Small Signal Analysis of the LCC-Type Parallel Resonant Converter

I. INTRODUCTION Small Signal Analyi of the LCC-Type Parallel Reonant Converter ISSA BATARSEH, Senior Member, IEEE C. MEGALEMOS Univerity of Central Florida M. SZNAIER, Member, IEEE Pennylvania State Univerity In thi paper, the mall ignal analyi of the LCC-type parallel reonant converter (LCC-PRC) operating in the continuou conduction mode i given. Thi analyi i baed on both the tate-plane diagram, which ha been uccefully ued to obtain the teady tate repone for reonant converter, and the Taylor erie expanion. Applying perturbation directly to the teady tate trajectory, a dicrete mall ignal model for the converter can be derived in term of the input voltage, witching frequency, and the converter tate variable. Baed on thi analyi, cloed-loop form olution for the input-to-output and control-to-output tranfer function are derived. It i hown that the theoretical and computer imulation reult are in full agreement. Manucript received February 7, 1994; revied January, 1995. IEEE Log No. T-AES///0451. Author addree I. Batareh and C. Megalemo, Dept. of Electrical and Computer Engineering, Univerity of Central Florida, Orlando, FL 816, email batareh@pegau.cc.ucf.edu; M. Sznaier, Electrical Engineering Department, Pennylvania State Univerity, Univerity Park, PA 1680. 0018-951/96/$10.00 c 1996 IEEE To achieve the deired ytem tability, the open-loop control-to-output frequency repone i ued to analytically deign the tranfer function of the control circuit. Furthermore, the characterization of the cloed-loop mall ignal performance due to the line voltage i neceary to predetermine the open-loop line-to-output frequency repone. Depending on the complexity of the converter topology and circuit component model ued for the converter power tage, thee frequency repone may be obtained experimentally, numerically, or analytically. Experimental meaurement of the converter frequency repone become neceary when the converter power tage i known a a black box. On the other hand, if the circuit topology a well a the control technique of the converter power tage are known, then the power tage repone can be imulated numerically, thu avoiding the cotly contruction of the converter power tage. However, due to highly intenive computation, thi method i time conuming. Hence, it i worthwhile to derive the analytical olution o that exceive time and expene in the experimental approach can be avoided. It i more efficient to utilize analytical olution for the power tage repone in the ytematic deign of a control circuit. To avoid cotly contruction and debugging of converter prototype baed on the trial and error approach, analytical method to obtain the mall ignal frequency repone for dc-to-dc converter have been developed [1 4]. The analytical mall ignal frequency repone help a circuit deigner chooe a proper compenation circuit or debug the deign of the converter power tage from it teady tate characteritic o that the mall ignal frequency repone of the revied converter and the elected compenation circuit are compatible for attaining the deired ytem tability. For many year, publihed work in thi area ha been focued on the pulewidth modulation (PWM) converter [1 1]. In recent year, however, the mall ignal analyi of dc-to-dc reonant converter have been invetigated by many reearcher in thi field [1 4]. Thu far, the analyi to achieve the mall ignal behavior i limited to econd-order erie and parallel reonant converter (PRC) [1 ], with little work done on high-order reonant converter [, 4]. Thi analyi i complicated in it mathematic and obcure in it phyical inight into the converter operation due to a tringent time-domain analyi. An alternative analytical method i needed to eaily viualize the mall ignal behavior around the converter teady tate trajectory and to allow further generalization to a larger cla of reonant converter. The teady tate analye for numerou reonant converter topologie have been thoroughly invetigated in the literature. The teady tate repone of thee 70 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

converter are known and in mot cae have been repreentedbycloedtrajectorieinthetate-plane diagram [19, 1, 8 1]. Since the tate-plane diagram of reonant converter in general conit of only a few well defined imple geometric curve in the tate-plane, it i poible to develop a generalized computer program from which the frequency repone of a reonant converter can be derived once it tate-plane diagram i pecified. Hence, thi technique can be generalized to any reonant converter topology once it tate-plane trajectory i known. Moreover, by uing the tate-plane diagram approach, more phyical inight into the converter dynamic behavior can be obtained. Thi i becaue the perturbation i done graphically on the tate-plane diagram with the actual diplacement of the converter tate variable, a well a input and control parameter, being oberved. The analyi method wa firt ued to achieve the dicrete mall ignal model of the conventional erie reonant converter (SRC) and PRC under variable witching frequency control in [19] and [] repectively. In thi analyi technique, the initial perturbation on the converter tate variable, the control and ource input, were made on the tate-plane diagram to produce the perturbed tate trajectorie. Then, uing the geometrical relation between the teady tate and the perturbed trajectorie and applying the uperpoition theorem, the perturbation repone can be written a a linear combination of thoe initial perturbation. Oberving the difference between the teady-tate and the perturbed trajectorie that are contructed on the ame tate-plane, we can attain better undertanding of the converter mall ignal behavior. The dicrete mall ignal model of thee reonant converter are developed without the time-averaging of the output waveform. When the converter operate in the continuou conduction mode (CCM), the dicrete mall ignal model can correctly predict mall ignal frequency repone becaue the ame et of the perturbed tate exit at any given time and thee perturbed tate poe their continuitie throughout the converter operation. Thi technique i applied to obtain the frequency repone for the well-known LCC-type PRC. It teady tate analyi ha been thoroughly analyzed in the literature [5 8]. Once the mall ignal model of the converter i obtained, then it cloed-loop compenation can be properly deigned and imulated before actually building the converter. The teady tate analyi and the derivation of the tate-plane diagram for the LCC-type PRC i briefly dicued in Section II. Section III provide the mathematical development of the mall ignal analyi. Uing Taylor erie expanion coupled with the ymmetry in teady tate tate-plane repone, we derive the line-to-output and the control-to-output tranfer function. Conequently, the frequency repone can be computed from thee function. Fig. 1. Simplified circuit of LCC-type PRC. Since it i alway poible to decompoe any order dc-to-dc reonant converter into many decoupled two-dimenional tate-plane by uing tate-variable tranformation technique, the mall ignal analyi preented here i baed on two-dimenional tate-plane diagram [9]. By uing the tate-plane diagram approach, more phyical inight into the converter dynamic behavior can be obtained. Thi i becaue the perturbation i done graphically on the tate-plane diagram with the actual diplacement of the converter tate variable and input and control parameter oberved. To verify our theoretical work, computer imulation reult are preented in Section IV and are compared to the theoretical one. II. STEADY STATE ANALYSIS In order to analytically obtain the mall ignal repone of a reonant converter, it i neceary that it dc operating condition be known. Uing thi teady tate trajectory, a geometrical approach can be ued to derive the mall ignal frequency repone model of the converter. In thi ection, teady tate analyi that i baed on the tate-plane diagram will be briefly dicued. The objective here i to expre the teady tate parameter of the converter in term of the initial witching point of the tate-plane trajectory. Once uch expreion are obtained, then perturbation ignal on thee tate-variable parameter are injected directly into the tate-plane diagram. The idealized circuit of the LCC-type PRC (PRC-LCC) i hown in Fig. 1. The detailed teady tate analyi of the converter can be found in [6 9]. Throughout the analyi, it i aumed that the reonant circuit i lole and the witching device and diode are ideal. When the converter operate under CCM with 50% duty ratio over one witching period T, it can be hown that the effect of witching of tranitor/diode pair can be repreented by an equivalent quare wave voltage ource v (t) with magnitude +V g and V g. Moreover, auming very large L 0, the input current to the full-bridge rectifier, i E (t), may be preented by a dependent current ource of magnitude +I 0 and I 0, depending on whether BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 70

Fig.. Equivalent model of Fig. 1. v cp (t) i greater or le than zero, repectively. Hence, the reultant equivalent circuit model i hown in Fig. from which the following differential equation are obtained, di l (t) dt dv cp (t) = 1 L [v (t) v c (t) v cp (t)] (1) = 1 [i dt C l (t) i E (t)] p () dv c (t) = 1 l (t) dt C () Defining a new tate variable, v c (t)=v cp (t)+v c (t), then (1) () can be rewritten a follow, di l dt = 1 L [v (t) v c (t)] (4) # dv c dt = 1 C T i l (t) i E (t) (1 + C p =C ) where C T i the total capacitance given by 1 C T = 1 C p + 1 C dv c di l = L C T i l i E (1 + C p =C ) v v c From (4) and (5), the tate-plane equation in term of v c (t) andi 1 (t) i given by, (6) can be expreed in term of the normalized tate-plane equation in the following form, where v nc (t)=v c (t)=v g dv nc (t) di nl (t) = i nl (t) i0 ne (t) v n (t) v nc (t) v n (t)=v (t)=v g (5) (6) (7) i nl (t)=z 0 i l (t)=v g i ne (t)=z 0 i E (t)=v g # ine 1 0 (t)=!0 i! ne (t) 0 and the characteritic impedance Z 0, and reonant frequencie and are given by L Z 0 =,! C 0 = p 1 and = p 1 T LCT LC Fig.. Typical tate-plane trajectory for LCC-PRC. It can be hown that the olution of (7) conit of four circular arc in the i nl v nc tate-plane with each ingular point [v n,ine 0 ] correpond to one of the following interval, 8 (+V ng, In0 0 >< ) t 0 t<t 1 [v n,ine 0 ]= (+V ng,+in0 0 ) t 1 t<t 0 + T ==t ( V > ng,+in0 0 ) t t<t ( V ng, In0 0 ) t t<t 0 + T (8) where V ng = v g =V g. In teady tate, V ng =1. Typical i nl v nc tate-plane diagram i illutrated in Fig.. Detailed derivation of thi figure i given in [8]. In thi figure, 1 i the angular diplacement of the trajectory when T 1 i turned on with v cp < 0, i the angular diplacement of the trajectory when T 1 i conducting with v cp > 0, and i the angular diplacement when D 1 i conducting. Thee angle are related to the witching and reonant frequencie by, = 1 + + = T = f 0 f ¼ = ¼ f n (9) where f and f 0 are the witching and reonant frequencie, repectively, and the normalized frequency i defined by f n = f =f 0 with f 0 = =¼. The olution of (7) conit of four circular arc in the v nc i nl tate-plane given by (v nc v n ) +(i nl i ne ) = nmi (10) where i =1,,,and4andV nmi are the radii which are given by ½ Vnm1 for t 0 t<t 1 or t 0 + T = t V nmi = V nm for t 1 t<t 0 + T = or t t<t 0 + T (11) From the tate-plane diagram, it can be hown that q V nm1 = (V ng v nc (t 0 )) +(i nl (t 0 )+I n0 ) (1) V nm = q (i nl (t 1 ) I n0 ) + Vng (1) 704 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

The dc output voltage V 0 i obtained by taking the average value of the reonant capacitor voltage, v cp, to yield V 0 = 1 #!0 [(! 1)V g +Z 0 i l (t 1 )] 0 (14) where 1 = (t t 0 ) The converter gain i obtained by normalizing (14) to give M = V 0 = 1 #!0 [( V g! 1)+i nl (t 1 )] 0 (15) Since the teady tate analyi hown here i only for the PRC-LCC operating in the natural commutation mode, i nl (t 0 ) mut be poitive and it can be expreed a i nl (t 0 )= 1 In0 + In0 1 vnc (t 0 ) I n0 (16) For the converter deign, it i more convenient to expre the output voltage and converter gain in term of the initial witching point v nc (t 0 )ori nl (t 0 )intead of i nl (t 1 ). From the tate-plane diagram given in Fig., it can be hown that the following relation can be obtained, i nl (t 1 )= v nc (t 0 ) In0 0 (17) Subtituting (17) into (15), the converter gain become M = 1!0 # ( 1) v nc (t 0 ) In0 0 (18) Finally, we need to expre the witching angle 1,,, and in term of the initial witching point [v nc (t 0 ),i nl (t 0 )]. Thi can be accomplihed from the tate-plane diagram given in Fig. to obtain the following relation, 1 =tan 1 i nl (t 1 )+I0 n0 v nc (t 1 ) i tan 1 nl (t 0 )+I0 n0 v nc (t 0 ) (19) + = ¼ i tan 1 nl (t 1 ) I0 n0 v nc (t 1 ) i tan 1 nl (t 0 )+I0 n0 1+v nc (t 0 ) (0) Uing (9), (19), and (0), the following functional expreion for can be obtained a follow, = [v nc (t 0 ),i nl (t 0 ),In0 0, = ] (1) For the converter deign, given =,wemay calculate In0 0, M, and once the initial witching point [v nc (t 0 ),i nl (t 0 )] i known. In the following ection, baed on the teady tate equation derived in Section II, we derive the mall ignal model by injecting perturbation to the tate-variable of the tate-plane. III. Fig. 4. Converter block diagram under line and control ignal perturbation. SMALL SIGNAL ANALYSIS A hown in Section II, when the converter repone i in teady tate, the trajectory of it tate variable form a unique cloed contour for every witching period with table dc value. However, the converter trajectory can deviate from thi unique contour due to diturbance on the line and control input. In uch a cae, the trajectory become a perturbed trajectory. Under mall ignal perturbation, the excurion of the perturbed trajectory from the teady tate trajectory can be phyically oberved on the tate-plane diagram [, ]. Phyically, the mall ignal excurion from the teady tate trajectory can be oberved a the perturbed trajectory periodically hrink and expand around the teady tate trajectory at the frequency of perturbation. Major ource of perturbation are due to the input and frequency variation. Fig. 4 how a implified block diagram for a cloed-loop converter ytem under line voltage v g, and control ignal (witching frequency) f, perturbation. Baed on thi model, we derive cloed-form olution for the line-to-output tranfer function H 1 () and control-to-output tranfer function H () which are defined a follow, H 1 () =ˆv 0 ˆv g H () =ˆv 0 ˆf (a) (b) In thi ection, baed on the tate-plane diagram and the generalized analyi preented in [1], the mall ignal analyi for the PRC-LCC when operated in the CCM i derived. By introducing perturbation to the converter tate variable, controlled witching frequency, and the input voltage at the beginning of a half witching period, we can obtain the perturbed repone at the beginning of the next half witching period by uing Taylor erie expanion. Baed on thi technique, a cloed form olution for H 1 () andh () i derived from which we obtain the frequency repone. BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 705

Fig. 5. Perturbed teady tate trajectory of Fig.. Under mall ignal perturbation, we define kth and (k + 1)th a trajectorie in the firt and econd half witching cycle of the tate-plane diagram of Fig. 4, repectively, a hown in Fig. 5. Uing thi figure, the tate-variable at the end of the half witching cycle can be related to the tate-variable at the beginning of the half witching cycle, the normalized voltage V ng, and the normalized witching frequency f n.the beginning and the final normalized tate variable at the kth half witching cycle are defined a follow i nl (k) reonant current at the beginning of the kth half witching cycle, v nc (k) reonant voltage at the beginning of the kth half witching cycle, f n (k) witching frequency for the kth half witching cycle, I n0 (k) output current for the kth half witching cycle, i nl (k + 1) reonant current at the end of kth half witching cycle, v nc (k + 1) reonant voltage at the end of kth half witching cycle, I n0 (k + 1) output current for the (k +1)th half witching cycle. In thi analyi, ˆP(k) andˆp(k + 1) are defined a the perturbation vector at the teady tate turn-on intant of tranitor T 1 and T, repectively, which are defined a follow ˆP(k)=[ˆ{ nl (k)ˆv nc (k)î n0 (k)] T () ˆP(k +1)=[ˆ{ nl (k +1)ˆv nc (k +1)Î n0 (k +1)] T (4) The perturbation vector given in () and (4) repreent the perturbation repone at the end of the econd half and firt half of the witching cycle, repectively. In another word, the perturbation repone at the end of the on-time interval of tranitor T 1,ˆ{ nl (k +1), ˆv nc (k +1) andˆ{ n0 (k) are defined a the perturbation vector ˆP(k +1). Hence, the perturbation tate ˆ{ nl (k) andˆv nc (k) canbe conidered a the initial tate perturbation at the beginning of the on-time interval of tranitor T 1.The perturbation tate ˆ{ nl (t) andˆv nc (t) are continuou time function within the given half witching period. However, only ˆ{ nl (k +1) and ˆv nc (k +1) are ued a the updated perturbation for the following half witching period a hown in Fig. 5. On the other hand, the perturbation due to input voltage and output current, ˆv ng (t) andˆ{ n0 (t), are conidered to be contant within each half witching period and are updated only at the end of the half witching period. Thi i valid under the aumption that the input and output filter time contant are large a compared with the operating witching period T. Moreover, the perturbation vector ˆP(k) i aumed to be mall enough o that when applying Taylor erie expanion econd and higher order term will be neglected a i hown in the next ection. Two et of dicrete tate equation can be obtained geometrically; one from the trajectory in even portion of the witching period and the other in the odd half witching period. Thee et of tate equation can be repreented a two vector-matrix dicrete tate equation having different coefficient matrice. Since the teady tate trajectory i ymmetrical about the origin, pecial tranformation on the perturbed tate variable can be applied to both et of equation to obtain a cloed-form vector-matrix tate equation which i valid in any half witching period. Conequently, mall ignal frequency repone can be calculated from the cloed-form tate equation. The perturbation repone ˆ{ nl (k +1) and ˆv nc (k + 1) are derived by uing the geometry of the tate-plane diagram and the application of Taylor erie expanion. Thee olution can then be ued to obtain the perturbation repone ˆ{ n0 (k + 1) from the output equation which i derived from the circuit topology a hown in (5). Due to the preence of the full-bridge rectifier at the output circuit of the PRC-LCC, the output equation i 0 (t) igivenby di L 0 0 dt = jv cp (t)j i 0 R 0 (5) From (5) and the tate-plane diagram, the dicrete model for the kth half witching cycle, which i located at the firt half of the witching cycle, can be expreed a follow, i nl (k +1)=In0 l (k)+v nm in(¼ + 4 ) (6) v nc (k +1)=v ng (k)+v nm co(¼ + 4 ) (7) In0 0 (k +1)=I0 n0 (k)+ Z # 0!0 L 0 ¼ f n (k) 1 LL0!0 # V nc (k) I 0 n0 (k) ¼R 0 I0 n0 (k) L 0 f n (k) (8) 706 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

In order to find a complete dicrete tate model that applie for both the firt and econd half witching cycle, new tate variable need to be defined [1] in term of the abolute value of the converter tate variable. Hence, a cloed-form olution for the upper and lower witching cycle can be obtained. It i clear from the tate-plane diagram hown in Fig. 5 that the abolute value of the tate variable mut be ued a follow y(k)=ji nl (k)j = i nl (k) =jv nc (k)j = v nc (k) y(k +1)=ji nl (k +1)j = i nl (k +1) x(k +1)=jv nc (k +1)j = v nc (k +1) Baed on thee new variable, the perturbation vector P(k) andp(k + 1) may redefined a hown in (9) and (0), repectively, 1 0 0 Q(k)=P T 6 (k) 40 1 7 05 0 0 1 =[y(k) In0 0 (k)] (9) 1 0 0 Q(k +1)=P T 6 (k +1) 40 1 7 05 0 0 1 =[y(k +1) x(k +1) In0 0 (k +1)] (0) By applying the Taylor erie expanion into the teady tate dicrete model decribed by (6) (8) and neglecting the higher order term, we obtain the repone for ŷ(k +1), ˆx(k +1), and Î n0 (k +1) a hown in (1), (), and (), repectively, ŷ(k +1)= @F 1 @y(k) ŷ(k)+ @F 1 @ ˆ+ @F 1 @I 0 n0 (k) Î0 n0 (k) + @F 1 @f n (k) ˆf n (k)+ @F 1 @V vg (k) ˆV ng (k) (1) ˆx(k +1)= @F @y(k) ŷ(k)+ @F @ ˆ+ @F @I 0 n0 (k) Î0 n0 (k) + @F @f n (k) ˆf n (k)+ @F @V vg (k) ˆV ng (k) () Î 0 n0 (k +1)= @F @y(k) ŷ(k)+ @F @ ˆ+ @F @I 0 n0 (k) Î0 n0 (k) + @F @f n (k) ˆf n (k)+ @F @V vg (k) ˆV ng (k) () The partial derivative hown in (1) () are derived in the Appendix. In a more compact form, (1) () may be repreented a follow, Q(k +1)=AQ(k)+Bˆf n (k)+cˆv ng (k) (4) where a 11 a 1 a 1 6 7 6 A = 4a 1 a a 5, B = 4 a 1 a a b 1 b b 7 6 5, C = 4 c 1 c c 7 5 The coefficient of matrice A, B, andc are given in the Appendix. To find the mall ignal frequency repone of the converter, the dicrete mall ignal model of (4) need to be repreented in the frequency domain uing z-tranformation a hown in (5), Q(z)=(zI A) 1 Bˆf n (z)+(zi A) 1 CˆV ng (z) (5) The line-to-output tranfer function in the z-domain can be obtained by letting ˆf n (z) be zero in (5) to obtain H 1 ()= ˆV n0 (z) ˆV ng (z) = Q p 6 4 7!0 5 [0 0 1][zI A] 1 C where Q p i the quality factor defined by, (6) Q p = R 0 (7) Z 0 Similarly, the control-to-output tranfer function in the z-domain can be obtained by etting ˆV ng (z) tozeroin (5) and uing normalized factor a hown in (8), H ()= ˆV n0 (z) ˆf n (z) = Q p 6 4 7!0 5 [0 0 1][zI A] 1 B (8) The Bode plot, both magnitude and phae, of the line-to-output tranfer function H 1 (), and the control-to-output tranfer function H (), baed on (6) and (8), are hown in Fig. 6 and 7, repectively. IV. SIMULATION RESULTS By uing the Ppice imulation program, the frequency repone for both tranfer function are obtained to verify the theoretical work. Under mall ignal perturbation, the driving ource in the equivalent circuit i amplitude modulated ignal for the line-to-output repone and frequency modulated ignal for the control-to-output repone. A. Line-to-Output Frequency Repone The etup for the line-to-output repone i imple a hown in the imulated circuit of Fig. 8. The BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 707

Fig. 6. Line-to-output frequency repone. Theoretical reult olid line. Simulated reult broken line. Fig. 7. Control-to-output frequency repone. Theoretical reult olid line. Simulated reult broken line. converter value are baed on a deign example for the following value DC input voltage V g =10V Load current I 0 =5A Output voltage V 0 =15V Switching frequency f = 100 khz Capacitor ratio C p =C =05 The converter component value are given a follow R 0 =, L 0 =0mH, L =15 ¹H, C p =16 ¹, C =16 ¹ Source v A (t) i a 50% duty ratio quare wave ignal with witching frequency of 100 khz and amplitude of 10 V. Source v F (t) generate the mall ignal Fig. 8. Simulated circuit for line-to-output repone. perturbation with dc offet of 10 V (v F (t) = 10+ in(¼f m t)), where f m i the mall ignal frequency between 10 Hz and 100 khz. Source v E (t) ia dependent ignal which generate the quare wave 708 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

Fig. 9. Simulated waveform for 4 khz perturbation ignal for Fig. 8. v(1) input (reference) voltage. v(5,) output voltage. Fig. 11. Simulated waveform for 10 khz perturbation ignal for Fig. 10. v(6) input (reference) voltage. v(5,) output voltage. Source E v E (t)=v z (t) where the zener voltage of the diode are et to 10 V. Fig. 11 how the output repone due to the control frequency perturbation at 10 khz. The imulation reult are alo plotted in Fig. 7. Thee reult how that the frequency repone from both the theory and imulation are in good agreement, epecially at low frequencie. Fig. 10. Simulated circuit for control-to-output repone. amplitude modulated ignal with peak amplitude of 10 V (v E (t)=01v F (t)v A (t)). Fig. 9 how the output repone at perturbation ignal of 4 khz. The imulation reult for the line-to-output frequency repone i hown in Fig. 6. B. Control-to-Output Frequency Repone The imulated circuit for the control-to-output repone i hown in Fig. 10 with the ame converter component given in Fig. 9. In Fig. 10, the ource v A (t) i frequency modulated ignal, v F (t) iuedaa mall ignal reference ource, and v E (t) i a dependent ource that generate the quare wave ingle frequency modulated ignal. Thee ource are defined a follow. Source A v A (t)=a 1 in[¼f c t + B in(¼f m t)where A 1 500 V, f c 100 khz (witching frequency), f m Small ignal frequency (10 Hz<f m < 100 khz), f 10%f c (witching frequency deviation), B Modulation index (B = f=f m ) Source F v F (t)=a co(¼f m t)where A 005 V (mall ignal amplitude), f m Perturbation frequency (10 Hz<f m < 100 khz) V. CONCLUSION The mall ignal analyi for the LCC-PRC operating in the CCM ha been preented. The cloed form olution of the mall ignal repone can be obtained from the tate-plane diagram with the application of Taylor erie expanion. Two related tranfer function under line and control perturbation were derived line-to-output and control-to-output tranfer function. Uing z-tranformation, frequency repone for thee two tranfer function were derived. Finally, imulated reult for the magnitude and phae repone of the tranfer function were reported and compared to verify the theoretical approach. It wa hown that the imulated reult are in good agreement with the theoretical reult, epecially at low frequencie. Since accurate knowledge of the tranfer function i more critical cloe to the cro-over frequency (i.e., for magnitude cloe to one), the relatively high error at high frequencie poe no problem. APPENDIX A. Derivation of Partial Derivative for () The coefficient of matrice A, B, andc aociated with ŷ(k +1) @F 1 @y(k) = a 11 = @Vnm @y(k) in 4 + V nm co 4 @ @y(k) @ 1 @y(k) + @ @y(k) BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 709

@F 1 @ = a 1 = @Vnm @ in 4 + V nm co 4 @ @ @ 1 @ + @ @ @F 1 @I n0 (k) = a 1 = 1+ @V nm @In0 0 (k) in 4 + V nm co 4 @F 1 ¼ @f n (k) = b 1 = V nm co 4 Fn @F 1 @V ng (k) = c 1 = @ @In0 0 (k) @ 1 @In0 0 (k) + @ @In0 0 (k) @V nm @V ng (k) in 4 + V nm co 4 @ @V ng (k) @ 1 @V ng (k) + @ @V ng (k) B. Derivation of Partial Derivative for () The coefficient of matrice A, B, andc aociated with ˆx(k +1) @F @y(k) = a 1 = @V nm @y(k) co 4 V nm in 4 ³ @ @y(k) @ 1 @y(k) + @ @y(k) @F @ = a = @V nm @ co 4 V nm in 4 ³ @ @ @ 1 @ + @ @ @F @I n0 (k) = a = @V nm @I 0 n0 (k) co 4 @ V nm in 4 @In0 0 (k) @ 1 @In0 0 (k) + @ @In0 0 (k) @F @f n (k) = b = V nm in 4 ¼ F n @F @V ng (k) = c =1+ @V nm @V ng (k) co 4 V nm in 4 @ @V ng (k) @ 1 @V ng (k) + @ @V ng (k) C. Derivation of Partial Derivative for (4) The coefficient of matrice A, B, andc aociated with În0 0 (k +1) @F @y(k) = a 1 = Z 0 L 0!0 # ³ @ @y(k) @ 1 @y(k) @F @ = a = Z 0 L 0 + L 1 L 0!0 @F @I 0 n0 (k) = a =1 Z 0 L 0 L 1 L 0 @F @V ng (k) = b = Z 0 L 0 @F @V ng (k) = c = Z 0 L 0!0!0 D. Other Parameter # 1 In0 0 (k)!0 #!0!0 # ³ @ @ @ 1 @ # @ I 0 n0 (k) ¼R 0 L 0 F n # ¼ F n # @ @In0 0 (k) @ 1 @In0 0 (k) + ¼R 0 I0 n0 L 0 F n @V ng (k) @ 1 @V ng (k) Below are other parameter relevant for the derivation of the coefficient of () and (). 1) Partial Derivative for Paramter V nm @V nm @y(k) = y(k)+i0 n0 (k) V nm @V nm @ = 1 V nm @V nm @In0 0 (k) = y(k)+i0 n0 (k) V nm @V nm @V ng (k) = +1 V nm ) Partial Derivative for Parameter 1 @ 1 @y(k) = Hxy(k)+In0 0 (k)) @ 1 @ = @ 1 @V ng (k) = nm1 nm I 0 n0 (k) + I0 n0 (k) nm1 I0 n0 (k) (+1)In0 0 (k) V nm1 nm I 0 n0 (k) + I0 n0 (k) (+1) Vnm1 nm1 710 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

@ 1 @I 0 n0 (k) = Vnm1 Vnm1 In0 0 (k) nm1 y(k)+ x (k) In0 0(k) I 0 n0 (k) + I0 n0 (k) nm I 0 n0 (k) nm I 0 n0 (k) + I0 n0 (k) ) Partial Derivative for Parameter @ @y(k) = +1 nm1 @ @ = y(k) I0 n0 (k) Vnm1 @ @y(k) = +1 Vnm1 @ @ = y(k) I0 n0 (k) Vnm1 4) Partial Derivative for Parameter @ @y(k) = In0 0 (k) I0 n0 (k) (y(k)+in0 0 (k)) nm Vnm1 @ @ = In0 0 (k) I0 n0 (k) (+1)In0 0 (k) V nm1 ++I0 n0 (k) @ @I 0 n0 (k) = @ @V ng (k) = nm I0 n0 (k) + In0 0 +1 (k) Vnm1 nm Vnm Vnm1 In0 0 (k) I0 n0 (k) y(k)+ x (k) In0 0(k) nm nm Vnm1 In0 0 (k) I0 n0 (k) (+1) Vnm1 # REFERENCES [1] Lau, B. Y. (1988) Small-ignal input-to-output frequency repone of witching converter. In IEEE Power Electronic Specialit Conference Record, (Apr. 1988), 155 16. [] Brown, A. R., and Middlebrook, R. D. (1981) Sampled data modeling of witching regulator. In IEEE Power Electronic Specialit Conference Record, 1981, 49 69. [] Redl, R., Molnar, B., and Sokal, N. O. (1984) Small-ignal dynamic analyi of regulated cla-e dc/dc converter. In IEEE Power Electronic Specialit Conference Record, 1984, 6 71. [4] Middlebrook, R. D., and Cuk, S. (1976) A general unified approach to modeling witching converter power tage. In IEEE Power Electronic Specialit Conference Record, 1976, 18 4. [5] King, R. J., and Stuart, T. A. (1985) Small-ignal model for the erie reonant converter. IEEE Tranaction on Aeropace and Electronic Sytem, AES-1, (May 1985), 01 19. [6] Panov, Y. V., and Lee, F. C. (1994) A novel control evaluation technique for reonant converter. In IEEE Power Electronic Specialit Conference Record, 1994, 01 08. [7] Verghee, G. C., Elbuluk, M. E., and Kaakian, J. G. (1986) A general approach to ampled-data modeling for power electronic circuit. IEEE Tranaction on Power Electronic, PE-1 (Apr. 1986), 76 89. [8] Vlatkovic, V., Sabate, J., Ridley, R., Lee, F. C., and Cho, B. (199) Small-ignal analyi of the phae-hifted PWM converter. IEEE Tranaction on Power Electronic, 7, 1 (Jan. 199), 18 15. [9] Vorperian, V., Tymerki, R., and Lee, F. C. (1985) Equivalent circuit for reonant and PWM witche. IEEE Tranaction on Power Electronic, PE-4, (Apr. 1985), 05 14. [10] Middlebrook, R. D., and Cuk, S. (198) Advance in Switched-Mode Power Converion, Vol.1 and. Telaco Paadena, CA, 198. [11] Sander, S. R., Noworolki, J. M., Liu, X. Z., and Verghee, G. C. (1990) Generalized averaging method for power converion circuit. In IEEE Power Electronic Specialit Conference Record, June 1990, 40. [1] Vorperian, V. (1990) Simplified analyi of PWM converter uing the model of the PWM witch Part I and II. IEEE Tranaction on Aeropace and Electronic Sytem, 6, (1990), 490 505. [1] Vorperian, V., and Cuk, S. (198) Small ignal analyi of reonant converter. In IEEE Power Electronic Specialit Conference Record, Albuquerque, NM, June 6 9, 198, 69 8. [14] Witulki, A. R., and Erickon, R. W. (1987) Small ignal ac equivalent circuit modelling of the erie reonant converter. In IEEE Power Electronic Specialit Conference Record, 1987, 69 704. BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 711

[15] Siri, K. (1991) Small ignal analyi of reonant converter and control approache for parallel-connected converter ytem. Ph.D. diertation, Univerity of Illinoi at Chicago, Department of Electrical Engineering and Computer Science, Chicago, IL, July 4, 1991. [16] Siri, K., Lee, C. Q., and Fang, S. J. (1990) Frequency repone of reonant converter. In Proceeding of IEEE Indutrial Electronic Conference, Pacific Grove, CA, Nov. 1990, Vol. II, 944 949. [17] Elbuluk, M. E., Verghee, G. C., and Kaakian, J. G. (1988) Sampled-data modeling and digital control of reonant converter. IEEE Tranaction on Power Electronic, (July 1988), 44 54. [18] Vorperian, V. (1989) Approximate mall-ignal analyi of the erie and the parallel reonant converter. IEEE Tranaction on Power Electronic, 4 (Jan. 1989), 15 4. [19] Siri, K., Fang, S. J., and Lee, C. Q. (1991) State-plane approach to frequency repone of reonant converter. IEE Proceeding, Pt.-G (Electronic Circuit and Sytem), 18, 5 (Oct. 1991), 557 56. [0] Witulki, A. R., and Erickon, R. W. (1987) Small ignal ac equivalent circuit modeling of the erie reonant converter. In IEEE Power Electronic Specialit Conference Record, 1987, 69 704. [1] Batareh, I., and Siri, K. (199) Generalized approach to the mall ignal modeling of dc-to-dc reonant converter. IEEE Tranaction on Aeropace and Electronic Sytem, 9, (July 199), 894 909. [] Siri, K., Batareh, I., and Lee, C. Q. (199) Frequency repone for the conventional parallel reonant converter baed on the tate-plane diagram. IEEE Tranaction on Circuit and Sytem, 40, 1(Jan. 199), 4. [] Yang, E. X., and Lee, F. C. (199) Small ignal modeling of LLC-type erie reonant converter. HFPC 9, San Diego, CA, May 7, 199, 186 197. [4] Yang, E. X., Lee, F. C., and Jovanovic, M. (199) Small ignal modeling of LCC-type parallel reonant converter. In IEEE Power Electronic Specialit Conference Record, 199, 941 948. [5] Steigerwald, R. L. (1988) A comparion of half-bridge reonant converter topologie. IEEE Tranaction on Power Electronic, (Apr. 1988), 174 18. [6] Bhat, A. K. S. (199) Analyi and deign of a erie-parallel reonant converter. IEEE Tranaction on Power Electronic, 8, 1 (Jan. 199), 1 11. [7] Batareh, I., Liu, R., Lee, C. Q., and Upadhyay, A. K. (1990) Theoretical and experimental tudie of the LCC-type parallel reonant converter. IEEE Tranaction on Power Electronic, 5, (Apr. 1990), 140 150. [8] Batareh, I., and Lee, C. Q. (1989) High frequency high order parallel reonant converter. IEEE Tranaction on Power Electronic, 6, 4 (Nov. 1989), 485 495. [9] Batareh, I. (1990) Analyi and deign of high order parallel reonant converter. Ph.D. diertation, Univerity of Illinoi at Chicago, June 1990. [0] Oruganti, R., and Lee, F. C. (1985) Reonant power proceing Part I tate-plane analyi. IEEE Tranaction on Indutry Application, (Nov. 1985), 144 1460. [1] Oruganti, R., and Lee, F. C. (1985) Reonant power proceing Part II method of control. IEEE Tranaction on Indutry Application, (Nov. 1985), 1461 1471. 71 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL., NO. APRIL 1996

Ia Batareh (S 8 M 91 SM 9) wa born in Jordan on January 9, 1961. He received hi B.S., M.S., and Ph.D. from the Univerity of Illinoi at Chicago in 198, 1985, and 1990, repectively, all in electrical engineering. He wa a viiting Aitant Profeor in the Electrical Engineering Department at Purdue Univerity Calumet, from 1989 to 1990. In Augut 1991, he joined the Department of Electrical and Computer Engineering at the Univerity of Central Florida in Orlando, a an Aitant Profeor. He i currently engaged in reearch in the area of power electronic. Hi reearch interet include PWM and high frequency reonant converter, power factor correction circuit, and mall ignal modeling of dc-to-dc reonant converter. Dr. Batareh i a member of Tau Beta Pi and Eta Kappa Nu. Preently erving a Chairman of the IEEE-PE/IAS/PEL Orlando Chapter. From 1991 199 he erved a advior to Eta Kappa Nu at the Univerity of Central Florida. He i a regitered Profeional Engineer in Florida. He ha erved on the program committee of IEEE APEC, PESC, IECON and IAS. Mario Sznaier (M XX) received the Ingeniero Electronico and Ingeniero en Sitema de Computacion degree from the Univeridad de la Republica, Uruguay in 198 and 1984, repectively, and the MSEE and Ph.D. degree from the Univerity of Wahington, Seattle, in 1986 and 1989, repectively. He pent 1990 a a Reearch Fellow in Electrical Engineering at California Intitute of Technology. From 1991 to 199 he wa an Aitant Profeor of Electrical Engineering at the Univerity of Central Florida. In 199 he joined the Department of Electrical Engineering at Pennylvania State Univerity, where he i currently an Aitant Profeor. Hi reearch interet include multiobjective robut control, l 1 and H 1 control theory, application of robut control to power electronic, and legged locomotion. In 199 Dr. Sznaier wa awarded a National Science Foundation Reearch Initiation Award for hi reearch on robut control of ytem under mixed time/frequency domain performance pecification. He i a member of SIAM, Tau Beta Pi, and Eta Kappa Nu. From 199 to 199 he erved a the faculty advior to the IEEE Student Branch at UCF, and from 199 he ha been a co-advior to Penn State IEEE Student Branch. He erved in the program committee of the 1994 IEEE CDC and i a member of the Control Sytem Society Conference Editorial Board. C. Magalemo photograph and biography not available. BATARSEH ET AL. SMALL SIGNAL ANALYSIS OF THE LCC-TYPE PARALLEL RESONANT CONVERTER 71