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Transcript of Principles of Electric Circuits - Floyd© Copyright 2006 Prentice-Hall Chapter 11.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Chapter 11
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Summary
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Summary
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Summary
The period and frequency are reciprocals of each other.
Summary
Period and frequency
Tf
1 and f
T1
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.)
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Summary
Sinusoidal voltages are produced by ac generators and electronic oscillators.
Summary
Sinusoidal 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
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Function generators
Function selection
Frequency
Output level (amplitude)
DC offset CMOS output
RangeAdjust
Duty cycle
Typical controls:
Outputs
Readout
Sine Sq ua re Tria ng le
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Sine wave voltage and current values
The peak-to-peak voltage is 40 V.
The rms voltage is 14.1 V.
VPP
Vrms
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Sine wave voltage and current values
The average value for the sinusoidal voltage is 12.7 V.
Vavg
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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 .2
00 2
24
4
3 2
34
5 4
7
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Phase shiftVo
ltag
e (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
v = 30 V sin ( 45o)
…and the equation has a negative phase shift
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Phase shiftVo
ltag
e (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
v = 30 V sin ( + 45o)
…and the equation has a positive phase shift
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
00 9 0
9 0
1 801 80 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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.)
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Pulse definitions
Am p litu d e
Pu lsewid th
Ba se line
Am p litu d 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 ilin g (fa lling ) e d g e
Le a d ing (fa lling ) e d g e
T ra ilin g (rising ) e d g e
Ideal pulses
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Pulse definitions
Non-ideal pulses
A0 .9 A
0 .1A
tr tt
fW
t t
0 .5 A
A
(a ) (b )Rise a nd fa ll tim e s Pu lse w id th
Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Harmonics
A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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 ND 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 ND Am p
Vo lts/Di 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 1C h 2
Line
Trig g e rc irc uits
Trig g e rleve l a ndslo p e
Tim e b a se
Se c /Div
Ho rizo nta lp o sitio n
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 sectionTrigger section
Dig ita lo nly
Ana lo go nly
Summary
Oscilloscopes 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
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 ND Am p
Vo lts/Di v
Ve rtic a lp o sitio n
Vertic a l s ec tio n
Exte rna ltrig g e r
A CDC
Ext
Trig g e rso urc e
Exte rna l trig g e rc o up ling
C h 1C 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 se
Se 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 n ly)
D ig ita lTo d isp la y se c tio no n ly
H orizon ta l sec tion
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
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 ND 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 ND 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 1C h 2
Line
Trig g e rc irc uits
T rig g e rleve l a ndslo p e
Tim e b a se
Se c /Div
Ho rizo nta lp o sitio n
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 sectionTrigger section
Dig ita lo nly
Ana lo go nly
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Summary
Oscilloscopes
HO 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 PE
Ð +
LEVEL
SO URC E
C H 1
C H 2
EXT
LINE
TRIG C O UP
DC A C
D ISPLAY
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
C H 1 C H 2 EXT TRIG
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 PE
Ð +
LEVEL
SO URC E
C H 1
C H 2
EXT
LINE
TRIG C O UP
DC A C
D ISPLAY
IN TEN SITY
PRO BE C O M P5 V
Vertical
HO RIZO N TALVERTIC A L TRIG G ER
5 s 5 ns
PO SITIO N
C H 1 C H 2 EXT TRIG
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 PE
Ð +
LEVEL
SO URC E
C H 1
C H 2
EXT
LINE
TRIG C O UP
DC A C
D ISPLAY
IN TEN SITY
PRO BE C O M P5 V
Horizontal
HO 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 PE
Ð +
LEVEL
SO URC E
C H 1
C H 2
EXT
LINE
TRIG C O UP
DC A C
D ISPLAY
IN TEN SITY
PRO BE C O M P5 V
Trigger
HO 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 PE
Ð +
LEVEL
SO URC E
C H 1
C H 2
EXT
LINE
TRIG C O UP
DC A C
D ISPLAY
IN TEN SITY
PRO BE C O M P5 V
Display
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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.
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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).
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Quiz
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Quiz
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
7. The number of radians in 90o are
a. /2
b.
c. 2/3
d. 2
Quiz
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Quiz
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
Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall
Chapter 11Chapter 11Quiz
Answers:
1. b
2. a
3. c
4. d
5. b
6. b
7. a
8. c
9. a
10. b