ee 221-mutually coupled circuits

26
EE 221 Circuits II Chapter 13 Magnetically Coupled Circuits 1

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Page 1: Ee 221-mutually coupled circuits

EE 221

Circuits II

Chapter 13

Magnetically Coupled Circuits

1

Page 2: Ee 221-mutually coupled circuits

Magnetically Coupled Circuits

13.1 What is a transformer?

13.2 Mutual Inductance

13.3 Energy in a Coupled Circuit

13.4 Linear Transformers

13.5 Ideal Transformers

13.6 Applications

2

Page 3: Ee 221-mutually coupled circuits

13.1 What is a transformer?

It is an electrical device designed on the basis of

the concept of magnetic coupling.

It uses magnetically coupled coils to transfer

energy from one circuit to another.

It is the key circuit element for stepping up or

stepping down ac voltages and currents,

impedance matching, isolation, etc….

3

Page 4: Ee 221-mutually coupled circuits

13.2 Mutual Inductance

It is the ability of one inductor to induce a voltage across a neighboring inductor,

Mutual inductance is measured in henrys (H).

4

dt

diMv 1

212 dt

diMv 2

121 dt

diLv 1

11 dt

diLv 2

22

Page 5: Ee 221-mutually coupled circuits

13.2 Mutual Inductance

If a current enters the dotted terminal of the first coil, the reference polarity of the voltage in the second coil is positive at the dotted terminal.

5 Illustration of the dot convention.

Page 6: Ee 221-mutually coupled circuits

13.2 Mutual Inductance

)connection aiding-(series

221 MLLL

)connection opposing-(series

221 MLLL

6

Series connection of two mutually coupled inductors; (a) series-aiding connection, (b) series-opposing connection.

)()( 21dt

diM

dt

diL

dt

diM

dt

diLv )()( 21

dt

diM

dt

diL

dt

diM

dt

diLv

Page 7: Ee 221-mutually coupled circuits

Problem 1:

7

Page 8: Ee 221-mutually coupled circuits

13.2 Mutual Inductance

8

Time-domain analysis of a circuit containing coupled coils.

Frequency-domain analysis of a circuit containing coupled coils

Page 9: Ee 221-mutually coupled circuits

13.2 Mutual Inductance

Example 1

Calculate the phasor currents I1 and I2 in the circuit shown below.

A149.2I A;4.49.13I 21

9

Ans:

Page 10: Ee 221-mutually coupled circuits

13.3 Energy in a Coupled Circuit (1)

The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0≤ k ≤1.

21LL

Mk

10

• The instantaneous energy stored in the circuit is given by

• (+) if both currents enter or leave the dotted sides

• (-) if one current enters the dotted side and the other current leaves dotted side.

21

2

22

2

112

1

2

1iMiiLiLw

Page 11: Ee 221-mutually coupled circuits

13.3 Energy in a Coupled Circuit

Example 2

Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = 1sec if v = 60cos(4t +30°) V.

11

Ans: k=0.56; w(1)=20.73J

Page 12: Ee 221-mutually coupled circuits

Problem 20:

12

Page 13: Ee 221-mutually coupled circuits

13.4 Linear Transformer – Reflected Impedance

13

Linear Transformer refers to a case where the

inductances L1 and L2 and mutual inductance M are

constant.

1V

2V

ref

L

in

L

ZLjRZRLj

MLjRZ

I

V

MIjILjIZR

MIjILjIRV

11

22

2

11

1

12222

21111

)(

)(

)()(

)(

Page 14: Ee 221-mutually coupled circuits

13.4 Linear Transformer

A1.1135.0I ;1.5314.100Z 1in

14

Example 3

In the circuit below, calculate the input impedance and current I1. Take Z1=60-j100Ω, Z2=30+j40Ω, and ZL=80+j60Ω.

Ans:

Page 15: Ee 221-mutually coupled circuits

13.4 Linear and Ideal Transformer

15

Voltage Ratio (ignore resistances)

Current Ratio

In case of unity coupling (i.e., k =1) and very large

inductances,

2

211

2

)()( MZLjLj

MZj

V

V

L

L

LZLj

Mj

I

I

21

2

1

2

1

21

1

2

L

L

ZLj

ZLLj

V

V

L

L

2

1

2

12

1

2

L

L

Lj

LLj

I

I

Page 16: Ee 221-mutually coupled circuits

13.4 Ideal Transformer

16

The inductance L is proportional to the number of turns

squared.

In case of an ideal Transformer,

nN

N

aN

aN

L

L

ZLj

ZLLj

V

V

1

2

2

1

2

2

1

2

21

221

1

2

nN

N

L

L

Lj

LLj

I

I 1

2

1

2

1

2

12

1

2

L=10-3N2r2/(228r+254l)

Page 17: Ee 221-mutually coupled circuits

13.5 Ideal Transformer

nN

Nn

N

N 1

I

I

V

V

2

1

1

2

1

2

1

2

17

An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite self-inductances.

(a) Ideal Transformer

(b) Circuit symbol

V2>V1 → step-up transformer

V2<V1 → step-down transformer

Page 18: Ee 221-mutually coupled circuits

13.5 Ideal Transformer

Example 4

An ideal transformer is rated at 2400/120V, 9.6 kVA, and has 50 turns on the secondary side.

Calculate:

(a) the turns ratio,

(b) the number of turns on the primary side, and

(c) the current ratings for the primary and secondary windings.

18

Ans:

(a) This is a step-down transformer, n=0.05

(b) N1 = 1000 turns

(c) I1 = 4A and I2 = 80A

Page 19: Ee 221-mutually coupled circuits

Problem 36

19

Page 20: Ee 221-mutually coupled circuits

13.6 Applications

Transformer as an isolation device to isolate ac supply from a rectifier

20

Page 21: Ee 221-mutually coupled circuits

13.6 Applications

Transformer as an isolation device to isolate dc between two amplifier stages.

21

Page 22: Ee 221-mutually coupled circuits

13.6 Applications

22

Transformer as a matching device

Using an ideal transformer to match the speaker to the amplifier

Equivalent circuit

Page 23: Ee 221-mutually coupled circuits

13.6 Applications

Example 5

Calculate the turns ratio of an ideal transformer required to match a 100Ω load to a source with internal impedance of 2.5kΩ. Find the load voltage when the source voltage is 30V.

23

Ans: n = 0.2; VL = 3V

Page 24: Ee 221-mutually coupled circuits

13.6 Practical Electric Utility transformers

24

• Used to step-up or step-down the voltage

Page 25: Ee 221-mutually coupled circuits

Ideal Auto-Transformer

Auto-transformers are used in cases where the voltage

ratio is less than 2.

Note that there is only one winding, the primary and

secondary side share part of this winding.

There is no electrical isolation between the primary and

secondary sides.

The apparent power rating of an auto-transformer is often

much higher than a two-winding transformer of the same

size (see example 13.10)

Page 26: Ee 221-mutually coupled circuits

3-Phase Transformer