# N V V L. R a L I. Transformer Equation Notes

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1 Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 wite-up. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions we use. To o the eivtion, we will use the figue pictue below: figue 1 As seen figue 1, the tnsfome hs two uctos: souce o pimy ucto n lo o secony ucto. Ech ucto loop is seies n hs some esistnce. The esistnce of the souce loop is epesente by n the esistnce of the lo loop is epesente by. The souce hs n put voltge of. The voltge the lo loop is mesue coss n is uce the lo ucto by the souce ucto. The cuent though the souce loop is given by. The cuent though the lo loop is. When you o ny eivtion, it is best to know wht esult you e tyg to chieve. The iel equtions fo tnsfome tems of the figue bove e: whee is constnt n is the numbe of tuns on ech ucto the tnsfome. Theefoe, is the tuns on the souce ucto n is the numbe of tuns on the lo ucto. is the put impence. t is the stey stte impence o esistnce of the souce ucto coil when it is couple with the lo coil the tnsfome.

2 nucto mpence figue We e gog to use the loop equtions fo the cicuit to eive ou fl esult. f the two cicuits e septe, s epicte figue, the stey stte loop equtions e simply: n 0 0. ote:. Wht effect oes puttg them togethe tnsfome hve? We will ssume tht the effect of the tnsfome is to cete some mutul uctnce,. The mutul uctnce is the net mount tht the souce ucto pushes on the lo ucto n vis ves. We o not know wht is, but we o know it is n uctnce n tht it woks gst the pimy n lo uctnces. We cn see tht becuse the cuents e gog opposite iections. This ie is pictue figue 3 below: figue 3 ow we cn wite ou loop equtions g: 1 ] n ] We will use these equtions to eive ll the eltionships bout tnsfomes, but fist, we nee to futhe efe. We know epens on both n. We lso know tht el tnsfome, thee will be some fluence of one ucto on the othe, but thee will lso be some losses ue to the fct tht they e connecte though meium such s i o ion. Theefoe, we will efe tems of constnt, k, the couplg

3 coefficient. We will let k tke on numbe fom 0 to 1. When k is 1, the fluence of one ucto the tnsfome on the othe is unelisticlly iel. This mens tht ll of the uctnce fom one pulls own on the othe. We cll this pefect tnsfome. themticlly, we will efe by the followg expession: k. The eson fo this exct choice will become evient lte when we o the eivtion. Fg the nput mpence Fist we wnt to f n expession fo, the net impence of the souce ucto the tnsfome. This impence is the combe fluence of n. We know tht whteve is, it must be the esistnce of the souce ucto the cicuit. Theefoe, we know the totl impence of the cicuit must be. T T T f we cn f n expession fo, then we will hve the put impence. We cn use the souce loop eqution to f this: ow we hve n expession fo tems of the souce uctnce n the mutul uctnce, but it is still tems of complex impences. We cn use the secon loop eqution n few ssumptions to simplify thgs. 1 ] ow ou efition fo begs to mke sense. ecll tht we efe with the expession: k. Befoe we contue, we must mke out fist ssumption. Assumption 1: Assume the tnsfome is pefectly couple. k1 ow we cn substitute fo n simplify.

4 ow we nee to mke ou next ssumption: Assumption : Assume tht is smll compe to the vlue of n. We coul lso mke the followg ssumption with the sme conclusion: Assumption b: Assume we e t high fequencies. ote tht genel uctnce gets bigge t high fequencies. We nee one of these conitions to be tue, so we cn elimte the tem n simplify the eqution.. We next efe constnt n cll it. We now hve the esie eltionship fo n. Cuent eltion Deivg the emg eltionships is fily stight fow. Fom loop eqution n the efition of, we hve: k Ag ssumg tht hs miml contibution n tht the tnsfome is pefectly couple k1, this becomes: 1. We now hve the cuent eltion. ote it is the vese of the othes.

5 oltge eltion The voltge eltion cn pply eithe to the souce voltge o to the put voltge epeng upon the size of the souce esistnce. Cse 1 : is smll figue 4 When is smll, we cn use the souce voltge s the voltge the eltionship. Fist we use the loop equtions fo the two cicuits. f is smll, we cn ignoe the voltge op coss the esisto n this becomes: n ow we cn substitute ou eltionships fo n the cuent tio: This is ou voltge eltionship.

6 Cse : is lge figue 5 When is lge, we nee to use to f the voltge eltionship ste of. n this cse we cn use iectly. umbe of Tuns eltionship The tuns eltionship epens moe on the physicl ttibutes of the tnsfome thn on the cicuit itself. We will ssume tht ou tnsfomes uctnce is eteme by the eqution given expeiment 3. c n f we ssume tht both coils hve the sme ius, e wppe oun the sme coe, n hve bout the sme length, then this euces to: c c

7 Conclusion The iel tnsfome equtions we use e sufficient to esign tnsfome tht woks une cet constts. You lo esistnce shoul be smll eltive to the impences of the uctos the tnsfome, t lest t the fequency you wnt it to wok t. Also, you shoul expect the tnsfome to hve losses fom the iel vlue tht you clculte. You cn quntify those losses by expeimentlly etemg couplg coefficient fo you tnsfome. Also, thee e physicl constts on tnsfome esign which fluence whethe o not it will behve like the equtions ictte. These clue the coe size, the length of the coil n the coe mteil. To check if tnsfome cicuit you esign is wokg popely, you nee to hve consistent esults fo these thee eltionships:

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