Ac fundamentals 2

Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall

Chapter 11Chapter 11

AC Fundamentals


Chapter 11Chapter 11Summary

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.



A wave is a disturbance. Unlike water waves, electrical waves cannot be seen directly but they have similar characteristics. All periodic waves can be constructed from sine waves, which is why sine waves are fundamental.

Summary



Sine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current; the period is the time interval for one complete cycle.

Sine waves

0 V

1 0 V

- 1 0 V

1 5 V

- 1 5 V

- 2 0 V

t ( s )µ0 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 µs

A

T



The period 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



3.0 Hz

SummarySummary

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



The period and frequency are reciprocals of each other.

Summary

Period and frequency

Tf

1= and fT

1=

Thus, if you know one, you can easily find the other.

If the period is 50 µs, the frequency is 0.02 MHz = 20 kHz.

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



Sinusoidal voltages are produced by ac generators.

Summary

Sinusoidal voltage sourcesGeneration of a sine wave

N S

M o t i o n o f c o n d u c t o r C o n d u c t o r

B

C

D

AA

B

C

D

AB

B

C

D

AC

B

C

D

AD

When a conductor rotates in a uniform magnetic field with constant speed, 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



Generators convert mechanical energy to electrical energy. A stationary field alternator with a rotating armature is shown. 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)

N S

s l i p r i n g s

a r m a t u r eb r u s h e s

Small alternators may use a permanent magnet as shown here; other use field coils to produce the magnetic flux.



AC generator (alternator)

By increasing the number of poles, the number of cycles per revolution is increased. A four-pole generator will produce two complete cycles in each revolution.



Sine wave voltage and current values

There are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value, abbreviated as VP for a voltage waveform.

0 V

1 0 V

- 1 0 V

1 5 V

- 1 5 V

- 2 0 V

t ( s )µ0 2 5 3 7 . 5 5 0 . 0

2 0 V

The peak voltage of this waveform is 20 V.

VP



0 V

1 0 V

- 1 0 V

1 5 V

- 1 5 V

- 2 0 V

t ( s )µ0 2 5 3 7 . 5 5 0 . 0

2 0 V

The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to-peak is twice the peak value. The rms value is 0.707 times the peak value.


The peak-to-peak voltage is 40 V.

The rms voltage is 14.1 V.

VPP

Vrms



0 V

1 0 V

- 1 0 V

1 5 V

- 1 5 V

- 2 0 V

t ( s )µ0 2 5 3 7 . 5 5 0 . 0

2 0 V

For some purposes, the average value (actually the half-wave average) is used to specify the voltage or current. By definition, the average value is as 0.637 times the peak value.


The average value for the sinusoidal voltage is 12.7 V.

Vavg



Angular measurements can be made in degrees (o) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360o or 2π radians in one complete revolution.

Angular measurement

R

R

1 . 0

- 1 . 0

0 . 8

- 0 . 8

0 . 6

- 0 . 6

0 . 4

- 0 . 4

0 . 2

- 0 . 2

00 2 πππ

2π

4π4

3 π2

3π4

5 π4

7



Because there are 2π radians in one complete revolution and 360o in a revolution, the conversion between radians and degrees is easy to write. To find the number of radians:

2π radian = 360o

Therefore, 1 radian = 360o/ 2π

To find the number of degrees:

1 degree = 2π/ 360o

Angular measurement



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

where

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

θsinpVv =

Vp = θ =

Peak voltageAngle in rad or degrees

19.2 V



Sine wave equation

v = = 1 9 . 2 V V p s i nV p

9 0 °

5 0 °0 °= 5 0 °

V p

V p

= 2 5 V

A plot of the example in the previous slide (peak at 25 V) is shown. The instantaneous voltage at 50o is 19.2 V as previously calculated.



Phase shift

whereφ = Phase shift

The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously.

( )φθ ±= sinPVv



Phase shiftVo

ltag

e (V

)

2 7 0 ° 3 6 0 °0 ° 9 0 ° 1 8 0 °

4 0

4 5 ° 1 3 5 ° 2 2 5 ° 3 1 5 °0

A n g l e (° )

3 0

2 0

1 0

- 2 0

- 3 0

- 4 0

4 0 5 °

P e a k v o l t a g e

R e f e r e n c e

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

Example of a wave that lags the reference

v = 30 V sin (θ − 45o)

…and the equation has a negative phase shift



Phase shiftVo

ltag

e (

V)

2 7 0 ° 3 6 0 °0 ° 9 0 ° 1 8 0 °

4 0

4 5 ° 1 3 5 ° 2 2 5 ° 3 1 5 °0

A n g l e (° )

3 0

2 0

1 0

- 2 0

- 3 0

- 4 0

P e a k vo l t a g e

R e f e r e n c e

- 4 5 °- 1 0

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

Example of a wave that leads the reference

v = 30 V sin (θ + 45o)

…and the equation has a positive phase shift



00 9 0

9 0

1 8 01 8 0 3 6 0

The sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor.

Phasors



Phasors allow ac calculations to use basic trigonometry. The sine function in trigonometry is the ratio of the opposite side of a right triangle to the hypotenuse.

h y p o t e n u s e

θr i g h t a n g l e

o p p o s i t e s i d e

a d j a c e n t s i d e h y p o t e n u s eo p p o s i t e s i d e

s i n = θ

Phasors



The position of a phasor at any instant can be expressed as a positive angle, measured counterclockwise from 0° or as a negative angle equal to θ − 360°.

Phasors

positive angle of θ

negative angle of θ − 360°

phasor



Angular velocity of a phasor

When a phasor rotates through 360° or 2π radians, one complete cycle is traced out.The velocity of rotation is called the angular velocity (ω).

ω = 2πf

The instantaneous voltage at any point in time is given by v = Vpsin 2πf

(Note that this angular velocity is expressed in radians per second.)



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 θ.

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.



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 negative peak to positive peak.

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.



Radian

Phasor

Amplitude The maximum value of a voltage or current.

A unit of angular measurement. There are 2π radians in one complete 360o revolution.

Selected Key Terms

A representation of a sine wave in terms of its magnitude (amplitude) and direction (phase angle).

Ac fundamentals 2

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