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Chapter 8 – AC Circuits

electronics fundamentalscircuits, devices, and applications

THOMAS L. FLOYDDAVID M. BUCHLA

AC Circuits

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Alternating Voltage is a voltage that:1. Continuously varies in magnitude2. Periodically reverses in polarity

AC Circuits

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Symbol for a sinusoidal voltage source.

AC Circuits

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The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform.

Sine waves

Electrical sine waves are named from the mathematical function with the same shape.

AC Circuits

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Sinusoidal voltages are produced by ac generators and electronic oscillators.

Sinusoidal voltage sources

Generation of a sine wave

N S

M o tio n o f c o nd uc to r C o nd uc to r

B

C

D

AA

B

C

D

AB

B

C

D

AC

B

C

D

AD

When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated.

When the conductor is moving parallel with

the lines of flux, no voltage is induced. When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced.

B

C

D

A

AC Circuits

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Sine waves are characterized by the amplitude and period.

Sine waves

0 V

1 0 V

-1 0 V

1 5 V

-1 5 V

-2 0 V

t ( s )0 2 5 3 7 .5 5 0 .0

2 0 V

The amplitude (A) of this sine wave is 20 V

The period is 50.0 ms

A

T

1. The amplitude is the maximum value of a voltage or current

2. The period is the time interval for one complete cycle.

AC Circuits

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The period (T) of a sine wave can be measured between any two corresponding points on the waveform.

Sine waves

T T T T

T T

By contrast, the amplitude of a sine wave is only measured from the center to the maximum point.

A

A

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3.0 Hz

Frequency

Frequency ( f ) is the number of cycles that a sine wave completes in one second.

Frequency is measured in hertz (Hz).

If 3 cycles of a wave occur in one second, the frequency is 1.0 s

AC Circuits

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AC Circuits

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The period and frequency are reciprocals of each other.

Period and frequency

Tf

1 and

fT

1

If the period is 50 ms, the frequency is0.02 MHz = 20 kHz.

(The 1/x key on your calculator is handy for converting between f and T.)

AC Circuits

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Sine wave voltage and current values

0 V

1 0 V

-1 0 V

1 5 V

-1 5 V

-2 0 V

t ( s )0 2 5 3 7 .5 5 0 .0

2 0 V

The peak voltage of this waveform is 20 V.

VP

v

v

• Instantaneous value (v): Voltage or current at any point on the curve.

• Peak value (VP for voltage): The amplitude of a sine wave.

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Peak to peak value: Value from positive peak to negative peak. Equation =

Sine wave voltage and current values

RMS (root mean squared) value: Is the sinusoidal wave with the same heat value as a DC voltage source (known as the effective value)

PPP VV 2 PPP II 2

Prms VV 707.0 Prms II 707.0vmsp VV 414.1 vmsp II 414.1

0 V

1 0 V

-1 0 V

1 5 V

-1 5 V

-2 0 V

t ( s )0 2 5 3 7 .5 5 0 .0

2 0 V

AC Circuits

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0 V

1 0 V

-1 0 V

1 5 V

-1 5 V

-2 0 V

t ( s )0 2 5 3 7 .5 5 0 .0

2 0 V

Sine wave voltage and current values

The peak-to-peak voltage is 40 V.

The rms voltage is 14.1 V.

VPP

Vrms

VP = 20 volts

rmsP VV 414.1rmsPP VV 828.2

This is magnitude Vpp

AC Circuits

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Phase of a Sine Wave

Phase: Angular measurement that specifies the position on the sine wave relative to a reference point.

AC Circuits

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Phase shifts

Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave.

AC Circuits

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Phase shift – Lead/Lag

Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave.

A - LeadsB – Lags by 450

A - LagsB – Leads by 300

AC Circuits

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Phase shift

Volta

ge

(V)

2 70 360 0 90 180

40

45 135 225 3150

Ang le ( )

30

20

10

-20

-30

- 40

405

Pe a k vo lta g e

Re fe re nc e

Notice that a lagging sine wave is below the axis at 0o

Example of a wave that lags the reference (not on guided notes)

v = 30 V sin (q - 45o)

…and the equation has a negative phase shift

AC Circuits

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Phase shift

Volta

ge

(V)

2 70 3600 90 180

40

45 135 225 3150

Ang le ( )

30

20

10

-20

-30

-40

Pe a k vo lta g e

Re fe re nc e

- 45 -10

Notice that a leading sine wave is above the axis at 0o

Example of a wave that leads the reference (not on guided notes)

v = 30 V sin (q + 45o)

…and the equation has a positive phase shift

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An important application of phase-shifted sine waves is in electrical power systems. • Electrical utilities generate ac with three phases that are

separated by 120o.

PolyPhase power

120o

• 3-phase power is delivered to the user with three hot lines plus neutral. The voltage of each phase, with respect to neutral is 120 V.

120o 120o

0o

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Instantaneous values of a wave are shown as v or i.

The equation for the instantaneous voltage (v) of a sine wave is

Sine wave equation

If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is

sinpVv

whereVp = (q theta) =

Peak voltageAngle in rad or degrees

19.2 V

AC Circuits

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A certain sine wave has a positive-going zero crossing at 0° and an peak value of 40V. Calculate its instantaneous voltage for the degrees listed below for the sine wave below.

Sine wave equation

45°, 125°, 180°, 220°,325°

sinpVv

Vp = (q theta) =

Peak voltageAngle in rad or degrees

Volta

ge (V

)

40

0

30

20

10

-10

-20

-30

- 40

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00 9 0

9 0

1 801 80 3 60

Phasor (aka Phase Vector):Representation of a sine wave whose amplitude (A) and angular frequency (ω - omega) are a constant rate.

Phasors

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AC Circuits

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Power in resistive AC circuits

A sinusoidal voltage produces a sinusoidal current

AC Circuits

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Kirchhoff’s voltage law applies to AC circuits just like DC circuits

Power in resistive AC circuits

AC Circuits

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The power formulas are:

Power in resistive AC circuits

rms rms

2

2rms

rms

P V I

VP

R

P I R

ac or dcsource

Bulb

Power in AC circuits is calculated using RMS values for voltage and current.

0 V

0 V

The dc and the ac sources produce the same power to the bulb:

120 Vdc

170 Vp = 120 Vrms

WHY?

AC Circuits

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Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated?

Power in resistive AC circuits

2 228.3 V

100 rmsV

PR

Volta

ge (V

)

40

0

30

20

10

-10

-20

-30

- 40

Vrms = 0.707 x Vp = 0.707 x 40 V = 28.3 V

8 W

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• Generators convert rotational energy to electrical energy.

• The armature has an induced voltage, which is connected through slip rings and brushes to a load.

• The armature loops are wound on a magnetic core (not shown for simplicity).

AC generator (alternator)

Small alternators may use a permanent magnet

Others use field coils to produce the magnetic flux.

AC Circuits

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AC Circuits

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AC generator (alternator)

• Increasing the number of poles increases the number of cycles per revolution.

• A four-pole generator will produce two complete cycles in each revolution.

AC Circuits

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Output Frequency of an AC Generator

120Ns

f f – frequency (Hz)

N – number of poless - speed in RPM

1

2

3

4

AC Circuits

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• In vehicles, alternators generate ac, which is converted to dc for operating electrical devices and charging the battery.

• AC is more efficient to produce and can be easily regulated, hence it is generated and converted to DC by diodes.

Alternators

Diode plate

Rotor

Stator coils

Housing

Slip ringsDiodes

The output is taken from the rotor through the slip rings.

AC Circuits

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There are two major classifications of ac motors. 1. Induction motor 2. Synchronous motor. 3. Both types use a rotating field in the stator windings.

AC Motors

Induction motors work because current is induced in the rotor by the changing current in the stator. This current creates a magnetic field that reacts with the moving field of the stator, which develops a torque and causes the rotor to turn.

Synchronous motors have a magnet for the rotor. In small motors, this can be a permanent magnet, which keeps up with the rotating field of the stator. Large motors use an electromagnet in the rotor, with external dc supplied to generate the magnetic field.

AC Circuits

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Rotating the stator produces a net magnetic field

AC Circuits

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Rotating the stator produces a net magnetic field

AC Circuits

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Induction vs. Stator Motors

Squirrel Cage Rotor

AC Circuits

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Induction vs. Stator Motors

AC Circuits

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Pulse definitions

Am p litud e

Pu lsewid th

Ba se line

Am p litud e

Pu lsewid th

Ba se line

(a ) Po sitive -g o ing p u lse (b ) N e g a tive -g o ing p u lse

Le a d ing (rising ) e d g e

T ra iling (fa lling ) e d g e

Le a d ing (fa lling ) e d g e

T ra iling (rising ) e d g e

Ideal pulses

AC Circuits

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Repetitive pulse waveforms

• Periodic waveforms repeat at fixed intervals.• Pulse repetition frequency: Rate at which the pulses

repeat.• Duty Cycle – Ratio of pulse width (tw) to the period (T)

Percent Duty Cycle = 100

T

tw

Vavg = baseline = (duty cycle)( amplitude)

AC Circuits

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Nonsinusoidal Wave Forms

• Define terms on page 359

• Rise time:

• Fall time:

• Pulse width:

AC Circuits

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Voltage ramps

Ramp – Linear increase or decrease in voltage or current.

Slope = t

Ior

t

V

Xaxis

Yaxis

AC Circuits

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Triangular and Sawtooth waves

Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease)

Triangular waveforms have positive-going and negative-going ramps of equal duration (same slope either increasing or decreasing).

The sawtooth waveform consists of two ramps, one of much longer duration than the other. (unequal slopes in either direction).

T T

T T

AC Circuits

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OscilloscopeA device that traces the graph of a measured electrical signal on its screen.

AC Circuits

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Video on Osciloscope

• http://www.cleanvideosearch.com/media/action/yt/watch?v=8VEg6L2QG5o

• The name of video is: AC vs Dc explain how to use an oscilloscope

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Reading an Oscilloscope

AC Circuits

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Calculate for each wave: Period, Peak, Peak to Peak, RMS

T= 20 msVp = 1.5 vVpp = 3.0 vRMS = 1.06 V

500 mv 0.5 ms

6 v 12 v 15 µs300 µs

T= 3.0 msVp = 1250 mvVpp = 2500 mvRMS = 833.8 mV

T= 6000 µs; 6 msVp = 20.4 vVpp = 40.8 vRMS = 14.4 V T= 30 µs

Vp = 24 vVpp = 48 vRMS = 16.97 V

AC Circuits

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Sine wave

Alternating current

Period (T)

Frequency (f)

Hertz

Current that reverses direction in response to a change in source voltage polarity.

The time interval for one complete cycle of a periodic waveform.

A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q.

Selected Key Terms

A measure of the rate of change of a periodic function; the number of cycles completed in 1 s.

The unit of frequency. One hertz equals one cycle per second.

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Instantaneous value

Peak value

Peak-to-peak value

rms value

The voltage or current value of a waveform at its maximum positive or negative points.

The voltage or current value of a waveform measured from its minimum to its maximum points.

The voltage or current value of a waveform at a given instant in time.

Selected Key Terms

The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to 0.707 times the peak value. rms stands for root mean square.

AC Circuits

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Quiz

1. In North America, the frequency of ac utility voltage is 60 Hz. The period is

a. 8.3 ms

b. 16.7 ms

c. 60 ms

d. 60 s

AC Circuits

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Quiz

2. The amplitude of a sine wave is measured

a. at the maximum point

b. between the minimum and maximum points

c. at the midpoint

d. anywhere on the wave

AC Circuits

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5. The time base of an oscilloscope is determined by the setting of the

a. vertical controls

b. horizontal controls

c. trigger controls

d. none of the above

Quiz

AC Circuits

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6. A sawtooth waveform has

a. equal positive and negative going ramps

b. two ramps - one much longer than the other

c. two equal pulses

d. two unequal pulses

Quiz

AC Circuits

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8. For the waveform shown, the same power would be delivered to a load with a dc voltage of

a. 21.2 V

b. 37.8 V

c. 42.4 V

d. 60.0 V

Quiz

0 V

3 0 V

-3 0 V

4 5 V

-4 5 V

-6 0 V

t ( s )0 2 5 3 7 .5 5 0 .0

6 0 V

AC Circuits

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Quiz

10. A control on the oscilloscope that is used to set the desired number of cycles of a wave on the display is

a. volts per division control

b. time per division control

c. trigger level control

d. horizontal position control