R. W. Erickon Department of Electrical, Computer, and Energy Engineering Univerity of Colorado, Boulder
Cloed-loop buck converter example: Section 9.5.4
In ECEN 5797, we ued the CCM mall ignal model to deign a feedback compenator:
CCM deign: theoretical loop gain
Simulation of loop gain uing CCM-DCM model
Simulated loop gain T CCM: load reitance R = 3 ohm DCM: load reitance R = 5 ohm
Simulation of Gvg Effect of DCM on open- and cloed-loop diturbance in v g
High-Frequency Inductor Dynamic in DCM Our model o far make the aumption L = 0, o that the DCM inductor doe not contribute dynamic to the mall ignal tranfer function. See Cuk, PhD thei, 976 Thi i the mot commonly ued model for DCM, and it i adequate for the majority of practical ytem. In practice, ome deviation i oberved at frequencie approaching the witching frequency. The literature contain example of DCM model that include inductor dynamic: Ue averaged witch model without letting L = 0 (Vorperian) Modify averaging derivation to treat the average inductor current a an independent tate (Sun et al) Employ a ampled-data analyi (without averaging) to compute the perturbation in inductor current that arie from a perturbation in duty cycle (Makimovic) The fundamental quetion: I the inductor current an independent tate? Doe it lead to pole and zeroe? What i the mechanim by which the inductor exhibit dynamic in DCM? We will anwer thee quetion now. The anwer expoe the limitation of averaging at frequencie approaching the witching frequency. ECEN 5807, Spring 05
The tate of a network At a given point in time, the value of the tate variable depend on the previou hitory of the ytem, rather than the preent value of the ytem input To olve the differential equation of a ytem, the initial value of the tate variable mut be pecified If the ytem i linear, then the derivative of the tate variable are expreed a linear combination of the ytem independent input and tate variable themelve The phyical tate variable of a ytem are uually aociated with the torage of energy For a typical converter circuit, the phyical tate variable are the inductor current and capacitor voltage i L (t) i pk In DCM, i the inductor current an independent tate variable? v L (t) T = d (t) v g (t) T + d (t) v(t) T =0 v L (t) v g v g L v L 0 t 0 ECEN 5807, Spring 05 v
Steady tate DCM waveform i L Inductor current c Control ignal (gate drive) dt d T T ECEN 5807, Spring 05 3
Perturbation in control ignal: DCM waveform i L Inductor current c Control ignal (gate drive) Perturbation in control ignal dt d T T Perturbation in inductor current ECEN 5807, Spring 05 4
Dicuion A perturbation in the control ignal caue a perturbation in the inductor current. The inductor current perturbation doe not end when the control perturbation end, but intead extend for time D T The inductor current perturbation end before the end of the witching period, and there i no influence on the next witching period. i L c So the average inductor current doe not exhibit dynamic that extend over multiple witching period. But the inductor current behave a an independent tate within one witching period. Modeling thi phenomenon i beyond the limit of traditional averaging. Correct modeling of thi repone require a ampled data model. dt d T T ECEN 5807, Spring 05 5
Introduction to converter ampled-data modeling A pule-width modulator perform ampling of it input control ignal u(t): Sampling occur at the tranitor turn-off tranition The gate drive ignal and duty cycle convey no information about u(t), except at the ampling intant Sampling of the PWM can be modeled uing traditional ampled-data theory If the perturbation i mall (mall-ignal aumption), then the ampling intant i eentially the ame in every witching period dˆ T δ c ĉ ( t ) t p PWM ramp t p dˆt u + uˆ Control input Gate drive ignal Perturbation from Steady tate Repreentation of ampled duty cycle perturbation PWM ampling occur at t p (i.e. at dt, periodically, in each witching period) ECEN 5807, Spring 05 6
Uniform ampling of a ignal v(t) The ampled ignal v*(t) ha a nonzero value only at the ampling intant t = nt Mathematically, we repreent v*(t) uing impule function whoe area i equal to the ampled value of v(t) A replica of v(t) can be recovered from v*(t) by lowpa filtering. v(t) v*(t) v(t) T Sampler v*(t) Sampler period T t t v + *( t) = v( t) δ ( t nt ) Unit impule (Dirac) ECEN 5807, Spring 05 7
The impule function δ(t) (t) δ(t) Δt area = t The impule a a limit of rectangular pule (t): Pule width i Δt Pule height i /Δt Pule area i Impule i obtained in limit a Δt goe to zero Propertie of the impule function: + + δ ( t) dt = Area = v t) δ ( t t ) dt = v( t ) Sifting property: ( Sampling of v(t) Laplace tranform of impule = Integral of impule i unit tep function ECEN 5807, Spring 05 8
Sampling in the frequency domain v + *( t) = v( t) δ ( t nt ) Repreentation of ampled ignal v*(t) uing um of impule function v( ) + t = v( t) e dt Definition of Laplace tranform v *( ) + t = v*( t) e dt Laplace tranform of v*(t) v *( ) = v( jkω ) T + k = Aliaing: Laplace tranform of ampled ignal i um of frequency-hifted term ECEN 5807, Spring 05 9
X v X(t) = v(t) X v () = Z 0B@ X Z X n= (t XnT) X v (t)e t dt Now evaluate the Laplace tranform of v : v () = Z 0 X v(t) T B@ e jk! t CA e t dt v () = T = T v () = T Z X k= X k= X k= Z Z 0 ECEN 5807, Spring 05 k= 0B@ Z X CA CA v(t)e jk! t e t dt v(t)e ( v( jk! ) jk! )t dt Proof of L(v*(t)) Z Exponential Fourier erie: X X (t nt) = C k e jk! t n= k=! = T = f 0 C k = T So Z T/ T/ B@ X (t nt) CA e jk! t dt = T n= X (t nt) = T n= X k= e jk! t
v () = T X Dicuion: L(v*(t)) k= Aliaing: The ubtraction term ( jkω ) contitute frequency hifting The pectrum of v() i hifted and repeated X Z v( jk! ) v*() additionally contain factor of /T ECEN 5807, Spring 05
Sampled data ytem v(t) A/D v*(t) D/A v o (t) Analog-todigital converter Digital-to-analog converter v(t) T v*(t) H v o (t) Sampler Zero-order hold ECEN 5807, Spring 05
The Zero-Order Hold (ZOH) v*(t) H v o (t) δ(t) H v o (t) Zero-order hold Zero-order hold v*(t) t v o ( t) = t t T δ ( τ ) dτ v o (t) nt (n+)t (n+)t T = ampling period /T = ampling frequency t ECEN 5807, Spring 05
ZOH: Frequency Domain Define tranfer function a impule repone: u(t) H Zero-order hold v o (t) Impule repone: v o ( t) = t t T δ ( τ ) dτ The definite integral ha Laplace tranform: H = e T ECEN 5807, Spring 05 3
ZOH: Sampled Data Sytem Example v(t) T Sampler v*(t) H Zero-order hold H v o (t) e = T v *( ) = v( jkω ) T + k = T T e e v o( ) = v*( ) = v( jkω ) T + k= Conider only low-frequency ignal: v o T e ( ) v( ) T Sytem tranfer function = v v o e = T T ECEN 5807, Spring 05 4
ZOH: Frequency Repone Let = jω jωt jωt / jωt / e jωt / e e in( ωt / ) jωt / jωt / jωt f = MHz Zero-order hold: frequency repone 0 = e j = ωt / ωt / Zero-Order Hold magnitude and phae repone e = inc( ωt / ) e 0 magnitude [db] -0-40 -60-80 H / T e = T T MATLAB file: zohfr.m -00 0 0 3 0 4 0 5 0 6 0 7 0 phae [deg] -50-00 -50 ECEN 5807, Spring 05 0 0 3 0 4 5 0 5 0 6 0 7 frequency [Hz]
The Padé Approximation Zero-order hold: frequency repone e T ω p + ω T e T p + ω p t -order Pade approximation ω p = T f f p = = Tπ π f = MHz magnitude [db] 0 0-0 -40-60 -80 Zero-Order Hold magnitude and phae repone MATLAB file: zohfr.m -00 0 0 3 0 4 0 5 0 6 0 7 0 phae [deg] -50-00 -50 0 0 3 0 4 0 5 0 6 0 7 N5807 Intro to Converter Sampled-Data Modeling frequency [Hz] ECEN 5807, Spring 05 6
Sampled-Data Model of Buck Converter Control Loop V g d L i + v + g D i + D v g C R v I d Sampled ignal PWM d u _ v ref V M G c + Sampling ECEN 5807, Spring 05 7
Sampled-Data Model of Converter Control Loop Equivalent hold G h () d T δ (t nt ), d = u T u G c () _ v + v ref Sampled-data model valid at all frequencie Equivalent hold decribe the converter mall-ignal repone to the ampled duty-cycle perturbation [Billy Lau, PESC 986] State-pace averaging or averaged-witch model are low-frequency continuou-time approximation to thi ampled-data model ECEN 5807, Spring 05 8
DCM High-Frequency Modeling i L c dt d T T ECEN 5807, Spring 05 9
ECEN 5807, Spring 05 0 Control to Inductor Current Tranfer Function + = + = + = k D T D T L jk d T e T L V V d e T L V V i ) ˆ( ) ˆ *( ) ( ˆ ω ) ˆ( ) ( ˆ d D T e D T L V V i D T L + ) ˆ( ) ( ˆ ω D T L V V d i L + + D T = ω D f f π = High-frequency pole due to the inductor current dynamic in DCM, ee (.77) in Section.3
Concluion PWM i a mall-ignal ampler with uniform ample rate The witching converter/regulator can be viewed a a ampled data ytem Duty cycle perturbation act a a tring of impule Converter repone can be modeled a an equivalent hold Averaged mall-ignal model can be viewed a low-frequency approximation to the equivalent hold function. A the frequency approache the witching frequency, converter dynamic within one witching period become ignificant. In DCM, thi caue the inductor dynamic (not preent at low frequencie) to appear In DCM at high frequencie, the inductor-current dynamic-repone i decribed by an equivalent hold that behave a a zero-order hold of length D T An approximate continuou-time model baed on the ampled-data model predict a highfrequency pole at frequency f /πd ECEN 5807, Spring 05