AC Circuits I
INDUCTORS AND TRANSFORMERS
1
AC Circuits I
INDUCTANCE

AC Circuits I

The ability of a component with a changing
current to induce a voltage across itself or a
nearby circuit by generating a changing magnetic
field
Inductor – a component designed to provide a
specific measure of inductance (coil)
2
INDUCTANCE
• The Effect of Varying Current on a Magnetic Field
μ m NI
B

I
AC Circuits I
– When current passes through an inductor (coil),
magnetic flux is generated
– Flux density is found as
B
where
B
m
NI
ℓ
= the flux density, in webers per square meter (Wb/m2)
= the permeability of the core material
= the ampere-turns product
= the length of the coil, in meters
3
INDUCTANCE

When an AC current is applied through the coil,
The magnetic field expands as current increases
 The magnetic field contracts as current decreases

AC Circuits I
4
FARADAY’S LAWS OF INDUCTION


Law 2: The voltage induced is
proportional to the rate of change in
magnetic flux encountered by the
wire.
dB
V
dt
AC Circuits I

Law 1: To induce a voltage across a
wire, there must be a relative motion
between the wire and the magnetic
field.
Law 3: When a wire is cut by 108
perpendicular lines of force (1Wb) per
second, 1 V is induced across that
wire.
Self-inductance
5
LENZ’S LAW

AC Circuits I

1834, Heinrich Lenz – derived the relationship
between a magnetic field and the voltage it induces
Lenz’s Law – an induced voltage always opposes its
source
6
LENZ’S LAW
An increase in the inductor current causes the magnetic field to
expand.

As the magnetic field expands, it cuts through the coil, inducing a
voltage.

The polarity of the voltage (CEMF) opposes the increase in
current.
AC Circuits I

7
LENZ’S LAW
An decrease in the inductor current causes the magnetic field to
collapse.

As the magnetic field collapses, it cuts through the coil, inducing
a voltage across the component.

The polarity of the voltage opposes the decrease in current.
8
AC Circuits I

INDUCED VOLTAGE
• Induced Voltage can be found as
AC Circuits I
di
vL  L
dt
where
vL = the instantaneous value of induced voltage
L
= the inductance of the coil, measured in henries (H)
di
= the instantaneous rate of change in inductor current
dt
(in amperes per second)
9
EXAMPLE

If
AC Circuits I
I (t )  6  sin( t ) mA
VL (t )  ?
VL (t )  6 cos(t ) mA
10
UNIT OF MEASURE – HENRY (H)
AC Circuits I
• Inductance is measured in volts per rate of change
in current
• When a change of 1A/s induces 1V across an
inductor, the amount of inductance is said to be 1 H
vL
L
 di 
 
 dt 
11
EXAMPLE

AC Circuits I
If current is changing at a rate of 45mA/s
through a 10mH inductor, Calculate the induced
voltage?
di
vL  L  10mH  45mA / s  0.45mV
dt
12
MAKE YOUR OWN INDUCTOR
The inductance of a coil can be characterized by the
following equation:
where
L
m
N2
A
ℓ
AC Circuits I
μm N A
L

2
=inductance
= the permeability of the core material
= the square of the number of turns
= cross-sectional area of the inductor core in cm2
= the length of the coil, in meters
13
SUMMARY
AC Circuits I
14
OVERVIEW
What is an inductor?
 What is the relationship between flux density
and current?
 How many Laws did Faraday postulate and what
are they?
 What is Lenz’s Law?
 What is the unit measure of inductance?
 Two inductors have identical physical
characteristics except that one of them has an air
core and the other has an iron core. Which one
will have the higher value of inductance?

AC Circuits I
15
THE PHASE RELATIONSHIP BETWEEN
INDUCTOR CURRENT AND VOLTAGE

Sine-Wave Values of
di
dt
reaches its maximum value when i = 0
AC Circuits I

16
THE PHASE RELATIONSHIP BETWEEN
INDUCTOR CURRENT AND VOLTAGE
 The



Voltage leads current by 90°
Current lags voltage by 90°
AC Circuits I
Phase Relationship Between Inductor
Voltage and Current
Ideal Inductors do not dissipate
power!!!! They store energy in a
magnetic field
17
CONNECTING INDUCTORS IN SERIES AND
PARALLEL
 Series-Connected
Inductors
where
Ln
= the highest-numbered inductor in the circuit
 Parallel-Connected
LT 
AC Circuits I
LT  L1  L2      Ln
Inductors
1
1
1
1

  
L1 L2
Ln
18
where
Ln
= the highest-numbered inductor in the circuit
Mutual Inductance
When one inductor is placed in close proximity to
another, the flux produced by each coil can induce a
voltage across the other
 Energy is transferred from one coil to another i.e.
coupled

AC Circuits I
19
Mutual Inductance (Continued)
 Amount of mutual inductance:
LM  k L1 L2

k
where
1
2
2
1
AC Circuits I
Coefficient of Coupling (k) – a measure of the degree of
coupling that takes place between two or more coils
= the amount of flux generated by L1
= the amount of 1 that passes through L2 at a
90° angle to the turns of the coil
20
Mutual Inductance (Continued)

The Effects of Mutual Inductance on LT
AC Circuits I
LT  L1  L2  2 LM
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INDUCTIVE REACTANCE (XL)

Vrms
XL 
or
I rms


X L  2fL
AC Circuits I

Inductive Reactance (XL) – the opposition (in ohms) that an
inductor presents to a changing current
Calculating the Value of XL
Inductors oppose
Current. For the
circuit shown,
But measured
resistance is 0.21Ω!!!!
How come!!!
Opposition 
Vrms 10 V

 10 kΩ
I rms 1 mA
22
XL and Ohm’s Law

Example: Calculate the total current below
AC Circuits I
I rms
V
12 V
 rms 
 12mA
X L 1 kΩ
f  100 Hz
V
12 V
I rms  rms 
 3.82 A
X L 2   100  5mH
23
RESISTANCE, REACTANCE AND
IMPEDANCE

Resistance is a static value.

Reactance is a dynamic value
Reactance is a function of frequency
 Reactance is an imaginary resistance

AC Circuits I

real resistance
Resistance and Reactance are similar…both
measured in ohms
 They however cannot be added together directly
 Must combine them into an impedance (Z)



Impedance is the total opposition to current in an ac
circuit, consisting of resistance and/or reactances.
More on this in the next chapter
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INDUCTIVE REACTANCE (XL)

Series and Parallel Reactances
AC Circuits I
25
REVIEW
What is mutual inductance?
 What do we mean when we say two components
are coupled?
 What is the Coefficient of Coupling (k) ?
 What is the opposition of current provided by the
inductor? and how would you measure it ?
 Why is the resistance of an inductor typically
low?
 What is impedance ?
 Show how the units of measure in 2 fL resolve
themselves to yield a result measured in ohms.

AC Circuits I
26
TRANSFORMERS

AC Circuits I
Nope, not this
kind of
transformer!!
27
Transformer
A
AC Circuits I
two-coil component that uses electromagnetic
induction to pass an ac signal from its input to
its output while providing dc isolation between
the two
28
Transformer Construction and Symbols

Construction - Two coils
Primary – input
 Secondary – output

AC Circuits I

Schematic Symbols
29
Transformer Classifications
AC Circuits I
30
DOT NOTATION
AC Circuits I
31
Transformer Operation

AC Circuits I

Changing magnetic field in the primary windings
induces a voltage in the secondary windings
Primary and secondary windings are not physically
connected
32
Turns Ratio
AC Circuits I
N P 320 turns 4


N S 80 turns 1
33
VOLTAGE AND CURRENT TRANSFORMER
RELATIONS

Voltage in the secondary winding is determined by the
primary voltage and turns ratio


For an ideal transformer power in the primary winding is
completely transfered to the secondary winding :
Therefore current varies inversely with turns ratio
AC Circuits I
NS
VS  VP
NP
N P VP

N S VS
PP  PS
VP
I PVP  I SVS  I S  I P
VS
IS NP
NP

 IS  IP
IP NS
NS
34
VOLTAGE AND CURRENT TRANSFORMER
RELATIONS

If a transformer as a 10:1 winding ratio
(stepdown)

AC Circuits I
Voltage in secondary winding is 10x smaller than
voltage in primary winding
 Current in secondary winding is 10x larger than
current in primary winding

If a transformer as a 1:10 winding ratio
(stepdown)
Voltage in secondary winding is 10x larger than
voltage in primary winding
 Current in secondary winding is 10x smaller than
current in primary winding

35
EXAMPLES

AC Circuits I
Determine the VS (Secondary winding voltage) for a
transformer whose NP/NS ratio is 7:1. Assume that VP
(Primary winding voltage is 120Vac(VRMS)
NS
1
VS  VP 
 120  17.143 VRMS
NP
7

Determine the secondary current (IS) of a transformer
whose IP is 100mA ,VP is 120VRMS and whose whose NP/NS
ratio is 15:1.
NS
1
VS  VP 
 120  8 VRMS
NP
15
IS  IP 
VP
120
 100mA 
 1 .5 A
VS
8
IS  IP 
NP
15
 100mA   1.5 A
NS
1
36
POWER TRANSFER
Ideal Conditions: PP
= PS

In Practice: Secondary power is always slightly lower
because of a number of power losses

Losses
or
IPVP = ISVS
Copper Loss (I2R loss)
 Resistance of the copper wire used in windings
 Loss Due to Eddy Currents
 Magnetic flux induces a current in the iron core called an
eddy current. This current travels in a circular through
the core. Resistance of the core causes further power loss
 Hysteresis Loss
 Energy expended to overcome Hysteresis loss
AC Circuits I


37
TRANSFORMERS
AC Circuits I
38
Primary Impedance (ZP)

 NP
Z P  Z S 
 NS



2
AC Circuits I

Proportional to the square of
the turns ratio
Can be derived from current
and voltage relations
where
ZS = the total opposition to current in the secondary
(generally assumed to equal the opposition
provided by the load)
39
TRANSFORMER AS AN IMPEDANCE
MATCHING CIRCUIT
Need to ensure that maximum power is delivered to
the load
 According to Maximum Power Transfer (MPT) this
only happens when RS = RL
 How can we achieve Maximum power transfer when
RS ≠ RL ?

AC Circuits I

Use a Transformer as an impedance matching circuit
(buffer)!
Set ZP to RS and set ZS to RL.
 The Transformer that will ensure MPT is one whose primary to
secondary ratios satisfy:

 NP

 NS
2

ZP
 
ZS

 NP
Z P  Z S 
 NS



2
40
TRANSFORMER AS AN IMPEDANCE
MATCHING CIRCUIT - EXAMPLE
NP

NS
ZP
100 10 5



ZS
4
2 1
AC Circuits I
Z P = Zin = RS = 100
 ZS = Zout = RL = 4
 NP:NS = 5:1

41
Center-Tapped Transformer

AC Circuits I
Voltage from S1 to center or S2 to center is half
VS (secondary winding voltage).
42
INDUCTORS - APPARENT POWER (PAPP)

Energy in an inductor is actually stored in the
electromagnetic field generated by the inductor



Power is dissipated only through resistance –
winding resistance, RW




Energy not dissipated
Called reactive power – units of measure: volt-amperereactive (VAR)
Energy dissipated as heat
Called true power – units of measure: watts (W)
Apparent Power – the combination of resistive
(true) and reactive (imaginary) power – units of
measure: volt-amperes (VA)
Example on board
INDUCTOR QUALITY (Q)

Is a figure of merit that indicates how close the
inductor comes to the power characteristics of an ideal
component
PX
Q
PRw
where
PX
= the reactive power of the component,
measured in VARs
PRw = the true power dissipation of the component,
measured in watts
INDUCTOR QUALITY (CONTINUED)
PX
I L2 X L
Q
 2
PRw I L Rw
or
XL
Q
Rw
TYPES OF INDUCTORS



Air-Core: low Q
Iron-core: higher Q, limited to low frequencies due to
power losses
Ferrite Core: highest Q, used in higher frequencies, low
power loss
TYPES OF INDUCTORS

Toroids
TYPES OF INDUCTORS

Chokes