1. Lecture 111 EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001

2. Lecture 112 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

3. Lecture 113 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

4. Lecture 114 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 where L 1 and L 2 are the self-inductances of the coils (inductors), and M is the mutual inductance between the two coils

5. Lecture 115 Mutually Coupled Coils i1(t)i1(t) + – v1(t)v1(t)L1L1 i2(t)i2(t) + – v2(t)v2(t)L2L2 M

6. Lecture 116 Class Example Extension Exercise E11.1

7. Lecture 117 Mutually Coupled Coils (AC) The frequency domain model of the coupled circuit is essentially identical to that of the time domain

8. Lecture 118 Class Examples Extension Exercise E11.2 Extension Exercise E11.3

9. Lecture 119 Source Input Impedance The source sees an input impedance, Z i, that is the sum of the primary impedance, and a reflected impedance, Z R, due to the secondary (load) side L1L1 L2L2 M ZLZL +–+– VSVS Z

10. Lecture 1110 Class Example Extension Exercise E11.4

11. Lecture 1111 Energy Analysis An energy analysis of the mutually coupled inductors provides an expression for the instantaneous stored energy The sign is positive (+) if currents are both entering (or leaving) the dots; sign is negative (-) if currents are otherwise

12. Lecture 1112 Quantifying the Coupling The mutual inductance, M, is in the range The coefficient of coupling (k) between two inductors is defined as –for k > 0.5, inductors are said to be tightly coupled –for k  0.5, coils are considered to be loosely coupled

13. Lecture 1113 Class Example Extension Exercise E11.5