ELECTRICAL CIRCUIT ET 201

1
ELECTRICAL CIRCUIT ET 201

• Define and explain characteristics of sinusoidal
  wave, phase relationships and phase shifting

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SINUSOIDAL ALTERNATING WAVEFORMS
(CHAPTER 1.1 1.4)
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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

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Sources of alternating current

• By rotating a magnetic field within a stationary
  coil
• By rotating a coil in a magnetic field

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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.

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Faradays and Lenzs Law involved in generating
a.c current

• Faradays 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? N
  Number of turn
• dt
  ? Magnetic Flux
• Lenzs 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.

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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.

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Sine Wave Characteristics
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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.

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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.

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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-to-frequency
  conversions are common in electronic calculations.

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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.

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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

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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 root-mean-square,
  and is determined by Vrms 0.707Vp or Irms
  0.707Ip

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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.

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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

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Peak Factor

• Crest or Peak or Amplitude Factor
• Peak factor is defined as the ratio of peak
  voltage to r.m.s value.

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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.

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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
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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

f Hz T seconds (s)
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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 2 cycles per
second
Frequency 1 cycle per
second
Frequency 21/2 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).

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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.

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13.2 Sinusoidal AC Voltage Characteristics
and Definitions
Angular velocity
Degree Radian
90 (p/180) x ( 90) p/2 rad
60 (p/180) x ( 60) p/3 rad
30 (p/180) x (30) p/6 rad
Radian Degree
p /3 (180 /p) x (p /3) 60
p (180 /p) x (p ) 180
3p /2 (180/p) x (3p /2) 270
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13.2 Sinusoidal AC Voltage Characteristics
and Definitions
Plotting a sine wave versus (a) degrees and (b)
radians.
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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
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13.2 Sinusoidal AC Voltage Characteristics
and Definitions

• Angular velocity
• Formula of angular velocity
• Since (?) is typically provided in
  radians/second, the angle ? obtained using ? ?t
  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 2p (360o).

(rad/s)
or
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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.
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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
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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
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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
• Calculate the area under the graph
• Average value area under the function in a
  period
• period
• 2. Use integral method
• 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.

Solution
For a sinus waveform , average value can be
calculate by
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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/v2 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

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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
and
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13.2 Sinusoidal AC Voltage Characteristics
and Definitions
Example 13.21 solution
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13.5 General Format for the Sinusoidal
Voltage or Current
The basic mathematical format for the sinusoidal
waveform is where Am peak value of the
waveform ? angle from the horizontal axis
volts or amperes
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
• General format for electrical quantities such as
  current and voltage is
• where and is the peak value
  of current and voltage while i(t) and v(t) is the
  instantaneous value of current and voltage.

a ?t
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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?,
For ? 0.8?
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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.
• Determine the time
• at which the magnitude
• is attained.

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13.5 General Format for the Sinusoidal
Voltage or Current
Example 13.9 - solution
Hence,
When v(t) 4 V,
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13.5 General Format for the Sinusoidal
Voltage or Current
Example 13.9 solution (contd)
(a) But a is in radian, so a must be calculate in
radian (b) Given, , so
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13.6 Phase Relationship

• Phase angle
• Phase angle is a shifted angle waveform from
  reference origin.
• Phase angle is been represent by symbol ? or F
• 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.

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13.6 Phase Relationship
The unshifted sinusoidal waveform is represented
by the expression
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13.6 Phase Relationship
Sinusoidal waveform which is shifted to the
right or left of 0 is represented by the
expression
where ? is the angle (in degrees or radians) that
the waveform has been shifted.
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13.6 Phase Relationship
If the wave form passes through the horizontal
axis with a positive-going (increasing with the
time) slope before 0
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13.6 Phase Relationship
If the waveform passes through the horizontal
axis with a positive-going slope after 0
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13.6 Phase Relationship
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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 ?) 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.

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13.6 Phase Relationship
Note sin (- a) - sin a cos(- a) cos a
Start at sin a position
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13.6 Phase Relationship

• If a sinusoidal expression should appear as
• the negative sign is associated with the sine
  portion of the expression, not the peak value Em
  , i.e.
• And, since

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13.6 Phase Relationship
Example 13.2
Determine the phase relationship between the
following waveforms
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13.6 Phase Relationship
Example 13.2 solution
i leads v by 40? or v lags i by 40?
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13.6 Phase Relationship
Example 13.2 solution (contd)
i leads v by 80? or v lags i by 80?
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13.6 Phase Relationship
Example 13.2 solution (contd)
i leads v by 110? or v lags i by 110?
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13.6 Phase Relationship
Example 13.2 solution (contd)
OR
v leads i by 160? Or i lags v by 160?
i leads v by 200? Or v lags i by 200?