Transcript Lecture 10 - UniMAP Portal

ELECTRICAL TECHNOLOGY
EET 103/4
 Define and explain sine wave, frequency,
amplitude, phase angle, complex number
 Define, analyze and calculate impedance,
inductance, phase shifting
 Explain and calculate active power,
reactive power, power factor
 Define, explain, and analyze Ohm’s law,
KCL, KVL, Source Transformation,
Thevenin theorem.
1
SINUSOIDAL ALTERNATING
WAVEFORMS
(CHAPTER 13)
2
13.1 Introduction
Alternating waveforms
The term alternating indicates only that the
waveform alternates between two prescribed
levels in a set time sequence.
3
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
 Generation
An ac generator (or alternator) powered by
water power, gas, or nuclear fusion is the primary
component in the energy-conversion process.
 The energy source turns a rotor (constructed of
alternating magnetic poles) inside a set of
windings housed in the stator (the stationary part
of the dynamo) and will induce voltage across the
windings of the stator.
d
eN
dt
4
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
 Generation
 Wind power and solar power energy are receiving
increased interest from various districts of the
world.
The turning propellers of the wind-power station are
connected directly to the shaft of an ac generator.
Light energy in the form of photons can be
absorbed by solar cells. Solar cells produce dc,
which can be electronically converted to ac with an
inverter.
 A function generator, as used in the lab, can
generate and control alternating waveforms.
5
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
 Definitions
 Waveform: The path traced by a quantity, such
as voltage, plotted as a function of some variable
such as time, position, degree, radius, temperature
and so on.
 Instantaneous value: The magnitude of a
waveform at any instant of time; denoted by the
lowercase letters (e1, e2).
Peak amplitude: The maximum value of the
waveform as measured from its average (or mean)
value, denoted by the uppercase letters Em (source
of voltage) and Vm (voltage drop across a load).
6
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
 Definitions
Peak value: The maximum instantaneous value
of a function as measured from zero-volt level.
 Peak-to-peak value: Denoted by Ep-p or Vp-p,
the full voltage between positive and negative
peaks of the waveform, that is, the sum of the
magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually
repeats itself after the same time interval.
7
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
8
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
 Definitions
Period (T): The time interval between
successive repetitions of a periodic waveform
(the period T1 = T2 = T3), as long as successive
similar points of the periodic waveform are used
in determining T
 Cycle: The portion of a waveform contained in
one period of time
Frequency: (Hertz) the number of cycles that
occur in 1 s
1
hertz, Hz 
f 
T
9
Heinrich Rudolph Hertz.
10
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
11
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
12
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Example 13.1
Determine:
(a) peak value
(b) instantaneous value at 0.3 s and 0.6 s
(c) peak-to-peak value
(d) period
(e) how many cycles are shown
(f) frequency
13
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Example 13.1 – solution
(a) 8 V; (b) -8 V at 3 s and 0 V at 0.6 s; (c) 16 V;
(d) 0.4 s; (e) 3.5 cycles; (f) 2.5 Hz
14
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Example 13.2
Find the period of periodic waveform with frequency of;
(a) 60 Hz
(b) 1000 Hz
Solution
(a)
1
1
T 
 16.67 ms
f 60
(b)
1
1
T 
 1 ms
f 1000
15
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Example 13.3
Determine the frequency of the following waveform
16
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Example 13.3 – solution
From the waveform;
T  20 ms
1
1
f  
 50 Hz
3
T 20 10
17
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Defined Polarities and Direction
The polarity and current direction will be for an instant
in time in the positive portion of the sinusoidal
waveform.
 In the figure, a lowercase letter is employed for
polarity and current direction to indicate that the
quantity is time dependent; that is, its magnitude will
change with time.
18
13.2 Sinusoidal ac Voltage
Characteristics and Definitions
Defined Polarities and Direction
19
13.4 The Sinusoidal Waveform
The sinusoidal waveform is the only alternating
waveform whose shape is unaffected by the
response characteristics of R, L, and C elements.
The voltage across (or current through) a resistor,
coil, or capacitor is sinusoidal in nature.
The unit of measurement for the horizontal axis is
the degree. A second unit of measurement
frequently used is the radian (rad).
1 rad  57.296  57.3
20
13.4 The Sinusoidal Waveform
 If we define x as the number of intervals of r (the
radius) around the circumference of a circle, then
C  2r  xr and we find x  2
Therefore, there are 2 rad around a 360° circle, as
shown in the figure.
21
13.4 The Sinusoidal Waveform
 The quantity  is the ratio of the circumference of
a circle to its diameter.
 For 180° and 360°, the two units of measurement
are related as follows:
22
13.4 The Sinusoidal Waveform
 The sinusoidal wave form can be derived
from the length of the vertical projection of a
radius vector rotating in a uniform circular
motion about a fixed point.
 The velocity with which the radius vector
rotates about the center, called the angular
velocity, can be determined from the
following equation:
23
13.4 The Sinusoidal Waveform
The angular velocity () is:


t
Since () is typically provided in radians per second, the
angle  obtained using  = t is usually in radians.
The time required to complete one revolution is
equal to the period (T) of the sinusoidal waveform.
The radians subtended in this time interval are 2.
2

T
or
  2f
24
13.4 The Sinusoidal Waveform
Example 13.6
Given  = 200 rad/s, determine how long it will take the
sinusoidal waveform to pass through an angle of 90
Solution
  90 


2
rad  t
  /2
t 
 7.85 ms
 200
25
13.4 The Sinusoidal Waveform
Example 13.7
Find the angle through which a sinusoidal waveform of
60 Hz will pass in a period of 5 ms.
Solution
  t  2ft  2  60  5 103  1.885 rad
 180 
  108
  1.885
  
26
13.5 General Format for the
Sinusoidal Voltage or Current
• The basic
mathematical format
for the sinusoidal
waveform is:
where:
Am is the peak value of
the waveform
 is the unit of measure
for the horizontal axis
27
13.5 General Format for the
Sinusoidal Voltage or Current
 The equation  = t states that the angle  through
which the rotating vector will pass is determined by
the angular velocity of the rotating vector and the
length of time the vector rotates.
 For a particular angular velocity (fixed ), the longer
the radius vector is permitted to rotate (that is, the
greater the value of t ), the greater will be the number
of degrees or radians through which the vector will
pass.
 The general format of a sine wave can also be as:
28
13.5 General Format for the
Sinusoidal Voltage or Current
 For electrical quantities such as current and voltage,
the general format is:
i  I m sin t  I m sin 
e  Em sin t  Em sin 
where: the capital letters with the subscript m
represent the amplitude, and the lower case letters i
and e represent the instantaneous value of current
and voltage, respectively, at any time t.
29
13.5 General Format for the
Sinusoidal Voltage or Current
Example 13.8
Given e = 5sin, determine e at  = 40 and  = 0.8.
Solution
For  = 40,
e  5 sin 40  3.21 V
For  = 0.8
 180 
  144
  0.8 
  
e  5 sin 144  2.94 V
30
13.5 General Format for the
Sinusoidal Voltage or Current
Example 13.9
(a) Determine the angle at which the magnitude of the
sinusoidal function v = 10 sin 377t is 4 V.
(b) Determine the time at which the magnitude is
attained.
31
13.5 General Format for the
Sinusoidal Voltage or Current
Example 13.9 - solution
Vm  10 V;   377 rad/s 
v  Vm sin t V
Hence,
v  10 sin 377t V
When v = 4 V,
4  10 sin 377t
Or;
4
sin 377t  sin  
 0.4
10
  sin 1 0.4  23.58
32
13.5 General Format for the
Sinusoidal Voltage or Current
Example 13.9 – solution (cont’d)
  377t  23.58  0.412 rad
0.412
t
 1.09 ms
377
33
13.6 Phase Relationship
The unshifted sinusoidal waveform is
represented by the expression:
a  Am sin t
t
34
13.6 Phase Relationship
Sinusoidal waveform which is shifted to the right
or left of 0° is represented by the expression:
a  Am sin t   
where  is the angle (in degrees or radians) that
the waveform has been shifted.
35
13.6 Phase Relationship
If the wave form passes through the horizontal
axis with a positive-going (increasing with the
time) slope before 0°:
a  Am sin t   
t
36
13.6 Phase Relationship
If the waveform passes through the horizontal
axis with a positive-going slope after 0°:
a  Am sin t   
t
37
13.6 Phase Relationship
The terms lead and lag are used to indicate
the relationship between two sinusoidal
waveforms of the same frequency plotted on
the same set of axes.
•
The cosine curve is said to lead the sine
curve by 90.
• The sine curve is said to lag the cosine
curve by 90.
•
90 is referred to as the phase angle
between
the two waveforms.
38
13.6 Phase Relationship
t
39
13.6 Phase Relationship
If a sinusoidal expression should appear as
e   Em sin t
the negative sign is associated with the sine
portion of the expression, not the peak value
Em , i.e.
e   Em sin t  e  Em  sin t 
And, since;

 sin t  sin t  180

 Em sin t  Em sin t  180


40
13.6 Phase Relationship
If a sinusoidal expression should appear as
e   Em sin t
the negative sign is associated with the sine
portion of the expression, not the peak value
Em , i.e.
e   Em sin t  e  Em  sin t 
And, since;

 sin t  sin t  180

 Em sin t  Em sin t  180


41
13.6 Phase Relationship
Example 13.2
Determine the phase relationship between the
following waveforms;

(a) v  10 sin t  30

i  5 sin t  70




v  10 sin t  20 
(b) i  15 sin t  60




v  3 sin t  10 
(c) i  2 cos t  10



v  2 sin t  10 
(d) i   sin t  30

42
13.6 Phase Relationship
Example 13.2 – solution

(a) v  10 sin t  30

i  5 sin t  70



i leads v by 40
Or
v lags i by 40
43
13.6 Phase Relationship
Example 13.2 – solution (cont’d)


v  10 sin t  20 
(b) i  15 sin t  60

i leads v by 80
Or
v lags i by 80
44
13.6 Phase Relationship
Example 13.2 – solution (cont’d)


v  3 sin t  10 
(c) i  2 cos t  10

i leads v by 110
Or
v lags i by 110
45
13.6 Phase Relationship
Example 13.2 – solution (cont’d)


v  2 sin t  10 
(d) i   sin t  30

v leads i by 160
Or
i lags v by 160
OR
i leads v by 200
Or
v lags i by 200
46
13.6 Phase Relationship
Example 13.2 – solution (cont’d)
47
13.7 Average Value
• Understanding the
average value using
a sand analogy:
– The average height
of the sand is that
height obtained if the
distance form one
end to the other is
maintained while the
sand is leveled off.
48
13.7 Average Value
The algebraic sum of the areas must be
determined, since some area contributions will
be from below the horizontal axis.
Area above the axis is assigned a positive sign and
area below the axis is assigned a negative sign.
The average value of any current or voltage is
the value indicated on a dc meter – over a
complete cycle the average value is the
equivalent dc value.
49
13.7 Average Value

Area of a  Am  sin tdt  2 Am
a  Am sin t
aaverage 
0
Area of a


2 Am

 0.632 Am
Average value
t
50
13.7 Effective (rms) Value
T
I (rms) 

0
i 2 t dt
T
i t   I m sin t  I m
2
I (rms) 
it   I m sin t
2
2

T
0
2
1

 2 1  2 cos t 

2 1
I m  1  2 cos t  dt
2

T
I (rms)
Im

2
51
13.7 Effective (rms) Value
Example 13.21
The 120 V dc source delivers 3.6 W to the load. Find
Em and Im of the ac source, if the same power is to be
delivered to the load.
52
13.7 Effective (rms) Value
Example 13.21 – solution
P
3.6
I dc 

 30 mA
Edc 120
Edc I dc  P  3.6 W
Erms
Em
 Edc 
2
and
I rms
Im
 I dc 
2
Em  2Edc  1.414 120  169.7 V
I m  2I dc  1.414  30  42.43 mA
53
13.7 Effective (rms) Value
Example 13.21 – solution
Erms
Em
 Edc 
2
Em  2Erms
 1.414 120
I rms
Im
 I dc 
2
 169.7 V
54