Volume 48, Number 3, 007 3 Describing Function Analysis of the Voltage Source Resonant nverter with Pulse Amplitue Moulation Anrás KELEMEN, Nimró KUTAS Abstract: Pulse amplitue moulation (PAM is a wiely use power control metho of inuction heating inverters with resonant loa. n this case the inverter switching strategy can focus on the achievement of zero-voltage (ZVS an zero-current (ZCS switching moes of the power evices, essential conitions of the high-frequency operation. The optimal switching moe, with minimum switching losses at a given loa power, epens on the resonant loa topology an is achieve by means of fast phase-locke loops. The aim of this paper is the analysis of a high-frequency PAM voltage inverter, assuming operation with optimal switching efine for a serial resonant loa an sinusoial loa current. Presence of lossless snubbers an transistor parasitic capacitors is taken into account an non-linearity of inverter external characteristics is being emonstrate. Describing function an a nonlinear inverter moel are erive in orer to confirm stability of this operation moe. Keywors: voltage-source inverter, ZVS, ZCS, inuction-heating converter, escribing function. NTRODUCTON nuction heating converters are use to prouce a high-intensity alternative magnetic fiel in an inuctor. The workpiece place in the inuctor is heate up by ey-current losses prouce by the magnetic fiel at a frequency efine by technological requirements. Using for this purpose a DClink converter with loa-resonant inverter is a common solution. Main control tasks relate to this process are the switching moe control of the loa-resonant inverter an control of the heating power. A certain egree of coupling is always present between these tasks as a result of the fact that the inverter has to be operate close to the resonance frequency of the loa. Attempts for ecoupling to a certain extent must be supporte by the use of auxiliary converters or pulse ensity moulation []. Auxiliary converters can eliver power to the inverter in iscretize energy packages [] or can perform the magnitue moulation of the DC-link state variables. Moulation of the DC-link current by means of a controlle rectifier is the usual power control solution in case of current inverters with parallel or LCC resonant loa. n case of high-frequency voltage inverters with serial or LLC loas, built with power MOS or GBT transistors, the DC-link voltage can be controlle by means of a twoquarant DC-to-DC converter, resulting in the heating power control by pulse amplitue moulation (PAM of the inverter output voltage. Thus, switching of the power evices can be carrie out in an optimal moe from the point of view of power transfer efficiency an minimization of switching losses. However, the result is poor ynamics of the power control in comparison with control methos by means of the inverter, ue to the large time-constants of the DC-link circuits. Attempts to minimize these time constants result in systems prone to unstable operation ue to the negative incremental input resistance of the converter with constant output power. Therefore, a careful analysis of
4 ACTA ELECTROTEHNCA the power control loop has to be consiere uner the given inverter operation moe. This paper has the aim to construct the moel of the converter with voltage-source inverter (Fig. an pulse amplitue moulation, assuming an optimal switching strategy of the inverter transistors (Fig.. This means, that transistors from an inverter iagonal (S, S 3 or S, S 4 are turne off at the minimum loa current that still allows achievement of ZVS turn-on of the other iagonal, in the presence of lossless snubber capacitors an transistor parasitic rain-source capacitors. The result is zero voltage switching at both turn-on an turn-off an almost zero current turn-off. Analysis of the converter operation is carrie out assuming the presence of a PLL loop, which is able to perform the above control function, an which is much faster than the power controller. n the first part of the paper, the mathematical moel of the converter is erive. n the secon part the stability analysis is performe using the escribing function metho, followe by simulation an practical results. DESCRBNG FUNCTON OF THE NVERTER OUTPUT VOLTAGE The inverter circuit from Fig. has been consiere, with lossless snubber capacitors an series resonant loa moele by a sinusoial current source. Describing function of the inverter output voltage is erive in orer to represent its non-linear epenence on the inverter output current. n the above conitions, waveforms of Fig. : Resonant loa current an half-brige voltage waveforms in case of optimal ZVS. the inverter output current an of the halfbrige voltage are shown in Fig.. The current through the series resonant loa is given by (. i α γ ( Ls Ls max sin ( nvestigation is starte from the moment when S 3 is turne off, assuming that Ci C an uc3 + uc4 U ct. n the (0..γ interval, the i Ls current flows through C 3 an C 4 : uc3 uc4 ubn ils C C t t ( t u BN n the α ( 0, γ interval, α 0 α ϕ γ ϕ ( ( sin ( ( cos ( α γ cos γ From the ( u γ U conition, BN. (3 U. (4 For α (, γ +, u ( cos ( α γ cos γ,(5 BN U + + an in case of α ( γ, +, u 0 BN. The uab U ubn inverter output voltage becomes: Fig. : Voltage-source inverter with series resonant loa.
Volume 48, Number 3, 007 5 U ωc U u AB U ωc + U ( cos( α γ ( cos( α γ + ( 0, γ ( γ, (, + γ ( + γ, (6 The first term of the Fourier ecomposition is erive taking into account the u ( α u ( α symmetry: AB + half-wave U ( + ( ωc b γ sin γ U ( + (7 yiels: AB Using ( U from (4 b sin γ γ (8 For small angles, it is suitable to use γ sin γ, in this case b U ( sin γ U + a + ( sinγ γ (0 U sin γ Thus, the funamental-frequency inverter output voltage an its complex representation become: cos uab A ωt + ϕ, ( j ϕ jωt + AB AB + AB q ( u Ae e u ju e jωt ( j + ϕ uab Re Ae Asinϕ a U sin γ (3 j + ϕ uab q m Ae Acos ϕ (4 b U ( + n the above relations A a + b UAB U ( + (5 sin γ tgϕ, (6 + cos γ an sin γ result from (4, where i + i. (7 Ls Lsq The complex representation of the inverter output current is: γ i e γ + j γ (8 j Ls sin cos ( Using the above equations, the relationship between the inverter output +. (9 voltage an the loa current can be represente by the escribing function Similarly: (3 efine as the ratio of the complex representations of their funamental harmonic components [3], [4]: N ( u U ( j sinγ ( sin γ + j( + cos γ ( sin γ + cos γ AB ils Ls max j U + + (9 The exact escribing function can be erive using the above efinition an (0, (. a U sin γ γ cos γ U b ( γ sin γ (0, yieling (
6 ACTA ELECTROTEHNCA ( ω N U γ sinγ, + + j Ls max ( n (9 an (, γ f ( ω equation (4. For γ 0 it hols that through N ( 4 U. (3 The phase shift between the funamental harmonics of the inverter output voltage an current results ϕ( uab, ils arg ( N( Ls max, ω γ sinγ, (4 ϕ arctg sin γ with graphic representation shown in Fig.3. Fig. 4: The amplitue of the output voltage funamental harmonic versus the amplitue of the output current. Fig. 5: System structure with nonlinear part represente by the escribing function (, an linear part represente by the linear resonant loa. Fig. 3: Phase shift versus the control angle γ in case of optimal control. The nonlinear behavior of the inverter output voltage versus the output current is epicte in Fig. 4. t can be notice that the output voltage amplitue grows in a nonlinear manner with growth of the output current amplitue. The natural question arises, whether it is possible to use the optimal control strategy without generating unesire oscillations, i.e. without getting to even stable limit cycles. n orer to investigate the possibility of limit cycles in the close-loop system from Fig. 5, the nonlinear part of the system is replace by the amplitue an frequency-epenent escribing function (. One shoul bear in min that the nonlinear relationship between the inverter output voltage an current results both from the existence of an intrinsic feeback loop ue to the presence of the lossless snubber capacitors, an from the presence of the fast control loop of the switching angle γ. The transfer function of the serial loa represents the linear part of the system an has the form: Lsmax ( jω jωcs G( jω, UABmax ( jω ω CL s s + jωrc s s (5 where Rs, Ls, C s enote the parameters of the serial resonant inuction heating loa. A limit cycle exists if N ω G jω (6 family of ( (, The Nyquist plot of G( jω an a Ni ( Lsmax, γ ( ω N( Ls max, ω sin γ + j sin γ + j ( + cos γ U (7
Volume 48, Number 3, 007 7 loci is rawn for fixe ω values an variable. A practical case stuy has been mae for an inuction heating converter with power control by PAM an the following inverter an loa circuit parameters (reuce to the primary of a k 4: matching transformer: R s 4 Ω, L s 64 uh, C s 0.04 uf, C 0 nf, U 00 V c. (8 As shown in Fig.6, there exists no intersection between the plots of G( jω an N (, i Lsmax γ ( ω. n conclusion no limit cycle can be preicte by escribing function analysis in the conitions create by the optimal switching angle control. averaging metho. The structure of the converter moel is shown in Fig.7. The input an output filter circuit parameters of the two-quarant DC-DC converter are: L.8 mh, C 00 uf respectively L 800 uh, C 00 uf. (9 The response of the resonant capacitor tank voltage on a step variation of u Csmax set value, obtaine using the Matlab-Simulink moel of the system from Fig.7 with inverter parameters given by (8, is presente in Fig.8. The P voltage controller parameters in this case are:k p 0-5, T i 0.33 ms. The rectifier output voltage is U 500V. r Fig. 6: Loci of G(jω an Ni( Lsmax,jω. THE CONVERTER MODEL The moel of the PAM converter with optimal inverter control has been erive base on the -q moel of the resonant loa [5], the escribing function introuce above an the moel of the DC-DC converter obtaine by the classical state-space Fig. 7: nuction heating converter moel with optimal (ZVS an close to ZCS switching moe control of the inverter. Fig. 8: Response of the system from Fig.7 with circuit parameters (8 an (9 on a step variation of U Csref. One can notice that the ynamics of the system is governe by the DC-DC converter ue to the large time constants of the filters. The waveforms from Fig.9, resulte in case of a resonant loa with high quality factor (Q 6, representing R s.5 Ω, show an unstable behavior of the system. For higher frequencies the quality factor increases, resulting in an increase gain of the voltage control loop. However, weaker magnetic coupling between the inuctor an the workpiece yiels less variation of the quality factor uring the heating process, aing some benefits from the point of view of voltage controller tuning. Tight control of the optimal switching conitions becomes more emaning at high-frequency operation.
8 ACTA ELECTROTEHNCA PLL loop can be observe in the first stage, followe by the voltage control to a 65% set value. The whole process lasts less than 00ms, which is acceptable from technological point of view. CONCLUSONS Fig. 9: U Csref step response of the converter from Fig.7 in case of a high quality factor loa (Q6. The unstable behavior from Fig.9 is associate with the increase gain of the control loop ue to the high quality factor an is not relate to the nonlinear relationship between the inverter output voltage an current. PRACTCAL RESULTS Base on the results presente in this paper, a high-frequency PAM inuctionheating converter has been evelope for jobharening application, with main parameters f N 300 khz, P HF_N 40 kw, U Cs_N 600 V. The inverter is built with power MOS transistor moules. Fast start-up is an important requirement in case of short heating processes [6]. Fig.0 shows the amplitue of the resonant capacitor voltage (u Csmax an the VCO control voltage (u VCO uring start-up of the above converter. Catching process of the uvco ucsmax The paper eals with stability analysis an control system esign for high-frequency inuction heating converters with resonant loa. The optimal switching strategy has been consiere in presence of a lossless snubber. The result is zero voltage turn-on an turnoff, reucing to a minimum the current switche off by the inverter transistors. The escribing function analysis le to the conclusion that the system is stable in case of optimal control. Unstable behavior can be observe in case of high quality factor loas, which is not relate to the optimal control law. Base on these results, a 400kHz an 40kW high frequency converter with pulse amplitue moulation has been evelope. REFERENCES. Fujita, H., Akagi, H., Pulse-Density-Moulate Power Control of a 4 kw, 450 khz Voltage- Source nverter for nuction Melting Applications, EEE Trans. n. Applic., Vol. 3, Mar./Apr. 996, pp. 79-86.. Toorov, T., Majarov, N., New Type of nverter Power Supplies for nuction Heating, HS-98- nternational nuction Heating Seminar, pp. 55-6. 3. Voicu, M.; "Tehnici e analiză a stabilităţii sistemelor automate", Eitura Tehnică Bucureşti - 986. 4. Levine, W.S, eitor; "The control hanbook"; CRC Press, 996. 5. Kelemen, A., Kutasi, N., nuction-heating voltage inverter with hybri LLC resonant loa, the D-Q moel, Pollack Perioica Vol., No., 007, pp. 7 37. 6. Kelemen, A., Kutasi, N., Szekely,., Voltagesource inuction-heating inverter- fast start-up consierations, OPTM006, Vol.., pp.79-85. Fig. 0: Fast start-up of the HF converter.
Volume 48, Number 3, 007 9 Anrás KELEMEN Nimró KUTAS Department of Electrical Engineering Sapientia Hungarian University of Transylvania Calea Sighişoarei /C 540553 Târgu Mureş, Romania Tel: +40-35-403-030 Fax:+40-65-06- pkelemen@rslink.ro kutasi@rslink.ro Anrás Kelemen is university lecturer, at Sapientia Hungarian University of Transylvania. His research interests inclue moeling an control of power converters, inuction heating equipments an power transistor rive circuits. Nimró Kutasi is university lecturer, at Sapientia Hungarian University of Transylvania. His research interests are moeling an control of hybri systems, inuction heating equipments an preictive control of hybri systems.