XIV International PhD Workshop OWD 2012, October Lumped Parameter Model of a Resistance Spot Welding DC-DC converter

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1 XIV International PhD Workshop OWD, 3 October Lmped Parameter Model of a Resistance Spot Welding DC-DC converter Martin Petrn, University of Maribor (prof. dr. Drago Dolinar, University of Maribor) Abstract This paper presents a lmped parameter model of a laboratory resistance spot welding (RSW) DC-DC converter. In the paper individal converter components modeling approaches are discssed in detail. In the iron core model of the isolation transformer the nonlinear hysteresis is taken into accont sing the Jiles-Atherton model. The calclated vales of the proposed model are validated and show good agreement with measrements from the laboratory RSW system.. Introdction High power DC-DC converters are sccessflly applied in resistance spot welding systems which are widely sed in the indstry for the efficient material joining process []-[3]. On one hand, the se of new materials in the welding process continosly reqires higher welding crrents and higher otpt powers of the RSW converters, while on the other hand the atomation of the welding process and the manfactring economy dictate size redctions of converters. For example, in the atomotive indstry the RSW transformers are monted on a moving robot arm that reqires high moving dynamics. Conseqently, the size and the weight of the RSW transformer are very important. Besides efficiency and power density of the converter, the DC welding crrent dynamics is also crcial, as the rise time of the DC welding crrent shold be as short as possible []. The analyzed laboratory RSW converter operates with a freqency of khz and can reach the otpt powers p to 6 kva with otpt DC crrents p to ka. De to the high power losses and high power density of the converter, the RSW system is water cooled. The RSW system consists of an inpt 3-phase rectifier that is connected to the grid, H- bridge inverter, welding transformer, and a center tapped fll-wave otpt rectifier, monted on the transformer. The schematic presentation of the laboratory RSW system is shown in Fig.. grid a b c inpt rectifier H-bridge inverter transformer with otpt rectifier D s D s L L R L load Fig.. Schematic presentation of analyzed RSW system. The main idea of this paper is to develop a simple, time efficient and accrate parametrical model of presented laboratory RSW DC-DC converter. The advantage of sch model is that the impact of individal parameters can be analyzed with adeqate model parameter variations. The operation of the RSW system with varios control techniqes can also be analyzed and evalated. This paper is organized as follows. The modeling approach of the laboratory RSW system is presented in Section. Detailed modeling of individal components for the presented RSW converter is discssed in sbsections. Sbsection. deals with the inpt rectifier model. In Sbsection. the model of H-bridge inverter is derived. De to active switching of the H-bridge, control variables are considered, which adeqately vary the topology of the H-bridge circit. In Sbsection.3 a nonlinear lmped parameter model for the transformer and otpt rectifier is presented. The hysteresis nonlinearity is taken into accont with the se of incremental permeabilities, which are determined sing the Jiles-Atherton hysteresis model. The reslts are presented in Section 3, where calclated vales are verified with measrements of a laboratory system for RSW. Finally, Sections 3 and 4 provide discssion and conclsion of the performed work.. Model of a RSW DC-DC converter. Inpt Rectifier The RSW DC-DC converter is connected to the 3-phase grid with a 3-phase fll wave inpt rectifier (Fig. ). Assming an ideal symmetric and independent grid and ideal rectifier diodes, the i w 393

2 otpt voltage of the inpt rectifier can be expressed by (), 3 sin( ) sin( ) 3 () 4 sin( t ) 3 f rect t t where f represents RMS vale of the line voltage and ω can be expressed with the freqency f of the grid voltage (). f () The voltage is applied to the DC-link of the RSW converter.. H-bridge The H-bridge is the only active part of the converter. It spplies the step down transformer with a plse width modlated voltage of the DClink. With the control of the 4 active transistor switches, 6 different trigger combinations of the transistors are possible (Fig. ). Reqired operation of the DC-DC converter is achieved with adeqate seqence of triggered transistor combinations. These combinations change the electric circit topology of the converter. The RSW DC-DC converter is hard switched with three trigger combinations: ) transistors S and S 4 are ON, the otpt voltage of the H-bridge is, ) transistors S and S 3 are ON, the otpt voltage of the H-bridge is -, 3) all transistors are OFF. S D S D energy flows from the DC-link into the transformer. b) All transistors are not condcting (Fig. 3 b)). Hence the crrent throgh the transformer winding is always continos, this forces the crrent trogh diodes D and D 3. While the crrent is still positive and decreasing to zero, the voltage on the primary winding is negative. The energy that was accmlated in the magnetic field of the transformer flows back to the DC-link. c) Transistors S and S 3 are condcting (Fig. 3 c)). Conseqently the voltage on the primary winding is negative and the crrent in the transformer winding is negative and decreasing, the energy flows from the DC-link into the transformer. d) All transistors are not condcting (Fig. 3 d)). This state is similar to state b), bt different diode pair is active becase the crrent in the winding is negative. The crrent flows trogh diodes D and D 4. While the crrent is negative and increasing to zero, the voltage on the primary winding is positive and the accmlated energy in the magnetic field of the transformer flows back to the DC-link. a) b) R on ip primary winding D3 =f( ) R on4 D =f( ) inpt rectifier model S3 D3 S4 D4 H-bridge model transformer model Fig.. The H-bridge sbmodel with link variables. Althogh there are three trigger combinations, the operation states of the converter are for, becase different diode pairs can condct the crrent when all transistors are closed. This depends on the primary crrent vale, as shown in Fig. 3. Diodes D and D 3 are condcting when the crrent is positive, otherwise diodes D and D 4 are condcting. Hence the RSW converter is in one of the following states: a) Transistors S and S 4 are condcting (Fig. 3 a)). Conseqently the voltage on the primary winding is positive and the crrent in the transformer winding is positive and increasing. The c) d) R on3 L ip d D =f( ) Fig.3. Operation states of the converter. R on D4 =f( ) Eqivalent circits for the 4 H-bridge operation states are shown in Fig. 3, where and are 394

3 applied voltage and corresponding crrent of the inpt rectifier,,, and are resistance, capacitance, voltage and crrent of the H-bridge DC-link and R on, R on, R on3 and R on4 are condcting resistances of the transistors S, S, S 3 and S 4, respectively. The crrent is the crrent in the H- bridge, while D, D, D3 and D4 are voltage drops on the H-bridge diodes. The diode drops are modeled with crrent driven voltage sorces which are based on the nonlinear diode characteristics (Fig. 4). D [V] i D [A] 8 x 4 Fig.4. Different diode characteristics, that can be sed for the diode models. Other parameters in the circit belong to transformer primary winding, where and are applied voltage corresponding primary crrent, and are resistance and leakage indctance of the primary winding and the parameter represents the nonlinear indctance of the iron core. Detailed modeling of the transformer is described in the next sbsection. While the primary winding of the transformer is connected to the H-bridge, the electrical parameters of the primary winding also inflence the operation of the H-bridge (Fig. ). In order to derive the model of the H-bridge apart from the transformer model, relations between variables, that link both models, are determined. The inpt model variable of the H- bridge model is the rectifier voltage and the otpt variable is the primary voltage, which will be linked with the transformer model (Fig, ). On the other hand the otpt voltage and the DC-link voltage also depend on the H-bridge crrent, which can be linked with the primary crrent (Fig. 3). The primary crrent depends on the applied primary voltage in the transformer model. With the crrent feedback from the transformer model, the operations of the H-bridge and the transformer model can be linked (Fig. ). The circit presented in Fig. 3 a) is described with (3), (4) and (5). rect irect RDC DC (3) i i dt (4) DC rect H CDC i R R Ron4 DC H on on4 p (5) According to Fig. 3 b)-d) only (5) changes with different states of the converter while (3) and (4) are the same for all of the states. For the states b), c) and d), (5) becomes (6), (7) and (8), respectively. DC p D D3 (6) DC ihron3 p (7) DC p D D4 (8) For the state variable of the H-bridge model DClink voltage and for the model otpt the primary voltage is chosen. With the introdction of additional switching variables, the H-bridge model can be expressed by (9) and (), d dt CDC RDC DC rect DC ip i R i R p DC H on3 H on3 DC3 DC4 (9) () where δ represents the relation between and and and de to H-bridge inverter switching. The switching variable δ always changes the sign when the polarity of the primary voltage switches. At the same time, the primary crrent changes direction in respect to the H-bridge crrent. Conseqently the vale of δ is changing between and -. The vales of variables λ, λ, γ and γ are always or as they switch individal elements into the circit according to the state of the converter (Fig. 5). 4 = 3 = a) c) b) d) Fig.5. The switching variables. The switching variables are determined from the control signals of the transistors. Transistor pairs S,S 4 and S,S 3 are triggered with signals α 4 and α 3, respectively (Fig. 4). The switching variables λ and λ are eqal to the trigger signals α 4 and α 3. Switching variables γ and γ depend on the working principle of passive switches (diodes). The states of the converter do not change random, bt the converter goes from state a) always to state b) and from state c) to state d). For the switching logic of the converter 395

4 therefore a memory element is needed to satisfy the switching behavior of the passive elements. The logic circit for the adeqate operation of the H- bridge model is shown in Fig. 6. S4 S3 S R S Q!Q NOT NOT AND AND OR OR Fig.6. Logic circit of the switching variables in Matlab.. Transformer with the otpt rectifier In the RSW DC-DC converter, a step down transformer with a center tapped fll wave otpt rectifier is sed. The transformer with the rectifier and the load is shown in Fig. 7, otpt rectifier transformer with load N p i s N s L s N s L s i s R s R s D4 d4 3 D3 d3 Ds s s Ds d4 d3 L L R L Fig.7. Electric circit of the transformer, otpt rectifier and load. where, R s, R s, and, L s, L s are resistances and leakage indctances of the windings, N p, N s and N s are nmber of trns of the windings, is the matrix of dynamic indctances of the iron core, Ds and Ds are voltage drops on the rectifier diodes and R L and L L are resistance in indctance of the load. Otpt rectifier diodes are also modeled with crrent driven voltage sorces Ds=f(i s) and Ds=f(i s). The electric circit eqations are of the transformer with the otpt rectifier and load are (). p Ds Ds Rp ip Rs RL R L i s Rs RL R L i s L Di p p Ls LL L L Di s Ls LL L L Di s i L delta delta () With the matrix the iron core nonlinearity is taken into accont on basis of incremental permeability [4], which represents the slope of the magnetisation crve (). B d () H The incremental permeability can be calclated for each time step of the simlation, either from the non-linear characteristic of the iron core B(H), or from varios hysteresis models [5]-[7](for example Jiles-Atherton, Preisach, Tellinen, Takacs,... ). In this work J-A model of the hysteresis is sed [5]. With calclated incremental permeability, individal relctances of the iron core can be calclated from the geometric properties of the core (3) R l m d A (3) Fe where l m is the average length and A Fe is the cross section of the magnetic path. In RSW converters C-type transformer cores are sed. The topology of the magnetic circit for the RSW transformer is shown in Fig. 8, where R m, R m and R δ are relctances of the iron core and the air gaps, F p, F s and F s are magnetomotive forces of the primary and secondary windings and Φ m is flx in the iron core. Φ m R m F p / Φ F s R δ Fp / R δ F s R m Fig.8. Magnetic circit of the C-type iron core. The magnetic sb-circit is solved in each time interval sing matrix method of flxes on the level of flx derivatives. The link between the electric () and magnetic circit is the matrix of incremental indctances (4), where C m represents incidence matrix, N F and N Ψ are matrices which contains the nmber of trns of the windings, and R is matrix of the incremental relctances. T d Ψ m m F L N C R C N (4) The matrix R is calclated from the branch relctances of the magnetic R sb-circit by (5). R C R C (5) T m m In this work a simple magnetic circit topology is sed. The magnetic topology can be also pgraded 396

5 with more magnetic paths if needed [4]. The matrices for the magnetic circit in Fig. 7 are (6)-(9) C C (6) m T ; m N Np Ns N F s, (7) N N (8) T Ψ F, R R R R (9) m δ m Using (4) and matrices (6)-(9), the matrix of incremental indctances can be derived () Np Ns N p NsN p R R R NN p s Ns NsN s L d () R R R NN p s NsNs Ns R R R The diagonal elements of the matrix represent the incremental self-indctance and the off-diagonal elements, the incremental mtal indctances of the windings. The signs of the mtal indctances depend on flx linkages generated by the corresponding windings nder positive crrent conditions. If the flx linkages of the corresponding windings enforce each other, the sign is positive, otherwise the sign is negative. This matrix is always symmetrical and is a link between the vector, which contains derivatives of flx linkages DΨ and the vector, which contains derivatives of winding crrents (). DΨ DI ps () To determine the incremental permeability for the next calclation step sing a nonlinear hysteresis model (), the vector which contains derivatives of magnetic flxes in the iron core is calclated by (). T m m F D ps DΦ C R C N I () Using ()-(5) and ()-(), the magnetic sbcircit is solved in each time interval step of simlation. 3. Reslts The data of the laboratory RSW transformer and the model parameter data are given in Table. De to some measrement restrictions not all parameter measrements were carried ot on the laboratory RSW converter (Tab. ). The missing parameters were adjsted in the model comparing the measrements and model responses. Frthermore, increased vales of winding and load resistances are sed in the model, becase of the skin effect of the crrents in the windings. The transistor condcting vales and diode characteristics were determined from the manfactrer datasheets de to measrements restrictions. The iron core nonlinearity was determined experimentally. The parameters of the J- A hysteresis model were obtained with differential evoltion, sing measrements in the no load condition of the RSW transformer [8],[9]. RSW DC-DC converter data Symbol Measred vale Model vale mf mf / mω N N, N 3 R 3.57 mω 5 mω L / μh R, R 3 45 μω 45 μω L, L 3 / nh R L μω 6 μω L L /.7 μh A Fe. m. m l m 34 mm 34 mm Tab.. The comparison of measred and calclated variables are shown in Fig. 9. The measrements and calclations were carried ot for dty ratio.75. The comparison shows good agreement between calclations and measrements, althogh not all model parameters were experimentally determined. With the fine tning of the model parameters, even better reslts can be achieved. [A] [V] i L [A] measrements calclated x Fig.9. 4Comparison of measred and calclated primary 5 khz crrent and voltage, and welding crrent. br [V] The detailed comparison of the reslts is shown in Fig.. The satration effect of the iron core is

6 visible from both; the calclated and measred primary crrent. This effect reslts in crrent spikes in primary crrent (Fig. ). When comparing the primary voltages, discrepancies are visible. The first reason for this are the parameters of the H- bridge, that were determined mostly on manfactrers datasheets. The given vales in the datasheets often deviate from the actal device vales becase of the manfactring tolerances. The second reason is the deviance of the grid voltages from the ideal voltages that are assmed in the model. The laboratory grid voltages are not ideally sinsoidal and it is possible that the RSW converter impacts the grid becase of the high operating power. Conseqently, the measred voltage ripple is bigger than the calclated (Fig. 9). [A] p [V] [A] [V] meritve 4 5 model - measrements calclated Fig.. Detailed comparison of calclated and measred vales. In the detailed comparison some discrepancies are also visible de to ringing of the measred primary voltage. This effect can be inclded in the model with more complex modeling of the semicondctors, where the dynamic effects are taken into accont. 4. Conclsion The presented model show good agreement with measred vales of an laboratory RSW converter. The advantages of the lmped parameter model is its balance between the simplicity and accracy along with the comptational performance. The impacts of individal parameters on the operation of RSW converter can be easily analyzed. Conseqently, the model can give sefl insight in operation when designing a DC-DC converter. It can also be applicable for monitoring or observer prposes. Frthermore, different control techniqes of the DC-DC converter can be analyzed and evalated with great insight in the operation principle of the converter. With frther development and more complex modeling of the eddy crrent effects and semicondctors, presented model cold also be sed for the optimization prposes of the RWS converters. Bibliography [] W. Li, E. Feng, D. Cerjanec, and G. A. Grzazinski, Energy Consmption in AC and MFDC Resistance Spot Welding, Sheet Metal Welding Conference XI, Sterling Heights, MI, May 4. [] B. Klopčič, D. Dolinar, and G. Štmberger, Advanced control of a resistance spot welding system, IEEE Trans. Power Electron., vol 3, no., pp. 44-5, 8 [3] G. Štmberger, B. Klopčič, K. Deželak, D. Dolinar, Prevention of Iron Core Satration in Mlti-Winding Transformers for DC-DC ers, IEEE Trans. Magn., vol. 46, no., pp ,. [4] A. D. Theocharis, J. Milias-Argitis, and T. Zacharias, Three-phase transformer model inclding magnetic hysteresis and eddy crrents effects, IEEE Trans. Power Delivery, vol. 4, no. 3, pp , Jl. 9. [5] F. R. Flginei, and A. Salvini, Softcompting for the identification of the Jiles-Atherton model parameters, IEEE Trans. Magn., vol. 4, no. 3, pp. -8, Mar. 5. [6] J. Tellinen, "A simple scalar model for magnetic hysteresis," IEEE Trans. Magn., vol.34, no.4, pp.-6, Jl 998 [7] K. Chwastek, A dynamic extension of the Takács model, Physica B, vol. 45, 38-, [8] M. Toman, G. Štmberger, and D. Dolinar, Parameter identification of the Jiles-Atherton hysteresis model sing differential evoltion, IEEE Trans. Magn., vol. 44, no. 6, pp. 98-, 8. [9] G. Štmberger, B. Polajžer, B. Štmberger, M. Toman, and D. Dolinar, Evalation of Experimental Methods for Determining the Magnetically Nonlinear Characteristics of Electromagnetic Devices, IEEE Trans. Magn., vol. 4, no., pp , Oct. 5. Athor: Martin Petrn University of Maribor, FERI Smetanova 7, SI- Maribor, Slovenia tel martin.petrn@ni-mb.si 398