Computation and Analysis of Dielectric Losses in MV Power Electronic Converter Insulation

Transcription

1 6 IEEE Proceedigs of the IEEE Eergy Coversio Cogress & Expo (ECCE USA 6), Milwaukee, WI, USA, September 8-, 6 Computatio ad Aalysis of Dielectric Losses i MV Power Electroic Coverter Isulatio T. Guillod, R. Färber, F. Krismer, C. Frack, J. W. Kolar This material is published i order to provide access to research results of the Power Electroic Systems Laboratory / D-ITET / ETH Zurich. Iteral or persoal use of this material is permitted. However, permissio to reprit/republish this material for advertisig or promotioal purposes or for creatig ew collective works for resale or redistributio must be obtaied from the copyright holder. By choosig to view this documet, you agree to all provisios of the copyright laws protectig it.

2 Computatio ad Aalysis of Dielectric Losses i MV Power Electroic Coverter Isulatio Thomas Guillod, Raphael Färber, Floria Krismer, Christia M. Frack, ad Joha W. Kolar Power Electroic Systems Laboratory (PES) High Voltage Laboratory (HVL) ETH Zurich ETH Zurich 89 Zurich, Switzerlad 89 Zurich, Switzerlad Abstract The ewly available Medium Voltage (MV) Silico- Carbide (SiC) devices eable a great extesio of the desig space of MV iverters. This icludes the utilizatio of uprecedeted blockig voltages, higher switchig frequecies, higher commutatio speeds, ad high temperature operatio. However, all these factors cosiderably icrease the isulatio stress. This paper details the computatio of dielectric losses, which are directly related to the isulatio stress ad ca be used for the isulatio desig ad diagostic. After a review of the method used to compute dielectric losses, scalable aalytical expressios are derived for the losses produced by PWM waveforms of DC- DC, DC-AC, ad multilevel DC-AC iverters. Fially, a Medium- Frequecy (MF) trasformer is aalyzed ad the impacts of the isulatio material ad the operatig temperature o the dielectric losses are discussed. It is foud that the isulatio losses ca represet a sigificat share (7 %) of the total trasformer losses. Idex Terms Power Electroics, Dielectrics ad Electrical Isulatio, Dielectric Losses, Medium-Frequecy Trasformers, Medium-Voltage Trasformers. I. INTRODUCTION The eed to itegrate reewable eergy sources ito the MV grid, to supply large loads from the MV grid, or to cotrol the power flow, has led to a rapid growth of the research ad idustry iterest for MV coected power electroic iverters, usig MF eergy coversio [] [3]. The desig of such coverters is ofte examied at system level [3], [4] ad compoet level [5] [8] while the isulatio coordiatio is ofte eglected [9], []. Accordig to [], isulatio failures, for example i coected magetic compoets, are already challegig i Low-Frequecy (LF) systems. I [] [4], it is show that the fast voltage trasiets geerated by power coverters further icrease the stresses applied to a dry-type isulatio system, leadig to icreased failure rates. I this cotext, the usage of ew SiC switches, which achieve higher voltage capabilities, higher switchig frequecies, ad higher switchig speeds, is particularly critical [], [7]. Moreover, due to the high power desity, elevated operatig temperatures usually apply to power coverters. For these reasos, the isulatio used i such iverters requires a careful examiatio i order to obtai a reliable system but also for achievig good thermal ad EMI performaces. Literature related to the evaluatio of isulatio stresses proposes the examiatio of partial discharges, breakdow voltages, field distributios, dielectric losses, etc. [9], [3] [5]. Amog these methods, the ispectio of dielectric losses is of particular iterest, sice dielectric loss desities are foud to be related to the lifetime ad the reliability of isulatio systems [4] [7] ad because the calculatio of dielectric losses is easier tha the computatio of partial discharge activities or breakdow voltages. Cosequetly, eve if the dielectric losses are usually small compared to the total losses of a coverter [9], their computatio is still very useful for classifyig isulatio stresses ad materials. Moreover, the olie or offlie moitorig of the isulatio losses allows the detectio of isulatio degradatio ad represets a valuable diagostic tool [6], [8], [9]. This paper examies i detail the computatio of isulatio losses i coverters. Fig. presets a typical workflow for the computatio of the losses where the depedecies betwee the differet parameters are show. The computatio of dielectric losses cosiders the applied waveforms, the geometry, the temperature distributio, the electric field, ad the material properties [5] [7]. The temperature distributio liks the computatio of the dielectric losses to the other losses (iduced currets, hysteresis, etc.) preset i the cosidered compoet. This paper is orgaized as follows. After the defiitio of the dielectric losses computatio i Sectio II, simple ad scalable expressios are derived for the evaluatio of the losses for typical coverter waveforms i Sectio III: PWM with costat duty cycle, siusoidal modulatio, ad multilevel siusoidal modulatio. Fially, i Sectio IV, the preseted aalysis method is applied to a MV/MF trasformer i order to evaluate the isulatio stress. II. DIELECTRIC LOSSES COMPUTATION The dielectric respose of a solid isulatio material depeds o various microphysical processes such as polarizatio (e.g. electroic, atomic, dipolar), coductio, charge trappig, etc. [5], [6]. I this paper, the polarizatio effects are cosidered to be liear with respect to the electric field, which holds approximately true for most of the materials used for MV isulatios [5], [7], []. With these assumptios, the dielectric respose of a isotropic material is fully described i the frequecy domai by the complex permittivity [5], [7], [9]: ε ( f,t ) = ε ( ε ( f,t ) jε ( f,t )), () which is depedet o the frequecy f ad the operatig temperature T. This model is ot applicable for modelig the respose of a material to a DC electric field. However, the DC coductio losses ca be eglected sice the AC losses are much larger at MF. Furthermore, this model is ot able to predict the oliearities ad the additioal losses appearig above the partial discharge iceptio voltage [].

3 FEM Simulatios Topology T [K] Measuremets P [kw/dm3] V/I Waveforms Material Sample Dielectric Losses Temperature P [kw/dm3] Copper ad Core Losses Time MF Trasformer [Novocotrol Alpha-A] Permittivity MEGA lik] T [K] E [kv/mm] [PES Electric Field f [Hz] Fig.. Workflow for the computatio of dielectric losses iside isulatio systems. The procedure is here show for a 4 kv MV/MF trasformer [] but ca be adapted to busbars, semicoductor packages, filters, etc. ε ta δ =, ε ε + ε ε Àε C ESR = C ε C, ε ε ε /ε ε Àε R ESR =. ε + ε f C f C ESR () (3) (4) PWM Voltage Switchig Trasitio..9 Dc/fs Vp.5 tr... /(fs) /(fs) Time. % - 9% ramp VDC Vp/VDC For practical applicatios, the real part of the permittivity (ε ) is ofte give together with the loss taget (ta δ). A equivalet circuit cosistig of the series coectio of a capacitor (C ESR ) ad a resistor (R ESR ) is also a commo model. The correspodig parameters are defied as: low-pass tr Time tr Fig.. PWM sigal with a costat duty cycle. Two differet approximatios of the switchig trasitio (rise time t r ): the ramp fuctio ad the step respose of a first order low-pass filter. III. D IELECTRIC L OSSES WITH PWM V OLTAGES The loss taget is a figure of merit for capacitors (losses compared to the stored eergy). Sice eergy storage is ot a desirable property of isulatio materials, the loss taget is ot a figure of merit for isulatio materials. Models based o the series equivalet circuit are frequecy-depedet eve for a costat complex permittivity. This meas that these models caot easily be used i the time domai ad lead to complex expressios for the dielectric losses. For these reasos, the complex permittivity represetatio i the frequecy domai is used i this paper. With the help of the complex permittivity, it is possible to express the time-averaged losses of a isulatio system excited with a periodic electric field as: Ñ µx P= ε ε f, T f E,RMS dv, (5) V A. PWM with Costat Duty Cycle where V is the isulatio volume, f the fudametal frequecy, ad E,RMS the RMS Fourier series coefficiets of the electric field orm. I case of a homogeeous ad isothermal dielectric placed betwee two electrodes, the expressio for the losses ca be simplified to: P= X P = C Expressio (6) facilitates the calculatio of the dielectric losses by meas of umerical computatios. It is, however, very difficult to idetify the ifluece of the differet parame ters, sice the frequecy-depedet material parameter ε f ad the amplitude of the voltage harmoic compoets appear i the product term of the ifiite sum. I order to fid a closed-form solutio, a costat value of ε f is cosidered i Subsectios III-A ad III-B, which, accordig to [], [] is a admissible assumptio for certai isulatio materials (i the cosidered MF rage), especially low-loss materials. With this hypothesis, ε f ca be take out of the sum (6), allowig the decouplig of the impact of the material from the excitatio. Afterwards, i Subsectio III-C, the impact of the frequecy depedece of the permittivity is aalyzed. X ε f, T f V,RMS, (6) where C is the vacuum capacitace (computed with ε ) associated with the geometry ad V,RMS are the RMS Fourier series coefficiets of the applied voltage. This expressio ca be exteded to systems with more tha two electrodes by usig the capacitace matrix. I steady state operatio, typical DC-DC coverters geerate PWM voltages with a costat duty cycle, accordig to Fig. [], [9]. Fig. proposes two approximatios for the switchig trasitio: the commoly used ramp fuctio ad the step respose of a first order low-pass filter. Sice the proposed calculatio of the dielectric losses is coducted i the frequecy domai, the ramp fuctio is foud to be less suitable sice it geerates a spectrum with side lobes. For this reaso, the step respose of the followig low-pass filter is used: l.9. G lp f =, fc =, (7) f t r +j fc

4 V,RMS /V,RMS Voltage Harmoics Voltage Harmoics (db voltage ) Power Harmoics (db power ) evelope ~/ = f/f s V,dB = log(v,rms /V,RMS) db/dec f c/f s V,RMS -4 db/dec 4 6 = f/f s P, db = log(p /P) db/dec f c/f s P -3 db/dec P,c /P [p.u.] Power Cumulative Sum = f/f s (c) = f/f s (d) P f c/f s P,c P Fig. 3. Voltage harmoics for PWM sigal with a costat duty cycle, evelope of the voltage harmoics (i db voltage ), (c) evelope of the power losses harmoics (i db power ), ad (d) cumulative sum of the power harmoics. The followig parameters are used: f s = khz, t r = s, ad D c =.5. With the chose duty cycle, oly the odd harmoics are o-zero. where t r is the % 9% rise time, cf. Fig.. The correspodig harmoic compoets (Fourier series) of the PWM voltage ca be computed as: ( ) si(d c ) Glp ( V,RMS = V DC fs ), (8) where D c is the duty cycle, V DC the amplitude, ad f s the switchig frequecy. From the voltage harmoics, the spectral compoets of the losses (cf. (6)) ca be expressed: P = ( ε C )( fs ) V,RMS. (9) The dielectric losses ca be computed as the sum of the spectral compoets: P = lim P,c, P,c = P. () = where P,c is the partial losses of the first harmoics ad P the total losses. Fig. 3 depicts the voltage harmoics (cf. (8)), the losses spectral compoets (cf. (9)), ad the cumulative sum of the losses (cf. ()). The voltage ad power harmoics are proportioal to / for frequecies below the corer frequecy. Therefore, the sum () oly coverges after the corer frequecy of the low-pass filter. This implies that a ifiitesimally short switchig trasitio would lead to ifiite losses. Therefore a model of the switchig trasitio is required (cf. (7)). Moreover, the sum P is much greater tha the fudametal harmoic P, showig that a fudametal frequecy aalysis is iappropriate. The calculatio of () is computatioally itesive sice it requires the summatio of may harmoics. Therefore, a closed-form approximatio is proposed (with the hypothesis ε ( f ) = cost.): P = ( ε C VDC) P, () P f ( s e γ ) l f c si(d c ), () f s where γ.57 is the Euler-Mascheroi costat ad P the ormalized losses. The derivatio of () requires elaborate calculatios, which are give i Appedix A. The accuracy of the approximatio is evaluated i Appedix C ad shows that the approximatio is valid (.% error). The error is also small (.5% error) if a ramp fuctio is used i place of the fs [Hz] Frequecy / Rise Time k k 5.5 D c =.5 t r f s > % k 3.5 μ t r [s] P [/s] P [/s] Duty Cycle f s = khz t r = s D c [p.u.] Fig. 4. Impact of the switchig frequecy, the rise time, ad the duty cycle o the dielectric losses. The excitatio is a PWM sigal with a costat duty cycle. All the power values are ormalized with respect to ε C V DC while the used parameters are give i the figures. TABLE I PWM WITH CONSTANT DUTY CYCLE. Param. Impact o P f s P f s l ( ) cost./f s t r P l(cost./t r ) D c P cost. V DC P V DC ε P ε low-pass filter step respose. This idicates that a exact model of the switchig trasitio is ot ecessarily required ad that the model preseted i (7) is valid. Fig. 4 depicts the results of () for differet rise times ad switchig frequecies. It ca be see that the impact of the frequecy is very strog while the losses icrease oly logarithmically with respect to the reciprocal of the rise time. Fig. 4 shows that the duty cycle has oly a mior impact o the dielectric losses. The impact of the differet parameters o the losses are summarized i Tab. I. B. PWM with Siusoidal Modulatio Sice may MV coverters are coected to the grid, it is also importat to evaluate the isulatio losses with LF ad MF excitatios (mixed-frequecy voltage stress) [3], [9], [8]. Sice the corer frequecy (cf. (7)) is much greater tha the grid frequecy, the umerical computatio of the dielectric losses requires a very high umber of harmoics (more tha 5 for a grid frequecy of 5Hz with rise times below s). This, especially, leads to a computatioally

5 Vp/VDC [p.u.] Full Bridge Iverter M i V p /f s /f s V 5 5 Time [ms] Multilevel Iverter Mi V V p c+ 5 5 Time [ms] TABLE II PWM WITH SINUSOIDAL MODULATION. Param. Impact o P Impact o P harm f s P cost. P f s l ( ) cost./f s f g P f g P cost. t r P cost. P l(cost./t r ) M i P M i P cost. c (f s = cost.) P cost. P (/ c )l(cost./ c ) c (f e = cost.) P cost. P / c Fig. 5. PWM waveform produced by a full-bridge iverter. Multilevel PWM waveform produced by c cascaded full-bridges (show for c = ). itesive situatio whe a volume itegratio of the losses is required (cf. (5)). It has bee foud that the scope of applicatio of () ca be exteded to iverters with siusoidal modulatio by meas of local averagig. For the voltage waveforms produced by a sigle-phase full-bridge iverter, cf. Fig. 5, the followig approximatio is derived (with the hypothesis ε ( f ) = cost.): P = P + P harm = ( ε C V DC)( P + P harm), (3) P = ( f g ) ( M i ) = f g M i, (4) P harm lim ɛ ɛ +ɛ D c (ξ) = M i si ( ( fe e γ )) l f c si(d c (ξ)) dξ, (5) f e ( ) ξ, f e = f s, (6) where f g is the grid frequecy, f s the switchig frequecy of the semicoductors, f e the effective switchig frequecy of the output AC sigal, ad M i the modulatio idex. Due to the limited voltage capabilities of MV semicoductors, multilevel coverter structures are ofte used [9], []. Fig. 5 shows a typical multilevel waveform produced by c cascaded full-bridges (with phase-shifted PWM carriers) producig c + levels. The local averagig of () is also applicable to multilevel waveforms where the effective switchig frequecy is f e = c f s. The local duty cycle, D c (ξ), is a piecewise defied fuctio. A case differetiatio is required due to the fact that, for low modulatio idexes, oly a reduced set of the available voltage levels is used. For this reaso, a simple expressio for D c (ξ) does ot exist. The accuracy of these approximatios is evaluated i Appedix C ad shows that the local averagig of the duty cycle is valid for both sigle ad cascaded iverters (3.4% error). Figs. 6 ad depict the losses for differet modulatio idexes. The losses at the grid frequecy are proportioal to the square of the modulatio idex ad idepedet of the umber of levels. The losses of the higher harmoics are ot strogly depedet o the modulatio idex. It has to be oted that P harm is much greater tha P. With c cascaded full-bridges, P harm oscillates c times whe evaluated over the modulatio idex. As explaied above, this is due to the piecewise defiitio of D c (ξ). Fig. 6(c) shows that the depedece of the losses o the switchig frequecy is similar to the case show i Fig. 4. The obtaied results reveal that the dielectric losses of the higher harmoics ca be greatly reduced with V DC P V DC P V DC ε P ε P ε multilevel iverters. This is due to the reductio of the voltage steps produced by multilevel waveforms, which is, however, mitigated by the icrease of the effective switchig frequecy. If the umber of levels is icreased with a costat effective switchig frequecy, a eve greater reductio of the losses results. This highlights oe advatage of multilevel iverters for MV applicatios []. The impact of the differet parameters o the losses (P ad P harm ) are summarized i Tab. II for cascaded fullbridges with phase-shifted PWM carriers. C. PWM with Frequecy-Depedet Materials Util ow, a costat ε ( f ) has bee assumed. This assumptio, however, may be iaccurate for some materials, especially for epoxy resis which are frequetly used for MV isulatios. I this paper, a typical MV isulatio epoxy resi has bee chose (Damisol 348, with glass trasitio temperature: T g = 36 C [3]). Fig. 7 shows the measured ε ad ε for the chose resi where the material parameters are strogly frequecy ad temperature depedet. The losses are particularly high for temperatures ear the glass trasitio temperature [4], [5]. The measuremets are doe with a disc-shaped specime usig a Novocotrol Alpha-A Aalyzer (cf. Fig. ) [6]. I Subsectios III-A ad III-B, it has bee show that the ifluece of the duty cycle ad the modulatio idex o the losses is moderate. For this reaso, a sigal with a costat duty cycle (D c =.5) is chose. With this assumptio, a closed-form approximatio ca be foud: P ( C V )( DC P add + P it), (7) P add = f s l( e γ) ε ( ) f s, (8) P it = f s l(f c) l(f s) ε ( f ) d ( l ( f )). (9) } {{ } ε it The derivatio of (7) is give i Appedix B. The proposed formula requires the measuremet of ε ( f ), which is ot always available ad difficult to measure (due to ε ε ) [6], [4]. However, the real ad imagiary parts of the permittivity are liked by the Kramers-Kroig relatios [5], [7]. These relatios ca be locally expressed as: ε ( f ) ε ( f ) l ( f ). () It has to be oted that this relatio is iaccurate for

6 Modulatio Idex / P Modulatio Idex / P harm Mi [p.u.].7 c = 4 3 c = [,5]. c = fg = 5 Hz fs = khz tr = s P harm [/s] fg = 5 Hz fs = khz tr = s P harm [/s] P [/s] Frequecy / P harm c = 3. c = c = 5 c = 4 c = 3 3 c = c = 3 3 c = 5 Mi [p.u.] fg = 5 Hz tr = s Mi =...7 k k k fs [Hz] (c) Fig. 6. Impact of the modulatio idex o the dielectric losses of the fudametal harmoic ad of the higher harmoics. (c) Impact of the switchig frequecy o the dielectric losses of the higher harmoics. The excitatio is a PWM sigal with siusoidal modulatio produced by c cascaded while the used parameters are give i the figures. full-bridges. All the power values are ormalized with respect to ε C VDC k k 3. k Frequecy [Hz]. k 3. M 45 k k Frequecy [Hz] M khz. khz ε [p.u] ε [p.u] ε [p.u] MHz Tg khz khz khz MHz ε from measuremets khz ε from measuremets.5 95 Temperature [ C] ε from measuremets ε from measuremets ε [p.u] Temperature [ C] Temperature [ C] (c) Tg Temperature [ C] (d) Fig. 7. ε f, T ad ε f, T measured for the epoxy resi Damisol 348 for typical operatig temperatures. (c) Measured ε f, T ad (d) ε f, T for a exteded temperature rage icludig the glass trasitio temperature Tg = 36 C. ε at T = C 3.65 ε at T = C.5 ε ε [p.u] ε [p.u] 3.6 ε diff fs 3.5 fc 3.45 k k k Frequecy [Hz] ε fs ε it fc ε ( f ) l( f ). M k k k Frequecy [Hz] M Fig. 8. Measured ε f ad ε f at C (blue curves). A approximatio of ε f, computed with the Kramers-Kroig relatios, is also show (red curve). The area ε (cf. (9)) ad the differece εdiff it (cf. (3)) are figures of merit for the losses produced by a material. quatifyig the loss peak associated with the glass trasitio temperature of the material (cf. Fig. 7). However, for the chose resi, the glass trasitio temperature is above the typical operatig temperature of power electroic compoets such that the Kramers-Kroig relatios ca be used. With (), the approximatio (7) ca be rewritte as: P C VDC P add + P it, () ε f, () P add = f s l eγ l f f s P it = f s ε f c ε f s. (3) {z } εdiff The area εit (cf. (9)) ad the differece εdiff (cf. (3)) are figures of merit for the losses produced by a material ad are show i Fig. 8. The measured imagiary part of the permittivity is compared to the value obtaied with (), showig that the Kramers-Kroig relatios hold true. The accuracy of the approximatios (7) ad () is evaluated i Appedix C. Both expressios are foud to be valid for the chose resi (7 % error). If the frequecy depedece of the material parameters is eglected (cf. ()), the error is greater tha 5 %, idicatig that the frequecy depedece of the permittivity must be cosidered for accurate computatios. IV. C ASE S TUDY: MV/MF T RANSFORMER With the help of the approximatios proposed above, the dielectric losses of differet compoets ca be examied. Sice it has bee idetified that the stresses are particularly high for MV/MF trasformers [8], [9], this compoet has bee chose. A. Trasformer Desig The proposed method is applied to the 4 kv/4 V, 5 khz, 5 kw MV/MF trasformer of the sigle-stage MV SiC SolidState Trasformer (SST) preseted i []. The trasformer is used iside a Dual Active Bridge (DAB) DC-DC coverter. This trasformer has bee chose sice it is subject to all idetified critical aspects, i.e. i particular high frequecy ad high voltage operatio. The characteristic properties of the trasformer are summarized i Tab. III. The voltage ad curret excitatios of the trasformer are computed usig phase shift modulatio (3 at omial power) [8]. The switchig trasitios feature Zero Voltage Switchig (ZVS) ad are modeled as preseted i Subsectio III-A. The curret depedecy of the switchig speed is cosidered accordig to [7].

7 Temperature Copper ad Core Losses.6.5 RMS Electric Field zoom.5. Dielectric Isulatio Losses zoom T [ C].4.3. P [kw/dm 3 ].5. E [kv/mm].6.4 P [kw/dm 3 ] 7. zoom.5 zoom. 6.. (c). (d) Fig. 9. Temperature distributio, core ad widigs loss desities, (c) RMS value of the electric field, ad (d) dielectric loss desity i the isulatio computed with the measured material parameters. The simulatios are coducted at rated coditios (5kW). Plosess [W] Trasformer Losses ruaway omial widig coolig core T [ C] Trasformer Temperature isulatio 6 core P out [kw] P out [kw] T g widig ruaway omial Fig.. Trasformer loss breakdow for differet load coditios. Widigs ad core hot spot temperatures. The omial load ad the load where a thermal ruaway occurs are show. The glass trasitio temperature of the isulatio material (T g = 36 C) is also idicated. Power TABLE III PARAMETERS OF THE MV/MF TRANSFORMER. 5kW Voltage ±4kV / ±4V Frequecy Rise time Widig 5kHz kw / 5kW / ZVS 6 : 6 turs / litz wire / shell-type Wire 3 µm / 38 7µm Core type ferrite N87 / E-core / 3mm Core widow 3 65mm Volume 3 4mm /.dm 3 Coolig Isulatio coductio / forced covectio / 5 C ambiet epoxy resi / Damisol 348 / 4mm thickess The trasformer has bee desiged followig the guidelies proposed i [6]. The magetic field, the electric field, ad temperature are computed with a D Fiite Elemet Method (FEM), accordig to the workflow preseted i Fig.. The spatial depedece of the losses ad the temperature depedece of the material parameters are cosidered. A iterative process is used i order to lik the temperature distributio with the losses. The widig losses (ski ad proximity effects) are computed accordig to [8], the core losses (improved geeralized Steimetz equatio) accordig to [9], ad the dielectric losses accordig to Subsectio III-C. B. Trasformer Losses The correspodig temperature, electric field, ad loss distributios are depicted i Fig. 9 at rated coditios (5kW). The hot spot temperature (5 C) is clearly located iside the widigs due to the thermal resistace of the isulatio. Due to the ihomogeeous electrode cofiguratio, the electric field is mostly located ear the MV widig. The maximal RMS electric field is i a typical rage for MV isulatio (.5kV/mm) [9]. Sice the isulatio losses are proportioal to the square of the electric field (cf. (5)), the dielectric losses are oly sigificat ear the surface of the medium voltages widig. The maximum dielectric loss desity is eve larger tha the core ad widig loss desities, which idicates that isulatio losses are critical. I order to get a better isight o the trasformer losses, the loss desities depicted i Fig. 9 are itegrated over the trasformer volume. Fig. shows the obtaied losses for differet load coditios ad Fig. depicts the correspodig hot spot temperatures. As expected, the voltage related losses (core, coolig, ad isulatio) are approximately load idepedet. Oly the widig losses, which are related to the curret, are strogly load depedet. At rated coditios (5kW), the total losses are 8W (efficiecy of 99.65%). C. Dielectric Losses At rated coditios (5kW), the total dielectric losses are 4W which represet 7% of the total losses. For the computatio of the dielectric losses the followig remarks should be cosidered: If the dielectric losses are eglected, a error of 7% results for the total losses ad C for the hot spot temperature. This meas that the dielectric losses caot be eglected for a accurate trasformer simulatio. If the dielectric losses are computed with a fudametal frequecy aalysis, a error of 9W (65%) results. Therefore, the actual rectagular voltage waveforms must be cosidered with a model of the switchig trasitio. If the approximatio of a frequecy-idepedet permittivity is doe, a error of 4.5W (3%) results for the isulatio losses. If the dielectric losses are computed without a thermal model, the error reaches 3.5W (5%). This illustrates the importace of the temperature ad frequecy depedece of the permittivity. A closer look at the isulatio losses (cf. Fig. ) shows a moderate icrease of these losses for loads betwee kw ad 33kW. This is due to the temperature variatio (cf. Fig. ) ad to the icreased switchig speed of the semicoductors (ZVS with icreased curret, cf. Tab. III). For loads exceedig 33kW (33% overload), a thermal ruaway occurs. The mechaism of the thermal ruaway

8 ca be explaied as follows. The hot spot temperature of the trasformer icreases whe evaluated over the load. This temperature icrease is maily produced by the widig losses which are strogly load depedet. Whe the hot spot temperature reaches the glass trasitio temperature of the isulatio material (T g = 36 C for the chose resi), a massive icrease of the dielectric losses occurs (cf. Fig. 7). Due to the high desity of isulatio losses (cf. Fig. 9), this leads to a thermal ruaway. Accordig to these results, the cosidered epoxy resi, which is a typical resi employed for low-frequecy systems, is less suitable at MF. This is explaied by the fact that that the dielectric are o-egligible at MF ad very high for operatig temperatures close to the glass trasitio temperature. This will be the case for most epoxy resis with typical glass trasitio temperatures ragig from 6 C to 4 C [6], [], [3]. For this reaso, materials with sigificatly lower or higher glass trasitio temperatures (compared to the operatig temperature) would be well-suited for the isulatio of MF trasformers. For example, the glass trasitio temperature of silicoe elastomers is below C, such that the loss peak ear the glass trasitio temperature is ot critical [3]. Furthermore, silicoe elastomers feature low dielectric losses at MF ad good thermal properties. V. CONCLUSION This paper derives closed-form aalytical expressios for calculatig the dielectric losses of the materials used i dry-type isulatio systems. The proposed approach applies to typical PWM voltages of DC-DC ad AC-DC coverters. A extesio for materials with frequecy-depedet losses is also derived. The proposed approximatios are verified with permittivity measuremets coducted for a typical MV isulatio epoxy resi at differet frequecies ad temperatures. It is show that fudametal frequecy aalysis iaccurately determies the dielectric losses ad that fiite rise ad fall times of the excitatio voltage limit the losses. A detailed ivestigatio reveals that the losses i the isulatio maily deped o the voltage amplitude, the switchig frequecy, ad the material parameters. The duty cycle ad the modulatio idex of PWM waveforms have oly a mior impact o the isulatio losses. Moreover, multilevel iverters exhibit a sigificat reductio of the losses. The described computatio techiques are applied to a 4kV/4V, 5kHz SST MF trasformer (sigle-stage coverter) with a frequecy ad temperature-depedet model. At omial operatio, the isulatio losses (epoxy resi) reach 7% of the total trasformer losses, which reveals that a accurate computatio of the dielectric losses is required for MV/MF applicatios. Moreover, the dielectric losses ca lead to a thermal ruaway of the trasformer. It is cocluded that typical low-frequecy isulatio materials are usuitable for MF excitatios. For this reaso, future research will ivestigate alterative isulatio cocepts ad materials, such as silicoe elastomers, for MF trasformers i more detail. APPENDIX A. PWM with Costat Duty Cycle This is the derivatio of (). From () ad (9), the ormalized losses P ca be writte as (with the hypothesis ε ( f ) = cost.): P P = ε C V = DC 4f s si (D c ) ( ) fs. (4) + f c The ifiite summatio ca be approximated by a fiite summatio (util the corer frequecy of the first order low-pass filter, cf. (7)). The followig summatio is derived (usig the double agle trigoometric formula): P f c /f s 4f s which ca be rewritte as: ( P f fc s /f s fc/fs cos(d c ), (5) ) ( ) cos(d c ). (6) A approximatio exists for the partial sum of the first term [3]. The secod term coverges quickly such that the fiite sum ca be approximated with the ifiite sum, for which a closed-form solutio exists [3]. This leads to: P f s ( l ( fc f s ) + γ + l( + cos(d c )) ), (7) which is equivalet to the expressio () (after usig the double agle trigoometric formula). B. PWM with Frequecy-Depedet Materials This is the derivatio of (7). From () ad (9), the ormalized losses P ca be writte as (D c =.5): P = P C V = DC 4f s ε ( f s ) si ( ) ( ) fs. (8) + f c Oly the odd harmoics are o-zero. Similar to Appedix A, the summatio is trucated at the corer frequecy: f P c /f s 4f s ε ( f s ). (9) (odd) This summatio ca be approximated with a itegral [33]. The iaccuracy of this approximatio ear the fudametal frequecy is compesated by a additioal term P add : P P add + 4f s l(f c) l(f s) ε ( f ) d ( l ( f )), (3) } {{ } P it where the factor / results from the fact that oly odd values of are cosidered. The term P is chose such that the add approximatio is equivalet to () for the special case of a frequecy-idepedet ε ( f ) (cf. (7) with D c =.5). This leads to: P add = f s (γ + ) l( + ) ε ( ) f s, (3) which, together with (3), is equal to the expressio (7).

9 TABLE IV CONSIDERED PARAMETER COMBINATIONS. PWM with Costat Duty Cycle f s [,]khz / t r [,]s / D c [.,.9] f s t r < %, i order to avoid quasi-triagular pulses PWM with Siusoidal Modulatio f g [5,6]Hz / c [,5] f s [,]khz / t r [,]s / M i [.,.] f s t r < %, i order to avoid quasi-triagular pulses PWM with Frequecy-Depedet Materials f s [,]khz / t r [,]s / D c =.5 T [,] C / Damisol 348 resi f s t r < %, i order to avoid quasi-triagular pulses TABLE V ACCURACY OF THE APPROXIMATIONS. PWM with Costat Duty Cycle () vs. (6).% (low-pass).5% (ramp) PWM with Siusoidal Modulatio (3) vs. (6).8% (low-pass) 3.4% (ramp) PWM with Frequecy-Depedet Materials () vs. (6) 53% (low-pass) 5% (ramp) (7) vs. (6) % (low-pass) 4% (ramp) () vs. (6) 7% (low-pass) 5% (ramp) C. Accuracies of the Proposed Approximatios The accuracy of the followig approximatios is cosidered: PWM with costat duty cycle (cf. ()), PWM with siusoidal modulatio (cf. (3)), frequecy-depedet ε ( f ) (cf. (7)), ad frequecy-depedet ε (cf. 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