Chapter 20 Quasi-Resonant Converters

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1 Chapter 0 Quasi-Resonant Converters Introduction 0.1 The zero-current-switching quasi-resonant switch cell Waveforms of the half-wave ZCS quasi-resonant switch cell 0.1. The average terminal waveforms The full-wave ZCS quasi-resonant switch cell 0. Resonant switch topologies 0..1 The zero-voltage-switching quasi-resonant switch 0.. The zero-voltage-switching multiresonant switch 0..3 Quasi-square-wave resonant switches 0.3 Ac modeling of quasi-resonant converters 0.4 Summary of key points Fundamentals of Power Electronics 1 Chapter 0: Quasi-Resonant Converters

2 The resonant switch concept A quite general idea: 1. PWM switch network is replaced by a resonant switch network. This leads to a quasi-resonant version of the original PWM converter Example: realization of the switch cell in the buck converter i L i PWM switch cell i v g v 1 Switch cell C R v v 1 Fundamentals of Power Electronics Chapter 0: Quasi-Resonant Converters

3 Two quasi-resonant switch cells D 1 Q 1 i r i D 1 i r i v 1 v 1r v 1 v 1r Q 1 Switch network Switch network Half-wave ZCS quasi-resonant switch cell Full-wave ZCS quasi-resonant switch cell Insert either of the above switch cells into the buck converter, to obtain a ZCS quasi-resonant version of the buck converter. and are small in value, and their resonant frequency f 0 is greater than the switching frequency f s. f 0 = 1 π = ω 0 π v g v 1 Switch cell i L i C R v Fundamentals of Power Electronics 3 Chapter 0: Quasi-Resonant Converters

4 0.1 The zero-current-switching quasi-resonant switch cell Tank inductor in series with transistor: transistor switches at zero crossings of inductor current waveform Tank capacitor in parallel with diode : diode switches at zero crossings of capacitor voltage waveform v 1 v 1r D 1 Q 1 Switch network i r i Two-quadrant switch is required: Half-wave ZCS quasi-resonant switch cell Half-wave: Q 1 and D 1 in series, transistor turns off at first zero crossing of current waveform D 1 i r i Full-wave: Q 1 and D 1 in parallel, transistor turns off at second zero crossing of current waveform Performances of half-wave and full-wave cells differ significantly. v 1 v 1r Q 1 Switch network Full-wave ZCS quasi-resonant switch cell Fundamentals of Power Electronics 4 Chapter 0: Quasi-Resonant Converters

5 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: i i Ts v 1 v 1 Ts In steady state, we can further approximate these quantities by their dc values: i I v 1 V 1 Modeling objective: find the average values of the terminal waveforms T s and T s Fundamentals of Power Electronics 5 Chapter 0: Quasi-Resonant Converters

6 The switch conversion ratio µ A generalization of the duty cycle d D 1 Q 1 i r 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. v 1 Ts v 1r Switch network Half-wave ZCS quasi-resonant switch cell i Ts If V/V g = M(d) for a PWM CCM converter, then V/V g = 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 i i Ts v 1 v 1 Ts In steady state: i I v 1 V 1 µ = Ts v 1r Ts = µ = V V 1 = I 1 I Ts i r Ts Fundamentals of Power Electronics 6 Chapter 0: Quasi-Resonant Converters

7 0.1.1 Waveforms of the half-wave ZCS quasi-resonant switch cell The half-wave ZCS quasi-resonant switch cell, driven by the terminal quantities v 1 Ts and i Ts. Waveforms: I D 1 Q 1 i r V 1 Subinterval: θ = ω 0 t v 1 Ts v 1r i Ts Switch network Half-wave ZCS quasi-resonant switch cell α β V c1 I δ ξ Each switching period contains four subintervals Conducting devices: ω 0 T s Q 1 Q 1 X D 1 D 1 Fundamentals of Power Electronics 7 Chapter 0: Quasi-Resonant Converters

8 Subinterval 1 Diode is initially conducting the filter inductor current I. Transistor Q 1 turns on, and the tank inductor current starts to increase. So all semiconductor devices conduct during this subinterval, and the circuit reduces to: Circuit equations: d = V 1 with i dt L 1 (0) = 0 r Solution: where R 0 = = V 1 t = ω 0 t V 1 R 0 V 1 I This subinterval ends when diode becomes reverse-biased. This occurs at time ω 0 t = α, when = I. (α)=α V 1 R 0 = I α = I R 0 V 1 Fundamentals of Power Electronics 8 Chapter 0: Quasi-Resonant Converters

9 Subinterval Diode is off. Transistor Q 1 conducts, and the tank inductor and tank capacitor ring sinusoidally. The circuit reduces to: V 1 The circuit equations are di L 1 (ω 0 t) r dt dv C (ω 0 t) r dt i c = V 1 (ω 0 t) (α)=0 = (ω 0 t)i (α)=i I The solution is (ω 0 t)=i V 1 R 0 sin ω 0 t α (ω 0 t)=v 1 1 cos ω 0 t α 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 R 0. Fundamentals of Power Electronics 9 Chapter 0: Quasi-Resonant Converters

10 Subinterval continued Peak inductor current: I 1pk = I V 1 R 0 (ω 0 t)=i V 1 R 0 sin ω 0 t α (ω 0 t)=v 1 1 cos ω 0 t α This subinterval ends at the first zero crossing of. Define β = angular length of subinterval. Then (α β)=i V 1 R 0 sin β = 0 sin β = I R 0 V 1 Must use care to select the correct branch of the arcsine function. Note (from the waveform) that β > π. I V 1 Subinterval: Hence α β ω 0 T s β = π sin 1 I R 0 V 1 π < sin1 x π I < V 1 R 0 δ ξ θ = ω 0 t Fundamentals of Power Electronics 10 Chapter 0: Quasi-Resonant Converters

11 Boundary of zero current switching If the requirement I < V 1 R 0 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 is (α β)=v c1 = V I R 0 V 1 Fundamentals of Power Electronics 11 Chapter 0: Quasi-Resonant Converters

12 Subinterval 3 All semiconductor devices are off. The circuit reduces to: I The circuit equations are dv C (ω 0 t) r =I dt (α β)=v c1 Subinterval 3 ends when the tank capacitor voltage reaches zero, and diode becomes forward-biased. Define δ = angular length of subinterval 3. Then (α β δ)=v c1 I R 0 δ =0 δ = V c1 = V I R 0 I R 0 I R 0 V 1 The solution is (ω 0 t)=v c1 I R 0 ω 0 t α β Fundamentals of Power Electronics 1 Chapter 0: Quasi-Resonant Converters

13 Subinterval 4 Subinterval 4, of angular length ξ, is identical to the diode conduction interval of the conventional PWM switch network. Diode conducts the filter inductor current I The tank capacitor voltage is equal to zero. Transistor Q 1 is off, and the input current 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. Fundamentals of Power Electronics 13 Chapter 0: Quasi-Resonant Converters

14 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 ω 0 T s = α β δ ξ = π f 0 = f π s F where the normalized switching frequency F is defined as F = f s f 0 So the minimum switching period is ω 0 T s α β δ Substitute previous solutions for subinterval lengths: π I R 0 π sin F V 1 I R 0 V I R 0 1 V 1 I R 0 V 1 Fundamentals of Power Electronics 14 Chapter 0: Quasi-Resonant Converters

15 0.1. The average terminal waveforms Averaged switch modeling: we need to determine the average values of and. The average switch input current is given by v 1 Ts v 1r D 1 Q 1 Switch network i r i Ts Ts = 1 T s t t T s dt = q 1 q T s q 1 and q are the areas under the current waveform during subintervals 1 and. q 1 is given by the triangle area formula: I Half-wave ZCS quasi-resonant switch cell q 1 q Ts q 1 = 0 α ω 0 dt = 1 α ω 0 I α ω 0 α β ω 0 t Fundamentals of Power Electronics 15 Chapter 0: Quasi-Resonant Converters

16 Charge arguments: computation of q q = α β ω 0 α ω 0 dt Node equation for subinterval : I q 1 q Ts =i C I Substitute: α ω 0 α β ω 0 t q = α β α ω 0 α β ω 0 ω 0 i C dt I dt α ω 0 Second term is integral of constant I : α β α ω 0 ω 0 β I dt = I ω0 V 1 i c Circuit during subinterval I Fundamentals of Power Electronics 16 Chapter 0: Quasi-Resonant Converters

17 Charge arguments continued q = α β α ω 0 α β ω 0 ω 0 i C dt I dt α ω 0 First term: integral of the capacitor current over subinterval. This can be related to the change in capacitor voltage : α β α ω 0 ω 0 α β i C dt = C ω v α 0 ω0 α β V c1 I Substitute results for the two integrals: q = CV c1 I δ β ω0 ξ α β ω 0 α ω 0 i C dt = C V c1 0 = CV c1 Substitute into expression for average switch input current: Ts = αi ω 0 T s CV c1 T s βi ω 0 T s Fundamentals of Power Electronics 17 Chapter 0: Quasi-Resonant Converters

18 Switch conversion ratio µ µ = Ts I = α ω 0 T s CV c1 I T s β ω 0 T s Eliminate α, β, V c1 using previous results: where µ = F 1 π 1 J s π sin 1 (J s ) 1 J s 1 1J s J s = I R 0 V 1 Fundamentals of Power Electronics 18 Chapter 0: Quasi-Resonant Converters

19 Analysis result: switch conversion ratio µ Switch conversion ratio: µ = F 1 π 1 J s π sin 1 (J s ) 1 J s 1 1J s with J s = I R 0 V 1 P1 J s 10 This is of the form µ = FP1 J s P1 J s = 1 π 1 J s π sin 1 (J s ) 1 J s 1 1J s J s Fundamentals of Power Electronics 19 Chapter 0: Quasi-Resonant Converters

20 Characteristics of the half-wave ZCS resonant switch ZCS boundary Switch characteristics: µ = FP1 J s max F boundary J s Mode boundary: J s 1 0. F = µ 1 J sf 4π µ Fundamentals of Power Electronics 0 Chapter 0: Quasi-Resonant Converters

21 Buck converter containing half-wave ZCS quasi-resonant switch Conversion ratio of the buck converter is (from inductor volt-second balance): M = V V g = µ ZCS boundary 1 For the buck converter, J s = IR 0 V g ZCS occurs when J s max F boundary I V g R 0 Output voltage varies over the range 0 V V g FIR 0 4π F = µ Fundamentals of Power Electronics 1 Chapter 0: Quasi-Resonant Converters

22 Boost converter example For the boost converter, M = V V g = 1 1µ I g L i J s = I R 0 = I gr 0 V 1 V I g = I 1µ V g D 1 v 1 C R V Half-wave ZCS equations: Q 1 µ = FP1 J s P1 J s = 1 π 1 J s π sin 1 (J s ) 1 J s 1 1J s Fundamentals of Power Electronics Chapter 0: Quasi-Resonant Converters

23 0.1.3 The full-wave ZCS quasi-resonant switch cell Half wave D 1 Q 1 i r i v 1 v 1r I V 1 Switch network Subinterval: θ = ω 0 t Half-wave ZCS quasi-resonant switch cell Full wave D 1 i r i v 1 v 1r Q 1 I V 1 Switch network Subinterval: θ = ω 0 t Full-wave ZCS quasi-resonant switch cell Fundamentals of Power Electronics 3 Chapter 0: Quasi-Resonant Converters

24 Analysis: full-wave ZCS Analysis in the full-wave case is nearly the same as in the half-wave case. The second subinterval ends at the second zero crossing of the tank inductor current waveform. The following quantities differ: β = π sin1 J s (half wave) π sin 1 J s (full wave) V c1 = V 1 1 1J s V 1 1 1J s (half wave) (full wave) In either case, µ is given by µ = Ts I = α ω 0 T s CV c1 I T s β ω 0 T s Fundamentals of Power Electronics 4 Chapter 0: Quasi-Resonant Converters

25 Full-wave cell: switch conversion ratio µ P 1 J s = 1 π 1 J s π sin 1 (J s ) 1 J s 1 1J s µ = FP 1 J s 1 ZCS boundary Full-wave case: P 1 can be approximated as 0.8 F = max F boundary so P 1 J s 1 µ F = f s f 0 J s µ Fundamentals of Power Electronics 5 Chapter 0: Quasi-Resonant Converters

26 0. Resonant switch topologies Basic ZCS switch cell: i r i SW v 1 v 1r Switch network SPST switch SW: ZCS quasi-resonant switch cell Voltage-bidirectional two-quadrant switch for half-wave cell Current-bidirectional two-quadrant switch for full-wave cell Connection of resonant elements: Can be connected in other ways that preserve high-frequency components of tank waveforms Fundamentals of Power Electronics 6 Chapter 0: Quasi-Resonant Converters

27 Connection of tank capacitor Connection of tank capacitor to two other points at ac ground. This simply changes the dc component of tank capacitor voltage. V g SW ZCS quasi-resonant switch i L C R V The ac highfrequency components of the tank waveforms are unchanged. V g SW i L C R V ZCS quasi-resonant switch Fundamentals of Power Electronics 7 Chapter 0: Quasi-Resonant Converters

28 A test to determine the topology of a resonant switch network Replace converter elements by their high-frequency equivalents: Independent voltage source V g : short circuit Filter capacitors: short circuits Filter inductors: open circuits The resonant switch network remains. If the converter contains a ZCS quasi-resonant switch, then the result of these operations is SW Fundamentals of Power Electronics 8 Chapter 0: Quasi-Resonant Converters

29 Zero-current and zero-voltage switching ZCS quasi-resonant switch: Tank inductor is in series with switch; hence SW switches at zero current Tank capacitor is in parallel with diode ; hence switches at zero voltage SW Discussion Zero voltage switching of eliminates switching loss arising from stored charge. Zero current switching of SW: device Q 1 and D 1 output capacitances lead to switching loss. In full-wave case, stored charge of diode D 1 leads to switching loss. Peak transistor current is (1 J s ) V g /R 0, or more than twice the PWM value. Fundamentals of Power Electronics 9 Chapter 0: Quasi-Resonant Converters

30 0..1 The zero-voltage-switching quasi-resonant switch cell When the previously-described operations are followed, then the converter reduces to SW A full-wave version based on the PWM buck converter: v Cr D 1 i Lr i L I V g v 1 Q 1 C R V Fundamentals of Power Electronics 30 Chapter 0: Quasi-Resonant Converters

31 ZVS quasi-resonant switch cell Switch conversion ratio Tank waveforms µ =1FP1 1 Js half-wave v Cr V 1 µ =1FP 1 1 Js full-wave Subinterval: θ = ω 0 t ZVS boundary i Lr I J s 1 peak transistor voltage V cr,pk =(1J s ) V 1 α β δ ξ A problem with the quasi-resonant ZVS switch cell: peak transistor voltage becomes very large when zero voltage switching is required for a large range of load currents. Conducting devices: X ω 0 T s D 1D Q 1 Q 1 Fundamentals of Power Electronics 31 Chapter 0: Quasi-Resonant Converters

32 0.. The ZVS multiresonant switch C s When the previously-described operations are followed, then the converter reduces to SW C d A half-wave version based on the PWM buck converter: C s i L I D 1 V g v 1 Q 1 C d C R V Fundamentals of Power Electronics 3 Chapter 0: Quasi-Resonant Converters

33 0..3 Quasi-square-wave resonant switches When the previouslydescribed operations are followed, then the converter reduces to ZCS SW ZVS SW Fundamentals of Power Electronics 33 Chapter 0: Quasi-Resonant Converters

34 A quasi-square-wave ZCS buck with input filter L f D 1 Q 1 L I V g C R V C f The basic ZCS QSW switch cell is restricted to 0 µ 0.5 Peak transistor current is equal to peak transistor current of PWM cell Peak transistor voltage is increased Zero-current switching in all semiconductor devices Fundamentals of Power Electronics 34 Chapter 0: Quasi-Resonant Converters

35 A quasi-square-wave ZVS buck D 1 i V g i v 1 Q 1 L C I R V V 1 V V 1 0 Conducting ω 0 T s devices: Q 1 X D 1 X ω 0 t The basic ZVS QSW switch cell is restricted to 0.5 µ 1 Peak transistor voltage is equal to peak transistor voltage of PWM cell Peak transistor current is increased Zero-voltage switching in all semiconductor devices Fundamentals of Power Electronics 35 Chapter 0: Quasi-Resonant Converters

36 0.3 Ac modeling of quasi-resonant converters Use averaged switch modeling technique: apply averaged PWM model, with d replaced by µ Buck example with full-wave ZCS quasi-resonant cell: Full-wave ZCS quasi-resonant switch cell µ = F D 1 i r i L i v g v 1 v 1r Q 1 C R v Gate driver Frequency modulator v c Fundamentals of Power Electronics 36 Chapter 0: Quasi-Resonant Converters

37 Small-signal ac model Averaged switch equations: Ts = µ v 1r Ts Linearize: = f s I f0 Ts = µ i r Ts Resulting ac model: = F s f 0 v 1r f s V 1 f 0 1 : F i r L v g v 1r f s I f0 f s V 1 f 0 C R v µ = F f s G m (s) v c Fundamentals of Power Electronics 37 Chapter 0: Quasi-Resonant Converters

38 Low-frequency model Tank dynamics occur only at frequency near or greater than switching frequency discard tank elements 1 : F L v g f s I f0 f s V 1 f 0 C R v f s G m (s) v c same as PWM buck, with d replaced by F Fundamentals of Power Electronics 38 Chapter 0: Quasi-Resonant Converters

39 Example : Half-wave ZCS quasi-resonant buck Now, µ depends on j s : µ= f s f 0 P1 j s j s =R 0 i r Ts v 1r Ts Half-wave ZCS quasi-resonant switch cell D 1 i r i L i v g v 1 v 1r Q 1 C R v Gate driver Frequency modulator v c Fundamentals of Power Electronics 39 Chapter 0: Quasi-Resonant Converters

40 Small-signal modeling Perturbation and linearization of µ(v 1r, i r, f s ): with µ=k v v 1r K i i r K c f s K v = µ j s R 0 I V 1 K i = µ j s R 0 V 1 µ j s = F s π f J s J s K c = µ 0 F s Linearized terminal equations of switch network: =µ I i r µ 0 =µ 0 v 1r µ V 1 Fundamentals of Power Electronics 40 Chapter 0: Quasi-Resonant Converters

41 Equivalent circuit model 1 : µ 0 i r L v g v 1r µ I µ V 1 C R v µ v 1r i r K v K i K c f s G m (s) v c Fundamentals of Power Electronics 41 Chapter 0: Quasi-Resonant Converters

42 Low frequency model: set tank elements to zero 1 : µ i L 0 µ V g v g µ I C R v µ v g i K v K i K c f s G m (s) v c Fundamentals of Power Electronics 4 Chapter 0: Quasi-Resonant Converters

43 Predicted small-signal transfer functions Half-wave ZCS buck G vg (s)=g g Q s ω 0 s ω 0 G vc (s)=g c Q s ω 0 s ω 0 Full-wave: poles and zeroes are same as PWM Half-wave: effective feedback reduces Q-factor and dc gains G g0 = µ 0 K v V g G c0 = ω 0 = Q = 1 K iv g R K c V g 1 K iv g R R 0 1 K iv g R 1 K iv g R R K iv g R R0 R 0 = Fundamentals of Power Electronics 43 Chapter 0: Quasi-Resonant Converters

44 0.4 Summary of key points 1. In a resonant switch converter, the switch network of a PWM converter is replaced by a switch network containing resonant elements. The resulting hybrid converter combines the properties of the resonant switch network and the parent PWM converter.. Analysis of a resonant switch cell involves determination of the switch conversion ratio µ. The resonant switch waveforms are determined, and are then averaged. The switch conversion ratio µ is a generalization of the PWM CCM duty cycle d. The results of the averaged analysis of PWM converters operating in CCM can be directly adapted to the related resonant switch converter, simply by replacing d with µ. 3. In the zero-current-switching quasi-resonant switch, diode operates with zero-voltage switching, while transistor Q 1 and diode D 1 operate with zero-current switching. Fundamentals of Power Electronics 44 Chapter 0: Quasi-Resonant Converters

45 Summary of key points 4. In the zero-voltage-switching quasi-resonant switch, the transistor Q 1 and diode D 1 operate with zero-voltage switching, while diode operates with zero-current switching. 5. Full-wave versions of the quasi-resonant switches exhibit very simple control characteristics: the conversion ratio µ is essentially independent of load current. However, these converters exhibit reduced efficiency at light load, due to the large circulating currents. In addition, significant switching loss is incurred due to the recovered charge of diode D Half-wave versions of the quasi-resonant switch exhibit conversion ratios that are strongly dependent on the load current. These converters typically operate with wide variations of switching frequency. 7. In the zero-voltage-switching multiresonant switch, all semiconductor devices operate with zero-voltage switching. In consequence, very low switching loss is observed. Fundamentals of Power Electronics 45 Chapter 0: Quasi-Resonant Converters

46 Summary of key points 8. In the quasi-square-wave zero-voltage-switching resonant switches, all semiconductor devices operate with zero-voltage switching, and with peak voltages equal to those of the parent PWM converter. The switch conversion ratio is restricted to the range 0.5 µ The small-signal ac models of converters containing resonant switches are similar to the small-signal models of their parent PWM converters. The averaged switch modeling approach can be employed to show that the quantity d is simply replaced by µ. 10.In the case of full-wave quasi-resonant switches, µ depends only on the switching frequency, and therefore the transfer function poles and zeroes are identical to those of the parent PWM converter. 11.In the case of half-wave quasi-resonant switches, as well as other types of resonant switches, the conversion ratio µ is a strong function of the switch terminal quantities v 1 and i. This leads to effective feedback, which modifies the poles, zeroes, and gains of the transfer functions. Fundamentals of Power Electronics 46 Chapter 0: Quasi-Resonant Converters

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