E E 2415Lecture 12Mutual Inductance and Ideal Transformers

Mutual Flux

• Strongly Coupled Circuits

Circuit Model - 1

1 1 1

2 2 2

L M iM L i

Circuit Model - 2

1 1 1

2 2 2

L M iM L i

Ideal Transformers (1/2)

V1 jX1 jX2

+V2-

Z2

I1 I21 : n~ ~

~ ~

Ideal Transformers (2/2)

The ideal transformer has no power loss and consumes no reactive power.

Impedances are reflected across the ideal transformer by the square of the turns ratio.

An Analogy

* *1 1 2 2V I V I 1 1 2 2f l f l

Ideal Transformer Example (1/2)

41600°

4160 : 460

I2~

j15

25 I1~

Ideal Transformer Example (2/2)

Ideal Transformer Example 2

Simplify Load Circuit

Simplify Load Circuit & Transform(0.5 0.5) ( 2) 0.8 0.4

0.5 1.5loadj jZ jj

Multiply by square of turns ratio

227 (0.8 0.4) 583.2 291.6hloadZ j j

Load as Seen on High Voltage Side

Calculations - 1

112,500 0 19.05 26.78

2.5 583.2 (4.0 291.6)I A

j

12 27 514.4 26.78I I A

1 1 583.2 291.6 12,420 0.214V I j V

12 460.1 0.21427VV V

* *2 2 1 1 236,700 26.57(211.7 105.8)

LoadS V I V I VAj kVA

Calculations - 2211.7 , 105.8

236.7Load Load

Load

P kW Q kVAR

and S kVA

cos(26.57 ) 100% 89.44 %PF lagging

Mutual Flux

• Weakly Coupled Circuits

Weak Coupling Example

1

2

100 10 6 40 4 2 4

j j Ij j I

1

2

6.75 38.42

6.04 115.02

I A

I A

1 1 2

2 2 2 1

0 100 0 10 6 4

0 2 12 8 4

I j I j I

I j I j I j I

2nd Weak Coupling Case

1

2

15 15 8 25180 08 25 5 5150 30j j I

j j I

1 24.180 133.40 11.67 116.50I A I A

1 1 2 1 2

2 2 2 1 1

0 180 0 10 25 15 8

0 150 30 20 5 25 8

j I j I I I j I

j I I j I I j I