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CouplingElementandCoupledcircuits
.
Coupledinductor

Coupledinductor
Magneticfluxisproducedbyeachcoilbythefunctions
Whereandarenonlinearfunctionofand
ByFaraday’slaw
Coupledinductor
Lineartime-invariantcoupledinductor
Ifthefluxisalinearfunctionofcurrents
and
Insinusoidsteady-state
NotethatthesignsofandarepositivebutthesignforMcanbe
Coupledinductor
DotsareoftenusedinthecircuittoindicatethesignofM

Coupledinductor
Coefficientofcoupling
Thecouplingcoefficientis
Ifthecoilsaredistanceawaykisverysmallandclosetozeroandequal
to1foraverytightcouplingsuchforatransformer.
Coupledinductor
Multi-windingInductorsandinductanceMatrix
Formorewindingsthefluxineachcoilare
areselfinductancesand
aremutualinductances
Inmatrixform
Coupledinductor
Fig3Three-windinginductor
Coupledinductor
Inducedvoltage
Theinducedvoltageintermcurrentvectorandtheinductancematrixis
Example1

samedirectionbutarenotthereforeispositivewhileand
arenegative.

Coupledinductor
Thecurrentsare
KVL
Byintegrationofvoltage
Therefore
H
Note
forparallelinductors