Review: Lecture 9
• Instantaneous and Average Power
p (t ) 
1
1
Vm I m cos ( v   i )  Vm I m cos (2 t   v   i )
2
2
• Maximum Average Power Transfer
*
Z L  RL  jX L  RTH  jX TH  ZTH
RL 
2
2
RTH
 X TH
 ZTH
• Effective or RMS Value
I 2rms 
Im
2
1
Peff  Vm I m cos (θv  θi )  Vrms I rms cos (θv  θi )
2
• Apparent Power and Power Factor
P  Vrms I rms cos (θv  θi )  S cos (θv  θi )
• Complex Power
S  V I   Vrms I rms  θv  θi 
• Conservation of AC Power
S
1
1
1
1
*
V I *  V ( I1  I 2* )  V I1*  V I 2*  S1  S2
2
2
2
2
Pmax 
VTH
2
8 RTH
Review: Inductors
Magnetic flux
Faraday’s law
Ampere’s law
d
dt

N: turns
Current
i
Three-dimensional magnetic field
di
dt
d
vN
dt
Oppose the change
L
Inductance:
i vs. v
d
d di
vN
N
dt
di dt
Len’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
Voltage induced
v1  N1
11
Magnetic flux
Links coil 1 only
d1
dt
1  11  12
Magnetically
coupled
12
Magnetic flux
link both
v2  N 2
M21
Express v2 in terms of the change of i1: v  N d12  N d12 di1
2
2
2
dt
di1 dt
d12
dt
Mutual inductance of
coil 2 with respect to
coil 1
Mutual Inductance
•
It is the ability of one inductor to induce a voltage across a neighboring
inductor, measured in henrys (H).
v2  M 21
di1
dt
The open-circuit mutual voltage
across coil 2
v1  M 12
di2
dt
The open-circuit mutual voltage
across coil 1
6
M12 and M21 ? The Upper Limit?
• The coupling coefficient, k, is a measure of the magnetic
coupling between two coils; 0≤ k ≤ 1.
M  k L1 L2
• The instantaneous energy stored in the circuit is given by
1 2 1 2
w  L1i1  L2i2  MI1 I 2
2
2
7
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.
8
Examples of the dot convention
Dot convention for coils in series; the sign indicates the polarity of the
mutual voltage; (a) series-aiding connection, (b) series-opposing
connection.
L  L1  L2  2M
(series - aiding connection )
L  L1  L2  2M
(series - opposing connection )
9
Mutual Inductance: Time and Frequency domains
KVL
Time-domain analysis of
a circuit containing
coupled coils.
i1 R1  L1
di1
di
 M 2  v1  0
dt
dt
i2 R2  L2
di2
di
 M 1  v2  0
dt
dt
Current leaves
the dot
Frequency-domain
analysis of a circuit
containing coupled coils
Z1I 1  jL2 I 1  jMI 2  V  0
Z L I 2  jL2 I 2  jMI1  0
10
Mutual Inductance: Example
Example 1
Calculate the phasor currents I1 and I2 in the circuit shown
below.
Ans: I1  13.01  49.39A; I 2  2.9114.04A
11
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
12
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.
13
Transformer
• It is generally a four-terminal device comprising two (or more)
magnetically coupled coils
Linear: wound on a magnetically
linear materials, air-core –
radio/TV…
Connected to the source
Connected to the load
V
 2M 2
Zin   R1  jL1  Z R , ZR 
is reflected impedance
I1
R2  jL2  Z L
14
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:
Zin  100.14  53.1; I1  0.5113.1A
15
Ideal Transformer
• An ideal transformer is a unity-coupled, lossless transformer in which the
primary and secondary coils have infinite self-inductances.
V2 N 2

n
V1 N1
Ideal Transformer (a) and Circuit symbol (b)
I 2 N1 1


I1 N 2 n
V2 > V1→ step-up transformer
V2< V1→ step-down transformer
16
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)
(b)
(c)
This is a step-down transformer, n=0.05
N1 = 1000 turns
I1 = 4A and I2 = 80A
17
Applications: Isolation device
• Transformer as an Isolation Device to isolate ac supply from a rectifier
18
Applications: Matching device
• Transformer as a Matching Device
Using an ideal transformer to match the speaker
to the amplifier
Equivalent circuit
19
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
20
Lecture 11 AC Circuits
Frequency Response
Appendix
Applications: Power distribution
• A typical power distribution system
23