R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder
18.2.2 DCM flyback converter v ac i ac EMI filter i g v g Flyback converter n : 1 L D 1 i v C R Q 1 D Operation in DCM: we found in Chapter 11 that the converter input port obeys Ohm s law with effective resistance = 2n 2 L/D 2 T s. Hence, simply connect input port to AC line. Fundamentals of Power Electronics 20 Chapter 18: PWM Rectifiers
Averaged large-signal model EMI filter i ac i g T Averaged model i T v ac v g 2n 2 L D 2 T s p T v T C R D Under steady-state conditions, operate with constant D Adjust D to control average power drawn from AC line Fundamentals of Power Electronics 21 Chapter 18: PWM Rectifiers
Converter design Select L small enough that DCM operation occurs throughout AC line cycle. DCM occurs provided that d 3 > 0, or d 2 <1D But d 2 =D v g nv Substitute and solve for D: D < 1 1 v g nv Converter operates in DCM in every switching period where above inequality is satisfied. i 1 Area q 1 i pk i 1 Ts t d 1 T s d 2 T s d 3 T s T s To obtain DCM at all points on input AC sinusoid: worst case is at maximum v g = V M : D < 1 1 V M nv Fundamentals of Power Electronics 22 Chapter 18: PWM Rectifiers
Choice of L to obtain DCM everywhere along AC sinusoid We have: D < 1 1 V M nv with V rms V ac,rms = R Re Substitute expression for to obtain Solve for L: D = 2nV V M L < L crit = L RTs RT s 4 1 nv V M 2 Worst-case design For variations in load resistance and ac input voltage, the worst case occurs at maximum load power and minimum ac input voltage. The inductance should be chosen as follows: L < L crit-min = R min T s 4 1 nv V M-min 2 Fundamentals of Power Electronics 23 Chapter 18: PWM Rectifiers
18.3 Control of the Current Waveform 18.3.1 Average current control Feedforward 18.3.2 Current programmed control 18.3.3 Critical conduction mode and hysteretic control 18.3.4 Nonlinear carrier control Fundamentals of Power Electronics 24 Chapter 18: PWM Rectifiers
18.3.1 Average current control i g Boost example Low frequency (average) component of input current is controlled to follow input voltage v g L Gate driver v Pulse width modulator Current reference v r v a R s i g Ts G c (s) Compensator Fundamentals of Power Electronics 25 Chapter 18: PWM Rectifiers
Block diagram Current reference derived from input voltage waveform v ac Multiplier allows control of emulated resistance value Compensation of current loop i ac v control Multiplier X i g v g v g R s i g v a Controller Boost converter L v r = k x v g v control v err Q 1 D 1 PWM G c (s) Compensator i C v R Fundamentals of Power Electronics 26 Chapter 18: PWM Rectifiers
The emulated resistance Current sensor has gain R s : i g i ac Boost converter L D 1 i v a =R s i g Ts v ac v g Q 1 C v R If loop is well designed, then: v a v r Multiplier: v r =k x v g v control Hence the emulated resistance is: = v g i g = v r k x v control v control Multiplier X v g R s i g v a v r = k x v g v control Controller v err PWM G c (s) Compensator which can be simplified to v a R s v control = R s k x v control Fundamentals of Power Electronics 27 Chapter 18: PWM Rectifiers
System model using LFR Average current control i g Ts Ideal rectifier (LFR) i Ts i ac p Ts v ac v g Ts C v Ts R = R s k x v control v control Fundamentals of Power Electronics 28 Chapter 18: PWM Rectifiers
Use of multiplier to control average power As discussed in Chapter 17, an output voltage feedback loop adjusts the emulated resistance such that the rectifier power equals the dc load power: P av = V 2 g,rms = P load v g v g i g Gate driver Pulse width modulator C v An analog multiplier introduces the dependence of on v. x y Multiplier k x xy v ref1 v control v a v err G c (s) Compensator G cv (s) v v ref2 Voltage reference Fundamentals of Power Electronics 29 Chapter 18: PWM Rectifiers
Feedforward Feedforward is sometimes used to cancel out disturbances in the input voltage v g. v g i g v To maintain a given power throughput P av, the reference voltage v ref1 should be Gate driver v ref 1 = P avv g R s 2 V g,rms v g Pulse width modulator Peak detector VM x z y multiplier k v xy z 2 v a v ref1 v control G c (s) Compensator G cv (s) v ref2 Voltage reference Fundamentals of Power Electronics 30 Chapter 18: PWM Rectifiers
Feedforward, continued Controller with feedforward produces the following reference: i g v ref 1 = k vv control v g V M 2 The average power is then given by P av = k vv control 2R s v g v g Gate driver Pulse width modulator v Peak detector VM x z y multiplier k v xy z 2 v ref1 v control v a G c (s) Compensator G cv (s) v ref2 Voltage reference Fundamentals of Power Electronics 31 Chapter 18: PWM Rectifiers
Modeling the inner wide-bandwidth average current controller Averaged (but not linearized) boost converter model: L i Ts i 1 Ts v g Ts v 1 Ts i 2 Ts v 2 Ts C R v Ts Averaged switch network In Chapter 7, we perturbed and linearized using the assumptions v g Ts = V g v g d=d d d'=d' d i Ts = i 1 Ts = I i v Ts = v 2 Ts = V v v 1 Ts = V 1 v 1 i 2 Ts = I 2 i 2 Problem: variations in v g, i 1, and d are not small. So we are faced with the design of a control system that exhibits significant nonlinear time-varying behavior. Fundamentals of Power Electronics 32 Chapter 18: PWM Rectifiers
Linearizing the equations of the boost rectifier When the rectifier operates near steady-state, it is true that v Ts = V v with v << V In the special case of the boost rectifier, this is sufficient to linearize the equations of the average current controller. The boost converter average inductor voltage is L substitute: L di g Ts dt di g Ts dt = v g Ts d'v d'v = v g Ts d'v d'v Fundamentals of Power Electronics 33 Chapter 18: PWM Rectifiers
Linearized boost rectifier model L di g Ts dt = v g Ts d'v d'v The nonlinear term is much smaller than the linear ac term. Hence, it can be discarded to obtain L Equivalent circuit: di g Ts dt = v g Ts d'v L i g Ts i g (s) d(s) = V sl v g Ts d'v Fundamentals of Power Electronics 34 Chapter 18: PWM Rectifiers
The quasi-static approximation The above approach is not sufficient to linearize the equations needed to design the rectifier averaged current controllers of buck-boost, Cuk, SEPIC, and other converter topologies. These are truly nonlinear timevarying systems. An approximate approach that is sometimes used in these cases: the quasi-static approximation Assume that the ac line variations are much slower than the converter dynamics, so that the rectifier always operates near equilibrium. The quiescent operating point changes slowly along the input sinusoid, and we can find the slowly-varying equilibrium duty ratio as in Section 18.2.1. The converter small-signal transfer functions derived in Chapters 7 and 8 are evaluated, using the time-varying operating point. The poles, zeroes, and gains vary slowly as the operating point varies. An average current controller is designed, that has a positive phase margin at each operating point. Fundamentals of Power Electronics 35 Chapter 18: PWM Rectifiers
Quasi-static approximation: discussion In the literature, several authors have reported success using this method Should be valid provided that the converter dynamics are suffieiently fast, such that the converter always operates near the assumed operating points No good condition on system parameters, which can justify the approximation, is presently known for the basic converter topologies It is well-understood in the field of control systems that, when the converter dynamics are not sufficiently fast, then the quasi-static approximation yields neither necessary nor sufficient conditions for stability. Such behavior can be observed in rectifier systems. Worstcase analysis to prove stability should employ simulations. Fundamentals of Power Electronics 36 Chapter 18: PWM Rectifiers
18.3.2 Current programmed control Current programmed control is a natural approach to obtain input resistor emulation: Peak transistor current is programmed to follow input voltage. Peak transistor current differs from average inductor current, because of inductor current ripple and artificial ramp. This leads to significant input current waveform distortion. v control i g v g v g Multiplier X i s i a Boost converter L D 1 Q 1 m a Clock 0 i c Comparator = k x v g v control Current-programmed controller i 2 T s C S Q R Latch v R Fundamentals of Power Electronics 37 Chapter 18: PWM Rectifiers
CPM boost converter: Static input characteristics i g Ts = v g Li 2 c f s V v g v g m a L i c 1 v g V m a v g L T s in DCM in CCM 1 Static input characteristics of CPM boost, with minimum slope compensation: Mode boundary: CCM occurs when or, i g Ts > T sv 2L i c > T sv L It is desired that v g V m a L V 1 v g V v g V Minimum slope compensation: 1 v g V i c = v 0.2 g 0 m a = V 2L Fundamentals of Power Electronics 38 Chapter 18: PWM Rectifiers R j g = i g base Ts V 0.8 0.6 0.4 = 0.1R base = 0.2R base = 0.33R base DCM = 0.5R base CCM m a = V 2L R base = 2L T s = R base = 2R base = 4R base = 0.0 0.2 0.4 0.6 0.8 1.0 v g V 10R base
Input current waveforms with current mode control i g Peak i g 1.0 0.8 0.6 0.4 0.2 0.0 Sinusoid = 0.1R base Re = 0.33Rbase = 2R base m a = V 2L R base = 2L T s ωt Substantial distortion can occur Can meet harmonic limits if the range of operating points is not too large Difficult to meet harmonic limits in a universal input supply Fundamentals of Power Electronics 39 Chapter 18: PWM Rectifiers
18.3.3 Critical conduction mode and hysteretic control Variable switching frequency schemes Hysteretic control Critical conduction mode (boundary between CCM and DCM) i g i g Hysteretic control ωt Critical conduction mode t on ωt Fundamentals of Power Electronics 40 Chapter 18: PWM Rectifiers
An implementation of critical conduction control EMI filter i g Boost converter i i ac L D 1 v ac v g Q 1 C v R v control v g Multiplier X R s va Zero current i g detector S Q v r = k x v g v control Comparator Controller R Latch Fundamentals of Power Electronics 41 Chapter 18: PWM Rectifiers
Pros and cons of critical conduction control Simple, low-cost controller ICs Low-frequency harmonics are very small, with constant transistor on-time (for boost converter) Small inductor Increased peak current Increased conduction loss, reduced switching loss Requires larger input filter Variable switching frequency smears out the current EMI spectrum Cannot synchronize converter switching frequencies Fundamentals of Power Electronics 42 Chapter 18: PWM Rectifiers
Analysis i g Transistor is on for fixed time t on Transistor off-time ends when inductor current reaches zero Ratio of v g to i g is = 2L t on t on ωt On time, as a function of load power and line voltage: t on = 4LP V M 2 Inductor volt-second balance: v g t on v g V t off =0 Solve for t off : t off = t on v g V v g Fundamentals of Power Electronics 43 Chapter 18: PWM Rectifiers
Switching frequency variations Solve for how the controller varies the switching frequency over the ac line period: 1 T s = t off t on T s = 4LP V M 2 1 v g V For sinusoidal line voltage variations, the switching frequency will therefore vary as follows: f s = 1 = V 2 M T s 4LP 1V M V sin (ωt) Minimum and maximum limits on switching frequency: max f s = V 2 M 4LP min f s = V 2 M 4LP 1V M V These equations can be used to select the value of the inductance L. Fundamentals of Power Electronics 44 Chapter 18: PWM Rectifiers
18.3.4 Nonlinear carrier control Can attain simple control of input current waveform without sensing the ac input voltage, and with operation in continuous conduction mode The integral of the sensed switch current (charge) is compared to a nonlinear carrier waveform (i.e., a nonlinear ramp), on a cycle-bycycle basis Carrier waveform depends on converter topology Very low harmonics in CCM. Waveform distortion occurs in DCM. Peak current mode control is also possible, with a different carrier Fundamentals of Power Electronics 45 Chapter 18: PWM Rectifiers
Controller block diagram Nonlinear carrier charge control of boost converter Boost converter i g L D 1 i s v g n : 1 i s Q 1 C v R C i i s /n v i Comparator Latch R Q v c v i v c Nonlinear carrier generator 0 T s S Q 0 v i dt s T s v control Clock Nonlinear-carrier charge controller Fundamentals of Power Electronics 46 Chapter 18: PWM Rectifiers
Derivation of NLC approach The average switch current is i s Ts = 1 T s t t T s i s (τ)dτ We could make the controller regulate the average switch current by Integrating the monitored switch current Resetting the integrator to zero at the beginning of each switching period Turning off the transistor when the integrator reaches a reference value In the controller diagram, the integrator follows this equation: i.e., v i = 1 C i 0 dt s v i (dt s )= i s T s nc i f s i s (τ) n dτ for 0 < t < dt s for interval 0 < t < T s Fundamentals of Power Electronics 47 Chapter 18: PWM Rectifiers
How to control the average switch current Input resistor emulation: i g Ts = v g Ts (v control ) Relate average switch current to input current (assuming CCM): i s Ts = d i g Ts Relate input voltage to output voltage (assuming CCM): v g Ts = d v Ts Substitute above equations to find how average switch current should be controlled: i s Ts = d 1d v Ts (v control ) Fundamentals of Power Electronics 48 Chapter 18: PWM Rectifiers
Implementation using nonlinear carrier Desired control, from previous slide: i s Ts = d 1d v Ts (v control ) Generate carrier waveform as follows (replace d by t/t s ): v c =v control t Ts 1 t T s for 0 t T s v c (t T s )=v c The controller switches the transistor off when the integrator voltage equals the carrier waveform. This leads to: v i (dt s )=v c (dt s )=v control d 1d i s Ts nc i f s = v control d 1d (v control )=d 1d v Ts i s Ts = v Ts nc i f s v control Fundamentals of Power Electronics 49 Chapter 18: PWM Rectifiers
Generating the parabolic carrier v control Removal of dc component v c Integrator with reset Integrator with reset Clock (one approach, suitable for discrete circuitry) Note that no separate multiplier circuit is needed Fundamentals of Power Electronics 50 Chapter 18: PWM Rectifiers
18.4 Single-phase converter systems containing ideal rectifiers It is usually desired that the output voltage v be regulated with high accuracy, using a wide-bandwidth feedback loop For a given constant load characteristic, the instantaneous load current and power are then also constant: p load =vi=vi The instantaneous input power of a single-phase ideal rectifier is not constant: with p ac =v g i g v g =V M sin (ωt) i g = v g so p ac = V 2 M sin R 2 ωt = V 2 M e 2 1 cos 2ωt Fundamentals of Power Electronics 51 Chapter 18: PWM Rectifiers
Power flow in single-phase ideal rectifier system Ideal rectifier is lossless, and contains no internal energy storage. Hence instantaneous input and output powers must be equal An energy storage element must be added Capacitor energy storage: instantaneous power flowing into capacitor is equal to difference between input and output powers: p C = de C dt = d 1 2 Cv 2 C dt = p ac p load Energy storage capacitor voltage must be allowed to vary, in accordance with this equation Fundamentals of Power Electronics 52 Chapter 18: PWM Rectifiers
Capacitor energy storage in 1ø system p ac P load v c = d 1 2 Cv 2 C dt = p ac p load Fundamentals of Power Electronics 53 Chapter 18: PWM Rectifiers t
Single-phase system with internal energy storage i g Ideal rectifier (LFR) i 2 p load = VI = P load v ac i ac v g p ac Ts C v C Dcdc converter v i load Energy storage capacitor Energy storage capacitor voltage v C must be independent of input and output voltage waveforms, so that it can vary according to = d 1 2 Cv 2 C dt = p ac p load This system is capable of Wide-bandwidth control of output voltage Wide-bandwidth control of input current waveform Internal independent energy storage Fundamentals of Power Electronics 54 Chapter 18: PWM Rectifiers
Hold up time Internal energy storage allows the system to function in other situations where the instantaneous input and output powers differ. A common example: continue to supply load power in spite of failure of ac line for short periods of time. Hold up time: the duration which the dc output voltage v remains regulated after v ac has become zero A typical hold-up time requirement: supply load for one complete missing ac line cycle, or 20 msec in a 50 Hz system During the hold-up time, the load power is supplied entirely by the energy storage capacitor Fundamentals of Power Electronics 55 Chapter 18: PWM Rectifiers
Energy storage element Instead of a capacitor, and inductor or higher-order LC network could store the necessary energy. But, inductors are not good energy-storage elements Example 100 V 100 µf capacitor 100 A 100 µh inductor each store 1 Joule of energy But the capacitor is considerably smaller, lighter, and less expensive So a single big capacitor is the best solution Fundamentals of Power Electronics 56 Chapter 18: PWM Rectifiers
Inrush current A problem caused by the large energy storage capacitor: the large inrush current observed during system startup, necessary to charge the capacitor to its equilibrium value. Boost converter is not capable of controlling this inrush current. Even with d = 0, a large current flows through the boost converter diode to the capacitor, as long as v < v g. Additional circuitry is needed to limit the magnitude of this inrush current. Converters having buck-boost characteristics are capable of controlling the inrush current. Unfortunately, these converters exhibit higher transistor stresses. Fundamentals of Power Electronics 57 Chapter 18: PWM Rectifiers
Universal input The capability to operate from the ac line voltages and frequencies found everywhere in the world: 50Hz and 60Hz Nominal rms line voltages of 100V to 260V: 100V, 110V, 115V, 120V, 132V, 200V, 220V, 230V, 240V, 260V Regardless of the input voltage and frequency, the near-ideal rectifier produces a constant nominal dc output voltage. With a boost converter, this voltage is 380 or 400V. Fundamentals of Power Electronics 58 Chapter 18: PWM Rectifiers
Low-frequency model of dc-dc converter Dc-dc converter produces well-regulated dc load voltage V. Load therefore draws constant current I. Load power is therefore the constant value P load = VI. To the extent that dc-dc converter losses can be neglected, then dc-dc converter input power is P load, regardless of capacitor voltage v c. Dc-dc converter input port behaves as a power sink. A low frequency converter model is i 2 p load = VI = P load i C v C P load V v load Energy storage capacitor Dc-dc converter Fundamentals of Power Electronics 59 Chapter 18: PWM Rectifiers
Low-frequency energy storage process, 1ø system A complete low-frequency system model: i g i 2 i ac p ac Ts p load = VI = P load i v ac v g C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dc-dc converter Difference between rectifier output power and dc-dc converter input power flows into capacitor In equilibrium, average rectifier and load powers must be equal But the system contains no mechanism to accomplish this An additional feeback loop is necessary, to adjust such that the rectifier average power is equal to the load power Fundamentals of Power Electronics 60 Chapter 18: PWM Rectifiers
Obtaining average power balance i g i 2 p load = VI = P load i ac p ac Ts i v ac v g C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dc-dc converter If the load power exceeds the average rectifier power, then there is a net discharge in capacitor energy and voltage over one ac line cycle. There is a net increase in capacitor charge when the reverse is true. This suggests that rectifier and load powers can be balanced by regulating the energy storage capacitor voltage. Fundamentals of Power Electronics 61 Chapter 18: PWM Rectifiers
A complete 1ø system containing three feedback loops v ac i g i ac v g Boost converter L D 1 Q 1 i 2 v C C DCDC Converter i Load v v control Multiplier X v g R s i g v a PWM d v ref1 = k x v g v control v err G c (s) Compensator Wide-bandwidth input current controller v Compensator and modulator v ref3 Wide-bandwidth output voltage controller v C Compensator v ref2 Low-bandwidth energy-storage capacitor voltage controller Fundamentals of Power Electronics 62 Chapter 18: PWM Rectifiers
Bandwidth of capacitor voltage loop The energy-storage-capacitor voltage feedback loop causes the dc component of v c to be equal to some reference value Average rectifier power is controlled by variation of. must not vary too quickly; otherwise, ac line current harmonics are generated Extreme limit: loop has infinite bandwidth, and v c is perfectly regulated to be equal to a constant reference value Energy storage capacitor voltage then does not change, and this capacitor does not store or release energy Instantaneous load and ac line powers are then equal Input current becomes i ac = p ac v ac = p load v ac = P load V M sin ωt Fundamentals of Power Electronics 63 Chapter 18: PWM Rectifiers
Input current waveform, extreme limit i ac = p ac v ac = p load v ac = P load V M sin ωt THD Power factor 0 v ac i ac t So bandwidth of capacitor voltage loop must be limited, and THD increases rapidly with increasing bandwidth Fundamentals of Power Electronics 64 Chapter 18: PWM Rectifiers
18.4.2 Modeling the outer low-bandwidth control system This loop maintains power balance, stabilizing the rectifier output voltage against variations in load power, ac line voltage, and component values The loop must be slow, to avoid introducing variations in at the harmonics of the ac line frequency Objective of our modeling efforts: low-frequency small-signal model that predicts transfer functions at frequencies below the ac line frequency Fundamentals of Power Electronics 65 Chapter 18: PWM Rectifiers
Large signal model averaged over switching period T s Ideal rectifier (LFR) i g Ts p Ts i 2 Ts v g Ts (v control ) C v Ts Load ac input dc output v control Ideal rectifier model, assuming that inner wide-bandwidth loop operates ideally High-frequency switching harmonics are removed via averaging Ac line-frequency harmonics are included in model Nonlinear and time-varying Fundamentals of Power Electronics 66 Chapter 18: PWM Rectifiers
Predictions of large-signal model If the input voltage is Ideal rectifier (LFR) i g Ts p Ts i 2 Ts v g = 2v g,rms sin ωt v g Ts (v control ) C v Ts Load Then the instantaneous power is: ac input v control dc output p Ts = 2 v g Ts (v control ) = 2 v g,rms (v control ) 1 cos 2ωt which contains a constant term plus a secondharmonic term Fundamentals of Power Electronics 67 Chapter 18: PWM Rectifiers
Separation of power source into its constant and time-varying components i 2 Ts V 2 g,rms cos R 2 2ωt e 2 V g,rms C v Ts Load Rectifier output port The second-harmonic variation in power leads to second-harmonic variations in the output voltage and current Fundamentals of Power Electronics 68 Chapter 18: PWM Rectifiers
Removal of even harmonics via averaging v v Ts v T2L t T 2L = 1 2 2π ω = π ω Fundamentals of Power Electronics 69 Chapter 18: PWM Rectifiers
Resulting averaged model i 2 T2L 2 V g,rms C v T2L Load Rectifier output port Time invariant model Power source is nonlinear Fundamentals of Power Electronics 70 Chapter 18: PWM Rectifiers
Perturbation and linearization The averaged model predicts that the rectifier output current is i 2 T2L = p T 2L v T2L = 2 v g,rms (v control ) v T2L = f v g,rms, v T2L, v control ) Let with v T2L = V v i 2 T2L = I 2 i 2 v g,rms = V g,rms v g,rms v control =V control v control V >> v I 2 >> i 2 V g,rms >> v g,rms V control >> v control Fundamentals of Power Electronics 71 Chapter 18: PWM Rectifiers
Linearized result where I 2 i 2 =g 2 v g,rms j 2 v v control r 2 g 2 = df v g,rms, V, V control ) dv g,rms v g,rms = V g,rms = 2 (V control ) V g,rms V 1 r 2 = df V g,rms, v T2L, V control ) = I 2 dv V T2L v = V T2L j 2 = df V g,rms, V, v control ) dv control v control = V control = 2 V g,rms VR 2 e (V control ) d (v control ) dv control v control = V control Fundamentals of Power Electronics 72 Chapter 18: PWM Rectifiers
Small-signal equivalent circuit i 2 g 2 v g,rms j 2 v control C v R r 2 Rectifier output port Predicted transfer functions Control-to-output v(s) v control (s) = j 2 R r 2 1 1sC R r 2 Line-to-output v(s) v g,rms (s) = g 2 R r 2 1 1sC R r 2 Fundamentals of Power Electronics 73 Chapter 18: PWM Rectifiers
Model parameters Table 18.1 Small-signal model parameters for several types of rectifier control schemes Controller type g 2 j 2 r 2 Average current control with feedforward, Fig. 18.14 Current-programmed control, Fig. 18.16 Nonlinear-carrier charge control of boost rectifier, Fig. 18.21 Boost with critical conduction mode control, Fig. 18.20 0 2P av VV g,rms 2P av VV g,rms 2P av VV g,rms P av VV control V 2 P av P av VV control V 2 P av P av VV control V 2 2P av P av VV control V 2 P av DCM buck-boost, flyback, SEPIC, or Cuk converters 2P av VV g,rms 2P av VD V 2 P av Fundamentals of Power Electronics 74 Chapter 18: PWM Rectifiers
Constant power load i g i 2 p load = VI = P load i ac p ac Ts i v ac v g C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dc-dc converter Rectifier and dc-dc converter operate with same average power Incremental resistance R of constant power load is negative, and is R = V 2 P av which is equal in magnitude and opposite in polarity to rectifier incremental output resistance r 2 for all controllers except NLC Fundamentals of Power Electronics 75 Chapter 18: PWM Rectifiers
Transfer functions with constant power load When r 2 = R, the parallel combination r 2 R becomes equal to zero. The small-signal transfer functions then reduce to v(s) v control (s) = j 2 sc v(s) v g,rms (s) = g 2 sc Fundamentals of Power Electronics 76 Chapter 18: PWM Rectifiers
18.5 RMS values of rectifier waveforms Doubly-modulated transistor current waveform, boost rectifier: i Q Computation of rms value of this waveform is complex and tedious Approximate here using double integral Generate tables of component rms and average currents for various rectifier converter topologies, and compare t Fundamentals of Power Electronics 77 Chapter 18: PWM Rectifiers
RMS transistor current RMS transistor current is i Q I Qrms = 1 Tac 0 T ac i Q 2 dt Express as sum of integrals over all switching periods contained in one ac line period: t T ac /T s I Qrms = 1 T 1 s i 2 Q dt Tac Ts n =1 nt s (n-1)t s Quantity in parentheses is the value of i Q2, averaged over the n th switching period. Fundamentals of Power Electronics 78 Chapter 18: PWM Rectifiers
Approximation of RMS expression T ac /T s n =1 I Qrms = 1 T 1 s i 2 Q dt Tac Ts nt s (n-1)t s When T s << T ac, then the summation can be approximated by an integral, which leads to the double-average: I Qrms 1 Tac T ac /T s nt s lim Ts T 1 0 s i 2 Q (τ)dτ Ts n=1 (n-1)t s = 1 1 i 2 Q (τ)dτdt Tac Ts 0 T ac t tt s = i Q 2 Ts T ac Fundamentals of Power Electronics 79 Chapter 18: PWM Rectifiers
18.5.1 Boost rectifier example For the boost converter, the transistor current i Q is equal to the input current when the transistor conducts, and is zero when the transistor is off. The average over one switching period of i Q2 is therefore If the input voltage is i Q 2 tt s t = 1 i T s T 2 Q dt s = di 2 ac v ac =V M sin ωt then the input current will be given by i ac = V M Re and the duty cycle will ideally be sin ωt V v ac = 1 1d (this neglects converter dynamics) Fundamentals of Power Electronics 80 Chapter 18: PWM Rectifiers
Boost rectifier example Duty cycle is therefore d=1 V M V Evaluate the first integral: i Q 2 = V 2 M T 2 s sin ωt Now plug this into the RMS formula: I Qrms = 1 Tac i Q 2 1 V M V sin ωt sin 2 ωt 0 T ac T s dt = 1 Tac 0 T ac 2 V M 2 1 V M V sin ωt sin 2 ωt dt I Qrms = 2 2 V M Tac 2 sin 2 ωt V M sin 3 V 0 T ac /2 ωt dt Fundamentals of Power Electronics 81 Chapter 18: PWM Rectifiers
Integration of powers of sin θ over complete half-cycle n 1 π 0 π sin n (θ)dθ 1 π 0 π sin n (θ)dθ = 2 2 4 6 (n 1) π 1 3 5 n 1 3 5 (n 1) 2 4 6 n if n is odd if n is even 1 2 π 2 1 2 3 4 3π 4 3 8 5 6 16 15π 15 48 Fundamentals of Power Electronics 82 Chapter 18: PWM Rectifiers
Boost example: transistor RMS current I Qrms = V M 2 1 8 3π V M V = I ac rms 1 8 3π V M V Transistor RMS current is minimized by choosing V as small as possible: V = V M. This leads to I Qrms = 0.39I ac rms When the dc output voltage is not too much greater than the peak ac input voltage, the boost rectifier exhibits very low transistor current. Efficiency of the boost rectifier is then quite high, and 95% is typical in a 1kW application. Fundamentals of Power Electronics 83 Chapter 18: PWM Rectifiers
Table of rectifier current stresses for various topologies Table 18. 3 Summary of rectifier current stresses for several converter topologies rms Average Peak CCM boost Transistor I ac rms 1 8 3π V M V I ac rms 2 2 π 1 π 8 V M I ac rms 2 V Diode I dc 16 3π V V M I dc 2 I dc V VM Inductor I ac rms I ac rms 2 2 π I ac rms 2 CCM flyback, with n:1 isolation transformer and input filter Transistor, xfmr primary L 1 I ac rms 1 8 3π V M nv I ac rms I ac rms 2 2 π I ac rms 2 2 π I ac rms 2 1 V n I ac rms 2 C 1 I 8 V M ac rms 3π nv 0 I ac rms 2 max 1, V M nv Diode, xfmr secondary I dc 3 2 16 3π nv V M I dc 2I dc 1 nv V M Fundamentals of Power Electronics 84 Chapter 18: PWM Rectifiers
Table of rectifier current stresses continued CCM SEPIC, nonisolated Transistor L 1 C 1 I ac rms 1 8 3π I ac rms V M I 2 2 ac rms V π I ac rms 2 1 V M V I ac rms 2 2 π I ac rms 8 3π V M V 0 I ac rms 2 I ac rms max 1, V M V L 2 Diode I ac rms V M V 3 2 I ac rms 2 I dc 3 2 16 3π V V M I dc V M V I ac rms V M V 2 2I dc 1 V V M CCM SEPIC, with n:1 isolation transformer transistor L 1 C 1, xfmr primary Diode, xfmr secondary I ac rms 1 8 3π I ac rms I 8 V M ac rms 3π nv V M nv I ac rms 2 2 π I ac rms 2 1 V M nv I ac rms 2 2 π I dc 3 2 16 3π nv V M I dc with, in all cases, I ac rms = I dc 2 V V M, ac input voltage = V M sin(ωt) dc output voltage = V I ac rms 2 Fundamentals of Power Electronics 85 Chapter 18: PWM Rectifiers 0 I ac rms 2I dc 2 max 1, n 1 nv V M
Comparison of rectifier topologies Boost converter Lowest transistor rms current, highest efficiency Isolated topologies are possible, with higher transistor stress No limiting of inrush current Output voltage must be greater than peak input voltage Buck-boost, SEPIC, and Cuk converters Higher transistor rms current, lower efficiency Isolated topologies are possible, without increased transistor stress Inrush current limiting is possible Output voltage can be greater than or less than peak input voltage Fundamentals of Power Electronics 86 Chapter 18: PWM Rectifiers
Comparison of rectifier topologies 1kW, 240Vrms example. Output voltage: 380Vdc. Input current: 4.2Arms Converter Transistor rms current Transistor voltage Diode rms current Transistor rms current, 120V Diode rms current, 120V Boost 2 A 380 V 3.6 A 6.6 A 5.1 A Nonisolated SEPIC Isolated SEPIC 5.5 A 719 V 4.85 A 9.8 A 6.1 A 5.5 A 719 V 36.4 A 11.4 A 42.5 A Isolated SEPIC example has 4:1 turns ratio, with 42V 23.8A dc load Fundamentals of Power Electronics 87 Chapter 18: PWM Rectifiers
18.6 Modeling losses and efficiency in CCM high-quality rectifiers Objective: extend procedure of Chapter 3, to predict the output voltage, duty cycle variations, and efficiency, of PWM CCM low harmonic rectifiers. Approach: Use the models developed in Chapter 3. Integrate over one ac line cycle to determine steady-state waveforms and average power. Boost example i L D g R L 1 i i g R L DR on D' : 1 V F i v g Q 1 C R v v g R v Dc-dc boost converter circuit Averaged dc model Fundamentals of Power Electronics 88 Chapter 18: PWM Rectifiers
Modeling the ac-dc boost rectifier Boost rectifier circuit v ac i g i ac v g R L L Q 1 D 1 i d C i v R controller Averaged model i g R L d R on d' : 1 V F i d i = I v g C R (large) v = V Fundamentals of Power Electronics 89 Chapter 18: PWM Rectifiers
Boost rectifier waveforms v g 300 i g 10 Typical waveforms 200 100 v g i g 8 6 4 2 (low frequency components) i g = v g 0 0 30 60 90 120 150 180 0 d 1 6 0.8 5 i d 0.6 0.4 4 3 2 i = I 0.2 1 0 0 30 60 90 120 150 180 0 0 30 60 90 120 150 180 ωt Fundamentals of Power Electronics 90 Chapter 18: PWM Rectifiers
Example: boost rectifier with MOSFET on-resistance i g d R on d' : 1 i d i = I v g C R (large) v = V Averaged model Inductor dynamics are neglected, a good approximation when the ac line variations are slow compared to the converter natural frequencies Fundamentals of Power Electronics 91 Chapter 18: PWM Rectifiers
18.6.1 Expression for controller duty cycle d Solve input side of model: i g dr on = v g d'v with i g = v g v g =V M sin ωt v g i g d R on d' : 1 i d i = I C R (large) v = V eliminate i g : v g dr on = v g d'v solve for d: d= v v g v v g R on Again, these expressions neglect converter dynamics, and assume that the converter always operates in CCM. Fundamentals of Power Electronics 92 Chapter 18: PWM Rectifiers
18.6.2 Expression for the dc load current Solve output side of model, using charge balance on capacitor C: I = i d T ac i d =d'i g =d' v g v g i g d R on d' : 1 i d i = I C R (large) v = V Butd is: d'= v g 1 R on v v g R on hence i d can be expressed as i d = v g 2 1 R on v v g R on Next, average i d over an ac line period, to find the dc load current I. Fundamentals of Power Electronics 93 Chapter 18: PWM Rectifiers
Dc load current I Now substitute v g = V M sin ωt, and integrate to find i d T ac : I = i d T ac = 2 T ac 0 T ac /2 1 2 V M This can be written in the normalized form v V M R on R on sin 2 ωt sin ωt dt I = 2 T ac 2 V M V 1 R sin 2 ωt on 1asin ωt 0 T ac /2 dt with a = V M V R on Fundamentals of Power Electronics 94 Chapter 18: PWM Rectifiers
Integration By waveform symmetry, we need only integrate from 0 to T ac /4. Also, make the substitution θ = ωt: I = V 2 M V 1 R π/2 on 2 sin 2 θ π 1asin θ dθ 0 This integral is obtained not only in the boost rectifier, but also in the buck-boost and other rectifier topologies. The solution is 4 π 0 π/2 sin 2 θ 1asin θ dθ = F(a)= 2 a 2 π 4 sin1 2a π a 2 cos 1 1a 2 a Result is in closed form a is a measure of the loss resistance relative to a is typically much smaller than unity Fundamentals of Power Electronics 95 Chapter 18: PWM Rectifiers
The integral F(a) 4 π 0 π/2 sin 2 θ 1asin θ dθ Approximation via polynomial: F(a) 1 0.862a 0.78a 2 For a 0.15, this approximate expression is within 0.1% of the exact value. If the a 2 term is omitted, then the accuracy drops to ±2% for a 0.15. The accuracy of F(a) coincides with the accuracy of the rectifier efficiency η. = F(a)= 2 a 2 π F(a) 0.85 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Fundamentals of Power Electronics 96 Chapter 18: PWM Rectifiers 1.15 1.1 1.05 1 0.95 0.9 4 sin1 2a π a 2 cos 1 a 1a 2 a
18.6.3 Solution for converter efficiency η Converter average input power is P in = p in Tac = V M 2 2 Average load power is P out = VI = V V M 2 V 1 R on F(a) 2 with a = V M V R on So the efficiency is η = P out P in = 1 R on F(a) Polynomial approximation: η 1 R on 1 0.862 V M V R on 0.78 V M V R on 2 Fundamentals of Power Electronics 97 Chapter 18: PWM Rectifiers
Boost rectifier efficiency η 1 0.95 R on / = 0.05 η = P out P in = 1 R on F(a) 0.9 0.85 0.8 0.75 R on / = 0.1 R on / = 0.15 R on / = 0.2 0.0 0.2 0.4 0.6 0.8 1.0 V M /V To obtain high efficiency, choose V slightly larger than V M Efficiencies in the range 90% to 95% can then be obtained, even with R on as high as 0.2 Losses other than MOSFET on-resistance are not included here Fundamentals of Power Electronics 98 Chapter 18: PWM Rectifiers
18.6.4 Design example Let us design for a given efficiency. Consider the following specifications: Output voltage 390 V Output power 500 W rms input voltage 120 V Efficiency 95% Assume that losses other than the MOSFET conduction loss are negligible. Average input power is P in = P out η = 500 W 0.95 = 526 W Then the emulated resistance is 2 = V g, rms = P in (120 V)2 526 W = 27.4 Ω Fundamentals of Power Electronics 99 Chapter 18: PWM Rectifiers
Design example η Also, 1 V M V = 120 2 V 390 V = 0.435 95% efficiency with V M /V = 0.435 occurs with R on / 0.075. 0.95 0.9 0.85 0.8 R on / = 0.05 R on / = 0.1 R on / = 0.15 R on / = 0.2 So we require a MOSFET with on resistance of R on (0.075) = (0.075) (27.4 Ω)=2Ω 0.75 0.0 0.2 0.4 0.6 0.8 1.0 V M /V Fundamentals of Power Electronics 100 Chapter 18: PWM Rectifiers