# Chapter 6: Converter circuits

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1 Chapter 6. Converter Circuits 6.1. Circuit manipulations 6.2. A short list of converters 6.3. Transformer isolation 6.4. Converter evaluation and design 6.5. Summary of key points Where do the boost, buck-boost, and other converters originate? How can we obtain a converter having given desired properties? What converters are possible? How can we obtain transformer isolation in a converter? For a given application, which converter is best? 1

2 6.1. Circuit Manipulations 1 L 2 C R Begin with buck converter: derived in Chapter 1 from first principles Switch changes dc component, low-pass filter removes switching harmonics Conversion ratio is M = D 2

3 Inversion of source and load Interchange power input and output ports of a converter Buck converter example 2 = D 1 Port 1 Port 2 1 L Power flow 3

4 Inversion of source and load Interchange power source and load: Port 1 Port 2 1 L Power flow 2 = D 1 1 = 1 D 2 4

5 Realization of switches as in Chapter 4 Reversal of power flow requires new realization of switches Transistor conducts when switch is in position 2 Interchange of D and D Port 1 Port Power flow L 1 = 1 D' 2 Inversion of buck converter yields boost converter 5

6 Cascade connection of converters Converter 1 Converter 2 1 = M 1 (D) 1 1 = M 2 (D) D 1 = M 1 (D) g = M(D)=M 1 (D)M 2 (D) = M 2 (D) 1 6

7 Example: buck cascaded by boost 1 L 1 L C C 2 R { { Buck converter Boost converter 1 = D 1 = 1 1D = D 1D 7

8 Buck cascaded by boost: simplification of internal filter Remove capacitor C 1 1 L 1 L C 2 R Combine inductors L 1 and L 2 1 L i L Noninverting buck-boost converter 8

9 Noninverting buck-boost converter 1 L i L subinterval 1 subinterval 2 i L i L 9

10 Reversal of output voltage polarity subinterval 1 subinterval 2 i L noninverting buck-boost i L i L i L inverting buck-boost 10

11 Reduction of number of switches: inverting buck-boost Subinterval 1 Subinterval 2 i L i L One side of inductor always connected to ground hence, only one SPDT switch needed: 1 2 i L = D g 1D 11

12 Discussion: cascade connections Properties of buck-boost converter follow from its derivation as buck cascaded by boost Equivalent circuit model: buck 1:D transformer cascaded by boost D :1 transformer Pulsating input current of buck converter Pulsating output current of boost converter Other cascade connections are possible Cuk converter: boost cascaded by buck 12

13 Differential connection of load to obtain bipolar output voltage dc source load Converter 1 1 = M(D) D Converter Differential load voltage is = 1 2 The outputs 1 and 2 may both be positive, but the differential output voltage can be positive or negative. 2 = M(D') D' 15

14 Buck converter 1} Differential connection using two buck converters Converter #1 transistor driven with duty cycle D Converter #2 transistor driven with duty cycle complement D Differential load voltage is 1 = D D' 2 Simplify: Buck converter 2{ =(2D 1) 16

15 Conversion ratio M(D), differentially-connected buck converters =(2D 1) M(D) D 1 17

16 Buck converter 1} Simplification of filter circuit, differentially-connected buck converters Original circuit Bypass load directly with capacitor Buck converter 2{ 18

17 6.2. A short list of converters An infinite number of converters are possible, which contain switches embedded in a network of inductors and capacitors Two simple classes of converters are listed here: Single-input single-output converters containing a single inductor. The switching period is divided into two subintervals. This class contains eight converters. Single-input single-output converters containing two inductors. The switching period is divided into two subintervals. Several of the more interesting members of this class are listed. 24

18 Single-input single-output converters containing one inductor Use switches to connect inductor between source and load, in one manner during first subinterval and in another during second subinterval There are a limited number of ways to do this, so all possible combinations can be found After elimination of degenerate and redundant cases, eight converters are found: dc-dc converters buck boost buck-boost noninverting buck-boost dc-ac converters bridge ac-dc converters current-fed bridge Watkins-Johnson inverse of Watkins-Johnson 25

19 Converters producing a unipolar output voltage 1. Buck M(D)=D M(D) D 2. Boost M(D)= 1 1D M(D) D 26

20 Converters producing a unipolar output voltage 3. Buck-boost M(D)= D 1D D M(D) 4. Noninverting buck-boost M(D)= D 1D M(D) D 27

21 Several members of the class of two-inductor converters 1. Cuk M(D)= D 1D D M(D) 2. SEPIC M(D)= D 1D M(D) g D 30

22 Several members of the class of two-inductor converters 3. Inverse of SEPIC 1 M(D)= D 1D M(D) D 4. Buck 2 M(D)=D 2 1 M(D) D 31

23 6.3. Transformer isolation Objectives: Isolation of input and output ground connections, to meet safety requirements Reduction of transformer size by incorporating high frequency isolation transformer inside converter Minimization of current and voltage stresses when a large step-up or step-down conversion ratio is needed use transformer turns ratio Obtain multiple output voltages via multiple transformer secondary windings and multiple converter secondary circuits 32

24 A simple transformer model Multiple winding transformer Equivalent circuit model i 1 n 1 : n 2 i 2 i 1 i M i 1 ' n 1 : n 2 i 2 v 1 v 2 i 3 v 3 v 1 L M v 1 n = v 2 1 n = v 3 2 n = =n 1 i 1 'n 2 i 2 n 3 i 3... : n 3 v 2 i 3 v 3 : n 3 Ideal transformer 33

25 The magnetizing inductance L M Models magnetization of transformer core material Appears effectively in parallel with windings If all secondary windings are disconnected, then primary winding behaves as an inductor, equal to the magnetizing inductance At dc: magnetizing inductance tends to short-circuit. Transformers cannot pass dc voltages Transformer saturates when magnetizing current i M is too large Transformer core B-H characteristic B v 1 dt saturation slope L M H i M 34

26 olt-second balance in L M The magnetizing inductance is a real inductor, obeying di v 1 =L M M dt integrate: v 1 i M i M (0) = 1 t v L 1 (τ)dτ M 0 Magnetizing current is determined by integral of the applied winding voltage. The magnetizing current and the winding currents are independent quantities. olt-second balance applies: in steady-state, i M (T s ) = i M (0), and hence i 1 i M L M i 1 ' n 1 : n 2 : n 3 i 2 v 2 i 3 v 3 0= T 1 T s v 1 dt s 0 Ideal transformer 35

27 Transformer reset Transformer reset is the mechanism by which magnetizing inductance volt-second balance is obtained The need to reset the transformer volt-seconds to zero by the end of each switching period adds considerable complexity to converters To understand operation of transformer-isolated converters: replace transformer by equivalent circuit model containing magnetizing inductance analyze converter as usual, treating magnetizing inductance as any other inductor apply volt-second balance to all converter inductors, including magnetizing inductance 36

28 Full-bridge and half-bridge isolated buck converters Full-bridge isolated buck converter Q 1 D 1 Q 3 D 3 i 1 1 : n D 5 i D5 L i v T v s C R v Q2 D 2 Q4 D 4 : n D 6 37

29 Full-bridge, with transformer equivalent circuit Q 1 D 1 Q 3 D 3 i 1 i 1 ' i M 1 : n D 5 i D5 i L v T L M v s C R v Q2 D 2 Q4 D 4 : n Ideal D 6 i D6 Transformer model 38

30 Full-bridge: waveforms i M v T i v s i D5 I conducting devices: L M 0 0 n i 0 i L M n 0.5 i 0.5 i 0 0 DT s T s T s DT s 2T s Q 1 D Q 5 2 D 5 Q 4 D Q 6 3 D 6 D 5 D 6 0 t During first switching period: transistors Q 1 and Q 4 conduct for time DT s, applying voltseconds DT s to primary winding During next switching period: transistors Q 2 and Q 3 conduct for time DT s, applying voltseconds DT s to primary winding Transformer volt-second balance is obtained over two switching periods Effect of nonidealities? 39

31 Effect of nonidealities on transformer volt-second balance olt-seconds applied to primary winding during first switching period: ( (Q 1 and Q 4 forward voltage drops))( Q 1 and Q 4 conduction time) olt-seconds applied to primary winding during next switching period: ( (Q 2 and Q 3 forward voltage drops))( Q 2 and Q 3 conduction time) These volt-seconds never add to exactly zero. Net volt-seconds are applied to primary winding Magnetizing current slowly increases in magnitude Saturation can be prevented by placing a capacitor in series with primary, or by use of current programmed mode (Chapter 12) 40

32 Operation of secondary-side diodes : n D 5 i D5 v s L C i R v During second (D ) subinterval, both secondary-side diodes conduct v s : n n D 6 n Output filter inductor current divides approximately equally between diodes i D5 conducting devices: i i 0.5 i 0 t 0 DT s T s T s DT s 2T s Q 1 D Q 5 2 D 5 Q 4 D Q 6 3 D 6 D 5 D 6 0 Secondary amp-turns add to approximately zero Essentially no net magnetization of transformer core by secondary winding currents 41

33 olt-second balance on output filter inductor : n D 5 i D5 L i i I i v s C R v v s n n 0 0 : n D 6 i D5 i 0.5 i 0.5 i 0 t 0 DT s T s T s DT s 2T s = v s = nd conducting devices: Q 1 D Q 5 2 D 5 Q 4 D Q 6 3 D 6 D 5 D 6 M(D) = nd buck converter with turns ratio 42

34 Half-bridge isolated buck converter Q 1 D 1 C a i 1 1 : n D 3 i D3 L i v T v s C R v Q2 D 2 C b : n D 4 Replace transistors Q 3 and Q 4 with large capacitors oltage at capacitor centerpoint is 0.5 v s is reduced by a factor of two M = 0.5 nd 43

35 Forward converter n 1 : n 2 : n 3 D 2 L D 3 C R Q 1 D 1 Buck-derived transformer-isolated converter Single-transistor and two-transistor versions Maximum duty cycle is limited Transformer is reset while transistor is off 44

36 Forward converter with transformer equivalent circuit n 1 : n 2 : n 3 D 2 L i M i 1 ' L M v 1 v 2 v 3 D 3 v D3 C R Q 1 i 1 i 2 i 3 D 1 v Q1 45

37 Forward converter: waveforms v 1 i M n 1 n 2 0 Magnetizing current, in conjunction with diode D 1, operates in discontinuous conduction mode v D3 L M n 1 n 2 n 3 n 1 L M 0 Output filter inductor, in conjunction with diode D 3, may operate in either CCM or DCM 0 0 DT s D 2 T s D 3 T s T s t Conducting devices: Q 1 D 1 D 3 D 2 D 3 46

38 Subinterval 1: transistor conducts i M n 1 : n 2 : n 3 i 1 ' D 2 on L L M v 1 v 2 v 3 v D3 C R i 1 i 2 i 3 Q 1 on D 1 off 47

39 Subinterval 2: transformer reset n 1 : n 2 : n 3 L i M i 1 ' L M v 1 v 2 v 3 D 3 on v D3 C R i 1 i 2 = i M n1 /n2 i 3 Q 1 off D 1 on 48

40 Subinterval 3 n 1 : n 2 : n 3 L i M = 0 L M v 1 i 1 ' v 2 v 3 D 3 on v D3 C R i 1 i 2 i 3 Q 1 off D 1 off 49

41 Magnetizing inductance volt-second balance v 1 0 n 1 n 2 i M L M n 1 n 2 L M 0 v 1 = D D 2 n 1 /n 2 D 3 0 =0 50

42 Transformer reset From magnetizing current volt-second balance: Solve for D 2 : v 1 = D D 2 n 1 /n 2 D 3 0 =0 D 2 = n 2 n 1 D D 3 cannot be negative. But D 3 = 1 D D 2. Hence D 3 =1D D 2 0 D 3 =1D 1 n 2 0 n 1 Solve for D D 1 for n 1 = n 2 : 1 n D n 1 51

43 What happens when D > 0.5 i magnetizing current M waveforms, D < 0.5 for n 1 = n 2 DT s D 2 T s D 3 T s t i M D > 0.5 DT s D 2 T s t 2T s 52

44 Conversion ratio M(D) : n 3 D 2 L D 3 C R v D3 n 3 n DT s D 2 T s D 3 T s T s t v D3 = = n 3 n 1 D Conducting devices: Q 1 D 2 D 1 D 3 D 3 53

45 Maximum duty cycle vs. transistor voltage stress Maximum duty cycle limited to which can be increased by increasing the turns ratio n 2 / n 1. But this increases the peak transistor voltage: For n 1 = n 2 D 1 1 n 2 n 1 max v Q1 = 1 n 1 n 2 D 1 2 and max(v Q1 ) = 2 54

46 The two-transistor forward converter D 1 Q 1 D 3 L 1 : n D 4 C R D 2 Q 2 = nd D 1 2 max(v Q1 ) = max(v Q2 ) = 55

47 Push-pull isolated buck converter Q 1 v T 1 : n i D1 D 1 i L v s C R v T D 2 Q 2 = nd 0 D 1 56

48 Waveforms: push-pull i M v T i v s i D1 I Conducting devices: L M 0 0 n i 0 i L M n 0.5 i 0.5 i 0 0 DT s T s T s DT s 2T s Q 1 D 1 Q 2 D 1 D 1 D 2 D 2 D 2 0 t Used with low-voltage inputs Secondary-side circuit identical to full bridge As in full bridge, transformer volt-second balance is obtained over two switching periods Effect of nonidealities on transformer volt-second balance? Current programmed control can be used to mitigate transformer saturation problems. Duty cycle control not recommended. 57

49 Flyback converter buck-boost converter: Q 1 D 1 L construct inductor winding using two parallel wires: L Q 1 D 1 1:1 58

50 Derivation of flyback converter, cont. Isolate inductor windings: the flyback converter L M Q 1 D 1 1:1 Flyback converter having a 1:n turns ratio and positive output: L M 1:n D 1 C Q1 59

51 The flyback transformer i g L M Transformer model i 1:n v L Q 1 D 1 C i C R v A two-winding inductor Symbol is same as transformer, but function differs significantly from ideal transformer Energy is stored in magnetizing inductance Magnetizing inductance is relatively small Current does not simultaneously flow in primary and secondary windings Instantaneous winding voltages follow turns ratio Instantaneous (and rms) winding currents do not follow turns ratio Model as (small) magnetizing inductance in parallel with ideal transformer 60

52 Subinterval 1 Transformer model i g i 1:n i C v L = L M v L C R v i C = v R i g = i CCM: small ripple approximation leads to Q 1 on, D 1 off v L = i C = R i g = I 61

53 Subinterval 2 i g = 0 Transformer model i v L v/n 1:n i/n C i C R v v L = v n i C = i n v R i g =0 CCM: small ripple approximation leads to Q 1 off, D 1 on v L = n i C = I n R i g =0 62

54 CCM Flyback waveforms and solution v L olt-second balance: i C /n I/n /R v L = D D' n =0 Conversion ratio is M(D)= = n D D' Charge balance: i g /R I DT s 0 D'T s t i C = D R D' I n R =0 Dc component of magnetizing current is I = D'R n Dc component of source current is I g = i g = DI D' 0 Conducting devices: T s Q 1 D 1 63

55 Equivalent circuit model: CCM Flyback v L = D D' n =0 I g I i C = D R D' I n R =0 DI D' D'I n R D n I g = i g = DI D' 0 1 : D D' : n I g I R 64

56 Discussion: Flyback converter Widely used in low power and/or high voltage applications Low parts count Multiple outputs are easily obtained, with minimum additional parts Cross regulation is inferior to buck-derived isolated converters Often operated in discontinuous conduction mode DCM analysis: DCM buck-boost with turns ratio 65

57 Boost-derived isolated converters A wide variety of boost-derived isolated dc-dc converters can be derived, by inversion of source and load of buck-derived isolated converters: full-bridge and half-bridge isolated boost converters inverse of forward converter: the reverse converter push-pull boost-derived converter Of these, the full-bridge and push-pull boost-derived isolated converters are the most popular, and are briefly discussed here. 66

58 Full-bridge transformer-isolated boost-derived converter i L v L Q 1 Q 3 1 : n D 1 i o v T C R v : n Q 2 Q 4 D 2 Circuit topologies are equivalent to those of nonisolated boost converter With 1:1 turns ratio, inductor current i and output current i o waveforms are identical to nonisolated boost converter 67

59 Transformer reset mechanism v T v L i I i o Conducting devices: Q DT s Q 2 Q 3 Q 4 /n /n I/n D /n /n I/n D'T s DT s D'T s T s Q 1 Q 1 Q 2 T s Q 4 Q 2 Q 3 Q 4 Q 3 D 2 t As in full-bridge buck topology, transformer voltsecond balance is obtained over two switching periods. During first switching period: transistors Q 1 and Q 4 conduct for time DT s, applying volt-seconds DT s to secondary winding. During next switching period: transistors Q 2 and Q 3 conduct for time DT s, applying volt-seconds DT s to secondary winding.

60 Conversion ratio M(D) v L i I /n /n Application of volt-second balance to inductor voltage waveform: v L = D D' n = 0 Solve for M(D): Conducting devices: DT s Q 1 Q 2 Q 3 Q 4 D'T s DT s D'T s T s Q 1 D 1 Q 1 Q 2 T s Q 4 Q 2 Q 3 Q 4 Q 3 D 2 t M(D)= = n D' boost with turns ratio n 69

61 Push-pull boost-derived converter i o L v T i C R v L v T Q 1 1 : n D 1 Q 2 D 2 M(D)= = n D' 70

62 Push-pull converter based on Watkins-Johnson converter Q 1 1 : n D 1 C R Q 2 D 2 71

63 Isolated versions of the SEPIC and Cuk converter Basic nonisolated SEPIC L 1 C 1 D 1 L 2 C 2 R v Q 1 L 1 C 1 1 : n D 1 Isolated SEPIC i 1 i p i s C 2 R v Q 1 72

64 Isolated SEPIC i p i 1 L 1 C 1 i 1 i 2 i p 1 : n i s D 1 i 2 Q 1 L M = L 2 Ideal Transformer model C 2 R v i s i 1 0 (i 1 i 2 ) / n I 1 M(D)= = nd D' i 2 I 2 Conducting devices: DT s T s D'T s Q 1 D 1 t 73

65 Inverse SEPIC 1 Nonisolated inverse SEPIC 2 Isolated inverse SEPIC 1 : n C 1 L 2 D 1 C 2 R v Q 1 74

66 Obtaining isolation in the Cuk converter L 1 L 2 Nonisolated Cuk converter C 1 Q D 1 1 C 2 R v L 1 L 2 Split capacitor C 1 into series capacitors C 1a and C 1b Q 1 C 1a C 1b D 1 C 2 R v 75

67 Isolated Cuk converter L 1 L 2 Insert transformer between capacitors C 1a and C 1b Q 1 C 1a C 1b D 1 C 2 R v M(D)= = nd D' 1 : n Discussion Capacitors C 1a and C 1b ensure that no dc voltage is applied to transformer primary or secondary windings Transformer functions in conventional manner, with small magnetizing current and negligible energy storage within the magnetizing inductance 76

68 6.4. Converter evaluation and design For a given application, which converter topology is best? There is no ultimate converter, perfectly suited for all possible applications Trade studies Rough designs of several converter topologies to meet the given specifications An unbiased quantitative comparison of worst-case transistor currents and voltages, transformer size, etc. Comparison via switch stress, switch utilization, and semiconductor cost Spreadsheet design 77

69 Switch stress and switch utilization Largest single cost in a converter is usually the cost of the active semiconductor devices Conduction and switching losses associated with the active semiconductor devices often dominate the other sources of loss This suggests evaluating candidate converter approaches by comparing the voltage and current stresses imposed on the active semiconductor devices. Minimization of total switch stresses leads to reduced loss, and to minimization of the total silicon area required to realize the power devices of the converter. 78

70 Total active switch stress S In a converter having k active semiconductor devices, the total active switch stress S is defined as where S = k Σ j =1 j I j j is the peak voltage applied to switch j, I j is the rms current applied to switch j (peak current is also sometimes used). In a good design, the total active switch stress is minimized. 79

71 Active switch utilization U It is desired to minimize the total active switch stress, while maximizing the output power P load. The active switch utilization U is defined as U = P load S The active switch utilization is the converter output power obtained per unit of active switch stress. It is a converter figure-of-merit, which measures how well a converter utilizes its semiconductor devices. Active switch utilization is less than 1 in transformer-isolated converters, and is a quantity to be maximized. Converters having low switch utilizations require extra active silicon area, and operate with relatively low efficiency. Active switch utilization is a function of converter operating point. 80

72 CCM flyback example: Determination of S During subinterval 2, the transistor blocks voltage Q1,pk equal to plus the reflected load voltage: Q1,pk = n = D' L M Q1 1:n D 1 C Transistor current coincides with i g. RMS value is I Q1,rms = I D = P load D i g I Switch stress S is S = Q1,pk I Q1,rms = n I D DT s 0 D'T s t Conducting devices: 81 T s Q 1 D 1

73 CCM flyback example: Determination of U Express load power P load in terms of and I: 1 : D D' : n I g I R P load = D' I n Previously-derived expression for S: CCM flyback model S = Q1,pk I Q1,rms = n I D Hence switch utilization U is U = P load S = D' D 82

74 Flyback example: switch utilization U(D) For given,, P load, the designer can arbitrarily choose D. The turns ratio n must then be chosen according to n = D' g D U max U = at D = 1/3 Single operating point design: choose D = 1/3. small D leads to large transistor current 0.1 large D leads to large transistor voltage D 83

75 Comparison of switch utilizations of some common converters Table 6.1. Active switch utilizations of some common dc-dc converters, single operating point. Converter U(D) max U(D) max U(D) occurs at D = Buck Boost Buck-boost, flyback, nonisolated SEPIC, isolated SEPIC, nonisolated Cuk, isolated Cuk Forward, n 1 = n 2 Other isolated buck-derived converters (fullbridge, half-bridge, push-pull) Isolated boost-derived converters (full bridge, push-pull) D 1 1 D' D D' D 1 2 D D 2 2 D' 2 1D = = =

76 Switch utilization : Discussion Increasing the range of operating points leads to reduced switch utilization Buck converter can operate with high switch utilization (U approaching 1) when D is close to 1 Boost converter can operate with high switch utilization (U approaching ) when D is close to 1 Transformer isolation leads to reduced switch utilization Buck-derived transformer-isolated converters U should be designed to operate with D as large as other considerations allow transformer turns ratio can be chosen to optimize design 85

77 Switch utilization: Discussion Nonisolated and isolated versions of buck-boost, SEPIC, and Cuk converters U Single-operating-point optimum occurs at D = 1/3 Nonisolated converters have lower switch utilizations than buck or boost Isolation can be obtained without penalizing switch utilization 86

78 Active semiconductor cost vs. switch utilization semiconductor cost per kw output power = semiconductor device cost per rated ka voltage derating factor current derating factor converter switch utilization (semiconductor device cost per rated ka) = cost of device, divided by product of rated blocking voltage and rms current, in \$/ka. Typical values are less than \$1/kA (voltage derating factor) and (current derating factor) are required to obtain reliable operation. Typical derating factors are Typical cost of active semiconductor devices in an isolated dc-dc converter: \$1 - \$10 per kw of output power. 87

79 Converter design using computer spreadsheet Given ranges of and P load, as well as desired value of and other quantities such as switching frequency, ripple, etc., there are two basic engineering design tasks: Compare converter topologies and select the best for the given specifications Optimize the design of a given converter A computer spreadsheet is a very useful tool for this job. The results of the steady-state converter analyses of Chapters 1-6 can be entered, and detailed design investigations can be quickly performed: Evaluation of worst-case stresses over a range of operating points Evaluation of design tradeoffs 88

80 Spreadsheet design example Specifications Maximum input voltage Minimum input voltage Output voltage Maximum load power P load Minimum load power P load Switching frequency f s Maximum output ripple v W 20 W 100 khz 0.1 Input voltage: rectified 230 rms ±20% Regulated output of 15 Rated load power 200 W Must operate at 10% load Select switching frequency of 100 khz Output voltage ripple 0.1 Compare single-transistor forward and flyback converters in this application Specifications are entered at top of spreadsheet 89

81 Forward converter design, CCM n 1 : n 2 : n 3 D 2 L D 3 C R Q 1 D 1 Design variables Reset winding turns ratio n 2 /n 1 1 Turns ratio n 3 /n Inductor current ripple i 2A ref to sec Design for CCM at full load; may operate in DCM at light load 90

82 Flyback converter design, CCM 1:n D 1 L M C Q1 Design variables Turns ratio n 2 /n Inductor current ripple i 3 A ref to sec Design for CCM at full load; may operate in DCM at light load 91

83 Enter results of converter analysis into spreadsheet (Forward converter example) Maximum duty cycle occurs at minimum and maximum P load. Converter then operates in CCM, with D = n 1 n 3 g Inductor current ripple is Solve for L: i = D'T s 2L L = D'T s 2 i i is a design variable. For a given i, the equation above can be used to determine L. To ensure CCM operation at full load, i should be less than the full-load output current. C can be found in a similar manner. 92

84 Forward converter example, continued Check for DCM at light load. The solution of the buck converter operating in DCM is These equations apply equally well to the forward converter, provided that all quantities are referred to the transformer secondary side. Solve for D: D = = n 3 n K D 2 with K =2L / RT s, and R = 2 / P load 2 K 2 2n 3 n in DCM D = n 1 n 3 in CCM at a given operating point, the actual duty cycle is the small of the values calculated by the CCM and DCM equations above. Minimum D occurs at minimum P load and maximum. g 93

85 More regarding forward converter example Worst-case component stresses can now be evaluated. Peak transistor voltage is max v Q1 = 1 n 1 n 2 RMS transistor current is I Q1,rms = n 3 D I 2 i n 1 3 (this neglects transformer magnetizing current) Other component stresses can be found in a similar manner. Magnetics design is left for a later chapter. 2 n 3 n 1 D I 94

86 Results: forward and flyback converter spreadsheets Forward converter design, CCM Flyback converter design, CCM Design variables Design variables Reset winding turns ratio n 2 /n 1 1 Turns ratio n 2 /n Turns ratio n 3 /n Inductor current ripple i 3 A ref to sec Inductor current ripple i 2 A ref to sec Results Results Maximum duty cycle D Maximum duty cycle D Minimum D, at full load Minimum D, at full load Minimum D, at minimum load Minimum D, at minimum load Worst-case stresses Worst-case stresses Peak transistor voltage v Q1 780 Peak transistor voltage v Q1 510 Rms transistor current i Q A Rms transistor current i Q A Transistor utilization U Transistor utilization U Peak diode voltage v D2 49 Peak diode voltage v D1 64 Rms diode current i D2 9.1 A Rms diode current i D A Peak diode voltage v D3 49 Peak diode current i D A Rms diode current i D A Rms output capacitor current i C 1.15 A Rms output capacitor current i C 9.1 A 95

87 Discussion: transistor voltage Flyback converter Ideal peak transistor voltage: 510 Actual peak voltage will be higher, due to ringing causes by transformer leakage inductance An 800 or 1000 MOSFET would have an adequate design margin Forward converter Ideal peak transistor voltage: 780, 53% greater than flyback Few MOSFETs having voltage rating of over 1000 are available when ringing due to transformer leakage inductance is accounted for, this design will have an inadequate design margin Fix: use two-transistor forward converter, or change reset winding turns ratio A conclusion: reset mechanism of flyback is superior to forward 96

88 Discussion: rms transistor current Forward Flyback 1.13A worst-case transistor utilization A worst case, 22% higher than forward transistor utilization CCM flyback exhibits higher peak and rms currents. Currents in DCM flyback are even higher 97

89 Discussion: secondary-side diode and capacitor stresses Forward Flyback peak diode voltage 49 rms diode current 9.1A / 11.1A rms capacitor current 1.15A peak diode voltage 64 rms diode current 16.3A peak diode current 22.2A rms capacitor current 9.1A Secondary-side currents, especially capacitor currents, limit the practical application of the flyback converter to situations where the load current is not too great. 98

90 Summary of key points 1. The boost converter can be viewed as an inverse buck converter, while the buck-boost and Cuk converters arise from cascade connections of buck and boost converters. The properties of these converters are consistent with their origins. Ac outputs can be obtained by differential connection of the load. An infinite number of converters are possible, and several are listed in this chapter. 2. For understanding the operation of most converters containing transformers, the transformer can be modeled as a magnetizing inductance in parallel with an ideal transformer. The magnetizing inductance must obey all of the usual rules for inductors, including the principle of volt-second balance. 99

91 Summary of key points 3. The steady-state behavior of transformer-isolated converters may be understood by first replacing the transformer with the magnetizing-inductance-plus-ideal-transformer equivalent circuit. The techniques developed in the previous chapters can then be applied, including use of inductor volt-second balance and capacitor charge balance to find dc currents and voltages, use of equivalent circuits to model losses and efficiency, and analysis of the discontinuous conduction mode. 4. In the full-bridge, half-bridge, and push-pull isolated versions of the buck and/or boost converters, the transformer frequency is twice the output ripple frequency. The transformer is reset while it transfers energy: the applied voltage polarity alternates on successive switching periods. 100

92 Summary of key points 5. In the conventional forward converter, the transformer is reset while the transistor is off. The transformer magnetizing inductance operates in the discontinuous conduction mode, and the maximum duty cycle is limited. 6. The flyback converter is based on the buck-boost converter. The flyback transformer is actually a two-winding inductor, which stores and transfers energy. 7. The transformer turns ratio is an extra degree-of-freedom which the designer can choose to optimize the converter design. Use of a computer spreadsheet is an effective way to determine how the choice of turns ratio affects the component voltage and current stresses. 8. Total active switch stress, and active switch utilization, are two simplified figures-of-merit which can be used to compare the various converter circuits. 101

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