Chapter 19 Resonant Conversion

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1 Chapter 9 Resonant Conversion Introduction 9. Sinusoidal analysis of resonant converters 9. Examples Series resonant converter Parallel resonant converter 9.3 Exact characteristics of the series and parallel resonant converters 9.4 Soft switching Zero current switching Zero voltage switching The zero voltage transition converter 9.5 Load-dependent properties of resonant converters Fundamentals of Power Electronics Chapter 9: Resonant Conversion

2 Introduction to Resonant Conversion Resonant power converters contain resonant L-C networks whose voltage and current waveforms vary sinusoidally during one or more subintervals of each switching period. These sinusoidal variations are large in magnitude, and the small ripple approximation does not apply. Some types of resonant converters: Dc-to-high-frequency-ac inverters Resonant dc-dc converters Resonant inverters or rectifiers producing line-frequency ac Fundamentals of Power Electronics Chapter 9: Resonant Conversion

3 A basic class of resonant inverters Basic circuit dc source v g N S i s v s L N T C s C p i v Resistive load R Switch network Resonant tank network Several resonant tank networks L C s L L C s C p C p Series tank network Parallel tank network LCC tank network Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

4 Tank network responds only to fundamental component of switched waveforms Switch output voltage spectrum f s 3f s 5f s f Tank current and output voltage are essentially sinusoids at the switching frequency f s. Resonant tank response f s 3f s 5f s f Output can be controlled by variation of switching frequency, closer to or away from the tank resonant frequency Tank current spectrum f s 3f s 5f s f Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

5 Derivation of a resonant dc-dc converter Rectify and filter the output of a dc-high-frequency-ac inverter Transfer function H(s) dc source v g i s v s L C s i R v R i v R N S Switch network N T Resonant tank network The series resonant dc-dc converter N R Rectifier network N F Low-pass filter network dc load Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion

6 A series resonant link inverter Same as dc-dc series resonant converter, except output rectifiers are replaced with four-quadrant switches: dc source v g L C s v R i Switch network Resonant tank network Switch network Low-pass filter network ac load Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

7 Quasi-resonant converters In a conventional PWM converter, replace the PWM switch network with a switch network containing resonant elements. v g Buck converter example i v Switch network i v L i C R v Two switch networks: i v PWM switch network i v i v ZCS quasi-resonant switch network L r C r i v Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

8 Resonant conversion: advantages The chief advantage of resonant converters: reduced switching loss Zero-current switching Zero-voltage switching Turn-on or turn-off transitions of semiconductor devices can occur at zero crossings of tank voltage or current waveforms, thereby reducing or eliminating some of the switching loss mechanisms. Hence resonant converters can operate at higher switching frequencies than comparable PWM converters Zero-voltage switching also reduces converter-generated EMI Zero-current switching can be used to commutate SCRs In specialized applications, resonant networks may be unavoidable High voltage converters: significant transformer leakage inductance and winding capacitance leads to resonant network Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

9 Resonant conversion: disadvantages Can optimize performance at one operating point, but not with wide range of input voltage and load power variations Significant currents may circulate through the tank elements, even when the load is disconnected, leading to poor efficiency at light load Quasi-sinusoidal waveforms exhibit higher peak values than equivalent rectangular waveforms These considerations lead to increased conduction losses, which can offset the reduction in switching loss Resonant converters are usually controlled by variation of switching frequency. In some schemes, the range of switching frequencies can be very large Complexity of analysis Fundamentals of Power Electronics 9 Chapter 9: Resonant Conversion

10 Resonant conversion: Outline of discussion Simple steady-state analysis via sinusoidal approximation Simple and exact results for the series and parallel resonant converters Mechanisms of soft switching Circulating currents, and the dependence (or lack thereof) of conduction loss on load power Quasi-resonant converter topologies Steady-state analysis of quasi-resonant converters Ac modeling of quasi-resonant converters via averaged switch modeling Fundamentals of Power Electronics 0 Chapter 9: Resonant Conversion

11 9. Sinusoidal analysis of resonant converters A resonant dc-dc converter: Transfer function H(s) dc source v g i s v s L C s i R v R i v R N S Switch network N T Resonant tank network N R Rectifier network N F Low-pass filter network dc load If tank responds primarily to fundamental component of switch network output voltage waveform, then harmonics can be neglected. Let us model all ac waveforms by their fundamental components. Fundamentals of Power Electronics Chapter 9: Resonant Conversion

12 The sinusoidal approximation Switch output voltage spectrum f s 3f s 5f s f Tank current and output voltage are essentially sinusoids at the switching frequency f s. Resonant tank response f s 3f s 5f s f Neglect harmonics of switch output voltage waveform, and model only the fundamental component. Tank current spectrum Remaining ac waveforms can be found via phasor analysis. f s 3f s 5f s f Fundamentals of Power Electronics Chapter 9: Resonant Conversion

13 9.. Controlled switch network model N S i s V g 4 π V g Fundamental component v s v g v s v s t Switch network V g If the switch network produces a square wave, then its output voltage has the following Fourier series: v s = 4V g π Σ n =, 3, 5,... n sin (nω st) The fundamental component is v s = 4V g π sin (ω st)=v s sin (ω s t) So model switch network output port with voltage source of value v s Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

14 Model of switch network input port N S i s I s i g v g v s i s ω s t Switch network Assume that switch network output current is i s I s sin (ω s t ϕ s ) It is desired to model the dc component (average value) of the switch network input current. ϕ s i g Ts = T s T s T s / 0 0 T s / = π I s cos (ϕ s ) i g (τ)dτ I s sin (ω s τ ϕ s )dτ Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

15 Switch network: equivalent circuit v g I s π cos (ϕ s) 4V g π v s = sin (ω st) i s = I s sin (ω s t ϕ s ) Switch network converts dc to ac Dc components of input port waveforms are modeled Fundamental ac components of output port waveforms are modeled Model is power conservative: predicted average input and output powers are equal Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion

16 9.. Modeling the rectifier and capacitive filter networks i R i R i V v R v R v R i R ω s t N R Rectifier network N F Low-pass filter network dc load ϕ R V Assume large output filter capacitor, having small ripple. v R is a square wave, having zero crossings in phase with tank output current i R. If i R is a sinusoid: i R =I R sin (ω s t ϕ R ) Then v R has the following Fourier series: v R = 4V π Σ n =,3,5, n sin (nω st ϕ R ) Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

17 Sinusoidal approximation: rectifier Again, since tank responds only to fundamental components of applied waveforms, harmonics in v R can be neglected. v R becomes v R = 4V π sin (ω st ϕ R )=V R sin (ω s t ϕ R ) Actual waveforms with harmonics ignored V v R 4 π V v R fundamental i R ω s t i R ω s t V i R = v R R e R e = 8 π R ϕ R ϕ R Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

18 Rectifier dc output port model i R v R i R i Output capacitor charge balance: dc v R load current is equal to average rectified tank output current i R T s = I N R Rectifier network N F Low-pass filter network dc load Hence I = T T s / I R sin (ω s t ϕ R ) dt S 0 V v R = π I R i R ω s t V ϕ R Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

19 Equivalent circuit of rectifier Rectifier input port: Fundamental components of current and voltage are sinusoids that are in phase v R i R R e π I R V I R Hence rectifier presents a resistive load to tank network Effective resistance R e is R e = v R i R = 8 π V I R e = 8 π R Rectifier equivalent circuit With a resistive load R, this becomes R e = 8 π R = 0.806R Fundamentals of Power Electronics 9 Chapter 9: Resonant Conversion

20 9..3 Resonant tank network Transfer function H(s) i s i R v s Z i Resonant network v R R e Model of ac waveforms is now reduced to a linear circuit. Tank network is excited by effective sinusoidal voltage (switch network output port), and is load by effective resistive load (rectifier input port). Can solve for transfer function via conventional linear circuit analysis. Fundamentals of Power Electronics 0 Chapter 9: Resonant Conversion

21 Solution of tank network waveforms Transfer function: v R (s) v s (s) = H(s) Ratio of peak values of input and output voltages: v s i s Z i Transfer function H(s) Resonant network i R v R R e V R V s = H(s) s = jωs Solution for tank output current: i R (s)= v R(s) = H(s) v R e R s (s) e which has peak magnitude I R = H(s) s = jω s R e V s Fundamentals of Power Electronics Chapter 9: Resonant Conversion

22 9..4 Solution of converter voltage conversion ratio M = V/V g Transfer function H(s) i s i R I V g Z i Resonant network v R R e π I R V R I s π cos (ϕ s) 4V g π v s = sin (ω st) R e = 8 π R M = V V g = R π Re H(s) s = jωs 4 π V I I I R I R V R V R V s V s V g Eliminate R e : V V g = H(s) s = jωs Fundamentals of Power Electronics Chapter 9: Resonant Conversion

23 Conversion ratio M V V g = H(s) s = jωs So we have shown that the conversion ratio of a resonant converter, having switch and rectifier networks as in previous slides, is equal to the magnitude of the tank network transfer function. This transfer function is evaluated with the tank loaded by the effective rectifier input resistance R e. Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

24 9. Examples 9.. Series resonant converter transfer function H(s) dc source v g i s v s L C s i R v R i v R N S switch network N T resonant tank network N R rectifier network N F low-pass filter network dc load Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

25 Model: series resonant converter transfer function H(s) i s L C i R I V g Z i v R R e π I R V R I s π cos (ϕ s) 4V g π v s = sin (ω st) series tank network R e = 8 π R H(s)= = R e Z i (s) = R e R e sl sc s Q e ω 0 s Q e ω s 0 ω 0 ω 0 = LC =π f 0 R 0 = Q e = R 0 R e Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion L C M = H(jω s ) = Q e F F

26 Construction of Z i Z i ωc ωl f 0 R 0 R e Q e = R 0 / R e Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

27 Construction of H H Q e = R e / R 0 R e / R 0 f 0 ωr e C R e / ωl Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

28 9.. Subharmonic modes of the SRC switch output voltage spectrum Example: excitation of tank by third harmonic of switching frequency resonant tank response f s 3f s 5f s f Can now approximate v s by its third harmonic: v s v sn = 4V g nπ sin (nω st) tank current spectrum f s 3f s 5f s f Result of analysis: M = V V g = H(jnω s) n f s 3f s 5f s f Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

29 Subharmonic modes of SRC M 3 5 etc. 5 f 0 3 f 0 f 0 f s Fundamentals of Power Electronics 9 Chapter 9: Resonant Conversion

30 9..3 Parallel resonant dc-dc converter dc source v g i s i R i L v R v v s R C p N S switch network N T resonant tank network N R rectifier network N F low-pass filter network dc load Differs from series resonant converter as follows: Different tank network Rectifier is driven by sinusoidal voltage, and is connected to inductive-input low-pass filter Need a new model for rectifier and filter networks Fundamentals of Power Electronics 30 Chapter 9: Resonant Conversion

31 Model of uncontrolled rectifier with inductive filter network I i R i R i v R ω s t v R v R ϕ R I k N R rectifier network N F low-pass filter network dc load 4 π I i R fundamental Fundamental component of i R : v R i R = v R R e R e = π 8 R ω s t i R = 4I π sin (ω st ϕ R ) ϕ R Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

32 Effective resistance R e Again define R e = v R i R = πv R 4I In steady state, the dc output voltage V is equal to the average value of v R : V = T T s / V R sin (ω s t ϕ R ) dt S 0 = π V R For a resistive load, V = IR. The effective resistance R e can then be expressed R e = π 8 R =.337R Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

33 Equivalent circuit model of uncontrolled rectifier with inductive filter network i R I v R R e π V R V R R e = π 8 R Fundamentals of Power Electronics 33 Chapter 9: Resonant Conversion

34 Equivalent circuit model Parallel resonant dc-dc converter transfer function H(s) i s L i R I V g Z i C v R R e π V R V R I s π cos (ϕ s) 4V g π v s = sin (ω st) parallel tank network R e = π 8 R M = V V g = 8 π H(s) s = jωs H(s)= Z o(s) sl Z o (s)=sl sc R e Fundamentals of Power Electronics 34 Chapter 9: Resonant Conversion

35 Construction of Z o Z o R e R 0 Q e = R e / R 0 f 0 ωl ωc Fundamentals of Power Electronics 35 Chapter 9: Resonant Conversion

36 Construction of H H R e / R 0 Q e = R e / R 0 f 0 ω LC Fundamentals of Power Electronics 36 Chapter 9: Resonant Conversion

37 Dc conversion ratio of the PRC M = 8 π Z o (s) sl s = jω s = 8 π s Q e ω 0 s ω 0 = 8 π F Q F e s = jω s At resonance, this becomes M = 8 π R e R 0 = R R 0 PRC can step up the voltage, provided R > R 0 PRC can produce M approaching infinity, provided output current is limited to value less than V g / R 0 Fundamentals of Power Electronics 37 Chapter 9: Resonant Conversion

38 9.3 Exact characteristics of the series and parallel resonant dc-dc converters Define f 0 k < f s < f 0 k or k < F < k mode index k ξ = k ()k subharmonic index ξ ξ = 3 ξ = etc. k = 3 k = f 0 / 3 f 0 / k = f 0 k = 0 f s Fundamentals of Power Electronics 38 Chapter 9: Resonant Conversion

39 9.3. Exact characteristics of the series resonant converter Q D Q 3 D 3 L C : n V g R V Q D Q 4 D 4 Normalized load voltage and current: M = V nv g J = InR 0 V g Fundamentals of Power Electronics 39 Chapter 9: Resonant Conversion

40 Continuous conduction mode, SRC Tank current rings continuously for entire length of switching period Waveforms for type k CCM, odd k : v s V g V g i L D Q Q π Q π π D (k ) complete half-cycles γ Q ω s t Fundamentals of Power Electronics 40 Chapter 9: Resonant Conversion

41 Series resonant converter Waveforms for type k CCM, even k : v s V g V g i L D Q Q π π π D D Q ω s t D k complete half-cycles γ Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

42 Exact steady-state solution, CCM Series resonant converter M ξ sin γ Jγ ξ ()k cos γ = where M = V nv g J = InR 0 V g γ = ω 0T s = π F Output characteristic, i.e., the relation between M and J, is elliptical M is restricted to the range 0 M ξ Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

43 Control-plane characteristics For a resistive load, eliminate J and solve for M vs. γ M = ξ 4 tan Qγ γ Qγ () k ξ cos γ Qγ ξ 4 tan cos γ γ Qγ Exact, closed-form, valid for any CCM Fundamentals of Power Electronics 43 Chapter 9: Resonant Conversion

44 Discontinuous conduction mode Type k DCM: during each half-switching-period, the tank rings for k complete half-cycles. The output diodes then become reverse-biased for the remainder of the half-switching-period. v s V g V g i L Q π Q π π X Q ω s t D k complete half-cycles γ Fundamentals of Power Electronics 44 Chapter 9: Resonant Conversion

45 Steady-state solution: type k DCM, odd k M = k Conditions for operation in type k DCM, odd k : f s < f 0 k (k ) γ > J > (k ) γ Fundamentals of Power Electronics 45 Chapter 9: Resonant Conversion

46 Steady-state solution: type k DCM, even k J = k γ Conditions for operation in type k DCM, even k : f s < f 0 k k > M > k gyrator model, SRC operating in an even DCM: I g = gv V g g I g = gv g V g = k γr 0 Fundamentals of Power Electronics 46 Chapter 9: Resonant Conversion

47 Control plane characteristics, SRC M = V / V g Q = Q = F = f s / f 0 Q = Q = 0 Fundamentals of Power Electronics 47 Chapter 9: Resonant Conversion

48 Mode boundaries, SRC k = DCM M k = DCM k = CCM k = 0 CCM 0.4 k = 3 DCM k = 4 DCM etc. k = 3 CCM k = CCM Fundamentals of Power Electronics 48 Chapter 9: Resonant Conversion F

49 Output characteristics, SRC above resonance 6 5 F =.07 F =.05 F =.0 4 F =.0 J 3 F =.5 F = M Fundamentals of Power Electronics 49 Chapter 9: Resonant Conversion

50 Output characteristics, SRC below resonance J 3 F =.90 F =.93 F =.96 F =.0.5 F =.85.5 π 0 F =.5 F =.5 F =. F =.75 k = CCM k = DCM k = DCM 4 π M Fundamentals of Power Electronics 50 Chapter 9: Resonant Conversion

51 9.3. Exact characteristics of the parallel resonant converter Q D Q 3 D 3 L : n D 7 D 5 V g C R V Q D Q 4 D 4 D 8 D 6 Normalized load voltage and current: M = V nv g J = InR 0 V g Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion

52 Parallel resonant converter in CCM CCM closed-form solution v s V g M = γ ϕ sin (ϕ) cos γ γ γ V g ω 0 t i L cos cos γ J sin γ for 0 < γ < π ( ϕ = cos cos γ J sin γ for π < γ <π v C v C V = v C T s Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion

53 Parallel resonant converter in DCM Mode boundary v C J > J crit (γ) J < J crit (γ) for DCM for CCM J crit (γ)= sin (γ) sin γ 4 sin γ ω 0 t DCM equations M C0 = cos (β) J L0 = J sin (β) cos (α β) cos (α)= sin (α β) sin (α)(δ α)=j β δ = γ M = γ (J δ) i L D 5 D 8 D 5 D 8 D 5 D 8 D 5 D 8 D 6 D 7 D 6 D 7 D 6 D 7 D 6 D 7 α δ γ I β (require iteration) ω 0 t Fundamentals of Power Electronics 53 Chapter 9: Resonant Conversion I

54 Output characteristics of the PRC 3.0 J F = F = M Solid curves: CCM Shaded curves: DCM Fundamentals of Power Electronics 54 Chapter 9: Resonant Conversion

55 Control characteristics of the PRC with resistive load 3.0 Q = M = V/V g.5.0 Q = Q = Q = 0.5 Q = f s /f 0 Fundamentals of Power Electronics 55 Chapter 9: Resonant Conversion

56 9.4 Soft switching Soft switching can mitigate some of the mechanisms of switching loss and possibly reduce the generation of EMI Semiconductor devices are switched on or off at the zero crossing of their voltage or current waveforms: Zero-current switching: transistor turn-off transition occurs at zero current. Zero-current switching eliminates the switching loss caused by IGBT current tailing and by stray inductances. It can also be used to commutate SCR s. Zero-voltage switching: transistor turn-on transition occurs at zero voltage. Diodes may also operate with zero-voltage switching. Zero-voltage switching eliminates the switching loss induced by diode stored charge and device output capacitances. Zero-voltage switching is usually preferred in modern converters. Zero-voltage transition converters are modified PWM converters, in which an inductor charges and discharges the device capacitances. Zero-voltage switching is then obtained. Fundamentals of Power Electronics 56 Chapter 9: Resonant Conversion

57 9.4. Operation of the full bridge below resonance: Zero-current switching Series resonant converter example Q i Q v ds D Q 3 D 3 L C V g v s Q D Q 4 D 4 i s Operation below resonance: input tank current leads voltage Zero-current switching (ZCS) occurs Fundamentals of Power Electronics 57 Chapter 9: Resonant Conversion

58 Tank input impedance Operation below resonance: tank input impedance Z i is dominated by tank capacitor. Z i ωc ωl Z i is positive, and tank input current leads tank input voltage. R e R 0 f 0 Q e = R 0 /R e Zero crossing of the tank input current waveform i s occurs before the zero crossing of the voltage v s. Fundamentals of Power Electronics 58 Chapter 9: Resonant Conversion

59 Switch network waveforms, below resonance Zero-current switching v s V g v s t Q i Q v ds D Q 3 D 3 L C V g i s Q D Q 4 D 4 v s i s T s t β t β T s t Conduction sequence: Q D Q D Conducting devices: Hard turn-on of Q, Q 4 Q D Q D Q 4 D 4 Q 3 D 3 Soft turn-off of Q, Q 4 Hard turn-on of Q, Q 3 Soft turn-off of Q, Q 3 Q is turned off during D conduction interval, without loss Fundamentals of Power Electronics 59 Chapter 9: Resonant Conversion

60 ZCS turn-on transition: hard switching v ds V g i ds t Q i Q v ds D Q 3 D 3 v s L C Conducting devices: Hard turn-on of Q, Q 4 t β T s T s t β Q D Q D Q 4 D 4 Q 3 Soft turn-off of Q, Q 4 D 3 t Q D Q 4 D 4 i s Q turns on while D is conducting. Stored charge of D and of semiconductor output capacitances must be removed. Transistor turn-on transition is identical to hardswitched PWM, and switching loss occurs. Fundamentals of Power Electronics 60 Chapter 9: Resonant Conversion

61 9.4. Operation of the full bridge below resonance: Zero-voltage switching Series resonant converter example Q i Q v ds D Q 3 D 3 L C V g v s Q D Q 4 D 4 i s Operation above resonance: input tank current lags voltage Zero-voltage switching (ZVS) occurs Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

62 Tank input impedance Operation above resonance: tank input impedance Z i is dominated by tank inductor. Z i ωc ωl Z i is negative, and tank input current lags tank input voltage. R e R 0 f 0 Q e = R 0 /R e Zero crossing of the tank input current waveform i s occurs after the zero crossing of the voltage v s. Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

63 Switch network waveforms, above resonance Zero-voltage switching V g v s v s t Q i Q v ds D Q 3 D 3 L C V g v s i s Q D Q 4 D 4 i s Conducting devices: D t α Q Q 4 T s D 4 Q Soft turn-on of Q, Q 4 Hard turn-off of Q, Q 4 D D 3 Soft turn-on of Q, Q 3 Q 3 t Hard turn-off of Q, Q 3 Conduction sequence: D Q D Q Q is turned on during D conduction interval, without loss Fundamentals of Power Electronics 63 Chapter 9: Resonant Conversion

64 ZVS turn-off transition: hard switching? v ds V g i ds t Q i Q v ds D Q 3 D 3 v s L C Q D Q 4 D 4 i s Conducting devices: D t α Q Q 4 T s D 4 Q Soft turn-on of Q, Q 4 Hard turn-off of Q, Q 4 D D 3 Q 3 t When Q turns off, D must begin conducting. Voltage across Q must increase to V g. Transistor turn-off transition is identical to hard-switched PWM. Switching loss may occur (but see next slide). Fundamentals of Power Electronics 64 Chapter 9: Resonant Conversion

65 Soft switching at the ZVS turn-off transition V g v ds Q Q Conducting devices: D D C leg C leg Q v ds X D Q 4 Turn off Q, Q 4 D 3 V g C leg C leg Commutation interval D 3 D 4 t Fundamentals of Power Electronics 65 Chapter 9: Resonant Conversion Q 3 Q 4 v s i s L to remainder of converter Introduce small capacitors C leg across each device (or use device output capacitances). Introduce delay between turn-off of Q and turn-on of Q. Tank current i s charges and discharges C leg. Turn-off transition becomes lossless. During commutation interval, no devices conduct. So zero-voltage switching exhibits low switching loss: losses due to diode stored charge and device output capacitances are eliminated.

66 9.4.3 The zero-voltage transition converter Basic version based on full-bridge PWM buck converter Q Q 3 i c L c D C leg C leg D 3 V g Q D C leg v C leg D 4 Q 4 v V g Can obtain ZVS of all primaryside MOSFETs and diodes Can turn on Q at zero voltage Secondary-side diodes switch at zero-current, with loss Conducting devices: Q X D t Phase-shift control Turn off Q Commutation interval Fundamentals of Power Electronics 66 Chapter 9: Resonant Conversion

67 9.5 Load-dependent properties of resonant converters Resonant inverter design objectives:. Operate with a specified load characteristic and range of operating points With a nonlinear load, must properly match inverter output characteristic to load characteristic. Obtain zero-voltage switching or zero-current switching Preferably, obtain these properties at all loads Could allow ZVS property to be lost at light load, if necessary 3. Minimize transistor currents and conduction losses To obtain good efficiency at light load, the transistor current should scale proportionally to load current (in resonant converters, it often doesn t!) Fundamentals of Power Electronics 67 Chapter 9: Resonant Conversion

68 Topics of Discussion Section 9.5 Inverter output i-v characteristics Two theorems Dependence of transistor current on load current Dependence of zero-voltage/zero-current switching on load resistance Simple, intuitive frequency-domain approach to design of resonant converter Examples and interpretation Series Parallel LCC Fundamentals of Power Electronics 68 Chapter 9: Resonant Conversion

69 Inverter output characteristics Let H be the open-circuit (RÕ ) transfer function: v o (jω) v i (jω) R = H (jω) sinusoidal source v i i i Z i transfer function H(s) resonant network purely reactive Z o i o v o resistive load R and let Z o0 be the output impedance (with v i Õ short-circuit). Then, v o (jω)=h (jω) v i (jω) R R Z o0 (jω) This result can be rearranged to obtain v o i o Z o0 = H v i The output voltage magnitude is: v o = v o v o* = with R = v o H v i Z o0 / R / i o Hence, at a given frequency, the output characteristic (i.e., the relation between v o and i o ) of any resonant inverter of this class is elliptical. Fundamentals of Power Electronics 69 Chapter 9: Resonant Conversion

70 Inverter output characteristics General resonant inverter output characteristics are elliptical, of the form v o V oc I sc i o = i o I sc = H v i Z o0 I sc inverter output characteristic matched load R = Z o0 with V oc = H v i I sc = H v i Z o0 V oc V oc = H v i v o This result is valid provided that (i) the resonant network is purely reactive, and (ii) the load is purely resistive. Fundamentals of Power Electronics 70 Chapter 9: Resonant Conversion

71 Matching ellipse to application requirements Electronic ballast Electrosurgical generator i o inverter characteristic i o A 50Ω inverter characteristic 400W lamp characteristic matched load v o kv v o Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

72 Input impedance of the resonant tank network Transfer function H(s) Z i (s)=z i0 (s) R Z o0 (s) R Z o (s) = Z i (s) Z o0(s) R Z o (s) R Effective sinusoidal source v s i s Z i Resonant network Purely reactive Z o i v Effective resistive load R where Z i0 = v i i i R 0 Z i = v i i i R Z o0 = v o i o v i short circuit Z o = v o i o v i open circuit Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

73 Other relations Reciprocity Z i0 Z i = Z o0 Z o Tank transfer function H(s)= H (s) R Z o0 where H = v o(s) v i (s) R H = Z o0 Z i0 Z i If the tank network is purely reactive, then each of its impedances and transfer functions have zero real parts: * Z i0 =Z i0 * Z i =Z i * Z o0 =Z o0 * Z o =Z o * H =H Hence, the input impedance magnitude is R Z o0 Z i = Z i Z i* = Z i0 Fundamentals of Power Electronics 73 Chapter 9: Resonant Conversion R Z o

74 Z i0 and Z i for 3 common inverters Series L C s ω C s Z i ωl Z i0 (s) =sl sc s Z i Z o Z i0 Z i (s) = f Parallel L ω C p ωl Z i0 (s) =sl Z i C p Z o Z i0 Z i Z i (s) =sl sc p f LCC L C s ω C s ω C s ω C p ωl Z i0 (s) =sl sc s Z i C p Z o Z i0 Z i f Z i (s) =sl sc p sc s Fundamentals of Power Electronics 74 Chapter 9: Resonant Conversion

75 A Theorem relating transistor current variations to load resistance R Theorem : If the tank network is purely reactive, then its input impedance Z i is a monotonic function of the load resistance R. l l l l So as the load resistance R varies from 0 to, the resonant network input impedance Z i varies monotonically from the short-circuit value Z i0 to the open-circuit value Z i. The impedances Z i and Z i0 are easy to construct. If you want to minimize the circulating tank currents at light load, maximize Z i. Note: for many inverters, Z i < Z i0! The no-load transistor current is therefore greater than the short-circuit transistor current. Fundamentals of Power Electronics 75 Chapter 9: Resonant Conversion

76 Proof of Theorem Previously shown: Z i = Z i0 R Z o0 R Z o á Differentiate: d Z i dr = Z i0 Z o0 Z o R R Z o á Derivative has roots at: (i) R =0 (ii) R = (iii) Z o0 = Z o, or Z i0 = Z i So the resonant network input impedance is a monotonic function of R, over the range 0 < R <. In the special case Z i0 = Z i, Z i is independent of R. Fundamentals of Power Electronics 76 Chapter 9: Resonant Conversion

77 Example: Z i of LCC for f < f m, Z i increases with increasing R. for f > f m, Z i decreases with increasing R. at a given frequency f, Z i is a monotonic function of R. It s not necessary to draw the entire plot: just construct Z i0 and Z i. Z i f ω C s ω C s ω C p increasing R f 0 f m increasing R ωl f Fundamentals of Power Electronics 77 Chapter 9: Resonant Conversion

78 Discussion: LCC Z i0 and Z i both represent series resonant impedances, whose Bode diagrams are easily constructed. Z i0 and Z i intersect at frequency f m. For f < f m then Z i0 < Z i ; hence transistor current decreases as load current decreases For f > f m then Z i0 > Z i ; hence transistor current increases as load current decreases, and transistor current is greater than or equal to short-circuit current for all R Z i ω C s ω C p ω C s Z i Z i0 L C s C p f 0 f m Z i0 f L LCC example ωl Z i f 0 = π LC s f = π LC s C p f m = π LC s C p C s C p f Fundamentals of Power Electronics 78 Chapter 9: Resonant Conversion

79 Discussion -series and parallel Series L C s ω C s Z i ωl No-load transistor current = 0, both above and below resonance. Z i Z o Z i0 f ZCS below resonance, ZVS above resonance Parallel L Z i LCC L C s C p Z o ω C s ω C p Z i0 ω C s ω C p ωl Z i f ωl Above resonance: no-load transistor current is greater than short-circuit transistor current. ZVS. Below resonance: no-load transistor current is less than short-circuit current (for f <f m ), but determined by Z i. ZCS. Z i C p Z o Z i0 Z i f Fundamentals of Power Electronics 79 Chapter 9: Resonant Conversion

80 A Theorem relating the ZVS/ZCS boundary to load resistance R Theorem : If the tank network is purely reactive, then the boundary between zero-current switching and zero-voltage switching occurs when the load resistance R is equal to the critical value R crit, given by R crit = Z o0 Z i Z i0 It is assumed that zero-current switching (ZCS) occurs when the tank input impedance is capacitive in nature, while zero-voltage switching (ZVS) occurs when the tank is inductive in nature. This assumption gives a necessary but not sufficient condition for ZVS when significant semiconductor output capacitance is present. Fundamentals of Power Electronics 80 Chapter 9: Resonant Conversion

81 Proof of Theorem Previously shown: Z o0 Z i = Z R i Z o R If ZCS occurs when Z i is capacitive, while ZVS occurs when Z i is inductive, then the boundary is determined by Z i = 0. Hence, the critical load R crit is the resistance which causes the imaginary part of Z i to be zero: Im Z i (R crit ) =0 Note that Z i, Z o0, and Z o have zero real parts. Hence, Im Z i (R crit ) =Im Z i =Im Z i Solution for R crit yields Re Re Z o0 R crit Z o R crit Z o0z o R crit Z o R crit R crit = Z o0 Z i Z i0 Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

82 Discussion ÑTheorem R crit = Z o0 Z i Z i0 l l l l l Again, Z i, Z i0, and Z o0 are pure imaginary quantities. If Z i and Z i0 have the same phase (both inductive or both capacitive), then there is no real solution for R crit. Hence, if at a given frequency Z i and Z i0 are both capacitive, then ZCS occurs for all loads. If Z i and Z i0 are both inductive, then ZVS occurs for all loads. If Z i and Z i0 have opposite phase (one is capacitive and the other is inductive), then there is a real solution for R crit. The boundary between ZVS and ZCS operation is then given by R = R crit. Note that R = Z o0 corresponds to operation at matched load with maximum output power. The boundary is expressed in terms of this matched load impedance, and the ratio Z i / Z i0. Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

83 LCC example l For f > f, ZVS occurs for all R. l For f < f 0, ZCS occurs for all R. l l For f 0 < f < f, ZVS occurs for R< R crit, and ZCS occurs for R> R crit. Note that R = Z o0 corresponds to operation at matched load with maximum output power. The boundary is expressed in terms of this matched load impedance, and the ratio Z i / Z i0. Z i ω C s ω C p ω C s Z i0 f 0 f f ZCS ZCS: R>R crit for all R ZVS: R<R crit Z i Z i0 { f m ZVS for all R ωl Z i f R crit = Z o0 Z i Z i0 Fundamentals of Power Electronics 83 Chapter 9: Resonant Conversion

84 LCC example, continued R Z i R = 0 ZCS 30 increasing R R crit 0 Z o0 ZVS R = f 0 f m f -90 f 0 f f Typical dependence of R crit and matched-load impedance Z o0 on frequency f, LCC example. Typical dependence of tank input impedance phase vs. load R and frequency, LCC example. Fundamentals of Power Electronics 84 Chapter 9: Resonant Conversion

85 9.6 Summary of Key Points. The sinusoidal approximation allows a great deal of insight to be gained into the operation of resonant inverters and dcdc converters. The voltage conversion ratio of dcdc resonant converters can be directly related to the tank network transfer function. Other important converter properties, such as the output characteristics, dependence (or lack thereof) of transistor current on load current, and zero-voltageand zero-current-switching transitions, can also be understood using this approximation. The approximation is accurate provided that the effective Qfactor is sufficiently large, and provided that the switching frequency is sufficiently close to resonance.. Simple equivalent circuits are derived, which represent the fundamental components of the tank network waveforms, and the dc components of the dc terminal waveforms. Fundamentals of Power Electronics 85 Chapter 9: Resonant Conversion

86 Summary of key points 3. Exact solutions of the ideal dcdc series and parallel resonant converters are listed here as well. These solutions correctly predict the conversion ratios, for operation not only in the fundamental continuous conduction mode, but in discontinuous and subharmonic modes as well. 4. Zero-voltage switching mitigates the switching loss caused by diode recovered charge and semiconductor device output capacitances. When the objective is to minimize switching loss and EMI, it is preferable to operate each MOSFET and diode with zero-voltage switching. 5. Zero-current switching leads to natural commutation of SCRs, and can also mitigate the switching loss due to current tailing in IGBTs. Fundamentals of Power Electronics 86 Chapter 9: Resonant Conversion

87 Summary of key points 6. The input impedance magnitude Z i, and hence also the transistor current magnitude, are monotonic functions of the load resistance R. The dependence of the transistor conduction loss on the load current can be easily understood by simply plotting Z i in the limiting cases as R Õ and as R Õ 0, or Z i and Z i0. 7. The ZVS/ZCS boundary is also a simple function of Z i and Z i0. If ZVS occurs at open-circuit and at short-circuit, then ZVS occurs for all loads. If ZVS occurs at short-circuit, and ZCS occurs at open-circuit, then ZVS is obtained at matched load provided that Z i > Z i0. 8. The output characteristics of all resonant inverters considered here are elliptical, and are described completely by the open-circuit transfer function magnitude H, and the output impedance Z o0. These quantities can be chosen to match the output characteristics to the application requirements. Fundamentals of Power Electronics 87 Chapter 9: Resonant Conversion

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