chapter 14 inductance and rl circuits © goodheart-willcox co., inc.permission granted to reproduce...

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?