AC Circuits Analysis 1 L (Lecture notes)

```chapter 11
Inductors
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
The Basic Inductor
When a length of wire is formed into a coil., it
becomes a basic inductor. When there is current in
the inductor, a three-dimensional magnetic field is
created.
A change in current
causes the magnetic
S
N
field to change. This in
turn induces a voltage
across the inductor that
opposes the original
change in current.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
The Basic Inductor
One henry is the inductance of a coil when a current,
changing at a rate of one ampere per second, induces one
volt across the coil. Most coils are much smaller than 1 H.
The effect of inductance is greatly
magnified by adding turns and winding
them on a magnetic material. Large
inductors and transformers are wound
on a core to increase the inductance.
Magnetic core
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Faraday’s law was introduced in Chapter 7 and repeated
here because of its importance to inductors.
The amount of voltage induced in a coil is directly
proportional to the rate of change of the magnetic field
with respect to the coil.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Lenz’s law
Lenz’s law was also introduced in Chapter 7 and is an
extension of Faraday’s law, defining the direction of the
induced voltage:
When the current through a coil changes and an
induced voltage is created as a result of the changing
magnetic field, the direction of the induced voltage is
such that it always opposes the change in the current.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Lenz’s law
A basic circuit to demonstrate Lenz’s law is shown.
Initially, the SW is open and there is a small
current in the circuit through L and R1.
L
VS
SW
+
R1
R2


Electronics Fundamentals 8th edition
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+
Chapter 11
1
Lenz’s law
SW closes and immediately a voltage appears
across L that tends to oppose any change in current.
L

+
VS
+
SW
R1
R2


Electronics Fundamentals 8th edition
Floyd/Buchla
+
Initially, the meter
as before the switch
was closed.
Chapter 11
1
Lenz’s law
After a time, the current stabilizes at a higher level
(due to I2) as the voltage decays across the coil.
L
VS
SW
+
R1
R2


+
Later, the meter
current because of
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Types of inductors
There are a variety of inductors, depending on the
amount of inductance required and the application.
Some, with fine wires, are encapsulated and may
appear like a resistor.
Common symbols for inductors (coils) are
Air core
Electronics Fundamentals 8th edition
Floyd/Buchla
Iron core
Ferrite core
Variable
Chapter 11
1
Factors affecting inductance
Four factors affect the amount of inductance for a
coil. The equation for the inductance of a coil is
N 2 A
L
l
where
L = inductance in henries
N = number of turns of wire
 = permeability in H/m (same as Wb/At-m)
l = coil length on meters
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
What is the inductance of a 2 cm long, 150
turn coil wrapped on an low carbon steel core that
is 0.5 cm diameter? The permeability of low
carbon steel is 2.5 x104 H/m (Wb/At-m).
A  πr 2  π  0.0025 m   7.85 105 m2
2
N 2 A
L
l
2
150 t   2.5 104 Wb/At-m  7.85 105 m2 

0.02 m
 22 mH
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Series inductors
When inductors are connected in series, the total
inductance is the sum of the individual inductors.
The general equation for inductors in series is
LT  L1  L2  L3  ...Ln
If a 1.5 mH inductor is
connected in series with
an 680 H inductor, the
total inductance is 2.18 mH
Electronics Fundamentals 8th edition
Floyd/Buchla
L
1
L
2
1
.
5
m
H 6
8
0

H
Chapter 11
1
Parallel inductors
When inductors are connected in parallel, the total
inductance is smaller than the smallest one. The
general equation for inductors in parallel is
LT 
1
1 1 1
1
   ... 
L1 L2 L3
LT
The total inductance of two inductors is
LT 
1
1 1

L1 L2
…or you can use the product-over-sum rule.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Parallel inductors
If a 1.5 mH inductor is connected in
parallel with an 680 H inductor,
the total inductance is 468 H
L1
1.5m
H
Electronics Fundamentals 8th edition
Floyd/Buchla
L2
680
H
Chapter 11
1
Inductors in dc circuits
When an inductor is connected
in series with a resistor and dc
source, the current change is
exponential.
Vinitial
t
0
Inductor voltage after switch closure
Ifinal
R
L
0
Current after switch closure
Electronics Fundamentals 8th edition
Floyd/Buchla
t
Chapter 11
1
Universal exponential curves
L
τ
R
100%
95%
99%
Rising exponential
63%
60%
40%
37%
Falling exponential
20%
14%
5%
0
0
Electronics Fundamentals 8th edition
Floyd/Buchla
98%
86%
80%
Percent of final value
Specific values for
current and voltage
universal curve. For
an RL circuit, the
time constant is
1t
2%
2t
3t
4t
Number of time constants
1%
5t
Chapter 11
1
Inductive reactance
Inductive reactance is the opposition to
ac by an inductor. The equation for
inductive reactance is
X L  2πfL
The reactance of a 33 H inductor when a
frequency of 550 kHz is applied is 114 W
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Inductive reactance
When inductors are in series, the total reactance is the sum of the
individual reactances. That is,
X L(tot )  X L1  X L2  X L3    X Ln
Assume three 220 H inductors are in series with a 455 kHz
ac source. What is the total reactance?
The reactance of each inductor is
X L  2 fL  2  455 kHz  220 μH   629 W
X L(tot )  X L1  X L2  X L3
 629 W  629 W  629 W  1.89 kW
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Inductive reactance
When inductors are in parallel, the total reactance is the reciprocal of
the sum of the reciprocals of the individual reactances. That is,
X L(tot ) 
1
1
1
1
1


  
X L1 X L2 X L3
X Ln
If the three 220 H inductors from the last example are placed
in parallel with the 455 kHz ac source, what is the total
reactance?
The reactance of each inductor is 629 W
X L(tot ) 
Electronics Fundamentals 8th edition
Floyd/Buchla
1
1

 210 W
1
1
1
1
1
1


+
+
X L1 X L2 X L3 629 W 629 W 629 W
Chapter 11
1
Inductive phase shift
When a sine wave
is applied to an
inductor, there is a
phase shift between
voltage and current
such that voltage
current by 90o.
Electronics Fundamentals 8th edition
Floyd/Buchla
VL 0
90
I 0
Chapter 11
1
Power in an inductor
True Power: Ideally, inductors do not dissipate power.
However, a small amount of power is dissipated in
winding resistance given by the equation:
Ptrue = (Irms)2RW
Reactive Power: Reactive power is a measure of the rate
at which the inductor stores and returns energy. One form
of the reactive power equation is:
Pr=VrmsIrms
The unit for reactive power is the VAR.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Q of a coil
The quality factor (Q) of a coil is given by the ratio of
reactive power to true power.
I2XL
Q 2
I RW
For a series circuit, I cancels, leaving
XL
Q
RW
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
1. Assuming all other factors are the same, the inductance
of an inductor will be larger if
b. the area is made larger
c. the length is shorter
d. all of the above
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
2. The henry is defined as the inductance of a coil when
a. a constant current of one amp develops one volt.
b. one volt is induced due to a change in current of
one amp per second.
c. one amp is induced due to a change in voltage of
one volt.
d. the opposition to current is one ohm.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
3. The symbol for a ferrite core inductor is
a.
b.
c.
d.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
4. The symbol for a variable inductor is
a.
b.
c.
d.
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
5. The total inductance of a 270 H inductor connected in
series with a 1.2 mH inductor is
a. 220 H
b. 271 H
c. 599 H
d. 1.47 mH
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
6. The total inductance of a 270 H inductor connected in
parallel with a 1.2 mH inductor is
a. 220 H
b. 271 H
c. 599 H
d. 1.47 mH
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
8. For circuit shown, the time constant is
L
a. 270 ns
2
7
0
H
b. 270 s
c. 270 ms
V
S
1
0V
R
1
.0k
W
d. 3.70 s
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
10. If a sine wave from a function generator is applied to an
inductor, the current will
a. lag voltage by 90o
b. lag voltage by 45o
c. be in phase with the voltage
d. none of the above
Electronics Fundamentals 8th edition
Floyd/Buchla
Chapter 11
1
Quiz
Electronics Fundamentals 8th edition
Floyd/Buchla
1. d
6. a
2. b
7. b
3. d
8. a
4. c
9. c
5. d
10. a