by Christophe Basso, ON Semiconductor, Toulouse, France and Alain Laprade, ON Semiconductor, East Greenwich, R.I.

Transcription

1 Simpliied Analysis O A DCM Boost Converter Driving An ED String Part : Theoretial Analysis SSUE: February 03 by Christophe Basso, ON Semiondutor, Toulouse, Frane and Alain aprade, ON Semiondutor, East Greenwih,.. The ixed-requeny boost onverter lends itsel very well to driving an ED string at a onstant urrent. Working in disontinuous-ondution mode (DCM), the onverter an be eiiently used or ast dimming operation. n this regard, the perormane o a DCM-operated boost onverter will be superior to that o its ontinuous-ondution mode (CCM)-operated ounterpart beause DCM operation oers better transient response. As a result o this ast transient perormane, a boost onverter operating in DCM will rapidly reharge the put apaitor when the EDs are turned on, thus minimizing analog dimming o the EDs. However, the design o a DCM boost onverter involves ertain hallenges suh as properly stabilizing the design. Although small-signal models exist or perorming this type o analysis, some diiulty arises in trying to apply these models to the ED driver appliations. That s beause the a analysis o a boost onverter driving EDs diers rom that using a standard resistive load. As the series diodes impose both d and a loading onditions, deriving the inal transer untion is not a simple matter. Here in part o this two-part artile, we present an easier method or analysis. ather than using the lassial small-signal model o the DCM boost onverter, we develop a simpliied model based on the put urrent expression or the onverter and then use this model or analysis. n Part o this artile (Pratial Considerations), we will delve into an implemented solution and veriy measurement auray versus the theoretial derivation. A Boost Converter Powering The ED String Fig. represents a simpliied onstant-requeny peak-urrent-operated boost onverter driving an ED string. The put urrent is permanently monitored by a sense resistor sense. The voltage it develops goes to a ontrol iruit that ontinually adjusts the power swith on-time to deliver a onstant ED urrent. This is the ontrolled put variable. D in SW C r C i sense PWM ontrol Fig.. A boost onverter drives an ED string to deliver light. The put urrent is regulated to a setpoint value. 03 HowPower. All rights reserved. Page o 3

2 When lit, the ED string gives rise to a voltage aross its onneting terminals. This voltage depends on the individual ED tehnology-related threshold voltage T0 and its dynami resistane r d. The total drop aross the ED string is thus the sum o the threshold voltages denoted as Z while the dynami resistane r EDs represents the sum o the series dynami resistanes. Fig. shows the adopted equivalent iruit. n EDs in series A A A A v F r d T0 r EDs v Z n i n i r di T0 i K K v r F d T 0 K K Fig.. The series onnetion o EDs leads to a summing o their threshold voltages and the series onnetion o their individual dynami resistanes. t will be your duty to haraterize the string voltage drop and its total dynami resistane. To measure it, bias the ED string to its nominal urrent, F. One ED thermal stability is reahed, measure the total voltage drop aross the ED string,. Change the urrent to a slightly lower value F and measure the new voltage drop. From these values, you an alulate the total dynami resistane as: r EDs F F. () The Zener voltage is one o the string voltages minus r EDs times the urrent at this measurement point:. () Z EDs F et s assume we bias our ED string with a 00-mA urrent. The measured total drop is 7.5. We redue the urrent to 80 ma and the new drop is 6.4. The total dynami resistane is simply: r EDs Ω. (3) From equation, the Zener voltage is simply: (4) Z ooking bak to Fig., the ED string is plaed in series with a sense resistor sense. The total a resistane is thus the ombination o both elements: a reds sense. (5) 03 HowPower. All rights reserved. Page o 3

3 The equivalent d diagram simpliies quite a bit to that o Fig. 3. The d put voltage is made o the put urrent irulating in the resistor a plus the Zener voltage: n a, as the Zener voltage is onstant, the expression simpliies to: a z. (6) a s s. (7) a Z Fig. 3. The d sketh shows the equivalent Zener diode and its dynami resistane. A Simpliied Model The urrent soure atually represents the urrent taken rom the input soure and transmitted to the put with losses. This soure is saled up or down by the ontrol voltage, whih sets the indutor peak urrent on a yle-by-yle basis. The ontroller works by observing the indutor peak urrent through the urrentsense resistor i or the boost onverter swith. When the voltage aross i and the ontrol voltage math, the power swith is instruted to turn o. we now onsider an a diagram, the apaitor and its parasiti element ome bak as shown in Fig. 4. The Zener element alone does not play a role as its voltage remains onstant during a modulation: only its dynami resistane r EDs stays in plae, merged into a as desribed by equation HowPower. All rights reserved. Page 3 o 3

4 iˆo s Z s vˆo vˆ s r C a C Fig. 4. The a model uses the total resistane a assoiated with the apaitor model. From this drawing, it is possible to express the small-signal put voltage level when the ontrol voltage is modulated: ˆo ˆ v s i s Z s o. (8) As previously mentioned, the urrent-soure value depends on the ontrol and put voltages. To derive a small-signal equivalent model, we extrat the partial derivatives o with respet to the ontrol voltage and the put voltage : ˆ i s vˆ s vˆ s o o vˆ 0, vˆ 0 vˆ 0, vˆ 0 in o in Substituting equation 9 into equation 8, the latter an be rewritten:. (9) vˆos vˆs vˆoszs. (0) vˆin 0, vˆ 0 vˆin 0, vˆ 0 eerene [] (equation -, p. 49) has derived the d transer untion or the DCM boost onverter as: TswD d M. () in n this last expression, the resistane loading the onverter must be replaed by. The new expression then beomes: in TswD. () 03 HowPower. All rights reserved. Page 4 o 3

5 From this equation, we need to derive the duty ratio expression and the ontrol voltage. n the presene o a ompensation ramp, the ontrol voltage is no longer a ixed d voltage but a ramp whose slope aets the inal peak urrent point. Fig. 5 shows the resulting waveorm. The peak urrent value is reahed sooner than in the absene o a ramp, as i we would artiiially inrease the urrent-ontrol sense resistor i. t has the eet o dereasing the gain o the urrent-ontrol loop and damping the double poles in ontinuous-ondution mode. When the onverter transitions to DCM, the ramp is still present and must be aounted or. v t i S e i peak S n in id t DT sw Fig. 5. The peak urrent is not equal to the ontrol voltage divided by sense beause o the ompensation ramp. The equations are the ollowing ones, aounting or the saling ator i as the external ramp S e is a voltage ramp: peak Se DTsw. (3) i i A similar expression is derived involving the indutor urrent slope: peak DT sw in. (4) Solving or D, we have: D ST T e sw i sw in. (5) This expression or D is now injeted into equation and we solve or : T sw S in. (6) e i in 03 HowPower. All rights reserved. Page 5 o 3

6 To obtain the small-signal value, we will alulate partial derivatives with respet to the ontrol voltage, and put voltage as desribed in equation 0: d vˆ vˆ d vˆin, vˆ Tsw S e i in in (7) in vˆ vˆ, vˆ Tsw in Se i in in vˆ. (8) The expression in equation 8 haraterizes the impat o the small-signal modulation o v on the put urrent. Then, alulating the partial derivative with respet to : d vˆo vˆ d vˆin, vˆ Tsw S e i in in o (9) in vˆo vˆ 0, vˆ 0 sw in e i in T S in vˆ o. (0) This last equation expresses a urrent depending on a voltage multiplied by a oeiient having the dimension o a ondutane g. t is a voltage-ontrolled urrent-soure as drawn in Fig. 6. iˆo s vˆo vˆ s vˆo s r C a C Fig. 6 The equation 0 oeiient is a voltage-ontrolled urrent-soure, eetively a resistane. 03 HowPower. All rights reserved. Page 6 o 3

7 v The urrent diretion o the (s) O ontrolled urrent soure is reversed beause o the negative sign in equation 0. As suh, sine we have a urrent-soure driven by the voltage aross it, it is simply a resistor whose deinition is: T sw in S e i in. () in n this simpliied derivation, the urrent soure illustrates the energy that is absorbed rom the input soure and transmitted to the put. The urrent soure expression does not arry inormation related to the onverter operating mode. For instane, looking at equation 6, we do not know i the part operates at a ixed requeny, transmits energy to the put load during the on-time or during the o-time, and so on. aking suh inormation, the model will mask seond-order ontributions suh as a right-hal plane zero (HPZ), or instane. However, we know rom previous analysis that the HPZ still exists in DCM operation but sine it is relegated to high requenies, we an omit its presene in this ase. The beneit o this simpliied approah is that it enables you to quikly derive an approximate model that gives you the low-requeny behavior o the onsidered struture: d gain and pole/zero ombination. An alternative would be to use the small-signal model o the DCM urrent-mode boost onverter and arry the omplete analysis with a load made o the elements in Fig. 4. This model would give an exat result but would require more iterations and omplex equations. The Complete AC Model Now that we have derived all o our oeiients, we an update the model originally presented in Fig. 4. The updated shemati appears in Fig. 7. orresponds to the equation 0 oeiient and indues a urrent diretly proportional to the put voltage modulation. Z s r C vˆo vˆ s a C Fig. 7. This is the updated a model rom whih we will alulate the omplete transer untion. To derive the transer untion o interest, vˆo loading the urrent soure. t is deined as: v ˆ, we will simpliy the iruit by looking at the impedane Z Z s rc eq sc sr C eq sc r rc eq sc eq C 03 HowPower. All rights reserved. Page 7 o 3. ()

8 n the above equation, eq is the parallel ombination o a and : eq a a. (3) The omplete transer equation is thus the oeiient given by equation 8 multiplied by the resistane in equation 3 and ollowed by the pole/zero ombination rom equation : H0 H s s z s p. (4) where H in 0 Tsw in Se i in eq, (5) z, and (6) rc C p r C C eq. (7) Deriving The Operating Points Beore plotting the a untion, we need to express the operating points and the put urrent dependene on ontrol voltage. We know that the put voltage is equal to: We an substitute this deinition into equation : a Z. (8) Z a in sw Z a sw D T D T. (9) From this expression, we an solve or : atsw D in Z Zin in Z in. (30) a 03 HowPower. All rights reserved. Page 8 o 3

9 We an also replae the duty ratio D by its expression in equation 5. n this ase, the put urrent expression beomes an ugly but useul equation: atsw in Z in zin z in ST e sw T i swin. (3) a Knowing the ED string voltage Z and its dynami resistane r EDs, this expression allows us to predit the urrent delivered by the boost onverter. et s veriy these ormulas with a pratial example. Pratial Appliation n this setion, we present the example o a DCM boost onverter delivering a onstant urrent to an ED string with a voltage o. We ll use the ollowing iruit values in our alulations. 3.3μH Tsw μs i 50mΩ C.μF rc 4mΩ sense Ω reds 55Ω in Se 00k s Z To impose the onstant urrent, we assume the ontrol voltage to be 400 m. We an alulate the duty ratio rom equation 5: D ST T e sw i sw in 39.6%. (3) The put urrent is obtained rom equation 3: 3 atsw in Z in z in z in ST e sw T i swin 64mA. (33) a The put voltage quikly ollows: 3.85 r. (34) d sense Z 03 HowPower. All rights reserved. Page 9 o 3

10 The extra resistor alulated in equation is ound to be: Tsw in Se i in in 6.8Ω. (35) When paralleled with a, it beomes eq aording to equation 3: eq a Ω. (36) a We an now evaluate the stati gain, H 0 : in 0 log G0 0 log 0 log dB eq T sw in Se i in (37) The pole and zero are derived: z 8MHz and (38) rc C p r C C eq.6 khz. (39) A SPCE simulation an be run to hek the validity o the bias points. A large-signal auto-toggling urrentmode model derived in e. [], p. 6 was used. The shemati and the releted bias points appear in Fig. 8. n this shemati, to obtain the right dynami resistane and operating voltage, we used a simple shunt regulator mimiking the operation o a peret Zener diode. This peret diode exhibits a breakdown voltage Z o and its dynami resistane is 55 Ω. t should be noted that a simple - d soure would work or an a analysis, but would not work or any transient simulations suh as start-up. When an a sweep is run, SPCE linearizes the iruit around its operating point and generates a small-signal model. The results displayed in the shemati are not ar rom what we have obtained through the analytial analysis. The urrent in the sense resistor with a 0.4- ontrol voltage reahes.77 6mA, lose to that alulated in equation 33. The plant Bode plot is shown in Fig. 9. The d gain is lose to that alulated in equation 37 and the pole is loated at the orret loation (.6 khz). The phase that ontinues to drop is due to the high-requeny HPZ loated at high requenies. Our simpliied approah annot predit the presene o this HPZ. ts existene relates to the topology arrangement: a boost onverter irst stores the soure energy in the indutor during the on-time and dumps it into the load during the o-time. Any load ondition hanges, i.e. an inrease in the put urrent, must irst ramp through the indutor urrent beore it is delivered to the put. This delay inherent to the operating mode is modeled through an HPZ. This energy transer delay does not expliitly appear in equation 6, whih simply deines a urrent in relationship to ontrol voltage,. n DCM, however, the let-hal plane zero deined in equation 38 ours at a requeny signiiantly above the operation requeny F sw. 03 HowPower. All rights reserved. Page 0 o 3

11 t should be noted that we analyzed the put voltage even though we atually regulate the ED urrent. As we observe the voltage aross the sense resistane sense, the eedbak signal is that o saled down by the division ratio brought by r EDs and sense. The saling adjustment beomes: This urve is also represented in Fig. 8. sense 0og0 0og0 5.6 db. (40) sense reds u.0 9 in d parameters i=-0.5 v= m 8 a duty-yle 0 400m v PWM swith CM p 7 X3 PWMCM = 3.3u Fs = Meg i = i Se = 00k {} AC = 3.6 C.uF 6 3 4m sense 3.6 X F530M d sense X AMPSMP Fig. 8. The averaged model helps to veriy the operating bias points but also the a response. 03 HowPower. All rights reserved. Page o 3

12 db H 0, 3.dB.66 khz H 0, 5.6dB sense sense k 0k 00k Meg Hz Fig. 9. The Bode plot onirms the d gain and the pole loation. Conlusion Here in part o this artile, we have desribed the derivation o the small-signal response o the boost onverter driving an ED string. ather than implementing the omprehensive small-signal model o the DCM boost onverter, a simple equation was derived that desribes a irst-order response o the ED boost onverter operating in disontinuous ondution mode. Despite its inherent limitation to irst order, the answer obtained in a ew lines is suiient to stabilize the ontrol loop. n Part (Pratial Considerations), we will delve into an implemented solution and veriy empirial results versus the theoretial derivation. eerene C. Basso, Swith Mode Power Supplies: SPCE Simulations and Pratial Designs, MGraw-Hill 008, SBN Ab The Authors Christophe Basso is an appliation engineering diretor at ON Semiondutor in Toulouse, Frane. He has originated numerous integrated iruits among whih the NCP0X series has set new standards or low standby power onverters. SPCE simulation is also one o his avorite subjets and he has authored two books on the subjet. Christophe s latest work is Designing Control oops or inear and Swithing Power Supplies: A Tutorial Guide. Christophe reeived a BSEE-equivalent rom the Montpellier University, Frane and an MSEE rom the nstitut National Polytehnique de Toulouse, Frane. He holds 7 patents on power onversion and oten publishes papers in onerenes and trade magazines. Alain aprade reeived an MEng in eletrial engineering rom MGill University in Montréal in 984. Sine then he has worked in power supply development or ommerial, teleommuniation, automotive, and aerospae appliations as an appliations engineer. n 00, he reloated to hode sland to join ON Semiondutor s Analog Power Group New Produt Development team. 03 HowPower. All rights reserved. Page o 3

13 For urther reading on the design o ED drivers, see the HowPower Design Guide, selet the Advaned Searh option, go to Searh by Design Guide Category and selet ED ighting in the Popular Topis ategory. 03 HowPower. All rights reserved. Page 3 o 3