Transcript DC/AC Fundamentals: A Systems Approach

DC/AC Fundamentals: A Systems
Approach
Thomas L. Floyd
David M. Buchla
Inductors
Chapter 11
Ch.11 Summary
The Basic Inductor
When a length of wire is formed into a coil., it becomes
an inductor. When there is current in the inductor, a
three-dimensional magnetic field is created.
A change in current causes
the magnetic field to change.
This in turn induces a voltage
across the inductor that
opposes the original change
in current.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
S
-
N
+
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
The Basic Inductor
One henry (H) 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 coil values are far less than 1 H.
Large inductors and transformers
are wound around an iron core to
increase inductance.
Iron core
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Faraday’s Law
The voltage induced in a coil is directly proportional to
the rate of change of the magnetic field with respect to
the coil.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Lenz’s Law
When the current through a coil changes, 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.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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
-
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
+
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Lenz”s Law
SW closes and immediately a voltage appears
across L that tends to oppose any change in current.
L
-
+
VS
+
R1
-
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
SW
+
R2
Initially, the meter
reads same
current as before
the switch was
closed.
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Lenz’s Law
After a time, the current stabilizes at a higher level
(due to I2) as the voltage across the coil decays.
L
-
+
VS
+
R1
-
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
SW
+
R2
Later, the
meter reads a
higher current
because of the
load change.
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Inductor Characteristics
In addition to inductance, inductors have winding
resistance (RW), which is the resistance of the wire, and
winding capacitance (CW) between the turns. An equivalent
circuit for a practical inductor that includes these effects is
shown:
CW
Notice that the winding resistance
is in series with the coil and the
winding capacitance is in parallel
with both.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
RW
L
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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
Iron core
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Ferrite core
Variable
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Factors Affecting Inductors
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
m = permeability in Wb/At-m
l = coil length in meters
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Example
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  r 2  (0.0025 m) 2  7.85  10-5 m2
N 2A
L
l
(150)2 (2.5  10 - 4 Wb/At  m)(7.85  10 -5 m2 )

0.02 m
 22 mH
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Common Inductors
Inductors come in a variety of types and sizes. A few
common ones are illustrated here.
Encapsulated
Wirewound, high current
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Torroid coil
Variable
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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 + ... + Ln
If a 1.5 m inductor is
connected in series with an
680 H inductor, the total
inductance is 2.18 mH
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
L1
L2
1.5 mH
680 H
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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
1
LT 
1 1
1
+
+ ... +
L1 L2
Ln
The total inductance of two inductors is
LT 
1
1 1
+
L1 L2
…or you can use the product-over-sum rule.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Parallel Inductors
The total inductance in the parallel circuit shown
is 468 mH
L1
1.5 mH
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
680 H
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Inductors in DC Circuits
When an inductor is connected
in series with a resistor and a
dc source, current changes at
an exponential rate.
0
t
Inductor voltage after switch closure
Ifinal
R
VS
Vinitial
L
0
Current after switch closure
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
t
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Inductors in DC Circuits
VS
Exponential waveforms are
also generated when a square
wave source is connected to a
series RL circuit.
VL
R
VS
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
L
VR
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Universal Exponential Curves
L

R
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
100%
99%
98%
95%
86%
Percent of the final value
The exponential curves
show how the current in
an RL circuit increases (or
decreases) over five equal
periods, called time
constants. For an RL
circuit, the length of a time
constant is
80%
Rising exponential
63%
60%
40%
37%
Falling exponential
20%
14%
5%
0
0
1
2
3
2%
4
1%
5
Number of time constants
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Universal Exponential Curves
The universal curves can be applied to general formulas for the
current (or voltage) curves for RL circuits. The general current
formula is
i  IF + (Ii - IF )e -Rt / L
where
IF = final value of current
Ii = initial value of current
i = instantaneous value of current
e = Napier’s constant (approximately 2.71828)
The final current is greater than the initial current when the
inductive field is building, and less than the initial current when
the field is collapsing.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Inductive Reactance
Inductive reactance (XL) is the opposition of an
inductor to alternating current (ac). The equation
for inductive reactance is
XL  2fL
The reactance of a 33 mH inductor that is
operated at 550 kHz is 114 W
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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 L 2 + ... + X Ln
Assume three 220 mH inductors are in series with a 455 kHz ac
source. What is the total reactance?
The reactance of each inductor is
XL  2fL  2(455 kHz)(220 H)  629 Ω
X L( tot )  X L1 + X L 2 + ... + X Ln
 629 Ω + 629 Ω + 629 Ω  1.89 kW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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
+
+ ... +
X L1 X L 2
X Ln
If the three 220 mH 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. Using these values:
X L( tot ) 
1
1

 210 W
1
1
1
1
1
1
+
+
+
+
X L1 X L 2 X L 3 629 W 629 W 629 W
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Inductive Phase Shift
When a sine wave is
applied to an inductor,
there is a phase shift
between voltage and
current such that
voltage always leads
the current by 90o.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
VL 0
90
IL
0
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Power in an Inductor
True Power: The power that is dissipated in the winding
resistance of an inductor. One form of the true power equation
is:
Ptrue = (Irms)2RW
The unit of measure for true power is the volt-ampere (VA).
Reactive Power: The rate at which the inductor stores and
returns energy. One form of the reactive power equation is:
Pr = Vrms  Irms
The unit for reactive power is the volt-ampere-reactive (VAR).
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Q of a Coil
The quality factor (Q) of a coil equals the ratio of
reactive power to true power.
Pr
Q
Ptrue
I 2 XL
or Q  2
I RW
Since I2 appears in both the numerator and the denominator of
the right-hand fraction, it cancels, leaving:
Q
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
XL
RW
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Key Terms
Inductor
An electrical device formed by a wire wound
around a core having the property of inductance;
also known as a coil.
Winding
The loops or turns of wire in an inductor.
Induced
voltage
Voltage produced as a result of a changing
magnetic field.
Inductance
The property of an inductor whereby a change
in current causes the inductor to produce a
voltage that opposes the change in current.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Key Terms
Henry (H)
RL time
constant
The unit of inductance.
A fixed time interval set by the L and R values,
that determines the time response of a circuit. It
equals the ratio of L/R.
Inductive
reactance
The opposition of an inductor to sinusoidal
current, measured in ohms.
Quality
factor (Q)
The ratio of reactive power to true power for an
inductor.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
1. Assuming all other factors are the same, the
inductance of an inductor will be larger if
a. more turns are added
b. the area is made larger
c. the length is shorter
d. all of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
3. The symbol for a ferrite core inductor is
a.
b.
c.
d.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
4. The symbol for a variable inductor is
a.
b.
c.
d.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
5. The total inductance of a 270 mH inductor
connected in series with a 1.2 mH inductor is
a. 220 mH
b. 271 mH
c. 599 mH
d. 1.47 mH
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
6. The total inductance of a 270 mH inductor
connected in parallel with a 1.2 mH inductor is
a. 220 mH
b. 271 mH
c. 599 mH
d. 1.47 mH
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
7. When an inductor is connected through a series
resistor and switch to a dc voltage source, the voltage
across the resistor after the switch closes has the
shape of
a. a straight line
b. a rising exponential
c. a falling exponential
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
8. For circuit shown, the time constant is
L
a. 270 ns
2
7
0
H
b. 270 ms
c. 270 ms
V
S
1
0V
R
1
.0k
W
d. 3.70 s
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Quiz
9. For circuit shown, assume the period of the square
wave is 10 times longer than the time constant. The
shape of the voltage across L is
a.
b.
L
V
S
R
c.
d.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
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
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.11 Summary
Answers
1. d
6. a
2. b
7. b
3. d
8. a
4. c
9. c
5. d
10. a
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved