Chapter 17 The Ideal Rectifier 17.1 Properties of the ideal rectifier 17.2 Realization of a near-ideal rectifier 17.3 Single-phase converter systems employing ideal rectifiers 17.4 RMS values of rectifier waveforms 17.5 Ideal three-phase rectifiers Fundamentals of Power Electronics 1 Chapter 17: The Ideal Rectifier 17.1 Properties of the ideal rectifier It is desired that the rectifier present a resistive load to the ac power system. This leads to • unity power factor • ac line current has same waveshape as voltage vac(t) iac(t) iac(t)= Re + Re is called the emulated resistance vac(t) Re – Fundamentals of Power Electronics 2 Chapter 17: The Ideal Rectifier Control of power throughput 2 V ac,rms P = iac(t) av R (v ) e control + vac(t) Re(vcontrol) Power apparently “consumed” by Re is actually transferred to rectifier dc output port. To control the amount – of output power, it must be possible to adjust the value of R . e vcontrol Fundamentals of Power Electronics 3 Chapter 17: The Ideal Rectifier Output port model Ideal rectifier (LFR) The ideal rectifier is iac(t) i(t) lossless and contains + p(t) = v 2/R + no internal energy ac e storage. Hence, the v (t) R (v ) v(t) instantaneous input ac e control power equals the instantaneous output – – power. Since the ac dc instantaneous power is input output independent of the dc vcontrol load characteristics, the output port obeys a v2 (t) 2 power source p(t)= ac vac(t) characteristic. v(t)i(t)=p(t)= Re(vcontrol(t)) Re Fundamentals of Power Electronics 4 Chapter 17: The Ideal Rectifier The dependent power source i(t) i(t) i(t) v(t)i(t) = p(t) + + p(t) v(t) v(t) p(t) v(t) – – power power source sink i-v characteristic Fundamentals of Power Electronics 5 Chapter 17: The Ideal Rectifier Equations of the ideal rectifier / LFR Defining equations of the When connected to a ideal rectifier: resistive load of value R, the input and output rms voltages and currents are related as v (t) follows: i (t)= ac ac R (v ) e control V rms = R v(t)i(t)=p(t) Vac,rms Re 2 vac(t) p(t)= I ac,rms R Re(vcontrol(t)) = I rms Re Fundamentals of Power Electronics 6 Chapter 17: The Ideal Rectifier 17.2 Realization of a near-ideal rectifier Control the duty cycle of a dc-dc converter, such that the input current is proportional to the input voltage: dc–dc converter ig(t) 1 : M(d(t)) i(t) + + iac(t) vac(t) vg(t) v(t) C R – – d(t) ig Controller vg Fundamentals of Power Electronics 7 Chapter 17: The Ideal Rectifier Waveforms v (t) ac ig(t) VM t v(t) V iac(t) VM /Re t M(t) Mmin v (t) g V M v(t) v (t)=V sin (ωt) M(d(t)) = = V ac M ω vg(t) VM sin ( t) ω vg(t)=VM sin ( t) V M min = VM Fundamentals of Power Electronics 8 Chapter 17: The Ideal Rectifier Choice of converter v(t) M(d(t)) = = V M(t) ω vg(t) VM sin ( t) Mmin • To avoid distortion near line voltage zero crossings, converter should be capable of producing M(d(t)) approaching infinity • Above expression neglects converter dynamics • Boost, buck-boost, Cuk, SEPIC, and other converters with similar conversion ratios are suitable • We will see that the boost converter exhibits lowest transistor stresses. For this reason, it is most often chosen Fundamentals of Power Electronics 9 Chapter 17: The Ideal Rectifier Boost converter with controller to cause input current to follow input voltage Boost converter ig(t) i(t) + + iac(t) L D1 vac(t) vg(t) Q1 C v(t) R – – v (t) control vg(t) ig(t) PWM Rs Multiplier X va(t) – verr(t) + Gc(s) vref(t) = kx vg(t) vcontrol(t) Compensator Controller Fundamentals of Power Electronics 10 Chapter 17: The Ideal Rectifier Variation of duty cycle in boost rectifier v(t) M(d(t)) = = V ω vg(t) VM sin ( t) ≥ ≥ Since M 1 in the boost converter, it is required that V VM If the converter operates in CCM, then M(d(t)) = 1 1±d(t) The duty ratio should therefore follow v (t) d(t)=1± g in CCM V Fundamentals of Power Electronics 11 Chapter 17: The Ideal Rectifier CCM/DCM boundary, boost rectifier Inductor current ripple is v (t)d(t)T ∆i (t)= g s g 2L Low-frequency (average) component of inductor current waveform is vg(t) ig(t) = Ts Re The converter operates in CCM when ∆ ⇒ 2L ig(t) > ig(t) d(t)< Ts ReTs Substitute CCM expression for d(t): R < 2L for CCM e v (t) T 1± g s V Fundamentals of Power Electronics 12 Chapter 17: The Ideal Rectifier CCM/DCM boundary R < 2L for CCM e v (t) T 1± g s V Note that vg(t) varies with time, between 0 and VM. Hence, this equation may be satisfied at some points on the ac line cycle, and not at others. The converter always operates in CCM provided that 2L Re < Ts The converter always operates in DCM provided that R > 2L e V T 1± M s V For Re between these limits, the converter operates in DCM when vg(t) is near zero, and in CCM when vg(t) approaches VM. Fundamentals of Power Electronics 13 Chapter 17: The Ideal Rectifier Static input characteristics of the boost converter A plot of input current ig(t) vs input voltage vg(t), for various duty cycles d(t). In CCM, the boost converter equilibrium equation is v (t) g =1±d(t) V The input characteristic in DCM is found by solution of the averaged DCM model (Fig. 10.12(b)): i (t) g Solve for input current: + vg(t) p(t) i (t)= + v (t) + p R V g g – Re(d(t)) Re(d(t)) V ± vg(t) 2 vg(t) – with p(t)= Re(d(t)) Beware! This DCM Re(d) from 2L Chapter 10 is not the same as the Re(d(t)) = 2 d (t)Ts rectifier emulated resistance Re = vg/ig Fundamentals of Power Electronics 14 Chapter 17: The Ideal Rectifier Static input characteristics of the boost converter Now simplify DCM current expression, to obtain 2L vg(t) 2 vg(t) ig(t)1± = d (t) VTs V V CCM/DCM mode boundary, in terms of vg(t) and ig(t): 2L vg(t) vg(t) ig(t)> 1± VTs V V Fundamentals of Power Electronics 15 Chapter 17: The Ideal Rectifier Boost input characteristics with superimposed resistive characteristic 1 = 1 CCM: = 0 d = 0.8 = 0.6 = 0.4 = 0.2 d d d d d v (t) g =1±d(t) 0.75 V ) t ( DCM: g i s L 2L vg(t) 2 vg(t) 2 i (t)1± = d (t) VT 0.5 g R e VTs V V (t)/ v g )= t )= ( (t g i g j CCM when 0.25 CCM 2L vg(t) vg(t) ig(t)> 1± DCM VTs V V 0 0 0.25 0.5 0.75 1 v (t) m (t)= g g V Fundamentals of Power Electronics 16 Chapter 17: The Ideal Rectifier Re of the multiplying (average current) controller Current sensor gain Solve circuit to find Re: va(t)=ig(t)Rs when the error signal is small, Boost converter i(t) ig(t) ≈ va(t) vref (t) i (t) + + ac L D 1 multiplier equation vac(t) vg(t) Q1 C v(t) R vref (t)=k xvg(t)vcontrol(t) – – v (t) v (t) i (t) control g g then Re is PWM vref (t) Rs Multiplier X va(t) k v (t) vg(t) x control v (t) – err Re = = + Gc(s) vref(t) ig(t) va(t) = kx vg(t) vcontrol(t) Compensator Rs Controller simplify: Rs Re(vcontrol(t)) = k xvcontrol(t) Fundamentals of Power Electronics 17 Chapter 17: The Ideal Rectifier Low frequency system model 〈i (t)〉 Ideal rectifier (LFR) 〈i(t)〉 g Ts Ts + + iac(t) 〈p(t)〉 Ts v (t) 〈v (t)〉 R C 〈v(t)〉 R ac g Ts e Ts – – Re(t) Rs vcontrol(t) Re(t)= k x vcontrol(t) Rs Re(vcontrol(t)) = This model also applies to other k xvcontrol(t) converters that are controlled in the same manner, including buck-boost, Cuk, and SEPIC. Fundamentals of Power Electronics 18 Chapter 17: The Ideal Rectifier Open-loop DCM approach We found in Chapter 10 that the buck-boost, SEPIC, and Cuk converters, when operated open-loop in DCM, inherently behave as loss-free resistors. This suggests that they could also be used as near-ideal rectifiers, without need for a multiplying controller. Advantage: simple control Disadvantages: higher peak currents, larger input current EMI Like other DCM applications, this approach is usually restricted to low power (< 200W). The boost converter can also be operated in DCM as a low harmonic rectifier. Input characteristic is 2 vg(t) vg(t) ig(t) = + Ts Re Re v(t)±vg(t) Input current contains harmonics. If v is sufficiently greater than vg, then harmonics are small. Fundamentals of Power Electronics 19 Chapter 17: The Ideal Rectifier 17.3 Single-phase converter systems containing ideal rectifiers • It is usually desired that the output voltage v(t) 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: pload(t)=v(t)i(t)=VI • The instantaneous input power of a single-phase ideal rectifier is not constant: pac(t)=vg(t)ig(t) with vg(t) vg(t)=VM sin (ωt) ig(t)= Re 2 2 so V M 2 V M pac(t)= sin ωt = 1 ± cos 2ωt Re 2Re Fundamentals of Power Electronics 20 Chapter 17: The Ideal Rectifier Power flow in single-phase ideal rectifier system • Ideal rectifier is lossless, and contains no internal energy storage.