inductors chapter 11 thomas l. floyd david m. buchla dc/ac fundamentals: a systems approach
TRANSCRIPT
InductorsInductors
Chapter 11
Thomas L. Floyd
David M. Buchla
DC/AC Fundamentals: A Systems DC/AC Fundamentals: A Systems ApproachApproach
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
NS
Ch.11 Summary
The Basic Inductor
- +
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Large inductors and transformers are wound around an iron core to increase inductance.
Iron core
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.
Ch.11 Summary
The Basic Inductor
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil.
Ch.11 Summary
Faraday’s Law
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
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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.
Ch.11 Summary
Lenz’s Law
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.11 Summary
Lenz’s Law
-R1
SW
R2VS
L
+
- +
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
SW closes and immediately a voltage appears across L that tends to oppose any change in current.
Initially, the meter reads same current as before the switch was closed.
Ch.11 Summary
Lenz”s Law
-R1
SW
R2
L
+
- +
VS
+ -
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
After a time, the current stabilizes at a higher level (due to I2) as the voltage across the coil decays.
Later, the meter reads a higher current because of the load change.
Ch.11 Summary
Lenz’s Law
-R1
SW
R2
L
+VS
+ -
+-
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
Notice that the winding resistance is in series with the coil and the winding capacitance is in parallel with both.
Ch.11 Summary
Inductor Characteristics
CW
RWL
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Common symbols for inductors (coils) are
Air core Iron core Ferrite core Variable
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.
Ch.11 Summary
Types of Inductors
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Four factors affect the amount of inductance for a coil. The equation for the inductance of a coil is
where L = inductance in henries N = number of turns of wire m = permeability in Wb/At-m l = coil length in meters
Ch.11 Summary
Factors Affecting Inductors
l
ANL
2
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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).
22 mH
Ch.11 Summary
Example
2522 m 107.85m) (0.0025 - rA
--
m 0.02
)m 10m)(7.85Wb/At10(2.5(150) 2542
2
l
ANL
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Inductors come in a variety of types and sizes. A few common ones are illustrated here.
Ch.11 Summary
Common Inductors
Encapsulated Torroid coil VariableWirewound, high current
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
When inductors are connected in series, the total inductance is the sum of the individual inductors. The general equation for inductors in series is
2.18 mH
If a 1.5 m inductor is connected in series with an 680 H inductor, the total inductance is
Ch.11 Summary
Series Inductors
nT LLLL +++ ...21
L1 L2
1.5 mH 680 H
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
When inductors are connected in parallel, the total inductance is smaller than the smallest one. The general equation for inductors in parallel is
The total inductance of two inductors is
…or you can use the product-over-sum rule.
Ch.11 Summary
Parallel Inductors
n
T
LLL
L1
...11
1
21
+++
21
111
LL
LT
+
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The total inductance in the parallel circuit shown is 468 mH
Ch.11 Summary
Parallel Inductors
L1
1.5 mH 680 H
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
When an inductor is connected in series with a resistor and a dc source, current changes at an exponential rate.
Vinitial
Ifinal
Ch.11 Summary
Inductors in DC Circuits
R
LVS
t0Current after switch closure
t0 Inductor voltage after switch closure
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Exponential waveforms are also generated when a square wave source is connected to a series RL circuit.
VS
VL
VR
Ch.11 Summary
Inductors in DC Circuits
VS
R
L
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
100%
80%
60%
40%
20%
00 1 2 3 4 5
99%98%
95%
86%
63%
37%
14%
5%2% 1%
Per
cent
of t
he fi
nal v
alue
Number of time constants
Rising exponential
Falling exponential
Ch.11 Summary
Universal Exponential Curves
R
L
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The universal curves can be applied to general formulas for the current (or voltage) curves for RL circuits. The general current formula is
Ch.11 Summary
Universal Exponential Curves
LRtFiF eIIIi /)( --+
whereIF = final value of currentIi = initial value of currenti = instantaneous value of currente = 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 ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Inductive reactance (XL) is the opposition of an inductor to alternating current (ac). The equation for inductive reactance is
The reactance of a 33 mH inductor that is operated at 550 kHz is 114
Ch.11 Summary
Inductive Reactance
fLXL 2
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
When inductors are in series, the total reactance is the sum of the individual reactances. That is,
Assume three 220 mH inductors are in series with a 455 kHz ac source. What is the total reactance?
1.89 k
The reactance of each inductor is
Ch.11 Summary
Inductive Reactance
LnLLtotL XXXX +++ ...21)(
Ω 629 H) kHz)(220 (45522 fLXL
Ω 629 Ω 629 Ω 296
...21)(
++
+++ LnLLtotL XXXX
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
When inductors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. That is,
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?
210
The reactance of each inductor is 629 Using these values:
Ch.11 Summary
Inductive Reactance
LnLL
totL
XXX
X1
...11
1
21
)(
+++
+
+
++
XXX
X
LLL
totL
6291
6291
6291
1111
1
321
)(
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.11 Summary
Inductive Phase Shift
0
0
90
VL
IL
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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).
Ch.11 Summary
Power in an Inductor
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The quality factor (Q) of a coil equals the ratio of reactive power to true power.
Since I2 appears in both the numerator and the denominator of the right-hand fraction, it cancels, leaving:
Ch.11 Summary
Q of a Coil
ortrue
r
P
PQ
W
L
RI
XIQ
2
2
W
L
R
XQ
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
An electrical device formed by a wire wound around a core having the property of inductance; also known as a coil.
The loops or turns of wire in an inductor.
Voltage produced as a result of a changing magnetic field.
The property of an inductor whereby a change in current causes the inductor to produce a voltage that opposes the change in current.
Ch.11 Summary
Key TermsInductor
Winding
Induced voltage
Inductance
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
The opposition of an inductor to sinusoidal current, measured in ohms.
The unit of inductance.
The ratio of reactive power to true power for an inductor.
Ch.11 Summary
Key TermsHenry (H)
RL time constant
Inductive reactance
Quality factor (Q)
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
3. The symbol for a ferrite core inductor is
a.
b.
c.
d.
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
4. The symbol for a variable inductor is
a.
b.
c.
d.
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
8. For circuit shown, the time constant is
a. 270 ns
b. 270 ms
c. 270 ms
d. 3.70 s
RV S
L
270 H
1 .0 k10 V
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
RV S
La.
b.
c.
d.
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.11 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
1. d
2. b
3. d
4. c
5. d
6. a
7. b
8. a
9. c
10. a
Ch.11 Summary
Answers