Transcript ELECTRIC CIRCUITS I

ELECTRICAL CIRCUIT
ET 201
 Define and explain characteristics of
sinusoidal wave, phase relationships
and phase shifting
1
SINUSOIDAL ALTERNATING
WAVEFORMS
(CHAPTER 1.1 ~ 1.4)
2
Understand Alternating Current
• DIRECT CURRENT (DC) – IS WHEN THE CURRENT
FLOWS IN ONLY ONE DIRECTION. Constant flow of
electric charge
•
EX: BATTERY
• ALTERNATING CURRENT AC) – THE CURRENT
FLOWS IN ONE DIRECTION THEN THE OTHER.
• Electrical current whose magnitude and direction vary
cyclically, as opposed to direct current whose direction
remains constant.
• EX: OUTLETS
Sources of alternating current
• By rotating a magnetic field within a
stationary coil
• By rotating a coil in a magnetic field
Generation of Alternating
Current
• A voltage supplied by a battery or other
DC source has a certain polarity and
remains constant.
• Alternating Current (AC) varies in polarity
and amplitude.
• AC is an important part of electrical and
electronic systems.
Faraday’s and Lenz’s Law
involved in generating a.c current
• Faraday’s Laws of electromagnetic
Induction.
Induced electromotive field
Any change in the magnetic environment of a coil of wire will cause a
voltage (emf) to be "induced" in the coil
e.m.f, e = -N d
dt
.
N = Number of turn
 = Magnetic Flux
Lenz’s law
An electromagnetic field interacting with a conductor will generate
electrical current that induces a counter magnetic field that opposes
the magnetic field generating the current.
Sine Wave Characteristics
• The basis of an AC alternator is a loop of
wire rotated in a magnetic field.
• Slip rings and brushes make continuous
electrical connections to the rotating
conductor.
• The magnitude and polarity of the
generated voltage is shown on the
following slide.
Sine Wave Characteristics
Sine Wave Characteristics
• The sine wave at the
right consists of two,
opposite polarity,
alternations.
• Each alternation is
called a half cycle.
• Each half cycle has a
maximum value called
the peak value.
Sine Wave Characteristics
• Sine waves may represent voltage,
current, or some other parameter.
• The period of a sine wave is the time from
any given point on the cycle to the same
point on the following cycle.
• The period is measured in time (t), and in
most cases is measured in seconds or
fractions thereof.
Frequency
• The frequency of a sine wave is the
number of complete cycles that occur in
one second.
• Frequency is measured in hertz (Hz). One
hertz corresponds to one cycle per
second.
• Frequency and period have an inverse
relationship. t = 1/f, and f = 1/t.
• Frequency-to-period and period-tofrequency conversions are common in
electronic calculations.
Peak Value
• The peak value of a sine wave is the
maximum voltage (or current) it reaches.
• Peak voltages occur at two different points
in the cycle.
• One peak is positive, the other is negative.
• The positive peak occurs at 90º and the
negative peak at 270º.
• The positive and negative have equal
amplitudes.
Average Values
• The average value of any measured
quantity is the sum of all of the
intermediate values.
• The average value of a full sine wave is
zero.
• The average value of one-half cycle of a
sine wave is:
Vavg = 0.637Vp or Iavg = 0.637Ip
Chapter 6 -
13
rms Value
• One of the most important characteristics
of a sine wave is its rms or effective value.
• The rms value describes the sine wave in
terms of an equivalent dc voltage.
• The rms value of a sine wave produces
the same heating effect in a resistance as
an equal value of dc.
• The abbreviation rms stands for rootmean-square, and is determined by: Vrms =
0.707Vp or Irms = 0.707Ip
Chapter 6 -
14
Peak-to-Peak Value
• Another measurement used to describe sine waves are
their peak-to-peak values.
• The peak-to-peak value is the difference between the
two peak values.
Form Factor
• Form Factor is defined as the ratio of r.m.s
value to the average value.
• Form factor =
•
•
=
r.m.s value
= 0.707  peak value
average value
0.637  peak valur
1.11
Peak Factor
– Crest or Peak or Amplitude Factor
• Peak factor is defined as the ratio of peak
voltage to r.m.s value.
13.1 Introduction
Alternating waveforms
• Alternating signal is a signal that varies with respect to time.
• Alternating signal can be categories into ac voltage and ac
current.
• This voltage and current have positive and negative value.
18
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
• Voltage and current value is represent by vertical axis and time
represent by horizontal axis.
• In the first half, current or voltage will increase into maximum positive
value and come back to zero.
• Then in second half, current or voltage will increase into negative
maximum voltage and come back to zero.
• One complete waveform is called one cycle.
volts or amperes
units of time
19
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Defined Polarities and Direction
• The voltage 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.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Defined Polarities and Direction
• For a period of time, a voltage has one polarity, while for the
next equal period it reverses. A positive sign is applied if the
voltage is above the axis.
• For a current source, the direction in the symbol
corresponds with the positive region of the waveform.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
There are several specification in sinusoidal
waveform:
1. period
2. frequency
3. instantaneous value
4. peak value
5. peak to peak value
6. angular velocity
7. average value
8. effective value
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Period (T)
• Period is defines as the amount of time is take to go through
one cycle.
• Period for sinusoidal waveform is equal for each cycle.
Cycle
• The portion of a waveform contained in one period of time.
Frequency (f)
• Frequency is defines as number of cycles in one seconds.
• It can derives as
1
f = Hz
f 
T
hertz, Hz 
T = seconds (s)23
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
The cycles within T1, T2, and T3 may appear different in
the figure above, but they are all bounded by one period of time
and therefore satisfy the definition of a cycle.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Signal with lower frequency
Signal with higher frequency
Frequency = 1 cycle
Frequency = 21/2 cycles
per second
per second
Frequency = 2 cycles
per second
1 hertz (Hz) = 1 cycle per second (cps)
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Instantaneous value
• Instantaneous value is magnitude value of waveform at
one specific time.
• Symbol for instantaneous value of voltage is v(t) and
current is i(t).
v(0.1)  8 V
v(0.6)  0 V
v(1.1)  8 V
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Peak Value
• The maximum instantaneous value of a function as measured
from zero-volt level.
• For one complete cycle, there are two peak value that is
positive peak value and negative peak value.
• Symbol for peak value of voltage is Em or Vm and current is Im .
Peak value, Vm = 8 V
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Peak to peak value
• The full voltage between positive and negative peaks of the
waveform, that is, the sum of the magnitude of the positive and
negative peaks.
• Symbol for peak to peak value of voltage is Ep-p or Vp-p and
current is Ip-p
Peak to peak value, Vp-p = 16 V
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Angular velocity
• Angular velocity is the velocity with which the radius vector
rotates about the center.
• Symbol of angular speed is
and units is
radians/seconds (rad/s)
• Horizontal axis of waveform can be represent by time and
angular speed.

2 radian  360 
3600
1 radian 
 57.30 ,   3.142
2
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Angular velocity
Degree
Radian
90°
(π/180°) x ( 90°) = π/2 rad
60°
(π/180°) x ( 60°) = π/3 rad
30°
(π/180°) x (30°) = π/6 rad
Radian
Degree
π /3
(180° /π) x (π /3) = 60°
π
(180° /π) x (π ) = 180°
3π /2
(180°/π) x (3π /2) = 270°
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Plotting a sine wave versus (a) degrees and (b) radians.
13.2 Sinusoidal AC
Voltage
Characteristics and
Definitions
•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.
Waveform picture with respect to angular velocity
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Angular velocity
• Formula of angular velocity
distance(degrees or radians) 
angular degree,  

time(seconds)
t
  t
Since (ω) is typically provided in radians/second, the
angle ϴ obtained using ϴ = ωt is usually in radians.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Angular velocity
• The time required to complete one cycle is equal to the
period (T) of the sinusoidal waveform.
• One cycle in radian is equal to 2π (360o).
2

T
or
  2f
(rad/s)
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Angular velocity
Demonstrating the effect of  on the frequency f and period T.
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
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
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
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  1.885rad
 180 
  108
  1.885
  
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Average value
• Average value is average value for all instantaneous value in
half or one complete waveform cycle.
• It can be calculate in two ways:
1. Calculate the area under the graph:
Average value = area under the function in a period
period
2. Use integral method
T
1
average _ value   v (t ) dt
T 0
For a symmetry waveform, area upper section equal to area
under the section, so just take half of the period only.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Average value
• Example: Calculate the average value of the waveform below.
T
Vm
Solution:
1
average_ value   v(t )dt
T 0
Vm

1

v


m
sin d
0

2
 rad
For a sinus waveform , average value can
be calculate by
Vaverage 
Vm

 0.637Vm

vm


 sin d
0
 cos o

vm

2vm


 0.637vm volt
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Effective value
• The most common method of specifying the amount of sine wave of
voltage or current by relating it into dc voltage and current that will
produce the same heat effect.
• Effective value is the equivalent dc value of a sinusoidal current or
voltage, which is 1/√2 or 0.707 of its peak value.
• The equivalent dc value is called rms value or effective value.
• The formula of effective value for sine wave waveform is;
1
I rms 
I m  0.707I m
2
1
Erms 
Em  0.707Em
2
I m  2 I rms  1.414I rms
Em  2 Erms  1.414Erms
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
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.
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13.2
Sinusoidal AC Voltage
Characteristics and Definitions
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
42
13.2
Sinusoidal AC Voltage
Characteristics and Definitions
Example 13.21 – solution
Erms
Em
 Edc 
2
Em  2 Erms
 1.414120
 169.7 V
I rms
Im
 I dc 
2
I m  2 I rms
 1.414 30
 42.43 mA
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13.5
General Format for the
Sinusoidal Voltage or Current
The basic mathematical volts or amperes
format for the sinusoidal
waveform is:
where:
Am : peak value of the
waveform
 : angle from the
horizontal axis
Basic sine wave for current or voltage
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13.5
General Format for the
Sinusoidal Voltage or Current
• The general format of a sine wave can also be as:
α= ωt
• General format for electrical quantities such as current
and voltage is:
it   I m sin t  I m sin 
et   Em sin t  Em sin 
where: I m and Em is the peak value of current
and voltage while i(t) and v(t) is the instantaneous
value of current and voltage.
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13.5
General Format for the
Sinusoidal Voltage or Current
Example 13.8
Given e(t) = 5 sin, determine e(t) at  = 40 and  = 0.8.
Solution
For  = 40,
et   5 sin 40  3.21V
For  = 0.8
 180 
  144
  0.8 
  
et   5 sin 144  2.94 V
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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(t) = 10 sin 377t is 4 V.
(b) Determine the time
at which the magnitude
is attained.
47
13.5
General Format for the
Sinusoidal Voltage or Current
Example 13.9 - solution
Vm  10 V;   377 rad/s
vt   Vm sin t V
Hence,
vt   10sin 377t V
When v(t) = 4 V,
4  10 sin 377 t
4
sin 377 t  sin  
 0.4
10
1  sin 1 0.4  23.58
2  180  23.58  156.42
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13.5
General Format for the
Sinusoidal Voltage or Current
Example 13.9 – solution (cont’d)
(a)
But α is in radian, so α must be calculate in radian:
1  377t  23.58  0.412 rad
 2  156.42  2.73 rad
(b)
Given,   t

, so t 

0.412
t1 
 1.09ms
377
2.73
t2 
 7.24 ms
377
49
13.6 Phase Relationship
Phase angle
• Phase angle is a shifted angle waveform from reference
origin.
• Phase angle is been represent by symbol θ or Φ
• Units is degree ° or radian
• Two waveform is called in phase if its have a same
phase degree or different phase is zero
• Two waveform is called out of phase if its have a
different phase.
13.6 Phase Relationship
The unshifted sinusoidal waveform is
represented by the expression:
a  Am sin t
t
51
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.
52
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   
a  Am sin t   
t
53
13.6 Phase Relationship
If the waveform passes through the horizontal axis
with a positive-going slope after 0°:
a  Am sin t   
t
54
13.6 Phase Relationship
t


sin t  90  sin t    cost
2



sin t  cost  90  cos t  
2

55
13.6 Phase Relationship
• The terms leading and lagging are used to
indicate the relationship between two sinusoidal
waveforms of the same frequency f （or angular
velocity ω) 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.
56
13.6 Phase Relationship
+ cos
α
cos (α-90o)
sin (α+90o)
- sin α
Note:
sin (- α) = - sin α
cos(- α) = cos α
+ sin α
Start at + sin α position;
- cos α
cos  sin   90
sin   cos  90
 sin   sin   180
 cos  sin   270  sin   90
57
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


58
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  10sin t  20 
(b) i  15sin t  60



v  3 sin t  10 
(c) i  2 cos t  10



v  2 sin t  10 
(d) i   sin t  30

59
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
60
13.6 Phase Relationship
Example 13.2 – solution (cont’d)


v  10sin t  20 
(b) i  15sin t  60


i leads v by 80
or
v lags i by 80
61
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
62
13.6 Phase Relationship
Example 13.2 – solution (cont’d)


v  2 sin t  10 
(d) i   sin t  30


OR
v leads i by 160 i leads v by 200
Or
i lags v by 160
Or
v lags i by 200
63