td3 In general, the characteristics of the ZCS QR converter are the same as the classical counterpart except that the circuit is operated under zero-current switching. Circuits operating in several hundred kHz are very common. 10 Practical implementation of Quasi-resonant ZCS converters The above four topologies have been sketched using SW to replace the transistor and the associated diode. In practice, the way to replace the SW by power MOSFET is shown in Fig. 19. The circuits are full-wave versions. The switch consists of power MOSFET T, with its body diode Db, a series diode Ds and an anti-parallel diode Da. Db is inherent in the MOSFET. However, Db is usually very slow and with considerable reverse recovery, it is usually disabled by series connecting the MOSFET by Ds. Da is the basic element of the full-wave version. The circuits can simply be used in half-wave provided that Da is removed.


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(a) Buck (b) Boost

(c)Buck-Boost (d) Cuk Fig. 19 Practical implementation of the ZCS QR converters

11 References [1] Liu, K.H., Orgauganti, R., and Lee, F.C., “Resonant switches-topologies and characteristics”,

Proceedings of IEEE Power Electronics and Specialists Conference, 1985, pp. 106-116. [2] K.W.E. Cheng and P.D. Evans, ‘The unified theory of extended-period quasi-resonant

converter’, IEE Proceedings-Electr. Power Appl., Vol. 147, No. 2, March 2000, pp. 127-132. [3] R.L. Steigerwald, “High frequency resonant transistor DC/DC converters”, IEEE Trans. Power

Electronics, 1E-31 (2), pp. 181-191. [4] S.Y.R. Hui, K.W.E. Cheng and S.R.N. Prakash, “A fully soft-switched extended-period quasi-

resonant power-factor-correction circuits”, IEEE Transactions on Power Electronics Vol. 12, No. 5, Sep 1997. pp. 922-930.

[5] K.W.E.Cheng, “Classical switched mode and resonant power converters”, Hong Kong Polytechnic

University, ISBN 962-367-364-7. 12 Tutorials 1) Show that the voltage conversion ratio of a Buck ZCS QR converter is:

⎟⎠⎞

⎜⎝⎛ −

Φ++

Φ= )cos1(1

22αα

π o

S

in

of

fVV

2) How can ZCS be affected by parasitic inductance in the circuit board or wiring

connection? 3) A full-wave zero-current switching quasi-resonant buck converter is operated at the

following specification: Input voltage VS=50V, Output voltage Vo=20V Switching frequency fs=500kHz, Output current Io = 10A Calculate the requirement of the resonant component Lr and Cr. Estimate the power loss in the Lr if its series equivalent resistance is 0.1Ω.

(answ: Lr=0.637µH, Cr=0.127µF, loss=6W)

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ZCS notes