Transformers

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Mutual Inductance L1 dI Lm 2 dt + + - I1 Lm L1 L2 + L2 + I2 V1 - Lm dI1 dt V2 - I1 Lm L1 L2 + L1 dI 2 − Lm dt + + - L2 + I2 V1 - dI1 − Lm dt V2 - V1 = L1 dI1 dI ± Lm 2 dt dt dI1 dI 2 V2 = ± Lm + L2 dt dt ã Correct sign for mutual inductance found from Lenz’ law and dot convention ã Dot convention: current flowing into one dot will induce current flow out of second dot © Robert York, 2006 Transformers A transformer is just a special case where the mutual inductance is made as
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  ©Robert York, 2006 Mutual Inductance  L  m  L 1 L  2 +-+-  I  1 V  1 +- V   2  I   2 2 m dI  Ldt  1  L +- 2  L 1 m dI  Ldt  1 21 11 22 2 mm dI dI V L Ldt dt dI dI V L Ldt dt  = ±= ± +  L  m  L 1 L  2 +-+-  I  1 V  1 +- V   2  I   2 2 m dI  Ldt  − 1  L +- 2  L 1 m dI  Ldt  − ã Correct sign for mutual inductance found from Lenz’law and dot convention ã Dot convention: current flowing into  one dot willinduce current flow out  of second dot  ©Robert York, 2006 Transformers A transformer is just a special case where the mutual inductanceis made as large aspossible by allowing both coils to share the same fluxThis is usually achieved by winding them both on a common core of high permeabilitymaterial (soft iron or ferrite materials)  I  1 I   2 + 1 1 1 22 1 2 2 mm V j L I j L I V j L I j L I  ω ω ω ω  = += + + V  1 V   2 -- When there is no flux leakage, the mutual inductance is related to the primary andsecondary inductances as 1 2 m  L L L = For real transformers this can never be quite achieved, so we write 1 2 where 0 1 m  L k L L k  = < < coefficient of coupling   ©Robert York, 2006 Ideal Transformer 1:n 1 12 2 1 V N V N n = = If both coils share the sameflux, then Farady’slaw gives:  I  1 I   2 ++ V  1 V   2 As the permeability of the coreincreases, the relationship betweenthe primary and secondarycurrents approaches a limitingvalue set by the turns ratio: - 1 22 1  I N n I N  ⇒ = - These two relationships define an ideal transformer  . This is a fictitouselement (note that µ →∞ implies infinite inductancesmso the impedance matrix is infinite) but a real transformer approaches this behavior.An idea transformer has the following property when one winding is terminated:  I  1 I   2 1:n Z  L 1 1 2 221 2 1 2 ( / )( / )  Lin V N N V Z  Z  I N N I n = = = ++- V   2 V  1 -  ©Robert York, 2006 Transformer Equivalent Circuit Using the tee-equivalent for reciprocal networks, we find the followignequivalent circuit for mutual inductances or transformers 1 m  L L − 2 m  L L − m  L 1 122 2 11 2 1 m  L N  L N n L L k L L knL ∝∝ == = This can be cascaded with an ideal 1:1 transformer to simulate the fact that areal transformer has electrically isolated ports
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