Optimization of Power Transformer Design: Losses, Voltage Regulation and Tests

Transcription

1 Journal of Power and Energy Engineering, 17, 5, htt://wwwsirorg/journal/jee ISSN Online: ISSN Print: X Otimization of Power Transformer Design: Losses, Voltage Regulation and Tests MPharlen Chiekwe Mgunda Wright State University, Fairborn, OH, USA How to ite this aer: Mgunda, MC (17) Otimization of Power Transformer Design: Losses, Voltage Regulation and Tests Journal of Power and Energy Engineering, 5, htts://doiorg/1436/jee1754 Reeived: November 3, 16 Aeted: February, 17 Published: February 3, 17 Coyright 17 by author and Sientifi Researh Publishing In This work is liensed under the Creative Commons Attribution International Liense (CC BY 4) htt://reativeommonsorg/lienses/by/4/ Oen Aess Abstrat This aer resents the design aroah used in designing transformers mostly used in ower sulies and ower systems The aer will over theoretial riniles alied in analyzing magneti iruits to better understand the oeration of the transformer Sine well-designed transformer is suosed to meet the seifiations of the environment that it is going to be used, there is a need to onfirm after the transformer is built to make sure that it is going to oerate effiiently and without a failure Therefore, this aer will also resent the traditional methods used to test transformers in the industries to make sure that it will oerate within the resribed loading limits and voltages This aer will over the transformer losses in detail inluding the test methods used to alulate namelate arameters for ower transformers used in ower systems Keywords TTR, Hysteresis Loss, Eddy Current, Fringing Fator, Curie Temerature 1 Introdution There are some similarities between eletri iruit and magneti iruit Magneti flux faes oosition due to relutane of magneti material while eletri urrent flow faes oosition due to the resistane of the iruit Both magneti flux and eletri urrent require driving fores, thus magnetomotive fore in magneti iruit and voltage in eletri iruit Transformer is one of the omonents mostly used in eletrial and eletroni systems Transformers are lassified based on their aliation For examle, in eletroni industries, the transformers an be lassified in two main ategories, thus low frequeny transformers and high frequeny transformers There are also lassified aording to their use, for examle, imedane mathing transfor- DOI: 1436/jee1754 February 3, 17

2 mers, voltage transformers and urrent transformers Imedane mathing transformers are ommonly used in eletroni iruits Their main urose is to make sure that there is maximum ower transfer from one stage of the iruit to the final stage Therefore, their design aroahes are based on the imedanes of the eletroni devie; for examle, audio ower amlifier and seaker, their rinial of oeration is to maintain the same value of imedane on the outut of one stage of the iruit to the inut of another stage of the iruit Thus Z = Z (111) am seaker Therefore, their turn s ratio an be alulated as follows, N r Z am = (11) Z seaker Current transformers are ommonly used in ower systems for metering and rotetion Their main urose is to ste down urrent levels roortionally to the rimary urrent of ower systems Most of the devies for metering and rotetion relays oerate on low urrents Both utility distribution and transmission systems oerate on very high urrents It is therefore for this reason that you need urrent transformers to ste down the urrents Voltage transformers are ommonly used in eletronis system s ower sulies and ower system s substations Their main urose is to transform the voltages from one level to another Voltage transformers are mainly lassified as ste-u transformers and ste-down transformers The design work in this aer overs mainly the voltage transformers The relationshi of turns-ration, urrent and voltages for the transformer is as shown in the equations below N1 V1 I = = (113) N V I 1 where N 1 and N are rimary and seondary windings; V 1 and V are rimary and seondary voltages; I 1 and I are rimary and seondary urrents resetively 11 Magnetism Understanding of the riniles of magnetism led to a major breakthrough in transformer design Hans Christian Oersted disovered that urrent flowing in the ondutor rodues magneti field This disovery heled Andre -Marie Amere to ome u with an idea that material magnetism results from loalized urrents [1] Amere observed that multile numbers of small urrent loos oriented aroriately indued magneti fields assoiated with magneti materials and ermanent magnets [] 1 Magneti Materials Transformer ores are made from materials with low relutane to the flow of magneti flux Transformer ores are suosed to give easiness to the flow of 46

3 magneti flux Magneti material an be desribed as soft materials when it an easily be magnetized and demagnetized Hard magneti materials are those whih annot be easily magnetized and demagnetized [1] Magneti material is one of the most imortant arts of the transformer The transformer ores are onstruted from magneti materials There are different tyes of magneti materials Transformer Design Engineers are therefore suosed to know the roerties of different tyes of magneti materials Iron is the most ommon material used in ore onstrution In some ases, small amount of silion is added to iron in order to imrove magneti roerties whih redue ore losses [] Other materials used in ore onstrution are: nikel, obalt and iron alloys (ermalloys) and iron oxides (ferrites) These materials are lassified in a grou of ferromagneti materials [] In summary, magneti ores have the following roerties, whih need to be onsidered when making a deision for seleting ore material for a seifi transformer design: resistivity, relative ermeability, saturation flux density, Hysteresis and Eddy urrent losses er unit volume, Curie temerature and uer oerating frequeny [1] Magneti Theory Sine transformers work on the riniles of magnetism, it is therefore imortant that this aer must introdue theoretial analysis of eletro-magnetism Magneti theory lays a vital role in transformer design 1 Amere s Law Amere disovery roved that a varying urrent, i( t ) flowing through a ondutor reated a varying magneti field, H( t ) around that ondutor [] Amere s law is then summarized using the formula whih resents the arameters of the losed loo [] N x H x d I = J d S = i s n = i i i i N = en n= 1 (11) di is the vetor length in the diretion of the ath, J is the urrent density, S is the surfae where urrent I is flowing, and i en is enlosed urrent in the ath C [] If a oil has N turns and the urrent flowing through it is I, then amere s Law an be alied to the oil using the formula: Farady s Law x H d I = Ni (1) This law states that a time varying magneti flux flowing through a losed loo generates voltage in that loo [] d ( ) dλ Nφ dφ d Ni N di di v( t) = = = N = N = = L dt dt dt dt dt dt (1) 47

4 where λ is flux linkage, N is the number of turns of the oil, φ is magneti flux, is relutane of magneti iruit and L is the indutane of the oil and i is the urrent The formula an further be substituted by flux density, magneti field intensity and ermeability as follows: db dh µ AN d i v( t) = NA = NAµ = () dt dt l dt Aording to Faraday s Law, flux density is deendent to the alied voltage to the rimary winding The volt-seonds, λ π alied to the rimary winding for half yle [1]; π λ = vtd t (3) π The hange in the flux value in the magneti ore will be; λ B = π (4) NA ( ) Therefore, to ahieve the required hange in flux density, the number of turns of rimary winding must be; λ N = π (5) BA The urrent in the indutor at a given seifi time an be reresented by the formula: 1 t 1 ωt it ( ) = vt d i( ) vd( t) i( ) L + = ω + ωl (6) 3 Lenz s Law This is another imortant law in magneti iruits Lenz noted that the voltage reated by a time-varying magneti flux had a diretion that indued eriodi urrent in the losed loo whih also indued magneti flux that oosed the hange in the initial flux Inreasing the initial magneti flux the indued urrent reates flux whih ooses the inrease of the initial flux and vie-versa [] 3 Magneti Ciruit Magneti iruit is somehow analogous to eletri iruit While as eletri iruit needs voltage to fore urrent flow in a losed iruit, magneti iruit requires magneto-motive fore (mmf) to ause magneti flux flow in magneti iruit Eletrial iruit faes an oosition to a urrent flow due to the resistanes of the iruit, while in a magneti iruit magneti flux faes oosition due to the relutane of the magneti ore and air ga Figure 1 shows a simle two winding transformer with a ore having an air ga Figure shows a simle magneti iruit of the transformer in Figure 1 Notie that the relutane of the magneti ore is in series with the relutane of the air ga Pratially the air ga has a very high relutane desite the very small length in the ga From the iruit diagram in Figure, R is ore relutane, R g is the air gag relutane and φ is magneti flux 48

5 Figure 1 Two winding transformer with air-ga Figure Magneti iruit of two winding transformer 31 Magnetomotive Fore (mmf) Magneti iruit requires a soure of energy to reate a flow of magneti flux in the magneti iruit Magneto-motive fore, is the soure of that energy whih makes magneti flux to flow in the magneti iruit Magneto-motive fore is reated by urrent flowing in the oil Magneto-motive fore is diretly roortional to the urrent flowing in the oil and the number of turns of the oil = Ni (311) The unit used to measure mmf is Aturns Inreasing either urrent or number of turns will inrease the mmf whih in return will inrease the flow of magneti flux in magneti iruit 3 Magneti Flux As stated earlier, magneti flux in the magneti iruit is analogous to the ur- 49

6 rent flowing in eletrial iruit Magneti flux on the surfae area of s is given by [] where B is flux er unit area x B s φ = d S (31) The unit used to measure magneti flux is weber (wb) 33 Magneti Field Intensity Magneto-motive fore of the oil an be defined as er unit length of the oil This tye of measure is desribed as magneti field intensity, H The unit used to measure magneti field intensity is Ams er meter, (A/m) Therefore, H an be resented by a formula: Ni A H = l m (331) where l the length of the oil in meters, N is the number of turns of the oil and i is the urrent flowing in the oil 34 Magneti Flux Density Magneti flux density is analogous to urrent density in eletrial iruit It is the amount of magneti flux er unit ross-setional area Magneti flux density an be alulated as follows: φ B = (341) A where, φ is the total flux in magneti iruit and A is the ross-setional area of magneti ath Magneti flux an also be alulated in relation to magneti field intensity; µ Ni B = µ H = µ rµ H = (34) l 7 H µ = 4π 1 m r (343) µ = µ µ (344) where, µ is ermeability of free sae and µ r is the relative ermeability As the urrent is inreased in the oil, the flux density in the ore inreases exonentially The rate of the inrease in flux density in the ore is very small when a urrent is inreased u to a ertain seified value The ore goes into saturation The flux density is said to reah its saturation oint, B s Therefore, the flux at saturation oint is, φ s = AB s (345) 35 Magneti Flux Linkage It is the sum of flux enlosed by individual turn of wound around the magneti ore [] Flux linkage, λ an be resented by the formulas below; 5

7 µ AN I λ = Nφ = NAB = NAµ H = = Li( Vs) (351) l Note that the unit for flux linkage is Volt-seond 36 Relutane and Permeability Magneti iruit has some oosition to the flow of magneti flux the same way it is with resistane in the eletrial iruit The oosition to magneti flux flow in the magneti iruit is all relutane, The easiness of magneti flux to flow in a magneti iruit is alled ermeane, P whih is analogous to ondutivity in eletrial iruit Relutane has, therefore an inverse relationshi to the magneti material ermeability and ross-setional area of the magneti ath Relutane is also diretly roortional to the length of magneti iruit Thus, relutane of magneti iruit an be alulated by []: 1 l = = (361) P µ A where l is magneti ore length, A is ross-setional area of the ore, P ermeane of magneti iruit The unit for relutane is turns er Henry, (turns/h) Based on the formula for relutane, we an drive the formula for ermeane as follows; 1 µ A P = = (36) l The unit for ermeane is Henry er turns, (H/turns) We an also drive formulas from for magneti flux from Equations (361) and (36), thus: 1 µ A Ni φ = = P = (363) l The magneti ore indutane, L an then be alulated using relutane of the ore N µ rµ AN L = = PN = (364) l Therefore, indutive reatane is; X 37 Air Ga Relutane L r π fl l πfµ µ AN = = (365) Some magneti ore may have an air ga by design or beause of some joints made in laminated ore The effetive ross-setional area of the air ga is not the same as the ore ross-setional area beause the magneti flux does not take a straight ath in the air ga Making and assumtion that the ross-setional area of an air ga is the same as that of the ore; we an visualize the relutanes of the ore and the air ga to 51

8 be in series with eah other Therefore, the total magneti iruit relutane is; l l g = T + = g µ µ A + (375) µ A r g Sine we made an assumtion that both air ga and ore have the same ross-setional area, and then we an alulate the total relutane of magneti iruit using the equation; l µ rl g T = 1 + (376) µ rµ A l Note that the term in the arenthesis in Equation (366) is defined as ga fator, F g [] Ga fator an also be defined as the ratio of the total iruit relutane to the total relutane of the ore, thus; F g + g T g µ rlg = = = 1+ = 1 + l (377) Based on the Equation (377), there is a linear relationshi between the length of the air ga and the ga fator Figure 3 shows the relationshi between length of the ga and the ga fator From the Equation (377) we an then ome u with a relationshi of the ore relative ermeability and air ga fator alled effetive relative ermeability, µ re ; µ r µ re µ re = = F µ g rlg 1+ l (378) Figure 3 Relationshi between hanging air ga and fringing fator 5

9 Sine ermeability of free sae is very low, introduing an air ga redues effetive ermeability of a magneti iruit The advantage of an air ga in a magneti iruit is that it rodues stable effetive ermeability and relutane whih in turn rodues reditable and stable indutane [] Figure 4 demonstrates the relationshi between the air ga and effetive ermeability The length of the ga an therefore be alulated using the equation: l g µ AN l = (379) L µ r From Equation (369) we an drive the equation for indutane for the ore with air ga µ AN L = l lg + µ r (371) As the ga length inreases, the indutane of the oil redues as shown in Figure 5 It must be noted that the indutane of the ga is diretly roortional to the effetive ermeability, µ e As shown in Figure 4, effetive ermeability dislays a grahial exonential derease as the air-ga of the ore inreases The same grahial exonential derease in indutane is dislayed in Figure 5 beause of the aforementioned relationshi between Indutane and effetive ermeability From Equation (379) we an then drive an equation to alulate the number of turns of the windings: Figure 4 Effet of air ga on effetive ermeability of the iruit 53

10 Figure 5 Effet of air ga on indutane l L lg + µ r N = (371) µ A 38 Fringing Flux Magneti flux lines have a tendeny to reel eah other when they are assing through a nonmagneti material When magneti iruit has an air ga, the magneti lines tend to bulge outwards, thus inreases the ross-setional area of the effetive air ga whih in returns redues flux density [] The bulged magneti flux lines in an air ga are desribed as fringing flux Sine the sum of fringing flux and the air ga flux must equal to the flux in the magneti ore, fringing relutane and air ga relutane must be viewed as being in arallel Therefore, magneti flux in the ore is: φ = φg + φf (381) where φ g air is ga magneti flux and φ f is fringing magneti flux Sine fringing and ga relutanes are in arallel and in series with ore relutanes, total magneti iruit relutane an be resented by an equation: g f T = + (38) + But ( ) g f l f f = (383) µ A f 54

11 lg g = (384) µ A g l = (385) µ µ A Substituting Equation (38) with Equations (383), (384) and (385) and then fatorize, we get the final equation; r l µ rlg 1 T = 1 + (386) µ Al rµ A l f g 1+ Al g f Therefore, the indutane of the magneti iruit taking into onsideration fringing effet will be []; L f N = = N µ l 1+ l T l µ µ A r g r 1 Al f 1+ Al g g f (387) The ratio of the indutane of the air ga with fringing flux lines to that of without fringing flux lines is desribed as fringing fator, F f Figure 6 shows the effet of fringing on indutane Lf Al f g F = f 1 L = + Al (388) f Figure 6 Effet of air ga and fringing on indutane 55

12 Based on the Equation (387), and Equation (388) the total magneti iruit indutane inreases with an inrease in air ga Beause of the inrease in the indutane due to the air ga, likewise, the number of turns required will be inreased too Thus; ll g N = (389) µ AF To avoid saturation, the maximum number of rimary windings of the transformer with air ga must be; N 1max f = Bklg µ I (381) max 4 Curie Temerature, T Temerature hanges have an effet on magneti material Sin alignment of atoms in a magneti material auses magnetization There is disintegration of ferromagneti domains in a magneti material when a temerature is above seified ritial temerature alled Curie temerature When this temerature is assed, the ferromagneti harateristis of the material hanges and adots aramagneti harateristis with relative ermeability, µ r 5 = [] Currie temerature is ruial when designing transformers whih will be used in high temerature environment Seletion of ore material of high Curie temerature an hel to solve the roblem The hange in ermeability er unit hange in temerature is defined as Temerature Coeffiient, TC Temerature Coeffiient an be reresented by the formula; µ T µ T TC = (41) µ T T ( ) T where µ T is the ermeability at temerature T and at temerature T µ T is the ermeability 5 Transformer Core and Condutor Sizing Priniles The transformer ore and winding ondutor seletions deends mainly on the VA rating of the transformer 51 Skin Effet There are also some fators whih may affet the size seletion of the winding ondutor, like frequeny due to skin effet At very high frequeny, the urrent flowing in the ondutor has a tendeny of onentrating on the surfae of the ondutor and not in the inner art of the ondutor The skin deth, δ w of a ondutor with resistivity, ρ w is alulated by; ρw 1 δ w = = (511) πµµ r f πµσ f Therefore, as the frequeny inreases, the urrent density on the surfae of the 56

13 ondutor inreases and the skin deth dereases [], thus; ( ) Hm Jm = = πfµσ Hm (51) δ w The maximum urrent flowing in the ondutor whose urrent density is I m J m ; Jmδ wh = (513) The rodut of δ w and h is the effetive ross-setional area of the ondutor, A e The resistane of the ondutor with length l is; w R w ρwlw lw = = π µρwf (514) hδ h w Both the indutive reatane and the DC resistane are the same and are roortional to the square root of frequeny, Rw f The time-average ower loss in winding stri of length l w is []; Hmh ρwjmawl w hlwhm D = rms w w = = µρw P I R = R π f (515) 4 The indutane then an be alulated, sine we have the value of indutive reatane and frequeny, thus; X L wlw lw w L = ρ µρ ω = ωhδ = h πf (516) w The resistivity of a winding ondutor is also affeted by temerature Conduting materials have a tendeny to inrease in resistivity due to inrease in temerature It is therefore to onsider the effet of temerature on a winding more eseially on transformers whih will be oerating at high temeratures [] Thus; ( ) ρ = ρt 1 α T T + (517) where α is temerature oeffiient of a ondutor and a ondutor at temerature, T 5 Winding Condutor and Core Sizing ρ T is the resistivity of The size of ondutor to be used in winding the transformer must be aable to urry ontinuous full load urrent at the seified oerating frequeny and temerature Amerian Wire Gauge, AWG is ommonly used in industries to define the wire sizes Using the Amerian Wire Gauge, the diameter of the wire an be alulated using the formula []: dn 36 AWG 39 [ mm] = 17 9 (51) By knowing the diameter and the length of wire we would like to use, we an therefore alulate the total DC resistane of the wire we are going to use 57

14 R wd ρwlw 4ρwlw = = (5) A πd w 53 Winding Insulation Transformer winding wire has a layer of insulation It is therefore neessary to also alulate the thikness of the insulation sine it will also ouy some sae in the ore window of the transformer It is there, the bare ondutor ross-setional area is divided by the rosssetional area of the insulated ondutor This ratio is defined as wire insulation fator, K i Aw Ki = (531) A For a round ondutor, the wire insulation fator an be alulated by; wi d i Ki = di + t (53) where d i is the diameter of uninsulated ondutor and t is the thikness of the insulation When winding the transformer, not all sae is filled with insulation and a wire There is some are okets within the winding The ratio of the area filled by insulated wire to the area filled by both insulated wire and air okets is defined as air fator K a Awi Ka = (533) A + A wi The windings of the transformer are not diretly wound on the ore Most of the transformer s windings are made on the bobbin It is neessary to determine the area ouied by the bobbin so that it an be added to the area ouied by the windings and the air The total volume of windings, air and bobbin will hel to size the area of the window required for the windings The ratio of the area of the bobbin window to the area of the ore window is alled bobbin fator, K b [] Bobbin window area Wa Wb Wb Kb = = = 1 (534) Core window area W W It is also imortant to know that not entire area of the window is used by the windings There is always an area whih is not used The ration of the area of the window used to the total area of the ore window is alled window utilization fator, K u [] bare ondutor are A AN w K = u ore window area = W = W (535) The rodut of the window area and a winding ondutor are is alled, area rodut, A Area rodut is also known as ferrite-oer area rodut [] LImaxIrms A = AW a = (536) KJ B air a a u rms k a a 58

15 6 Otimum Transformer Design Proedures Transformers are suosed to be designed to meet seified load requirements, like VA rating, oerating voltages, temerature onditions, vetor grou and hase dislaement as for transformers used in ower systems, and high effiieny The designers must also onsider the eonomi ost of the tye of the transformer they design Transformer design engineers are suosed to minimize the ost without omromising quality Some of the best hints to be onsidered when designing transformers are: Consider to minimize transformer ore losses Consider the oerating flux density to avoid saturation Consider the oerating frequeny more eseially on high frequeny transformers Consider normal oerating temerature of the transformer to avoiding overheating whih may affet the life of the transformer 61 Transformer Winding Wire Seletion and Core Area Produt A well designed ore must be able to handle the maximum load ower with minimal losses It must be able to handle the maximum urrent without going into saturation More imortantly, the ore must be able to aommodate the windings flexibly Given the transformer ower rating, P = VI, eak flux density B k, and the oerating frequeny, f we an alulate the number of the rimary turns, N required V λmax N = = (611) K fb A B A f k k where K f = 444 for a sine wave The number of seondary turns an be alulated in relation to the voltage ratio and rimary turns, thus; Vs Ns = N (61) V We an now alulate the ross-setional area of the wire required for both rimary and seondary windings; A A w I = (613) J I J s sw = (614) where J is the urrent density of the tye of wire seleted The total ross-setional area to be ouied by both rimary and seondary windings an be alulated by the formula give below 1 IV IV s s KW u a = N Aw + NA s sw = + K f fbk A J J (615) 59

16 If the urrent density of the rimary wire is the same as seondary wire, the area rodut of the transformer is; A VI + VI s s LIL max = WA = = (616) K K JB f K JB a f u k u For an ideal transformer with sinusoidal voltages and urrents, K f = Notie that as the frequeny is inreased the area rodut of a transformer redues 6 Calulation of Window Area Used by the Windings The length of eah turn of the transformer winding an be alulated from the erimeter of the ore It must also be known that eah layer of the winding will have a different winding length For a round ore, the estimated length of eah turn of the winding for the first layer, l T layer1 is l = πd (61) Tlayer1 The erimeter of eah following layer is inreased roortionally to the diameter of the winding wire Therefore, the length of the seond layer, l T layer of the turn is; l Tlayer 4A = π D + π w max (6) where D is the diameter of the ore and A w is the ross-setional area of the rimary winding wire The total length of the rimary winding wire is; l = Nl (63) w T The total length of the seondary winding wire is; l = Nl (64) ws s Ts To minimize the margin of error, it is good to alulate the total length of eah winding layer and then add them u to find the total length of the rimary winding and the seondary winding The ross-setional area required by rimary winding: A Tw N Aw The ross-setional area required by seondary winding: A = (65) Tws NA s ws = (66) Sine the total ross-setional area assigned to all the windings is KW u a, the fration of the area ouied by the rimary windings is; ATw N Aw x = KW = KW (67) u a u a The fration of the area ouied by the seondary windings is ATws NA s ws x = s 1 x KW = KW = (68) u a u a 6

17 Therefore, rearranging Equations (67) and (68); 7 Transformer Losses and Phase Shift A A w ws xkw u a = (69) N xkw s u a = (61) N The ower sulied to the transformer is not the same as the outut ower Some of the inut ower in the transformer is lost in the transformer Sine the transformer has reative omonents, there is also a hase shift between the inut and outut voltage The list below is some of the auses of the transformer ower loss; Winding loss due to the resistane of the winding ondutors Reative loss due to the indutane of the windings and stray aaitane Eddy urrent loss due to the irulating urrents in the transformer ore Hysteresis loss due to the ontinuous realignment of magneti dioles in the ore 71 Transformer Power Loss Due to the Winding Resistane Knowing the length, ross-setional area and the tye of the wire to be used in the winding, the total DC resistane of the winding an be alulated [] R wd w u a s ρwlt ρwlt N = = (711) A x KW R wsd ρwlts ρwltsns ρwltsns = = = A xkw x KW ( 1 ) ws s u a u a (71) We an therefore alulate the d winding ower losses in both rimary and seondary windings sine we know the maximum oerating urrent of the transformer and DC resistane of the windings; P IρwlT N = I R = (713) x KW wd wd The seondary winding DC ower loss is; u a P Is ρwlts Ns Is ρwlts Ns = I R = = xkw x KW wsd s wsd ( 1 ) s u a u a (714) Therefore, the total DC winding ower loss for the transformer is; ρwt l NI NsI s PTd = Pwd + Pwsd = + KW u a x 1 x (715) For a two winding transformer; N Is = I (716) N s 61

18 By substituting get; I s in Equation (615) by Equation (616) and fatorize, we P Td ρwt lni 1 = KW u a x( 1 x) The derivative of Equation (617) will give the value of x whih will minimize the winding losses dp ρ 1 Td wl x T = = dx KW u a x( 1 x) The minimum DC winding loss ours when: (717) (718) By substituting 1 x = otimum (719) x in Equation (617) with the value of Equation (619); P Td _ otimum 4ρwT lni = (711) KW u a Substituting N in Equation (6) with Equation (611) 4ρ l λ I = (7111) wt max Td _ otimum KWB u a k A P As the eak flux density inreases, the otimum winding losses redue exonentially The minimum transformer winding losses are ahieved when; x NI VI = = N I + NI V I + VI s s s s NI s s VI s s xs = = N I + NI V I + VI s s s s (711) (7113) Figure 7 shows grahially the oint where otimum transformer losses are ahieved in relation to flux density The designer must also make sure that transformer must not be saturated when oerating at normal voltage Therefore, there must be a limit for maximum flux density Thus the maximum flux density an be alulated by; B max Vrms = (7114) 444A Nf where A e is the effetive ross-setional area of the ore Considering the DC omonents; e V B = + NI A (7115) ( ) rms max DC 444A Nf 7 Transformer Winding Caaitane Sine the windings of the transformer are insulated and that they are side by side searated by the insulation, they at like aaitors The transformer winding 6

19 Figure 7 Otimum transformer ower losses aaitane may have an effet on hanges in voltage hase angle and may also ontribute to surge henomena during system swithing [3] The aaitane between windings an be alulated by the formula; C gw επdh = m (71) tinsu ε where ε is ermittivity through sae, D m is mean diameter between two windings, H is the height of the winding, t insu is the thikness of the insulation between the ondutors and ε insu is the ermittivity of the insulation The aaitane between ylindrial ondutors and ground lane is [3] insu πhε C = 1 S Cosh R (7) where R is the radius of the ylindrial ondutor, H is the length of the ylindrial ondutor and S is the distane of enter of ylindrial ondutor from the lane The aaitane between the rimary and seondary windings is given by; C T ( ) insu πd m w tinsu εε + + = (73) t where D m is the average diameter of the winding, w is the width of the bare ondutor, t insu is the total thikness of the insulation Eletrial energy stored in a aaitor is; insu 63

20 W 1 a = CV (74) For the RF transformers, resonane frequeny, f r is ahieved when indutive reatane is equal to the aaitive reatane Thus f r 1 = (75) π ( LC) 73 Phase Shift The series elements like winding resistane and leakage reatane ause hase shift The out-of-hase omonent of voltage dro aross these series elements is usually aused by in-hase load urrent flowing through the leakage indutane or out-of-hase load urrent flowing through the winding resistane or a ombination [3] Eo Ro + jωlo = E R + R + jω L + L ( ) in w o o L Therefore, the transformer hase shift is; ( L + L ) ωl ω = 8 Transformer Core Losses 1 o 1 o L shift tan tan R o Ro + R w (731) (73) The main auses of the transformer ore losses are eddy urrent and Hysteresis losses The other losses in a transformer are stray losses whih are diffiult to alulate reisely due to their omlexity 81 Hysteresis Loss For the ore to be magnetized there is a ontinuous alignment and reversal of magneti dioles in the ore material This roess requires energy hene ontributes to the ore loss alled hysteresis loss The hysteresis loss is frequeny deendent loss Energy lost er yle is; 1yle π W = fa l HdB (811) where A is ross-setional area of the ore and l is the length of the ore Emirial equation (Steinmetz equation) ( ) 16 W = K fa l B (81) h h m where W h is hysteresis loss in watts, K h ( K h = 444 for a sine voltage) is a Hysteresis onstant, f is frequeny in Hertz and B m is maximum flux density 8 Eddy Currentlosses Based on Faraday s Law, hanging magneti flux indues voltage in a onduting material whih in return auses urrent flow in that material 64

21 e ind dφ = (81) dt Sine the ore is within the viinity of magneti flux, there will be an indued voltage in the ore whih will ause urrent irulation in the ore This irulating urrent will ause ower loss in the ore in form of heat due to the resistane of the ore The formula for alulating Eddy urrent is; AB f m E = (8) ρ P where A the ross-setional area of the ore is, ρ is the resistivity of the ore, B max and f is the frequeny There are other transformer losses alled stray losses Some of the stray losses may be aused by some of the magneti flux inked to the transformer tank Stray losses in a transformer are diffiult to estimate, hene this aer will only over hysteresis and eddy urrent losses for the ore Estimated total ore losses will the sum of hysteresis and eddy urrent losses, thus 83 Indutive Reatane Power When a voltage is alied to the windings, the magneti fields built u in the windings Some of the inut ower to the transformer is used u for this magneti field built-u The magneti field on the rimary are suosed to link the seondary winding so the seondary voltage is indued in the oil Not all the rimary magneti flux is linked to the seondary Therefore, this also ontributes to energy losses in the transformer 84 Estimated Total Core Losses That is, LleakageIk P L =, [ watt-seonds] (831) AB f P = P + P = K fb + (841) T h E h P 16 m max ρ a b = A l kf B (84) T m 9 Transformer Resistive and Reative Comonents Ciruit Reresentation Having alulated the transformer winding resistane, indutane and aaitane, they an now be reresented in a transformer iruit diagram Where R is rimary winding resistane, L is rimary leakage reatane, C is inter-turn rimary aaitane, Re is ore resistane, Cw is winding to winding aaitane, Ls is seondary leakage reatane, Rs is seondary winding resistane, and Cs is seondary inter-turn winding (Figure 8) 65

22 The traditional resentation of the transformer exludes the aaitane for easy alulation sine it is not easy to alulate the exat values of the transformer aaitane (Figure 9) Referred Transformer Resistane and Indutive Reatane To alulate the total resistane and the total indutive reatane of the transformer, both the seondary resistane and indutive reatane must be referred to the rimary or vise-versa (Figure 1, Figure 11) Figure 8 Transformer iruit showing resistanes, aaitanes and indutanes Figure 9 Transformer iruit omitting aaitive effets Figure 1 Showing rimary imedane referred to the seondary 66

23 Figure 11 Showing seondary imedane referred to rimary R eqp X eq = (911) nrs = (91) nxs where, n = N Ns, is the transformer turns ratio The rimary imedanes referred to the seondary an be translated by the formulas; 1 Voltage Regulation R eqs X X eqs R = (913) n meqs R eqs X = (914) n X m = (915) n R = (916) n As we have analyzed in hater 6, the transformers have series imedanes One the transformer is on load, there is a voltage dro in the transformer due to these series imedanes Therefore, the atual-] outut voltage is not exatly as the one assumed to be based on the turns ratio The atual outut voltage of the transformer is therefore deendent on the load urrent [1] Calulation and Phasor Reresentation of Voltage Regulation The omarison of the transformer outut voltage at no load to that of full load voltage is defined as Voltage Regulation, VR Vs V no-load sfull-load VR = 1% (111) V sfull-load Alying turns ratio and transformer rimary and seondary voltage relationshi, we an also alulate voltage regulation as follows, 67

24 V V n V sfull-load sfull-load 1% (11) Voltage regulation an also be reresented in the formula by the total transformer imedane and the load imedane Z st + Z nzt L 1% (113) It is imortant to have as small voltage regulation as ossible For the ideal transformer the voltage regulation, VR = The atual outut voltage on the seondary transformer outut terminals is Note that ( Req Is jx eqis ) V Vs = Req Is jx eqis (113) n + is the voltage dro, V d due to transformer total resistane and indutive reatane V s is the voltage measured at the outut terminals of the transformer The hase angle dislaement between inut and outut voltage of the transformer is; X α = R eq 1 eq tan (114) Figure 1 is the hasor diagram for the transformer with lagging ower fator (Figure 13) The voltage regulation is also affeted by the ower fator besides the loading Figure 14 shows the variation in voltage regulation at different loads and at a seified ower fator As shown in Figure 14, as the loading on the transformer inreases, while at the same time maintaining the onstant ower fator, the erentage voltage Figure 1 Phasor diagram for transformer voltages with lagging ower fator Figure 13 Phasor diagram for transformer voltages with lagging ower fator 68

25 Figure 14 Effet of ower fator on voltage regulation regulation of the transformer inreases linearly The amount of hange of erentage voltage regulation differs between leading ower fator and lagging ower fator The erentage voltage regulation hange for a leading ower fator is more than the erentage voltage regulation hange for a lagging ower fator for the same value of load hange 11 Transformer Effiieny Transformer designers may also seify how good the design is based on minimum ower loss The ratio of the outut ower from the transformer to the inut ower to the transformer is defined as effiieny, η Pout η = 1 (111) P in Sine the inut ower to the transformer is the some of the total transformer losses and the ower outut, the effiieny equation an also be reresented as; η = Pout 1 P + P (11) out total loss The total transformer losses P total loss are aused by the following: Winding losses due to winding resistanes, ( I R winding ) Hysteresis losses in the transformer ore Eddy urrent losses due to ore resistane, R and the urrents irulating in the ore Note that the total outut transformer Pout = ( VI s s os φ ) (113) 69

26 Therefore, the transformer effiieny is η = ( VI s s osφ ) ( VI osφ ) + Ploss s s (114) 1 Transformer Prototye Test Proesses Transformers go through different tyes of tests to verify that they will oerate aording to design seifiations This aer will disuss the more imortant tests all the transformers have to undergo regardless of their transformer lass 11 Transformer Turns Ratio Test, TTR Transformer turns ratio test is one of the most imortant tests Sine there is a diret relationshi between turns ratio and transformer voltage ratio, any huge era in turns ratio will be more damaging to the iruit sulied by suh a transformer beause the seondary voltage of the transformer will be different Therefore, every transformer will go under TTR test after being wound The iruit diagram in Figure 15 demonstrates how TTR test an be done Transformers xfmr1 and xfmr are both onneted in arallel and are sulied by voltages V soure Transformer xfmr1 is the transformer under taste Transformer xfmr has a known variable turns ratio Set the turns ratio of the transformer xfmr to the alulated turns ratio of the transformer xfmr1 Connet the transformers as shown in the iruit diagram in Figure 15 If the outut voltages of transformers xfmr1 and xfmr are the same then the transformer xfmr1 is wound orretly If the voltages are different then the transformer xfmr1 is not orretly wound 1 Oen Ciruit Test This test is used to measure transformer ore losses The measured values an be used to alulate the ative omonent and reative omonents of the transformer ore, R and X M resetively Figure 16 shows how this test is set u In the iruit diagram, xfmr1 is the transformer under test, A is ammeter, W is wattmeter with both urrent and voltage robes onneted to the iruit to measure ative ower, V is the voltmeter The rimary rated voltage, V in is alied to the transformer as shown in the iruit The oen iruit urrent in the rimary, I o is measured by ammeter A, oen iruit voltage, V o is measured by the voltmeter V and ative ower P o is measured by wattmeter W It must be noted that P o is the ative ore ower loss [1] Therefore, the imedane angle is Oen iruit imedane is P φ o = 1 o os VoIo Z o (11) Vo = (1) I o 7

27 Figure 15 Transformer turns ratio test iruit Figure 16 Oen iruit test Therefore, we an find the values of R and X M of the transformer ore from the short iruit results Thus R = Z φ (13) 13 Short Ciruit Test os ( ) ( ) o o X = Z sin φ (14) M o o This test is imortant beause it measures the winding ower losses in the transformer The measured values of the urrent and voltages an be used to alulate the ative and reative omonents of the windings Figure 17 shows the iruit set u of the test 71

28 Figure 17 Short iruit test The inut rimary voltage, V in is gradually inreased from zero volts until the ammeter; A reading reahes the maximum value of the urrent rating of the transformer The voltage reading, V s and wattmeter reading, P s are reord- V ed when the maximum urrent rating, I s is reahed The inut voltage s for the maximum urrent of the transformer is also reorded by voltmeter V The imedane angle is P φs = VsIs 1 s os Therefore, the rimary winding total imedane is Z (131) Vs = (1) I Therefore, we an find the values of R and X of the rimary winding of the transformer from the short iruit test results Thus s R = Z φ (133) os ( ) o ( ) X = Z sin φ (134) o We an do the same short iruit test on the seondary of the transformer to find the winding imedane, Z s 14 Winding Resistane Test The hanes to have the insulation of the windings srathed are high during the winding of the transformer It is therefore neessary to measure the resistane of the winding and omare it with the alulated one If the measured resistane is very low omared to the alulated one, then some of the turns are shorting due to the damaged winding insulation This test is not as reliable sine it uses DC voltage hene does not take into ount skin dee effet Although this test is not as aurate omared to TTR test, it is very easy to erform There are two main methods to measure the winding resistane It an be measure diretly using ohmmeter The winding resistane an also be measured using voltmeter and ammeter as shown in the iruit in Figure 18 V Rwd = (141) I 7

29 Figure 18 Ciruit for winding resistane measurement 13 Conlusions This aer has overed the most imortant onets of transformer rinials whih are utilized in transformer design The haters are listed so that the reader an easily go ste-by-ste to aly the tehnique to design her/his own transformer with a very high effiieny Most of the designs aroah used in this aer are also utilized in transformer industry The author has also inluded the most imortant tests for the rototye transformers There are several tests erformed by ower transformer manufaturers Most of these tests results are listed on the namelate of the transformer The author of this aer has just listed those tests whih will hel the designer to omare the alulated transformer effiieny and voltages to that of the tests results The grahs, iruit diagrams and hasor diagrams have been drawn using estimated transformer arameters to hel visional understanding of the design onets and alulated results of the transformer The atual measured values will likely be different from the simulated values sine some of the stray losses whih have less imat on the low frequeny transformer have not been onsidered Some of the effets whih have not been onsidered in the simulations are; roximity tests and winding aaitane effets The rograms whih have been used for simulations and iruit drawings in this aer are; MATLAB, Saber and Visio Aknowledgements The author of this aer would like to thank Dr Marian K Kazimierzuk, a rofessor in the Deartment of Eletrial and Comuter Engineering at Wright State University The author s knowledge in transformer design is the true refletion of the suort he got from Dr M K Kazimierzuk s lassroom and outside lassroom time Referenes [1] Chaman, SJ (1999) Eletrial Mahinery Fundamentals 4th Edition, MGraw Hill 73

30 Comanies In, New York [] Kazimierzuk, MK (9) High-Frequeny Magneti Comonents John Wiley & Sons, Hoboken [3] DelVehio, RM, Poulin, B, Feghali, PT, Shah, DM and Ahuja, R (1) Transformer Design Priniles nd Edition, CRC Press, Boa Raton htts://doiorg/111/ebk Submit or reommend next manusrit to SCIRP and we will rovide best servie for you: Aeting re-submission inquiries through , Faebook, LinkedIn, Twitter, et A wide seletion of journals (inlusive of 9 subjets, more than journals) Providing 4-hour high-quality servie User-friendly online submission system Fair and swift eer-review system Effiient tyesetting and roofreading roedure Dislay of the result of downloads and visits, as well as the number of ited artiles Maximum dissemination of your researh work Submit your manusrit at: htt://aersubmissionsirorg/ Or ontat jee@sirorg 74