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EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 11 1 Magnetically Coupled Networks • A new four-terminal element, the transformer, is introduced in this chapter • A transformer is composed of two closely spaced inductors, that is, two or more magnetically coupled coils – primary side is connected to the source – secondary side is connected to the load Lecture 11 2 Dot Convention • dot convention: dots are placed beside each coil (inductor) so that if the currents are entering (or leaving) both dotted terminals, then the fluxes add • right hand rule says that curling the fingers (of the right hand) around the coil in the direction of the current gives the direction of the magnetic flux based on the direction of the thumb • We need dots on the schematic to know how the coils are physically oriented wrt one another Lecture 11 3 Mutually Coupled Coils The following equations define the coupling between the two inductors assuming that each respective current enters the dot side which is also the positive voltage side d i1 d i2 v1 (t ) L1 M dt dt d i1 d i2 v 2 (t ) M L2 dt dt where L1 and L2 are the self-inductances of the coils (inductors), and M is the mutual inductance between the two coils Lecture 11 4 Mutually Coupled Coils M i1(t) + v1(t) – L1 i2(t) L2 + v2(t) – d i1 d i2 v1 (t ) L1 M dt dt d i1 d i2 v 2 (t ) M L2 dt dt Lecture 11 5 Class Example • Extension Exercise E11.1 Lecture 11 6 Mutually Coupled Coils (AC) • The frequency domain model of the coupled circuit is essentially identical to that of the time domain V1 j L1 I1 j M I 2 V2 j M I1 j L2 I 2 Lecture 11 7 Class Examples • Extension Exercise E11.2 • Extension Exercise E11.3 Lecture 11 8 Source Input Impedance The source sees an input impedance, Zi, that is the sum of the primary impedance, and a reflected impedance, ZR, due to the secondary (load) side VS Zi Z P Z R Z P f Z L IP M Z VS + – L1 Lecture 11 L2 ZL 9 Class Example • Extension Exercise E11.4 Lecture 11 10 Energy Analysis • An energy analysis of the mutually coupled inductors provides an expression for the instantaneous stored energy w(t ) L1 i1 (t ) L2 i2 (t ) M i1 (t ) i2 (t ) 1 2 2 1 2 2 • The sign is positive (+) if currents are both entering (or leaving) the dots; sign is negative (-) if currents are otherwise Lecture 11 11 Quantifying the Coupling • The mutual inductance, M, is in the range 0 M L1 L2 • The coefficient of coupling (k) between two inductors is defined as 0 k 1 L1 L2 M – for k > 0.5, inductors are said to be tightly coupled – for k 0.5, coils are considered to be loosely coupled Lecture 11 12 Class Example • Extension Exercise E11.5 Lecture 11 13