EE 221 Circuits II. Chapter 13 Magnetically Coupled Circuits

EE 221

Circuits II

Chapter 13

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

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,

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).

dt di M v₂  ₂₁ 1 dt di M v₁  ₁₂ 2 dt di L v₁  ₁ 1 dt di L v₂  ₂ 2

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.

13.2 Mutual Inductance

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

connection. ) ( ) ( _{1 2} dt di M dt di L dt di M dt di L v     ( ₁ ) ( ₂ ) dt di M dt di L dt di M dt di L v    

13.2 Mutual Inductance

13.2 Mutual Inductance

Example 1

Calculate the phasor currents I₁ and I₂ in the circuit shown below. A 14 9 . 2 I A; 4 . 49 . 13 I₁    ₂    Ans:

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.

2 1

L

L

M

k



• 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.

2 1 2 2 2 2 1 1

2

1

2

1

i

Mi

i

L

i

L

w







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.

13.4 Linear Transformer – Reflected Impedance

 Linear Transformer refers to a case where the

inductances L₁ and L₂ and mutual inductance M are constant.   1 V   2 V ref in L Z L j R M L j R Z V MI j I L j I Z R MI j I L j I R V                  1 1 2 1 1 1 2 2 2 2 2 1 1 1 1 ) ( ) ( ) ( ) (        

13.4 Linear Transformer

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

13.4 Linear and Ideal Transformer

 Voltage Ratio (ignore resistances)

 Current Ratio

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

inductances, 2 2 1 1 2 ) ( ) ( j L Z M L j MZ j V V L L        L Z L j M j I I   2 1 2   1 2 1 2 1 1 2 L L Z L j Z L L j V V L L     2 1 2 1 2 1 2 L L L j L L j I I    

13.4 Ideal Transformer

 The inductance L is proportional to the number of turns

squared.

 In case of an ideal Transformer,

n N N aN aN L L Z L j Z L L j V V _{    } 1 2 2 1 2 2 1 2 2 1 2 2 1 1 2   n N N L L L j L L j I I 1 2 1 2 1 2 1 2 1 2       L=10-3_N2_r2_/(228r+254l)

13.5 Ideal Transformer

 An ideal transformer is a unity-coupled, lossless transformer

in which the primary and secondary coils have infinite self-inductances.

(a) Ideal Transformer

V2>V1 → step-up transformer V2<V1 → step-down transformer

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.

Ans:

(a) This is a step-down transformer, n=0.05 (b) N1 = 1000 turns

13.6 Applications

 Transformer as an isolation device to isolate ac supply

13.6 Applications

 Transformer as an isolation device to isolate dc

13.6 Applications

 Transformer as a matching device

Using an ideal transformer to match the speaker to the amplifier

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.

13.6 Practical Electric Utility transformers

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)