1 Analysis of heat transfer in a single-phase transformer

Transcription

1 Assignent -7 Analysis of heat transr in a single-phase transforer The goal of the first assignent is to study the ipleentation of equivalent circuit ethod (ECM) and finite eleent ethod (FEM) for an electroagnetic device with a siple geoetry. A steadystate heat transr proble is analysed for a single-phase shell-type of transforer with difrent geoetric proportions. This analysis supposes to give answers about What geoetric proportions between the electric circuit and agnetic gives the highest transrred power and which the highest efficiency? What is the difrence between ECM and FEM estiations and what ight be the reasons? How to interpret the results: transrred power, heat power, cooling and teperature rise: are they reasonable?. Getting started Quick progressing guide Tools for calculation: FEMM 4. for finite eleent Analysis (FEA) and MATLAB for equivalent circuit analysis (ECA). Create project ap, open LUA and M-script, and redefine geoetric data for analysis object according to.. Execute files, when is done find a recently created txt file fro project ap. Load txt-files to MATLAB by using load coand. Find out which colun belongs to current density and use this value to obtain transrred power (eq..8) and heat power (eq..9). lot the graphs according to.8 and carry on discussion on the ain questions by support of the appendix. otice that It is assued that both priary and secondary winding have the sae current density. Even though the heat transr odels exclude the heat losses in the end-turns and the axial cooling surfaces (along z-axis), the total heat or power losses need to consider the conductor losses in the end turns. Fro theral analysis point of view, by assuing that the axial cooling takes care of axial power losses and does not contribute neither the heating nor the cooling of the lainated part of the transforer, then the theral situation in lainated part of the transforer reains unchanged. The LUA-script considers 50% slot size by default. You activate script by selecting Open Lua Script and terinate the FE process by pushing Break, if everything goes fine then you can follow the process in lua console that ay take up to w inutes.. Analysis object Five difrent sizes for EI laination is given (in table below) where each of the have two difrent stack height, which are ½ and /3 of the width of laination. The volue of Ltr x Wtr x Hc defines the part of transforer where the power losses are calculated and the lateral surfaces Hc x (Ltr + Wtr) define the (only) cooling area. EIE0 Design of Electrical Machines, IEA, 06

2 Assignent -7 Laination type Length Ltr [] Width Wtr [] Stack Height Hc [] EI EI EI EI EI EI EI EI EI EI H c W tr L tr.3 rogra structure The FE odel of transforer and odelling process for heat transr analysis is written in lua script, which has to be open in Fe 4., and the atching EC odel is written in script for Matlab. These scripts have the sae structure and structured output. On the basis of the outcoe the goal of the further calculation is to find an optial relation between an electric and a agnetic circuit in the transforer i.e. priary and secondary winding in a lainated core. araeterization Length Geoetric proportions Magnetic loading Material properties Geoetric odelling Derive geoetry in respect with paraeters araetric change Sensitivity study roportion between agnetic and electric circuits D Finite Eleent Method Heat transr (Mirage) D Equivalent circuit ethod Theral circuits (Matlab) Objective Estiate electric loading Transforer specification Figure. rogra structure that is ipleented in the Matlab script and the LUA script for the heat transr analysis. The exaple of the transforer has diensions: Ltr=00e-3 x Wtr=80e-3 x h_c=50e-3 as well as the thickness of insulation (bobbin) ins=e-3 (all geoetric diensions in etres). ote that you can select new diensions that you like fro table in section.. The agnetic loading (Bc=.4T) is unchanged and assued to be hoogeneous that gives the sae loss density in the whole core. The goal of the coputation routine is to estiate current loading and the corresponding conductor losses within the theral liit. The teperature dependence of the conductor resistivity is taken into account. EIE0 Design of Electrical Machines, IEA, 06

3 Assignent 3-7 Initialization Cooling conditions Magnetic: B -> p Electric: J k,ρ 0 -> p cu Find teperature Coil hot-spot ax Coil average ave Target target - ax 0.05 iter ax_iter Result visualization Teperature plot (bp) Electric loading (txt) Obtain new values ρ= ρ 0(+α( ave- 0)) p cu=0.5 ρ (J k+) Kf iter=iter+ Obtain current density if target - ax < - 40 then J k+=j k x 0.5 if target - ax > 40 then J k+=j k x else J k+=j k W( target - ax) Figure. Iterative hot-spot coputation loop. Flowchart of the iterative current density estiation where the target hot spot teperature of the coil is focused. The hot-spot teperature is obtained fro a line that is defined through the cross-section of a winding. The weight factor W is chosen so that the converging process is as well as fast as stabile. This flowchart does not give a solution if the core losses only give the target teperature or ore than that The theral odels: EC odel and FE odel Are two diensional (D), which eans that heat transport along z axis and the heat generated in/dissipated fro the end turns are neglected Assue the sae loss density in the priary and the secondary windings Assue naturally cooled sides h= W/K and abient teperature of 40 0 C.4 Geoetry paraeterization The geoetric odel for shell type of transforer is forulated so that transforer length and insulation thickness together with proportion factors will define the whole geoetry. Additional paraeters, such as proportion factors specify the geoetric proportions between the difrent parts of the transforer. Ks (variable) proportion between the slot length and core lib length Kw (=0.5) width proportions between priary and secondary winding The length of the leg in the agnetic core is expressed by using the relative slot length k s and the length needed to for a one electroagnetic pole. For the single-phase transforer either for core and shell types the nuber of poles is p =. l l tr c k s (. ) p Siilarly the length of the slot or the available area for the windings is forulated according to the length of a single electroagnetic pole and the relative slot length. EIE0 Design of Electrical Machines, IEA, 06

4 Assignent 4-7 l tr ls ks (. ) p The width of the slot is given by the total width of the transforer w tr inus the width of the agnetic core yokes. The width of the agnetic back-core (yoke) equals to the length of the agnetic leg-core. The slot width for the shell type of transforer is w s w l (.3 ) tr c Figure.3 The geoetry paraeterization of the shell type of transforer. The upper figure shows the agnetic flux flow plane (xy-plane) and the lower figure shows the electric current flow plane (xz-plane)..5 Size equations The size of transforer is related to the power capability of the electroagnetic device by the rational liits for the flow densities such as electric current density J, agnetic flux density B and the power loss density p. In an ideal transforer the agnetic coupling between the windings is perct. There is the sae core flux (t) that links each turn of each winding. Apparent power of an ideal lossless transforer is expressed as S U I (.4 ) here is assued sinusoidal variation of voltage and current and instead of rs values the peak values has been used instead. Considering the lossless electroagnetic circuit the voltage of the EIE0 Design of Electrical Machines, IEA, 06

5 Assignent 5-7 electric circuit can be directly linked to the induced back electrootive force (ef) and the agnetic flux in the agnetic circuit. u t U t e t t t d d cos (.5 ) dt dt Ideally it is assued that all the agnetic flux links with the winding and flux can be expressed directly fro the voltage that is applied to the winding. U (.6 ) t sin t sin t B A sin t The axiu value of agnetic flux Φ is related to the cross-section area of the pure agnetic conductor A and the axiu flux density B that is defined by the aterial ability to conduct agnetic flux and the agnetic saturation B sat. Siilar to the agnetic circuit, the flow in the electric circuit is defined by current I (axiu value) in the single turn which is related to the total Apere turns I i.e. agnetootive force (f), the cross-section area of the pure electric conductor A e and the axiu current density J that is defined by device s ability to conduct heat flow and the theral liit coil. I J A cos (.7 ) e t I t cos t cos t i By substituting the axiu values for voltage and current in the equation of the apparent power (eq..) the size of the transforer can be expressed through the agnetization frequency, the allowed current and flux density over the geoetrically interlinked circuit areas of A e and A.. S U I B J A A e (.8 ) The power of transforer depends on the electric loading J, the agnetic loading B and the agnetization frequency ω as well as the geoetry such as cross section areas of electric A e and agnetic circuit A. By assuing equal agnetootive forces in the priary and the secondary winding, thus the infinitely pereable core, the active power can be taken equal to the apparent power. The transrred power is less due to power losses. The axiu transrred power is deterined by the aount of power loss or heat power that the transforer can handle for given teperature liits. The power losses can be estiated according to the loss origin in difrent parts of the device. The total heat power is expressed as a su of losses, which are outcoe of power loss density and the geoetry of the electric as well as the agnetic circuit. loss A l p A l p (.9 ) e e e cu The conductor loss for the direct current cu is expressed through the power loss density, which depends on resistivity ρ and the current density square J, and the volue of the conductor V e. JAe le cu I R J Ael e J Ve (.0 ) Ae The reagnetization loss in the agnetic conductor for the syetric sinusoidal excitation can be found fro the specific loss data k at certain agnetization frequency and agnetic induction over the core volue V EIE0 Design of Electrical Machines, IEA, 06

6 Assignent 6-7 k B V p V, (. ) The specific core loss is calculated according to the polynoial curve fitting of the loss characteristics at 50 Hz..557B.936B p (. ) Finally, the efficiency can be found fro the input power and the loss power. loss (.3 ) Lua-script as well as -script EMK_task_.lua and EMK_task_., calculate hot spot teperature of the winding ax, average teperature of winding ave, core teperature cc, the specific copper loss pcu, axiu current density Jc and so on as a function of proportion Ks. Soe of the variables are not estiated with the theral equivalent circuit. for kh =,9, do x [] ax [ C] ave [ C] pcu [W/ 3 ] Jc [A/ ] Aw [ ] cc [ C] end The result of the coputations is written into file tp_heat_.txt and tp_heat_ec.txt..6 EC odel Equivalent circuit odel and odelling process is defined in EMK_task_.. The theral conductivity network consists of eleents that represent the heat dissipation in the syetric part of the transforer. The cooling condition through the natural convection is taken into account in the eleents 4 and. odal network represents the teperature over 9 node points. The copper losses are applied into node and 3, the core losses to node and the abient (rerence) teperature is described in nodes 5 and 9. 9 iter [-] Figure.4 The theral equivalent circuit of the single-phase shell-type transforer. Theral eleents in the end-turns are excluded in order to ake EC odel and FE odels coparable. EIE0 Design of Electrical Machines, IEA, 06

7 Assignent 7-7 Code that shows the forulation of theral equivalent circuit and the eleents in the circuit % theral equivalent circuit - topology atrix % [eleent() node(n) node() theral conductivity] Tec = [ 0.5*h_c * 0.5*w_s / (0.5*l_c/tc_+ins/tc_ins+0.5*l_w/tc_win); 3 0.5*h_c * 0.5*w_s / (0.5*l_w/tc_win+ins/tc_ins+0.5*l_w/tc_win); *h_c * 0.5*w_s / (0.5*l_w/tc_win+ins/tc_ins+0.5*l_c/tc_); *h_c * 0.5*w_s / (0.5*l_c/tc_+/Aconv * w_s / Wtr); *h_c * 0.5*l_c / ((0.5*w_s+0.5*l_c)/tc_); *h_c * l_w / ((0.5*w_s-ins)/tc_win+ins/tc_ins+0.5*l_c/tc_); *h_c * l_w / ((0.5*w_s-ins)/tc_win+ins/tc_ins+0.5*l_c/tc_); *h_c * 0.5*l_c / ((0.5*l_c+ins+0.5*l_w)/tc_); *h_c * 0.5*l_c / ((0.5*l_w+ins+0.5*l_w)/tc_); *h_c * 0.5*l_c / ((0.5*w_s+0.5*l_c)/tc_); *h_c * 0.5*Ltr / ((0.5*l_c)/tc_+/Aconv);]; lease pay attention to the definition of cooling surfaces (highlighted part of code) apart to conducting surfaces in the code!.7 FE odel Finite eleent odel and odelling process is defined in EMK_task_.lua. The xy-plane cross-section of a shell type of transforer is the base geoetry for the heat transr analysis. otice that the loop calculations are coented out in the code and the relative slot opening of 50% is selected initially..8 Assignent Based on the outcoe fro FE and EC odel (that takes at ca 5 inutes) carry out the following tasks and discuss the outcoe in relation to the target questions which were stated in the beginning of the assignent. Show Jc=f(ks) for these difrent odels Calculate a transrred power =f(ks) for these difrent odels Calculate copper and core losses loss=f(ks) Estiate the efficiency of the transforer based on the difrent odels Validate and discuss the odelled results by analysing supplier data of single phase transforer fro TRAMO-ETV AB or soe other copanies EIE0 Design of Electrical Machines, IEA, 06