Modelng and Desgn onsderatons of oupled Inductor onerters Guangyong Zhu Auscom Engneerng, Inc. A subsdary of ompal Electroncs, Inc. One Dell Way, Round Roc, TX 7868 Abstract-In ths part of the sequel on the modelng and analyss of coupled nductors and coupled nductor based multphase swtchng conerters, the recently deeloped symmetrcal coupled nductor model s frst extended to nclude the nductor wndng dc resstance (DR. The extended model s then used to analyze the nfluence of the couplng on the DR based current sensng schemes popularly used n mult-phase swtchng regulators. It s found that the tme-constant matchng condton n coupled nductor conerters needs to be modfed to nclude the couplng coeffcent. The proposed model s also used to dere the small-sgnal control-to-output transfer functon of the conerters ncorporatng coupled nductors, wth whch the effect of couplng on the dynamc behaors of the conerter power stage, such as resonant frequency and dampng factor, can be easly ealuated. I. INTRODUTION oupled nductor as a specal form of multple wndng coupled magnetc structure, or transformer, has been around snce the early years of electrcal engneerng. In modern hgh-frequency swtchng conerters, t has been used n many applcatons such as mult-output, cross-regulated conerters [1, 4], rpple cancellaton and multple magnetc component ntegraton [, 6], snubberng [3], and transformer/nductor ntegraton [5], etc. More recently, t also found applcatons n mult-phase buc regulators [7-18]. Moreoer, the basc coupled magnetc structure was extended to practcal mplementatons beyond three phases [1-13]. It s generally accepted that reerse couplng between the coupled wndngs s the preferred couplng scheme [7, 1], and fast transent response can be acheed especally when the control scheme allows multple phases to oerlap n case of fast load transent []. Some also made plausble argument that coupled nductor based conerter also mproes conerson effcency, but ths seems to be ald only under the assumpton that n order to achee the same transent response, a couplednductor based conerter needs to swtch only at a reduced swtchng frequency. It s noted that the phase current rpple cancellng effect at the output of a regular mult-phase swtchng regulator s lost n the coupled-nductor based one, leadng to hgher rpple oltage at the output []. Most of the modelng and analyss wors on coupled nductors were based on the well-establshed transformer models, such as the T- or π-equalent model [1, 14, 15, 19], and the cantleer model [6]. Although these models are largely ald, they are asymmetrcal relate to the roughly symmetrcal magnetc structure largely used n multphase Kunrong Wang Enterprse Systems Group Dell, Inc. One Dell Way, Round Roc, TX 7868 buc regulators because the magnetzng nductance appears only n one of the coupled wndngs. The asymmetry n the model usually maes the analyss ery cumbersome. An equalent nductance model based on the concept of self and mutual nductances n basc crcut theory was ntroduced n [7, 8], but t stll fell short n crystallzng the general relatonshp noled n coupled nductors. Wth the understandng that coupled nductors are magnetc structures wth loose couplng and that the couplng coeffcent plays an mportant role n the resultng crcut performance, a general symmetrcal and narant model for mult-wndng coupled nductors was recently dered []. It was based on the same self- and mutual-nductance concept as n [7, 8], but has the adantage n analyzng all the ey crcut behaors of the coupled nductor based mult-phase regulators, such as steadystate rpple currents, and transent response, etc. The model can be easly extended to multple (more than two wndng cases, and all the crcut parameters noled n the model, such as leaage nductances and the couplng coeffcent, can be easly determned based on termnal nductance measurements. In ths paper, the symmetrcal model presented n [] s frst extended to nclude the wndng dc resstance (DR. It s then used to analyze the nfluence of the couplng on the DR based current sensng schemes and to dere the small-sgnal control-to-output model of the conerters ncorporatng coupled nductors. II. REVIEW AND EXTENSION OF OUPED INDUTOR MODES Fg. 1 shows the well-nown general transformer model of two nersely coupled nductors. It has four parameters: magnetzng nductor M, turns rato n, leaage nductors 1 and, wth only three of them ndependent. Snce ths model s asymmetrcal n structure, t s unable to prode enough analytcal nsght nto how the couplng coeffcent affects a conerter s steady state and transent performances. A new symmetrcal transformer model for coupled nductors was deeloped n []. Whle t was dscussed extensely n [] that the symmetrcal model s easy to use and sutable for analyzng conerters utlzng coupled nductors, some practcal desgn consderatons n coupled nductor conerters were not dscussed, such as lossless current sensng based on nductor DR [1, ], and 978-1-444-4783-1/1/$5. 1 IEEE 7
the effect of couplng coeffcent on the small-sgnal model of the conerter. In ths secton, a symmetrcal transformer model wth non-zero wndng resstances wll be dered frst, whch wll sere as the bass for the dscussons n the subsequent sectons. For smplcty, two nersely coupled dentcal wndngs, each wth a self nductance of and a DR of R, as shown n Fg. (a, are assumed. The model can be analytcally dered from the followng expresson goernng two nersely coupled wndngs: + 1 1-1 1 n:1 M 4 + - 3 Fgure 1. A general transformer model of two nersely coupled nductors. 1 d1 d M + R1 dt dt, (1 d1 d M + + R dt dt where M s the mutual nductance. The expresson can also be re-arranged as: 1 d1 + R( 1 + dt, ( d 1 + + R( 1 + dt where s the leaage nductance defned as (1- and s the couplng coeffcent defned as M/. From (, one can easly come up wth a new symmetrcal equalent crcut as shown n Fg. (b. It can be seen that the model shown n Fg. (b taes on a qute dfferent form from the tradtonal transformer models where R s smply a seres resstor on both the prmary and secondary sdes of a transformer. The magnetzng nductor, whch leads to asymmetry n other model, does not explctly appear n the new model. It should also be ponted out that n Fg. (b s the nductance that s drectly measureable and can be obtaned by measurng across one wndng whle the other wndng s shorted. It s dfferent from the leaage nductances, 1 and, shown n Fg. 1. One of the dstncte features of the new equalent crcut model for two coupled nductors gen n Fg. (b s that t s ery conenent to dere the dynamc models of conerters utlzng coupled nductors. The condton for DR based current sensng can also be easly determned accordng to ( and Fg. (b. III. APPIATIONS OF THE DERIVED MODE 3.1 Inductor DR Based urrent Sensng n oupled Inductor onerters ossless nductor DR based current sensng has become a popular method n multphase, hgh current swtchng regulators such as those for PU, memory and graphcs [1, ]. The condton to accurately extract nductor current nformaton s well understood n conerters wth dscrete, or uncoupled, nductors. Howeer, such a condton, whch s of practcal engneerng alues, s not well establshed n conerters utlzng coupled nductors. + 1 - M 1 R 1 4 R 3 + - (a + 1 - - + 1 1 4 3 - + 1 R R + - (b Fgure. (a Two nersely coupled nductor wndngs wth seres resstors, and (b ts new equalent crcut model. Fg. 3(a shows the general lossless nductor DR current sensng crcut n a two-phase nterleaed buc conerter wth dscrete nductors. Tme-doman waeforms of the phase 1 nductor current, 1 (t, and the sensed oltage across the capactor, c1 (t, are gen n Fg. 3(b. It s well-nown that wth proper desgn,.e. when the tme constant matches between the nductor and the current sensng r, networ, or r/r, the nductor current waeform can be extracted from the oltage waeform across the sensng capactor,.e. c1 (t 1 (t*r. A two-phase buc conerter crcut usng the lossless nductor DR current sensng crcut wth nductors coupled to each other s also shown n Fg. 4 (a, together wth the tme-doman nductor wndng current waeforms [], 1 (t and (t, as well as oltage waeforms across the sensng 1 8
capactors, 1 (t and (t, n Fg. 4(b. It s obous that the sensed capactor oltage s no longer a replca of the wndng current n coupled nductors, no matter how the tme constants wth the nductor and the r, networ are selected, as the frequency of nductor current s twce the frequency of the oltages across the sensng capactors. In order to extend the lossless nductor DR current sensng technque to applcatons wth coupled nductors, the followng analyss s carred out n s-doman. Accordng to (, (1 r, (6 R (1 + R then (5 becomes or, ( I I ( V 1 + V R 1 + s, (7 ( ( t ( 1( t + ( t R 1 + t. (8 (a (a (b Fgure 3. ossless nductor current sensng wth uncoupled nductors: (a crcut dagram; (b nductor current and capactor oltage waeforms. V V 1 s V I V 1 1 + s + R I + R ( I1( s + I ( I + I 1, (3 and from Fg. 4(a, V1( s V V1, V. (4 1+ sr 1+ sr Hence, the sum of the oltages across the sensng capactors can be obtaned as I1( s + I + V 1( s V R 1+ s. (5 1+ sr R (1 + In the equaton aboe, f r and are selected such that (b Fgure 4. ossless nductor current sensng wth coupled nductors: (a crcut dagram; (b nductor current and capactor oltage waeforms. Equaton (6 s the new tme constant matchng condton for DR based current sensng n coupled nductor conerters, whch dffers from that n uncoupled nductor conerters. It s apparent that the couplng coeffcent has a sgnfcant mpact on the selecton of the r and alues. (7 and (8 show that once (6 s satsfed, the sum of the oltages across the two sensng capactors s the exact replca of the sum of the currents n both wndngs. 9
The concluson expressed n (7 and (8 aboe, was also dscoered n [16, 19]. The dfference s that the tme constant matchng condton obtaned n (6 ncludes the effect of the couplng coeffcent, whch s usually not close to 1 n coupled nductors, and a clear defnton of the leaage nductance gen n [] and (, and drectly measurable wth no need to conert or partton. It should be ponted out that een though the sensed capactor oltage does not represent the nddual nductor wndng current n coupled nductors, the result obtaned from (6, (7 and (8 s stll of sgnfcant practcal mportance. In many applcatons, obtanng the total current nformaton s crtcal to mplement output oltage postonng, or load-lne, and oer-current protecton. 3. Dynamc Modelng of oupled Inductor onerters Fgure 5 shows the equalent crcut of a two-phase coupled nductor buc conerter where the coupled nductors are replaced by the model gen n Fg. (b. If we denote and ph as the swtchng node or phase node oltages (referred to ground, and x1 and x as the oltages (referred to ground at the fcttous nodes x 1 and x, then from Fg. 5, we hae, where the symbol s used to denote the aerage of the correspondng arable. Fg. 6(a shows the same equalent crcut model of the conerter n Fg. 5, whle ts equalent aeragng crcut s gen n Fg. 6(b. Assumng d D + d ˆ( t, V + ˆ ( t and n Vn, where d ˆ( t s the small-sgnal duty cycle perturbaton and ˆ ( t s the resultng perturbaton on the output oltage, then the control-to-output transfer functon of the power stage, G d (s, can be dered as V Gd dˆ( s, (1 RVn (1 + (1 + sr R + R ( s + ξω s + ω where R + R ω Δ ( 1+, and R + R ( R + R ξ Δ + (1 + ( R ω // R R + are the resonant frequency and dampng factor, respectely. If R >> R and R >> R, whch s the case n most applcatons, (1 can be smplfed as and, 1 x1 x Fgure 5. An equalent crcut of a two-phase coupled nductor buc conerter. ph ph V, (9 V + + 1 ( ( + + ph ph V. (1 V In order to dere the dynamc model and the control-tooutput small sgnal transfer functon of the conerter n Fg. 5, aeragng modelng approach [3] s adopted n the followng dscusson. Accordng to (1, the aerage phase node oltage,, can be obtaned as x x (, (11 x1 x d 1+ n V Gd dˆ( s s + s R V (1 + (1 + s R n + (1 + ( R R +, (13 + (1 + and the resonant frequency of the conerter power stage transfer functon can be smplfed as (1 + ω. (14 (1 For comparson purposes, the resonant frequency of a two-phase buc conerter wth the nductors uncoupled (, ω ds, s ω ds ds, (15 where ds s the nductance of the uncoupled nductors. 1
+ - x1 x - + x 1 R R (a 1 R R Output current: I A Self nductance: 1 µh ouplng coeffcent:.6 eaage nductance:.64 µh Inductor DR: R 1 mω Sensng networ parameter: r 4 Ω,.1 µf Per-phase swtchng frequency: f s 5 Hz In ths case, the sensng tme-constant, r.4 ms, was chosen accordng to (6 to match the tme constant of the coupled nductor, /R (1+.4 ms. The smulated results of the nddual phase currents, 1 and, and summed phase current,, sensng capactor oltages, 1 and, and ther sum,, are shown n Fg. 7(a n steady-state operaton, and n Fg. 7(b durng load transent, respectely. It s obous that the sum of the oltages on the nddual sensng capactors replcates the summed phase currents exactly n both stuatons, and the scalng factor s R 1mΩ. (b Fgure 6. (a A smplfed conerter crcut model wth coupled nductors, and (b ts aerage crcut model. In order to compare the mult-phase power conerters wth and wthout the nductors coupled, the followng two dfferent desgn cases can be consdered: a desgn based on equal magnetc component sze, and b desgn based on equal transent respond speed. The former mples that ds of the uncoupled nductor equals the self nductance of each wndng n the coupled one, or ds, whle the latter means ds equals the leaage nductance of the coupled nductor, or ds. 1 c c1 c (a From (14 and (15, t s obsered that, regardless of ds or ds, there exsts: ω > ω ds (16 1 for any >. Equaton (16 means that, due to couplng, the resonant frequency n conerters wth coupled nductors s always greater than that n conerters wth uncoupled nductors, regardless of ds or ds. Ths concluson mples that t s possble to desgn a control loop wth hgher crossoer frequency, or bandwdth, n a regulator utlzng coupled nductors. IV. SIMUATION AND EXPERIMENTA VERIFIATION The DR based total current sensng scheme n coupled nductor conerters as dscussed n the preous secton was erfed through PSpce smulaton. The crcut parameters used n the smulaton, referred to Fg. 4(a, are lsted as follows: Input oltage: V n 1 V Output oltage: V o 1. V c c1 c (b Fgure 7. Smulated results of DR based total current sensng scheme: (a Steady-state; (b durng transents. The same test board as used n [] was used to measure the power conerter control-to-output transfer functon. The controller used on the test board shown n Fg. 8 s IS666, whch utlzes the so-called Robust Rpple Regulator (R 3 modulator. Accordng to the controller manufacturer [4], the small sgnal transfer functon, or modulaton gan, of the modulator, G m (s, s gen by, 11
G dˆ( s m V 5 5 1.5 1 ( V V 3.V 1, (14 comp n n + 5 fs 1.67s + 1 where V comp (s s the error compensator output and f s s the per-phase swtchng frequency of the conerter. The coupled nductor s 174-R3R9A from NE/Ton, whch specfes a typcal self nductance of 31 nh. The measured parameters (wth a short external wre connected to one termnal of each wndng to ft a current probe are 353.5 nh,.6, and 16.7 nh. Fgure 8. Pcture of the two-phase coupled nductor regulator test board. Fg. 9 shows the measured Bode-plot of the transfer functon from V comp (s to V (s. The magntude and phase s. frequency are compared wth the model predcton, G m (sg d (s, whch s also plotted n the same fgure. The followng measured parameters and operaton condtons were used n the theoretcal calculaton and expermental measurement: 6 Input oltage: V n 1 V Output oltage: V 1.15 V Output current: I A Inductor DR: R.4 mω Output capactors: 36x µf (M, R c.mω Measured equalent parastc loop resstance: 6.4 mω Per-phase swtchng frequency: f s 47 Hz Based on the nformaton aboe, the load resstance R 57.5 mω. The test board uses all ceramc capactors n parallel as the output capactor. The equalent ESR R s ery low. Snce R >>R and R >>R, (13 was adopted n the calculaton of the Bode-plot n Fg. 9. It can be seen n Fg. 9 that the low-frequency gan and phase between the predcted and measured results match well, whle the measured resonant frequency, ω, s slghtly hgher than predcted, whch s most probably the outcome of the capactance reducton effect under output bas oltage. The actual capactance of the hgh-alue X5R rated M output capactors s usually reduced by at least 5% under dc bas oltage. Wth the ncluson of the equalent parastc onboard loop resstance n the model, whch was actually measured wth the njecton of a current source, the resonant pea also matches well. Beyond ω, the predcted and measured results follow the same general trend, but the measured results roll off at a hgher frequency, whch s partly due to the same capactance reducton phenomenon and could be partly attrbuted to the naccuracy noled n the transfer functon of the complex modulator used n the controller. Fg. 1 also shows the Bode-plots of the conerter power stage control-to-output transfer functons, G d (s, wth and wthout the nductors coupled usng the same parameters gen aboe. The gan peas at ~1.3 Hz wth coupled nductors. Wth dscrete nductors, t peas at ~16.1 Hz when ds 16.7 nh and at ~1.5 Hz when ds 353.5 nh, both of whch s lower than that wth the coupled one. It confrms the predcton made n (16. 6 3 oupled nductors 3 Measured 3 3 6 6 9 Dscrete nductors ds 353.5nH Dscrete nductors ds 16.7nH 9 1 Predcted 1 15 15 18 1 1.1 3 1.1 4 1.1 5 1.1 6 fhz ( Fgure 9. Measured and predcted Bode-plots of the transfer functon G m(sg d(s 18 1 1. 1 3 1. 1 4 1. 1 5 1. 1 6 f Fgure 1. Bode-plots of the control-to-output transfer functons of the power stage wth and wthout nductors coupled. 1
V. ONUSIONS In ths paper, the preously deeloped symmetrcal coupled nductor model s frst extended to nclude the DR. The extended model s then used to analyze the nfluence of the couplng coeffcent on the DR based current sensng schemes popularly used n mult-phase swtchng regulators. It s found that the well-nown tme-constant matchng condton n uncoupled nductor conerters should be modfed to nclude the effect of when nductors are coupled, and only the summaton of the sensed phase currents s ald. Fnally, the proposed coupled nductor model s also used to dere the small-sgnal control-to-output model of the conerters ncorporatng coupled nductors. The deraton shows that the resonant frequency of the power stage s effectely ncreased through couplng, ndcatng n practcal applcatons a hgher bandwdth loop desgn s acheable n regulators ncorporatng coupled nductors. REFERENES [1] S. u, and R.D. Mddlebroo, oupled-nductor and other extensons of a new optmum topology swtchng dc-dc conerter, Adances n swtched-mode Power onerson, Vols. I and II, pp. 331-347. [] S. u, Swtched dc-to-dc conerter wth zero nput and output current rpple, Proc. of IEEE IAS Annual Meetng 1978, pp. 1131-1146. [3] G. W. Wester, An mproed push-pull oltage fed conerter usng a tapped output-flter nductor, Record of IEEE PES 1983, pp. 366-376. [4] S u, and Z. Zhang, "oupled-nductor analyss and desgn," Record of IEEE PES 1986, pp. 655-665. [5] W. hen, G. Hua, D. Sable, and F.. ee, Desgn of hgh effcency, low profle, low oltage conerter wth ntegrated magnetcs, Proc. of IEEE APE 1997, pp. 911-917. [6] D. Masmoc, R. Ercson, and. Gresbach, Modelng of crossregulaton n conerters contanng coupled nductors, Proc. of IEEE APE 1998, pp. 35-356. [7] P.-. Wong, Q. Wu, P. Xu, B. Yang, and F.. ee, Inestgatng couplng nductor n nterleang QSW VRM, Proc. of IEEE APE, pp. 973-978. [8] P.-. Wong, F.. ee, X. Ja, and J.D. an Wy, A noel modelng concept for mult-couplng core structures, Proc. of IEEE APE 1, pp. 1-18. [9] P.-. Wong, P. Xu, B. Yang, and F.. ee, Performance mproements of nterleang VRMs wth couplng nductors, IEEE Trans. on Power Electroncs, ol. 16, no. 4, July 1, pp. 499-57. [1] J.,.R. Sullan, and A Schultz, oupled-nductor desgn optmzaton for fast-response low-oltage dc-dc conerters, Proc. of IEEE APE, pp. 817-83. [11] J., A. Strataos, A. Schultz, and.r. Sullan, Usng coupled nductors to enhance transent performance of mult-phase buc conerters, Proc. of IEEE APE 4, pp. 189-193. [1] A.M. Schultz, and.r. Sullan, Voltage conerter wth coupled nducte wndngs and assocated methods, U.S. Patent 6,36,986, Mar. 6,, Volterra Semconductor orp. [13] A.V. edene, G.G. Guro, and R.M. Porter, Multple power conerter system usng combnng transformers, U.S. Patent 6,545,45 B1, Apr. 8, 3, Adanced Energy Industres, Inc. [14] W. Wu, N. ee, and G. Schuellen, Mult-phase buc conerter desgn wth two-phase coupled nductors, Proc. of IEEE APE 6, pp. 487-49. [15] J. Gallagher, oupled nductors mproe multphase buc effcency, Power Electroncs Technology Magazne, Jan. 6, pp. 36-4. [16] Y. Dong, M. Xu, and F.. ee, "DR current sensng method for acheng adapte oltage postonng (AVP n oltage regulators wth coupled nductors, Record of IEEE PES 6, pp. 1-7. [17] Y. Dong, F.. ee, and M. Xu, Ealuaton of coupled nductor oltage regulators, Proc. of IEEE APE 8, pp. 831-837. [18] Y. Dong, Y. Yang, F.. ee, and M. Xu, The short wndng path coupled nductor oltage regulators, Proc. of IEEE APE 8, pp. 1446-145. [19] S. Xao, W. Qu, T. Wu, and I. Batarseh, Inestgatng effects of magnetzng nductance on coupled-nductor oltage regulators, Proc. of IEEE APE 8, pp. 1569-1574. [] G. Zhu, B. McDonald, and K. Wang, Modelng and analyss of coupled nductors n power conerters, Proc. of IEEE APE 9, pp. 83-89. [1] D. Goder, System to protect swtch mode dc/dc conerters aganst oerload current, U.S. Patent 6,17,814, Oct. 3,, Swtch Power, Inc. [] X. Zhou, P. Xu, and F.. ee, "A hgh power densty, hgh effcency and fast transent oltage regulator module wth a noel current sensng and current sharng technque," Proc. of IEEE APE 1999, pp. 89-94. [3] V. Vorperan, Smplfed analyss of PWM conerters usng the model of the PWM swtch, part I: contnuous current mode, IEEE Trans. on Aerospace and Electronc Systems, ol. 6, no. 3, May 199, pp. 49-496. [4] Prate communcatons wth engneerng staff of Intersl orp., No. 9, aalable upon request. 13