Review: Lecture 9

• Instantaneous and Average Power

• Maximum Average Power Transfer

• Effective or RMS Value

• Apparent Power and Power Factor

• Complex Power

• Conservation of AC Power

) 2( cos 2

1 ) ( cos

2

1 )( ivmmivmm tIVIVtp

*

THTHTHLLL ZjXRjXR Z TH

2

TH

max 8

R

VP

TH

2

TH

2

THL Z XRR

2

I I m2

rms )θ(θIV) θ(θIVP ivrmsrmsivmmeff coscos2

1

) θ(θ S) θ(θ IVP ivivrmsrms coscos

ivrmsrms θθ I V IV S

2121212

1

2

1

2

1

2

1 S S I VI V) II (V I V S *****

Review: Inductors

Current

Three-dimensional magnetic field

Magnetic flux

dt

di

dt

d

i

dt

dNv

Oppose the change

Inductance: dt

di

di

dN

dt

dNv

vi vs.

L

N: turns

Faraday’s law

Len’s law

Ampere’s law

Lecture 10 AC Circuits Magnetically Coupled Circuit

Contents

• Mutual Inductance

• Energy in a Coupled Circuit

• What is a transformer?

• Linear Transformers

• Ideal Transformers

• Applications

4

Mutual Inductance

Magnetic flux Links coil 1 only 11

12 Magnetic flux link both

12111

dt

dNv 1

11

dt

dNv 12

22

Voltage induced

Magnetically coupled

Express v2 in terms of the change of i1:

dt

di

di

dN

dt

dNv 1

1

122

1222

M21 Mutual inductance of coil 2 with respect to coil 1

6

Mutual Inductance

• It is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H).

dt

diMv 1

212 dt

diMv 2

121

The open-circuit mutual voltage across coil 2

The open-circuit mutual voltage across coil 1

7

M12 and M21 ? The Upper Limit?

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

• The instantaneous energy stored in the circuit is given by

21LLkM

21

2

22

2

112

1

2

1IMIiLiLw

8

Mutual Inductance: Dot convention

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

Illustration of the dot convention.

9

Examples of the dot convention

)connection aiding-(series

221 MLLL

Dot convention for coils in series; the sign indicates the polarity of the mutual voltage; (a) series-aiding connection, (b) series-opposing connection.

)connection opposing-(series

221 MLLL

10

Mutual Inductance: Time and Frequency domains

Time-domain analysis of a circuit containing coupled coils.

Frequency-domain analysis of a circuit containing coupled coils

0121

111 vdt

diM

dt

diLRi 02

12222 v

dt

diM

dt

diLRi

KVL

022 VIIIZ 111 MjLj 01222 IIIZ MjLjL

Current leaves the dot

11

Mutual Inductance: Example

Example 1 Calculate the phasor currents I1 and I2 in the circuit shown

below.

A0414912 A;39490113 .... 21 IIAns:

12

Energy in a Coupled Circuit: Example

Example 2

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

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

13

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 elements for stepping up or stepping down ac voltages or currents, impedance matching, isolation, etc.

14

Transformer

• It is generally a four-terminal device comprising two (or more) magnetically coupled coils

impedance reflected is Z

Z, ZI

VZ

22

22

11

1

in

L

RRLjR

MLjR

Connected to the source Connected to the load

Linear: wound on a magnetically linear materials, air-core –

radio/TV…

15

Linear Transformer: Example

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: A1.1135.0I ;1.5314.100Z 1in

16

Ideal Transformer

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

Ideal Transformer (a) and Circuit symbol (b)

nN

Nn

N

N 1

I

I

V

V

2

1

1

2

1

2

1

2

V2 > V1→ step-up transformer

V2< V1→ step-down transformer

17

Ideal Transformer: Example

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

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

18

Applications: Isolation device

• Transformer as an Isolation Device to isolate ac supply from a rectifier

19

Applications: Matching device

• Transformer as a Matching Device

Using an ideal transformer to match the speaker to the amplifier

Equivalent circuit

20

Applications: Example

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.

Ans: n = 0.2; VL = 3V

Lecture 11 AC Circuits Frequency Response

Appendix

23

Applications: Power distribution

• A typical power distribution system