Chapter 11
Chapter 11
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Sine waves
The sinusoidal waveform (sine wave) is the fundamental
alternating current (ac) and alternating voltage waveform.
Electrical sine waves are
named from the
mathematical function
with the same shape.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
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.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Sine waves
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.
20 V
15 V
The amplitude (A)
of this sine wave
is 20 V
The period is 50.0 s
A
10 V
0V
t (s)
25
0
37.5
50.0
-10 V
-15 V
-20 V
Principles of Electric Circuits - Floyd
T
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Sine waves
The period of a sine wave can be measured between
any two corresponding points on the waveform.
TT T T
A
T
T
By contrast, the amplitude of a sine wave is only
measured from the center to the maximum point.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
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 3.0 Hz
1.0 s
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Period and frequency
The period and frequency are reciprocals of each other.
1
f 
T
and
T 
1
f
Thus, if you know one, you can easily find the other.
(The 1/x key on your calculator is handy for converting between f and T.)
If the period is 50 s, the frequency is 0.02 MHz = 20 kHz.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Sinusoidal voltage
Generation
of a sinesources
wave
Sinusoidal voltages are produced by ac generators and
electronic oscillators.
When a conductor rotates in a constant magnetic
field, a sinusoidal wave is generated.
C
N
D
B
S
A
B
C
D
A
Motion of conductor
Conduc tor
When
theisconductor
is moving parallel
with
When the
loop
moving perpendicular
to the
lines
flux, no voltage
is induced.
lines of the
flux,
theofmaximum
voltage
is induced.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
AC generator (alternator)
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).
Small alternators may use a
permanent magnet as shown
here; other use field coils to
produce the magnetic flux.
N
brushes
S
arm ature
slip rings
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 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 11
Readout
Function generators
Typical controls:
Function selection
Frequency
Range
Adjust
Output level (amplitude)
DC offset
CMOS output
Sine
Principles of Electric Circuits - Floyd
Square
Outputs
Duty cycle
Triangle
© Copyright 2006 Prentice-Hall
Chapter 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. 20 V
15 V
VP
10 V
The peak voltage of
this waveform is 20 V.
0V
t (s)
0
25
37.5
50.0
-10 V
-15 V
-20 V
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Sine wave voltage and current values
The voltage of a sine wave can also be specified as
either the peak-to-peak or the rms value. The peak-topeak is twice the peak value. The rms value is 0.707
times the peak value. 20 V
15 V
The peak-to-peak
voltage is 40 V.
The rms voltage
is 14.1 V.
10 V
Vrms
0V
0
VPP
t (s)
25
37.5
50.0
-10 V
-15 V
-20 V
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Sine wave voltage and current values
For some purposes, the average value (actually the halfwave average) is used to specify the voltage or current.
By definition, the average value is as 0.637 times the
peak value.
20 V
15 V
The average value for
the sinusoidal voltage
is 12.7 V.
10 V
Vavg
0V
t (s)
0
25
37.5
50.0
-10 V
-15 V
-20 V
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Angular measurement
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
2p radians in one complete revolution.
1.0
0.8
R
0.6
0.4
0.2
R
0
-0.2
0
p
p
4
2
3p
4
p
5p
4
3p
2
7p
4
2p
-0.4
-0.6
-0.8
-1.0
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Angular measurement
Because there are 2p 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:
rad 
2p rad
 degrees
360
To find the number of degrees, given the radians:
deg 
360
 rad
2p rad
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Sine wave equation
Instantaneous values of a wave are shown as v or i. The
equation for the instantaneous voltage (v) of a sine
wave is
v  V p sin 
where
Vp = Peak voltage
 = Angle in rad or degrees
If the peak voltage is 25 V, the instantaneous
voltage at 50 degrees is 19.2 V
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Sine wave equation
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.
90
Vp
Vp = 25 V
v = Vp sin = 19.2 V
= 50
0
50
Vp
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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.
v  VP sin   f 
where
f = Phase shift
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Example of a wave that lags the
reference
…and the equation
Phase shift
has a negative phase
shift
Referenc e
40
Peak voltage
30
v = 30 V sin ( - 45o)
Voltage (V)
20
10
0
0
45
90
135 180
225
270
315
360
405
-20
-30
- 40
Notice that a lagging sine
wave is below the axis at 0o
Angle ()
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Example of a wave that leads the
reference
Notice that a leading sine
Referenc e
wave is above the axis at 0o
Phase shift
40
Peak voltage
30
Voltage (V)
20
v = 30 V sin ( + 45o)
10
-45
0 0
-10
-20
-30
-40
45
90 135
180
225
270
315
360
…and the equation
has a positive phase
shift
Angle ()
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Phasors
The sine wave can be represented as the projection of a
vector rotating at a constant rate. This rotating vector is
called a phasor.
90
180
Principles of Electric Circuits - Floyd
0
0
90
180
360
© Copyright 2006 Prentice-Hall
Chapter 11
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.
hypotenuse
opposite side

right
angle
adjacent side
Principles of Electric Circuits - Floyd
o
p
p
o
s
i
t
e
s
i
d
e
s
i
n

=
h
y
p
o
t
e
n
u
s
e
© Copyright 2006 Prentice-Hall
Chapter 11
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.
positive angle of 
negative angle of  - 360
phasor
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Angular velocity of a phasor
When a phasor rotates through 360 or 2p radians, one
complete cycle is traced out.
The velocity of rotation is called the angular velocity ().
 = 2pf
(Note that this angular velocity is expressed in radians per second.)
The instantaneous voltage at any point in time is given by
v = Vpsin 2pf
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Superimposed dc and ac voltages
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.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Pulse definitions
Ideal pulses
Leading (rising) edge
Leading (falling) edge
Trailing (falling) edge
Trailing (rising) edge
Baseline
Am plitude
Am plitude
Baseline
Pulse
width
(a) Positive-going pulse
Principles of Electric Circuits - Floyd
Pulse
width
(b) Negative-going pulse
© Copyright 2006 Prentice-Hall
Chapter 11
Pulse definitions
Non-ideal pulses
A
0.9 A
A
0.5 A
0.1A
t
tr
t
tW
tf
(a) Rise and fall times
(b) Pulse width
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 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 negativegoing 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 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 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 11
Summary Display section
Vertical section
Signal coupling
Volts/Di v
The oscilloscope is divided into
From
Analog
Amp
four main sections.
vertic al only
Oscilloscopes
AC
DC
Ch 1
GND
Conversion/storage
sec tion
(Digital scopes only)
Vertical section
Signal coupling
Volts/Di v
AC
DC
Ch 1
GND
AC
Ch 2
DCAC
Vertical
Conversion/storage
position (Digital
scopes only)
Amp
Vertical
Trigger section
DC
Ch 2
Digital Display section
To display sec tion
only
Amp
GND position
Amp
GND
Conversion/storage
(Digital scopes only)
External trigger
coupling
External
trigger
AC
DC
External
trigger
AC
DC
Ch 1
Ext
Line
Intensity
Digital
only
Trigger
source
Trigger
source
Analog
only
Horizontal From horizontal sec tion
Conversion/storage
(Digital
scopes only)
section
Trigger
Horizontal Control and proc ess
section level and
Trigger section
External trigger
coupling
Intensity
(Digital scopes only)
slope
Trigger
and
Ch 1level
slope
Ch 2
Ch 2
Control and process
(Digital scopes
Sec/Di
v only)
Trigger
base
circuits
TimeTime
base
Sec /Div
Ext
Line
Trigger
circuits
Horizontal
position
Horizontal
position
AC
Power supply
Principles of Electric Circuits - Floyd
DC to all sec tions
AC
© Copyright 2006 Prentice-Hall
Vertical section
Chapter 11
Signal coupling
AC
DC
Ch 1
GND
AC
Ch 2
DC
GND
Volts/Di v
Display section
Amp
Conversion/storage
(Digital scopes only)
Vertical
position
Amp
Analog
only
Intensity
Conversion/storage
(Digital scopes only)
Digital
only
Horizontal
section
Trigger section
External trigger
coupling
External
trigger
Trigger
source
AC
DC
Ch 1
Ext
Line
Trigger
level and
slope
Ch 2
Control and process
(Digital scopes only)
Sec /Div
Trigger
circuits
Time base
Horizontal
position
AC
Power supply
Principles
Principles
ofof
Electric
Electric
Circuits
Circuits
- Floyd
- Floyd
DC to all sec tions
© Copyright 2006 Prentice-Hall
Chapter 11
Summary
Oscilloscopes
Display
Vertical Horizontal
Trigger
VERT
ICAL
VERT
ICAL
HORIZONT
AL
HORIZONT
AL
CH
HH
CH11 CH
CH22 BOT
BOT
TR
TIGGER
RIGGER
SLOPE
SLOPE
ÐÐ
POSIT
ION
POSIT
ION
POSIT
ION
POSIT
ION
VOLT
S/DIV
VOLT
S/DIV
VOLT
S/DIV
VOLT
S/DIV
++
POSIT
ION
POSIT
ION
LEVEL
LEVEL
SEC/DIV
SEC/DIV
SOUR
CE
SOUR
CE
CH
11
CH
CH
22
CH
55VV
22mmVV
55VV
22mmVV
COUPLING
COUPLING
COUPLING
COUPLING
AC-DC-GND
AC-DC-GND
AC-DC-GND
AC-DC-GND
5 5s s
5 5nsns
EXT
EXT
LINE
LINE
TR
TIG
RIGCOUP
COUP
DC
DC
DISPLAY
DISPLAY
PP
RR
OB
EECOMP
OB
COMP
55VV
CH
CH11
CH
CH22
AC
AC
EXT
EXTTRIG
TRIG
INT
INTENSIT
ENSITYY
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Selected Key Terms
Sine wave A type of waveform that follows a cyclic
sinusoidal pattern defined by the formula
y = A sin .
Alternating Current that reverses direction in response to a
current change in source voltage polarity.
Period (T) The time interval for one complete cycle of a
periodic waveform.
Frequency (f) A measure of the rate of change of a periodic
function; the number of cycles completed in 1 s.
Hertz The unit of frequency. One hertz equals one
cycle per second.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Selected Key Terms
Instantaneous The voltage or current value of a waveform at
value a given instant in time.
Peak value The voltage or current value of a waveform at
its maximum positive or negative points.
Peak-to-peak The voltage or current value of a waveform
value measured from its minimum to its maximum
points.
rms value 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 11
Selected Key Terms
Radian A unit of angular measurement. There are 2p
radians in one complete 360o revolution.
Phasor A representation of a sine wave in terms of its
magnitude (amplitude) and direction (phase angle).
Amplitude The maximum value of a voltage or current.
Pulse A type of waveform that consists of two equal
and opposite steps in voltage or current
separated by a time interval.
Harmonics The frequencies contained in a composite
waveform, which are integer multiples of the
pulse repetition frequency.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Quiz
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 11
Quiz
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Quiz
7. The number of radians in 90o are
a. p/2
b. p
c. 2p/3
d. 2p
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Quiz
8. For the waveform shown, the same power would be
delivered to a load with a dc voltage of
a. 21.2 V
60 V
45 V
b. 37.8 V
30 V
c. 42.4 V
0V
d. 60.0 V
t (s)
0
25
37.5
50.0
-30 V
-45 V
-60 V
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Quiz
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
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
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Chapter 11
Quiz
Answers:
Principles of Electric Circuits - Floyd
1. b
6. b
2. a
7. a
3. c
8. c
4. d
9. a
5. b
10. b
© Copyright 2006 Prentice-Hall