ECE 201 Circuit Theory I 1

Mutual Inductance

• Consider two circuits “linked” by a magnetic field (magnetically-coupled coils).

ECE 201 Circuit Theory I 2

“Self-Inductances” are L1 and L2.

The “Mutual” Inductance is M.

The voltage induced in one circuit is related to the time-varying current in the other circuit.

ECE 201 Circuit Theory I 3

Analysis

• Easiest with mesh-current method.

ECE 201 Circuit Theory I 4

Write the circuit equations in terms of the coil currents.

Arbitrarily assign the current directions.

There will be two voltages across each coil, the “self-induced” voltage, L(di/dt), and a “mutually induced” voltage, M(di/dt).

ECE 201 Circuit Theory I 5

Determination of Voltage Polarities

• “Dot convention”– Dots indicate the direction in which the coils

are wound.

ECE 201 Circuit Theory I 6

The Rule for using the Dot Convention

• When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal.

ECE 201 Circuit Theory I 7

Alternate Rule for the Dot Convention

• When the reference direction for a current leaves the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is negative at its dotted terminal.

ECE 201 Circuit Theory I 8

For this Example

• The voltage induced in coil 1 by the current in coil 2 is negative at the dotted terminal of coil 1, and is a voltage rise with respect to current i1.

ECE 201 Circuit Theory I 9

For this Example

• The voltage induced in coil 2 by the current in coil 1 is positive at the dotted terminal of coil 2, and is a voltage rise with respect to current i2.

ECE 201 Circuit Theory I 10

Write the Mesh Equations

1 2

1 1 1

2 1

2 2 2

0

0

g

di div i R L M

dt dtdi di

i R L Mdt dt

ECE 201 Circuit Theory I 11

Example 6.6, page 206

• Write a set of mesh equations that describe the circuit shown in terms of i1 and i2.

ECE 201 Circuit Theory I 12

1 2

1 1 2

2 1

2 2 1

4 20 5 8 20 8 5 0

16 20 60 20 8 16 0

g

g

g

didi dii i i i

dt dt dtdidi di

i i idt dt dt

ECE 201 Circuit Theory I 13

1

2 1 2 1

1

2 1 2 2

4 8 20 5 0

20 60 16 8 0

g g

g

di di i i i i i

dt dtd di

i i i i idt dt