A second order system

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1 The Link Between The Phase Magin And The Convete Tansient Response Chistophe BASSO ON Seionduto 4, ue Paul Mesplé BP TOULOUSE Cedex - Fane When designing a losed-loop syste, a swith-ode powe supply fo instane, a path is eated between the vaiable you want to onito and the ontol pin of you onvete. This ontol pin an be the peak uent set point in a uent-ode powe supply o the duty-yle input with a voltage-ode ontolle. If the onitoed vaiable deviates fo its iposed taget, the ontolle eats by eithe ineasing o deeasing the deliveed powe to the load via an aplified eo signal fed to its ontol pin. The powe stage, howeve, is affeted by a gain and a phase that ae fequeny dependent, H(s). To ake sue the esulting powe supply will behave pe the speified data, it is the designe task to shape the etun path G(s) to opensate fo the powe stage esponse at etain fequeny points. Aong the ipotant paaetes ae the d gain fo the sallest stati eo and the lowest output ipedane, but also the oss ove fequeny fo the equied esponse speed. At the oss ove point, whee the loop gain odule T(s) equals, the etuning signal will be affeted by a etain phase otation. If the signal etuns in phase with the ontol signal, we have onditions to fo an osillato, soething you want to avoid. To ake sue the signal does not etun in phase, that is to say with a 36 phase otation, you ust plan a etain aount of agin between the phase otation of T(s) at the oss ove fequeny and the 36 liit: this is the phase agin. Howeve, how uh phase agin should you ask to obine pefoane and stable answe? 45 as often found in the books, oe than that? Let us disove how uh though the following lines. A seond ode syste Figue shows a LC low-pass filte whee the esisto R illustates the losses in the netwok. This ahitetue ould be seen as a siplified lossy output filte of an unloaded buk onvete. In that ase, the input voltage V in epesents the aveage level of the squae wave signal pesent at the powe swith/feewheel diode athode juntion. Fo the pupose of this study, this aveage voltage will be a odulated and we ae looking fo the expession of the output voltage aoss the output apaito. We will then alulate the tansfe funtion H(s) = V out (s)/v in (s) of this stutue. 3 R {R} L {L} Vout Vin C {C} paaetes f=35k L=u C=/(4*3.459^*f^*L) w=({l}*{c})^-.5 Q= R=/((({C}/(4*{L}))^.5)**{Q}) Figue : a buk onvete an be epesented by a siple low-pass filte. The sipt at the botto autoates the alulation of R when hanging the Q value. Using Laplae notation, () desibes the tansfe funtion H(s) of this RLC netwok: = H s LCs + RCs + () By e-aanging the expession, it beoes possible to identify the quality oeffiient Q and the esonant fequeny.

2 s s s s + ζ Q = = H s () Whee, is the esonant fequeny, ζ (zeta) is the daping fato and Q, the quality oeffiient: C = (3), ζ = R (4), Q LC 4L = ζ (5) The idea is now to evaluate the esponse to a -V input step and hange the quality oeffiient values by tweaking the esisto R. This esisto is epesentative of the losses in the netwok suh as the Equivalent Seies Resisto of the induto fo instane. As you an ead in Figue, we have autoated the alulation of R whose value is evaluated aoding to the seleted Q. We ould also ultiply () by /s and alulate the invese Laplae tansfo to obtain the tepoal esponse. A SPICE siulation will be faste in ou ase. The esults appea in Figue. Plot vout#6, vout#5, vout#4, vout#3, vo ut in volts Q = 5 Q = Q =.77 Q =.5 Q <.5 ove daping Q =.5 itial daping Q >.5 unde daping Oveshoot = 65% Asyptotially stable Fast esponse and no oveshoot! 789 Q =. 5.u 5.u 5.u 35.u 45.u tie in seonds Figue : when Q is swept fo. to 5, the answe to a step is eithe slow (Q =.) but without oveshoot o faste, with a lage oveshoot fo big Q values. As one an see, low Q values lead to a opletely osillation fee esponse wheeas values above.5 give bith to oveshoots. As Q ineases, eaning less losses, the oveshoot gets lage. If Q would go to infinity, it would iply an undaped LC netwok keeping osillations going futhe to an exitation. Looking fo oots The study of () denoinato will eveal the oots fo whih H(s) goes to infinity. Matheatially, it oesponds to the following haateisti equation: s s + + = (6) Q Whee the oots oe easily as follows: ( Q ) s, s = ± 4 (7) Q

3 In (7), the te unde the squae oot an eithe be positive o negative, depending on the quality oeffiient value. Fo Q values below.5, the so-alled ovedaped ase, the te unde the squae oot eains positive and both oots s and s ae sepaated eal oots. The step esponse is sluggish as shown in Figue. When Q eahes.5, alled the itially daped ase, the oots ae still eal but ae now oinident. The step esponse is uh faste but still does not exhibit oveshoot. Now, if Q futhe gows, we ae in an undedaped ase and the oots weloe an iaginay potion that ineases as Q goes up: we have a fast step esponse now featuing oveshoot and osillations. If Q eahes infinity, the eal potion of oots s and s fades away and the syste feely osillates: thee is no oe daping (losses) bought by the eal tes. Analysing the tajetoy of these oots is alled oot lous analysis: it shows how the oots ae positioned in the s-plane and give indiation on how they ove in elationship to soe paaetes. It is Q in this exaple but it ould be the gain k of a syste whee, at soe point when k is ineased, the oots igate in the ight half plane and eate an instability. Figue 3 desibes the path taken by s and s as Q hanges. j Q = High Q Low Q Low Q σ Q =.5 High Q LHP RHP Figue 3: oot lous analysis helps to undestand how the oots ove in elationship to a seleted paaete, hee the quality oeffiient of ou LC netwok. Appoxiation of an open-loop esponse Based on what we have aleady dislosed, it would be inteesting to odel ou losed-loop d-d onvete with an equation whee a quality oeffiient te would appea. That way, we ould selet the paaete that affets this Q to shape the output esponse we ae looking fo: slow but without any oveshoot o, on the opposite, faste but aepting a little oveshoot. Let us stat the deivation poess by looking at Figue 4: 8 8. T(s) Phase ( ) Module (db) db f agt( s ) agt(f ) 5 ϕ k k k fequeny in hetz Figue 4: the open-loop esponse of a opensated buk onvete an be appoxiated to a seond-ode syste in the viinity of the oss ove fequeny. This figue shows the oplete loop gain T(s) ade of the onvete powe stage tansfe funtion H(s) futhe shaped by the opensato tansfe funtion, G(s). This exaple is dealing with a CCM buk onvete opeated

4 in voltage-ode ontol. In this figue, we onentate on the aea aound the oss ove fequeny f whih epesents one ipotant design paaete of the d-d onvete you ty to stabilize. Asyptotially looking at the uve within the fae eveals the effets of an oigin pole and a high fequeny pole. Matheatially, this appoxiation an be foulated by: s s + (8) In this appoxiated expession, we onside exta poles and zeos fa away fo f, natually liiting thei ipat on the tansfe funtion. Howeve, ou inteest lies in the esponse the d-d onvete is going to delive one its loop is losed. In othe tes, let us identify the losed-loop tansfe funtion deived fo (8). To + : obtain the losed expession, we an evaluate + = s s + + (9) Equation (9) appeas aanged in fo that ealls that of (). Theefoe, we an put it unde the failia fo of a seond ode syste as desibed by (): + = s s + + Q () The identifiation of the quality oeffiient Q and the esonant fequeny is staightfowad: and = (). Q = () We now have an equation that desibes the appoxiated losed-loop esponse of ou d-d and it inludes a quality oeffiient. The next step is to establish a elationship between the losed-loop Q and the key design paaete, the open-loop phase agin. Fist, based on (8), let us alulate the oss ove fequeny bought by the loation of the oigin pole and its assoiated high fequeny pole. At the oss ove point, we know that the T(s) odule equals. Theefoe: j j + = (3) Extating and e-aanging gives:

5 + 4 = (4) If we substitute ( ()) into (4), we obtain a Q-dependent oss ove fequeny : 4 ( + Q ) 4 = (5) Equation (5) shows us how the losed-loop quality oeffiient and the open-loop oss ove fequeny ae linked. It is ipotant fo this eak to be well undestood: Q epesents the esulting losed-loop esponse quality oeffiient based on the open-loop pole/zeo aangeent desibing the appoxiated open-loop opensated tansfe funtion T(s) in (8). To ontinue futhe with ou analysis, we evaluate the phase otation of T(s) at the oss ove fequeny: agt ( ) = tan + tan = tan (6) The phase agin ϕ epesents the distane between the total phase otation at the oss ove fequeny given by (6) and the 8 liit. In this ase, we puposely neglet the phase evesal bought by the opeational aplifie. Hene, we have: ϕ = + agt (7) Substituting (6) into (7), we obtain: ϕ = tan = tan (8) Reebeing ou fa fa away tigonoeti lasses (!), we have: tan x + tan = (9) x Thanks to (9), we an update ()(6): ϕ = tan () We have aleady defined the oss ove fequeny vesus the losed loop quality oeffiient in (5). If we apitalize on this definition in (), we have:

6 ϕ = tan ( Q ) () The next step is to extat the losed-loop quality oeffiient fo () and siplify the esult: ( ϕ) ( ϕ ) ( ϕ) ( ϕ ) 4 + tan os Q = = () tan sin This is it! We now have a elationship between ou ain design iteion the open-loop phase agin and the quality oeffiient ou loop will exhibit one losed. The best is to exploe the vaious Q diffeent phase agin hoies will bing though a gaph as poposed by Figue Q + tan( φ) 4 5 ϕ tan ( φ ) φ Figue 5: the gaph shows the evolution of the losed-loop quality oeffiient as you selet diffeent phase agin. If you want to obine speed and lak of oveshoot, Figue suggests a Q of.5. Reading the oesponding phase agin in Figue 5, we an see a design iteion of 76 satisfies this equest fo suh a Q, fa away fo the 45 found in the ajoity of text books! What does it ean then? In the esponse to a load step, one the loop is losed, the open-loop phase agin ostly affets the eovey shape and a little the undeshoot depth. Theefoe, it eally depends on the kind of esponse you ae looking fo o what the ustoe speifiations ipose on you design. If a fast eovey is needed and a little oveshoot aepted, then eduing the phase agin an be an option. On the ontay, if absolutely no oveshoots ae toleated, you have no hoie than ineasing the phase agin to the detient of the eovey speed. Whateve solution you selet, you have to ake sue that whateve the opeating onditions, input/output, tepeatue and noal paaeti vaiations (ESRs fo instane), the phase agin neve goes below 45. In othe wods, shooting fo a typial value aound 7 should beoe a good design patie. Tansient esponse and phase agin We have stabilized the buk onvete using one of the autoated siulation platfos desibed in Ref. []. The tehnique allows to keep the sae oss ove fequeny while playing on the phase agin only. The oveall shape is the sae as that pesented in Figue 4 with a -khz oss ove fequeny. The output is subjeted to step anging fo A to A in µs. The esults appea in Figue 6. The 76 phase agin gives a little oveshoot of.5% wheeas the 49 agin tiples that oveshoot, still easonable though given the vetial axis sale of V pe division. Howeve, you an obseve a faste eovey in the 49 phase ase (7 µs) vesus the 76 ase (7 µs). Why do we still have oveshoot with the 76 when theoy states thee should be none? It is beause (8) is a siplified view of the tansfe funtion in the viinity of the oss ove fequeny. As detailed in Ref. [], if you have thee o oe poles installed nea the oss ove fequeny, the Q fato

7 appoxiation we have been though does not wok anyoe and exta wok is equied. Nevetheless, as exeplified by Figue 6, a sall phase agin leads to a peaky losed-loop esponse ϕ = 36 ϕ = ϕ = 64 V out (t) V/div 4.96 ϕ = t 5 µs/div 8u Figue 6: the phase agin has been adjusted at diffeent values and it lealy affets the tansient esponse in both the eovey tie and the oveshoot above the 5-V taget. Conlusion The design of a powe onvete equies aes when it oes to loop ontol. Nueous text books just eoend to design fo a 45 phase agin without any explanations. This atile shows how to analytially deive a phase agin taget whih is supisingly highe than 45 and lose to 76. Despite soe appoxiations at the beginning of the study, the final esult is baked up by siulation esults that onfi the need fo a phase agin geate than the lassial 45 eoendations. Refeenes. C. Basso, Swith Mode Powe Supplies: SPICE Siulations and Patial Designs, MGaw-Hill, 8. R. Eikson, D. Maksiovi, Fundaentals of Powe Eletoni, Kluwes Aadei Pess,