Resonant Power Conversion

Resonant Power Conversion Prof. Bob Erickson Colorado Power Electronics Center Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

Outline. Introduction to resonant power conversion 2. Simple frequency-domain modeling of resonant converters with the fundamental approximation 3. The series and parallel resonant converters, and zero-voltage switching 4. Design techniques: shaping the tank characteristics to achieve desired output I-V characteristics, achieve zero-voltage switching, and improve light-load efficiency 5. A resonant converter that has found substantial recent commercial application: the L s -L p -C converter

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 2 Chapter 9: Resonant Conversion

A basic class of resonant inverters N S i s N T i Basic circuit dc source v g v s L C s C p 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

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 Tank current spectrum 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 f s 3f s 5f s f Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

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 N R Rectifier network N F Low-pass filter network dc load The series resonant dc-dc converter Fundamentals of Power Electronics 5 Chapter 9: Resonant Conversion

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

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

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

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 Tank current spectrum f s 3f s 5f s f Neglect harmonics of switch output voltage waveform, and model only the fundamental component. Remaining ac waveforms can be found via phasor analysis. f s 3f s 5f s f Fundamentals of Power Electronics 2 Chapter 9: Resonant Conversion

9.. Controlled switch network model N S i s V g 4 V g Fundamental component v s v g 2 2 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 n =, 3, 5,... n sin (n st) The fundamental component is v s = 4V g sin ( st)=v s sin ( s t) So model switch network output port with voltage source of value v s Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

Model of switch network input port N S i s I s i g v g 2 2 v s i s s t Switch network s Assume that switch network output current is i s I s sin ( s t s ) It is desired to model the dc component (average value) of the switch network input current. i g Ts = T 2 s T 2 s T s /2 0 0 T s /2 = 2 I s cos ( s ) i g ( )d I s sin ( s s )d Fundamentals of Power Electronics 4 Chapter 9: Resonant Conversion

Switch network: equivalent circuit v g 2I s cos ( s) 4V g v s = sin ( st) i s = I s sin ( s t 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

9..2 Modeling the rectifier and capacitive filter networks i R i R i V v R v R v R i R s t N R Rectifier network N F Low-pass filter network dc load 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 ( s t R ) Then v R has the following Fourier series: v R = 4V n =,3,5, n sin (n st R ) Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

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 sin ( st R )=V R sin ( s t R ) Actual waveforms with harmonics ignored V v R 4 V v R fundamental i R s t i R s t V i R = v R R e R e = 8 2 R R R Fundamentals of Power Electronics 7 Chapter 9: Resonant Conversion

Rectifier dc output port model i R v R i R i v R Output capacitor charge balance: dc 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 2 T s /2 I R sin ( s t R ) dt S 0 V v R = 2 I R i R s t V R Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion

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 2 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 2 V I R e = 8 R 2 Rectifier equivalent circuit With a resistive load R, this becomes R e = 8 2 R = 0.806R Fundamentals of Power Electronics 9 Chapter 9: Resonant Conversion

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 20 Chapter 9: Resonant Conversion

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 R V s = H(s) s = j s v s i s Z i Transfer function H(s) Resonant network i R v R R e 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 = j s R e V s Fundamentals of Power Electronics 2 Chapter 9: Resonant Conversion

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 2 I R V R 2I s cos ( s) 4V g v s = sin ( st) R e = 8 2 R M = V V g = R 2 Re H(s) s = j s 4 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 = j s Fundamentals of Power Electronics 22 Chapter 9: Resonant Conversion

Conversion ratio M V V g = H(s) s = j s 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 23 Chapter 9: Resonant Conversion

9.2 Examples 9.2. 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 24 Chapter 9: Resonant Conversion

Model: series resonant converter transfer function H(s) i s L C i R I V g Z i v R R e 2 I R V R 2I s cos ( s) 4V g v s = sin ( st) series tank network R e = 8 R 2 H(s)= = R e Z i (s) = R e R e sl sc s Q e 0 s Q e s 0 0 2 0 = LC =2 f 0 R 0 = Q e = R 0 R e L C M = H(j s ) = 2 Q e F 2 F Fundamentals of Power Electronics 25 Chapter 9: Resonant Conversion

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

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

9.2.2 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 n sin (n st) tank current spectrum f s 3f s 5f s f Result of analysis: M = V V g = H(jn s) n f s 3f s 5f s f Fundamentals of Power Electronics 28 Chapter 9: Resonant Conversion

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

9.2.3 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

Model of uncontrolled rectifier with inductive filter network I i R i R i v R s t v R v R R I k N R rectifier network N F low-pass filter network dc load 4 I v R i R = v R R e R e = 2 8 R i R fundamental s t Fundamental component of i R : i R = 4I sin ( st R ) R Fundamentals of Power Electronics 3 Chapter 9: Resonant Conversion

Effective resistance R e Again define R e = v R i R = V R 4I In steady state, the dc output voltage V is equal to the average value of v R : V = T 2 T s /2 V R sin ( s t R ) dt S 0 = 2 V R For a resistive load, V = IR. The effective resistance R e can then be expressed R e = 2 8 R =.2337R Fundamentals of Power Electronics 32 Chapter 9: Resonant Conversion

Equivalent circuit model of uncontrolled rectifier with inductive filter network i R I v R R e 2 V R V R R e = 2 8 R Fundamentals of Power Electronics 33 Chapter 9: Resonant Conversion

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 2 V R V R 2I s cos ( s) 4V g v s = sin ( st) parallel tank network R e = 2 8 R M = V V g = 8 2 H(s) s = j s H(s)= Z o(s) sl Z o (s)=sl sc R e Fundamentals of Power Electronics 34 Chapter 9: Resonant Conversion

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

Construction of H H R e / R 0 Q e = R e / R 0 f 0 2 LC Fundamentals of Power Electronics 36 Chapter 9: Resonant Conversion

Dc conversion ratio of the PRC M = 8 Z o (s) = 2 sl 8 s = j s 2 s Q e s 0 0 = 8 2 F 2 2 Q F 2 e 2 s = j s At resonance, this becomes M = 8 2 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

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 2 D 2 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

Tank input impedance Operation below resonance: tank input impedance Z i is dominated by tank capacitor. Z i C L 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

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 2 D 2 Q 4 D 4 v s i s T s 2 t t T s 2 t Conduction sequence: Q D Q 2 D 2 Conducting devices: Hard turn-on of Q, Q 4 Q D Q 2 D 2 Q 4 D 4 Q 3 D 3 Soft turn-off of Q, Q 4 Hard turn-on of Q 2, Q 3 Soft turn-off of Q 2, Q 3 Q is turned off during D conduction interval, without loss Fundamentals of Power Electronics 59 Chapter 9: Resonant Conversion

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 T s T s 2 t Q D 2 Q 2 D 2 Q 4 D 4 Q 3 Soft turn-off of Q, Q 4 D 3 t Q 2 D 2 Q 4 D 4 i s Q turns on while D 2 is conducting. Stored charge of D 2 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

9.4.2 Operation of the full bridge above 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 2 D 2 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

Tank input impedance Operation above resonance: tank input impedance Z i is dominated by tank inductor. Z i C L 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 62 Chapter 9: Resonant Conversion

Switch network waveforms, above resonance Zero-voltage switching Q i Q v ds D Q 3 D 3 L C v s Q 2 D 2 Q 4 D 4 i s Conduction sequence: D Q D 2 Q 2 Q is turned on during D conduction interval, without loss Fundamentals of Power Electronics 63 Chapter 9: Resonant Conversion

ZVS turn-off transition: hard switching? Q i Q v ds D Q 3 D 3 L C v s Q 2 D 2 Q 4 D 4 i s When Q turns off, D 2 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

Soft switching at the ZVS turn-off transition 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 2. 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. Fundamentals of Power Electronics 65 Chapter 9: Resonant Conversion

9.4 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 2. 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 doesnʼt!) Fundamentals of Power Electronics 2 Chapter 9: Resonant Conversion

Inverter output characteristics Let H be the open-circuit (R ) transfer function: H (s)= v(s) v s (s) R and let Z o0 be the output impedance (with v i short-circuit). Then, v(s)=h (s)v s (s) R R Z o0 (s) The output voltage magnitude is: v(j s ) 2 = H (j s ) 2 v s (j s ) 2 Z o0(j s ) 2 R 2 This result can be rearranged to obtain v(j s ) 2 i(j s ) 2 Z o0 (j s ) 2 = H (j s ) 2 v s (j s ) 2 Hence, at a given frequency, the output characteristic (i.e., the relationship between v and i ) of any resonant inverter of this class is elliptical. with R = v(j s) i(j s ) Fundamentals of Power Electronics 69 Chapter 9: Resonant Conversion

Inverter output characteristics General resonant inverter output characteristics are elliptical, of the form v(j s ) 2 V oc 2 i(j s) 2 2 = I sc i I sc = H v s Z o0 I sc 2 Inverter output characteristic Matched load R = Z o0 with V oc = H (j s ) v s (j s ) I sc = H (j s ) v s (j s ) Z o0 (j s ) = V oc Z o0 (j s ) V oc 2 V oc = H v s v 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

Matching ellipse to application requirements Electronic ballast Electrosurgical generator i o inverter characteristic i o 2A 50 inverter characteristic 400W lamp characteristic matched load v o 2kV v o Fundamentals of Power Electronics 6 Chapter 9: Resonant Conversion

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

Z i0 and Z i for 3 common inverters Series L C s C s Z i L Z i0 (s) =sl sc s Z i Z o Z i0 Z i (s) = f Parallel L C p L 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 C s C s C p L 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 9 Chapter 9: Resonant Conversion

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 2 = 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 R2 2 Z o0 2 2 Z i = Z i Z i* = Z i0 Fundamentals of Power Electronics 8 Chapter 9: Resonant Conversion R 2 Z o 2

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. 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 20 Chapter 9: Resonant Conversion

Proof of Theorem Previously shown: Z i 2 = Z i0 2 R Z o0 2 R Z o 2 Differentiate: 2 d Z i dr =2 Z i0 2 Z o0 2 Z o 2 R R 2 Z o 2 2 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 2 Chapter 9: Resonant Conversion

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. Itʼs not necessary to draw the entire plot: just construct Z i0 and Z i. Z i f C s C s C p increasing R f f m increasing R L f Fundamentals of Power Electronics 22 Chapter 9: Resonant Conversion

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. Z i C s C p C s f 0 f LCC example L 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 i0 L C s f m L Z i f 0 = 2 LC s f = 2 LC s C p f m = 2 LC s 2C p C s f Z i C p Z i0 C p Fundamentals of Power Electronics 23 Chapter 9: Resonant Conversion

Discussion -series and parallel Series L C s C s Z i L 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 C s C p Z i0 C s C p L Z i f L 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 24 Chapter 9: Resonant Conversion

A Theorem relating the ZVS/ZCS boundary to load resistance R Theorem 2: 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 25 Chapter 9: Resonant Conversion

Proof of Theorem 2 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 2 R crit 2 Z o 2 R crit R crit = Z o0 Z i Z i0 Fundamentals of Power Electronics 26 Chapter 9: Resonant Conversion

Discussion Theorem 2 R crit = Z o0 Z i Z i0 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 27 Chapter 9: Resonant Conversion

LCC example For f > f, ZVS occurs for all R. For f < f 0, ZCS occurs for all R. 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 C s C p 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 L Z i f R crit = Z o0 Z i Z i0 Fundamentals of Power Electronics 28 Chapter 9: Resonant Conversion

LCC example, continued R Z i 90 60 R = 0 ZCS 30 increasing R R crit 0 Z o0 ZVS -30-60 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 29 Chapter 9: Resonant Conversion

A popular subclass of tank circuits Tank network jx s jx p Tank Series branch reactance X s Shunt branch reactance X p Series L C Parallel L C LCC L C s C p LLC L s C L p Key equations Z i0 = jx s, X p H ( )= X p X s Z o0 ( )= jx sx p = jx X p X s H ( ) s R crit = Z o0 At f = f m : Z i = jx s X p Z i Z i0 = X p X s = 2 X p X s X s X p 2 2 M a J = b M = a Q e 2 b a = H ( ) = b = H ( ) R o Z o0 ( ) 2 X p X p X s = R o X s M = V out, J = I outr o, Q V in V e = R o in R e (output characteristic) (control characteristic)

Important frequencies of popular tank circuits Tank 0 m Series LC Parallel LC 2LC LCC LC s LC s C p LC s 2C p LLC L s C L s L p C L s L p 2 C

The LLC tank network DC-DC converter circuit Tank network input impedances Z i0 = sl s /sc Z i = s(l s L p ) /sc

LLC Tank input impedance plots Z i0, Z i Good characteristics obtained for f s > f m Switch current varies directly with load current ZVS obtained at most operating points including matched load

LLC output characteristic for f m < f s < f 0 R crit < Z o0 so ZVS region includes matched load

LLC: ZVS boundary in the region f 0 < f s < f R crit = R nf o0 n L R o0 = s C n = L p L s F = f s f F 2 n F 2 ZVS for R > R crit n = 4 ZCS for R < R crit

LLC Conversion Ratio M vs. Switching Frequency F, L p /L s = 4 F = f s f f = 2 (L s L p )C Q s = 0.25 0.35 Q s = 0 H Q s = R o0 R e R o0 = L s C 0.5 0.75.5 2 3 5 0 25 f s = f f s = f m f s = f 0

LLC Output plane characteristic, L p /L s = 4 f s < f F = n f < f s < f 0 F = Increasing F F = F = 0.5 Increasing F F = f s f f = 2 (L s L p )C f 0 = 2 L s C f 0 = n f Increasing F F = n f 0 < f s n = L p L s M = V V g J = IR o0 V g f m = f n n 2 F = 0

Summary. Simple models approximate the tank waveforms by their fundamental sinusoidal components, and that facilitate resonant converter design using conventional frequency response methods 2. Several theorems show how to employ frequency response plots to determine important properties: The inverter or dc-dc converter elliptical output characteristic How the transistor currents vary with load, and how to shape the tank impedances to improve light-load efficiency Dependence of the ZVS/ZCS boundary on load 3. The LLC tank network exhibits the following advantages: Buck-boost conversion ratio Wide range of operating points exhibiting zero-voltage switching, i.e., for f 0 < f s or for f < f s < f 0 with R e > R crit Transistor currents vary directly with load current for f m < f s

Selected References. R. L. Steigerwald, A Comparison of Half-Bridge Resonant Converter Topologies, IEEE APEC, 987 Record, pp. 35-44. 2. S. Johnson, Steady-State Analysis and Design of the Parallel Resonant Converter, M.S. Thesis, Univ. of Colo., 986. 3. R. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2 nd ed., Chapter 9, Springer, 200. 4. B. Yang, F. Lee, A. Zhang, and G. Huang, LLC Resonant Converter for Front-End DC/DC Conversion, IEEE APEC, 2002 Record, pp. 08-2.