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Chapter 14
Inductance and RL Circuits
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Objectives
• Define the terms inductor and inductance.
• Explain how inductance affects a current.
• Describe an RL circuit’s transient response.
• Define mutual inductance.
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Objectives
• Describe the effect of inductance in ac circuits.
• Explain and compare true power and apparent power.
• Use various measuring and computing methods to determine the values of currents and voltages in inductive circuits.
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Inductance
• Coils of wire• Inductors
– Reactors– Chokes
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Inductance (Cont.)
• Lenz’s law
• Symbol is L, unit is henry (H)
t
ILE
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Inductance (Cont.)
• Strong and weak magnetic fields
• Self induction
• Transient responses
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Producing Stronger and Weaker Magnetic Fields
• Stronger magnetic field produces stronger induced voltage
• Large inductors wound on laminated iron cores
• Small inductors have powdered iron or air cores
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Strength of Inductance
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Self Induction
• Expanding or collapsing magnetic field cutting across coil wires
• Counter emf induced
• Strength depends on– Number of turns of wire in coil– Link between length of coil and its diameter– Permeability of core
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Transient Responses
• Coil connected to dc voltage source
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Transient Responses (Cont.)
• Current builds up gradually when switch is closed
• Current decays gradually when switch opens
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RL Circuits
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RL Circuits (Cont.)
• L coil shorted through switch 2
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Transient Response Curves
• Graphs show charge and discharge circuits
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Transient Response Curves
• Time constant ()
– Inductance in henrys– Resistance in ohms
R
L
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Review
What is inductance?
The property in an electric circuit that resists a change in current
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Review
What is Lenz’s law?
The field created by induced current is of such a polarity that it opposes the field of the permanent magnet
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Review
What is a transient response?
The response of the current and voltage in a circuit after an instant change in applied voltage
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Review
What kind of circuit contains resistance and inductance?
RL circuit
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Mutual Inductance
• Two coils close together
• Coupling
• Unity coupling
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Mutual Inductance (Cont.)
– LM is mutual inductance
– k is coefficient or percentage of coupling
– L1 and L2 are inductances of coils
21M L LL k
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Phase Relationship in Series Inductance
• Without mutual inductance– LT = L1 + L2 + L3 …+ LN
• With mutual inductance– LT = L1 + L2 ± 2LM
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Phase Relationship in Parallel Inductance
• Without mutual inductance
• With mutual inductance
N321T LLLLL
1
1
1
1
1
21
21T LL
LLL
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Review
What kind of inductance is produced by two coils within magnetic reach of each other?
Mutual inductance
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Review
What is the degree to which the lines of force of one coil link with the windings of the second coil called?
Coupling
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Inductance in AC Circuits
• Inductance produces counter emf to oppose source voltage
• Reactance (X)
• Inductive reactance (XL)
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Induced Current and Voltage
• Counter emf is 180° out of phase with source voltage
• XL = 2fL
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Induced Current and Voltage (Cont.)
• As frequency or inductance increases, inductive reactance increases
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Power in Inductive Circuits
• True power– P = I × E
• Reactive power– Measured in volt-ampere-reactive (VAR)
• Apparent power– Measured in volt-amperes (VA)
– Apparent power = Eeff × Ieff
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Power Factor (PF)
• True power = Apparent power × cos • True power = Eeff × Ieff × cos
powerApparent
power True cos factor Power θ
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Power Factor Example
0 VA 1000
W 0 cos PF
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A Purely Resistive Circuit
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Resistance and Inductance in an AC Circuit
• Different traits for added inductor
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Resistance and Inductance in an AC Circuit (Cont.)
• Impedance (Z)• Pythagorean theorem
• Finding impedance
• Phase angle ()• cos = PF
22 bac
22 LXRZ
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Resistance and Inductance Example
• Given an impedance of 500 Ω and an applied ac voltage of 100 V, find the current
• True power = 100 V × 0.2 A × cos 53.1° = 12 W• Wattless power
A 0.2 500
V 100
Z
EI
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Ohm’s Law for AC Circuits
• E = I × Z
Z
EI
I
EZ
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Ohm’s Law Example
• Reactance of L– XL = 2 × 3.14 × 60 Hz × 8 H
– XL = 3014 Ω
• Impedance
– Z = 5000 Ω
22 ) (3000 ) (4000 Z
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Ohm’s Law Example (Cont.)
• Current
– I = 0.04 A
• Voltage drops– ER = 0.04 A × 4000 Ω = 160 V
• Source voltage
5000
V 200 I
V 120 3000 A 0.04 LXE
V 200 V) (120 V) (160 22 SE
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Ohm’s Law Example (Cont.)
• Phase angle
• Apparent power = 0.04 A × 200 V = 8 VA
• True power = 0.04 A × 200 V × 0.8
True power = 6.4 W
0.8 5000
4000 cos
Θ
o37Θ
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Parallel RL Circuits
• No phase difference among the elements
• Phase difference among total and branch currents
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Parallel RL Circuits (Cont.)
T
S
I
EZ
22 LRT II I
R
L
I
Iarctan
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Parallel RL Circuit Example
• Inductive reactance– XL = 6.28 × 100 Hz × 4 H = 2512 Ω
• Branch currentsA 0.067
1500
V 100
R
EI S
R
A 0.04 2512
V 100
L
SL X
EI
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Parallel RL Circuit Example (Cont.)
• Phase angle
– arctan 0.597 = 30.8°
• Total circuit current
0.597arctan A 0.067
A 0.04arctan θ
A 0.078 A) (0.04 A) (0.067 22 TI
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Parallel RL Circuit Example (Cont.)
• Impedance
1282 A 0.078
V 100 Z
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Inductive Heating
• A magnetic field generates hysteresis and eddy currents
• Amount of heat is due to the intensity of the magnetic field and frequency of applied ac
• Uses include induction stoves and annealing
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Review
What is opposition to an alternating current due to inductance or capacitance called?
Reactance
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Review
Because counter emf opposes source voltage, what is the phase relationship between the two?
The counter emf is 180 degrees out of phase with the source voltage
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Review
What is the relationship between inductive reactance and frequency or inductance?
They are in direct proportion
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Review
How is reactive power measured?
Volt-ampere-reactive (VAR)
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For Discussion
• Discuss the relationship between induction and frequency.
• Discuss the differences between apparent power and true power.
• How can an inductive circuit be completely 180 degrees out of phase?