floyd chap 11 ac fundamentals

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

Chapter 11Chapter 11

Chapter 11


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 )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 s

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

SummarySummaryFrequencyFrequency ( 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.

SummaryPeriod and frequency

Tf 1

and fT 1

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

If the period is 50 s, 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 and electronic oscillators.

SummarySinusoidal voltage sourcesGeneration of a sine wave

N S

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

BC

D

AA

BC

D

AB

BC

D

AC

BC

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



Generators convert rotational 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

slip ring s

a rm a tureb rushe 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.



Function generators

Function selectionFrequency

Output level (amplitude)DC offset CMOS output

RangeAdjust

Duty cycle

Typical controls:

Outputs

Readout

Sine Sq ua re Tria ng le


Chapter 11Chapter 11 RMS Value• root mean square or the effective value, • measure of the heating effect of the sine wave.

• The rms value of a sinusoidal voltage is equal to the dc voltage that produces the same amount of heat in a resistance as does the sinusoidal voltage.


Chapter 11Chapter 11 RMS Value


Chapter 11Chapter 11Average Value

• The average value of a sine wave taken over one complete cycle is always zero because the positive values (above the zero crossing) offset the negatIve values (below the zero crossing).


Chapter 11Chapter 11Average Value

• the average value of a sine wave is defined over a half-cycle rather than over a full cycle.

• The average value is the total area under the half-cycle curve divided by the dis- tance in radians of the curve along the horizontal axis.



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 )0 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 )0 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 )0 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 2 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 .20

0 2 2

4

4

3 2

34

5 4

7



Because there are 2 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, given the number of degrees:

degrees360

rad 2 rad

radrad 2

360 deg

To find the number of degrees, given the radians:

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

sinpVv

Vp = =

Peak voltageAngle in rad or degrees

19.2 V



Sine wave equation

v = = 19 .2 V V p sinVp

90

50 0= 50

V p

V p

= 25 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

where = 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.

sinPVv



Phase shiftVo

ltage

(V)

2 70 360 0 90 18 0

40

45 1 35 2 25 3 150

An g le ( )

302010

-20-30- 40

4 05

Pe a k vo lta g e

Re fe re 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 ( 45o)

…and the equation has a negative phase shift



Phase shiftVo

ltage

(V)

2 70 36 00 90 18 0

40

45 135 2 25 3 150

An g le ( )

302010

-20-30-40

Pe a k vo lta g e

Re fe re nc 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 ( + 45o)

…and the equation has a positive phase shift



00 9 0

9 0

1 80180 3 60

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 adjacent side.

h y p o te n u s e

r ig h t a n g le

o p p o s ite s id e

a d ja c e n t s id e h y p o te n u seo p p o s ite s id e

s in =

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 360.

Phasors

positive angle of

negative angle of 360

phasor



Angular velocity of a phasor

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

= 2f

The instantaneous voltage at any point in time is given by

v = Vpsin 2f

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



Frequently dc and ac voltages are together in a waveform. They can be added algebraically, to produce a composite waveform of an ac voltage “riding” on a dc level.

Superimposed dc and ac voltages



Pulse definitions

Am p litu d e

Pu lsewid th

Ba se lin e

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 in g p u lse

Le a d ing (rising ) e d g eT ra iling (fa lling ) e d g e

Le a d in g (fa lling ) e d g eT ra ilin g (risin g ) e d g e

Ideal pulses



Pulse definitions

Non-ideal pulses

A0 .9 A

0 .1A

tr t tf

W

t t

0 .5 A

A

(a ) (b )Rise a nd fa ll tim e s Pu lse wid th

Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level.



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 slope.

The sawtooth waveform consists of two ramps, one of much longer duration than the other.



Harmonics

All repetitive non-sinusoidal waveforms are composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies.Odd harmonics are frequencies that are odd multiples of the fundamental frequency.

Even harmonics are frequencies that are even multiples of the fundamental frequency.



Harmonics

A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).



C h 1

Exte rna ltrig g e r

C o nve rsio n/sto ra g e(Dig ita l sc o p e s o nly)

Sig na l c o up ling

A CDC G N D Am p

C h 2 C o nve rsio n/sto ra g e(Dig ita l sc o p e s o nly)

A CDC G N D Am p

Vo lts/D i v

Ve rtic a lp o sitio n

A CDC

Ext

Trig g e rso urc e

Exte rna l trig g e rc o up ling

C h 1 C h 2

Line

Trig g e rc irc u its

Trig g e rleve l a ndslo p e

Tim e b a seSe c /D iv

Ho rizo nta lp o sitio n

C o ntro l a nd p ro c e ss(D ig ita l sc o p e s o nly)

Inte nsity

A C

DC to a ll se c tio nsPo we r sup p ly

Vertical section

D isp lay section

H orizonta l sectionTrigger section

Dig ita lo n ly

Ana lo go nly

SummaryOscilloscopes The oscilloscope is divided into

four main sections.C h 1 C o nve rsio n/sto ra g e(Dig ita l sc o p e s o nly)

Signa l c o up ling

A CDC G ND Am p


A CDC G ND Am p

Vo lts/Di v


Vertic a l s ec tio n


A CDC

Ext

Trig g e rso urc e


C h 1 C h 2

Line

Trig g e rc irc uits

T rig g e rleve l a ndslo p e

AC

T r ig g e r s ec tio n Fro m ho rizo n ta l se c tio n

Fro mve rtic a l se c tio n

Inte nsity

Ana lo go nly

D isp la y se c t io n

Tim e b a seSe c /D iv


C o ntro l a nd p ro c e ss(D ig ita l sc o p e s o n ly)

D ig ita l To d isp la y se c tio no n ly

H orizo n ta l sec tion




C h 1


C o nve rsio n/sto ra g e(Dig ita l sc o p e s o nly)

Sig na l c o up ling

A CDC G ND Am p


A CDC G ND Am p

Vo lts/D i v


A CDC

Ext

Trig g e rso urc e


C h 1 C h 2

Line

Trig g e rc irc uits

Trig g e rleve l a ndslo p e

Tim e b a seSe c /Div


C o ntro l a nd p ro c e ss(Dig ita l sc o p e s o nly)

Inte nsity

A C

DC to a ll se c tio nsPo we r sup p ly

Vertical section

D isp lay section

H orizon ta l sectionTrigge r section

Dig ita lo nly

Ana lo go nly



OscilloscopesHO RIZO N TALVERTIC A L TRIG G ER

5 s 5 ns

PO SITIO N

C H 1 C H 2 EXT TRIG

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

C H 1 C H 2 BO TH

PO SITIO N

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

SEC /D IV

PO SITIO N

SLO PEÐ +

LEVEL

SO URC EC H 1C H 2EXT

LINE

TRIG C O UPDC A C

DISPLAY

IN TEN SITY

PRO BE C O M P5 V

HO RIZO N TALVERTIC A L TRIG G ER

5 s 5 ns

PO SITIO N


AC -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

C H 1 C H 2 BO TH

PO SITIO N

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

SEC /D IV

PO SITIO N

SLO PEÐ +

LEVEL


LINE

TRIG C O UPDC A C

DISPLAY

IN TEN SITY

PRO BE C O M P5 V

VerticalHO RIZO N TALVERTIC A L TRIG G ER

5 s 5 ns

PO SITIO N


AC -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

C H 1 C H 2 BO TH

PO SITIO N

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

SEC /D IV

PO SITIO N

SLO PEÐ +

LEVEL


LINE

TRIG C O UPDC A C

DISPLAY

IN TEN SITY

PRO BE C O M P5 V

HorizontalHO RIZO N TALVERTIC A L TRIG G ER

5 s 5 ns

PO SITIO N


A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

C H 1 C H 2 BO TH

PO SITIO N

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

SEC /D IV

PO SITIO N

SLO PEÐ +

LEVEL


LINE

TRIG C O UPDC A C

DISPLAY

IN TEN SITY

PRO BE C O M P5 V

TriggerHO RIZO N TALVERTIC A L TRIG G ER

5 s 5 ns

PO SITIO N


A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

C H 1 C H 2 BO TH

PO SITIO N

A C -DC -G ND

5 V 2 m V

VO LTS/D IV

C O UPLIN G

SEC /D IV

PO SITIO N

SLO PEÐ +

LEVEL


LINE

TRIG C O UPDC A C

D ISPLAY

IN TEN SITY

PRO BE C O M P5 V

Display



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

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 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.



Radian

Phasor

Amplitude

Pulse

Harmonics

The maximum value of a voltage or current.

A type of waveform that consists of two equal and opposite steps in voltage or current separated by a time interval.

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

Selected Key Terms

The frequencies contained in a composite waveform, which are integer multiples of the pulse repetition frequency.

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


Chapter 11Chapter 11Quiz

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



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



3. An example of an equation for a waveform that lags the reference is

a. v = 40 V sin ()

b. v = 100 V sin ( + 35o)

c. v = 5.0 V sin ( 27o)

d. v = 27 V



4. In the equation v = Vp sin , the letter v stands for the

a. peak value

b. average value

c. rms value

d. instantaneous value

Quiz



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



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



7. The number of radians in 90o are

a. /2

b.

c. 2/3

d. 2

Quiz



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



9. A square wave consists of

a. the fundamental and odd harmonics

b. the fundamental and even harmonics

c. the fundamental and all harmonics

d. only the fundamental

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



Answers:

1. b

2. a

3. c

4. d

5. b

6. b

7. a

8. c

9. a

10. b

floyd chap 11 ac fundamentals

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