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LECTURE 01-06
Chapter 10: Sinusoidal Steady-State Analysis
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Key Concepts
• Characteristics of Sinusoidal Functions
• Phasor Representation of Sinusoids
• Converting Between the Time and Frequency Domains
• Impedance and Admittance
• Reactance and Susceptance
• Parallel and Series Combinations in the Frequency Domain
• Determination of Forced Response using Phasors
• Application of Circuit Analysis Techniques in the Frequency
Domain
• Phasor Diagrams
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Lecture 01: Agenda
Introduction and Background
2. Characteristics of Sinusoids
1.
1.
2.
3.
Lagging and Leading
Converting Sines to Cosines
Practices
Forced Response to Sinusoidal Functions
3.
1.
2.
3.
4.
5.
6.
The Steady-State Response
A More Compact and User-Friendly Form
Observations
Example
Practices
Recap
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1. Introduction
• The response of a linear electric circuit is composed of two
parts
• The Natural Response: Short-lived transient response of a circuit to a
sudden change in its condition
• The Forced Response: Long-term steady state response of a circuit to any
independent sources present
• Up to this point, the only forced response we have considered is
due to DC sources.
• Another very common forcing function is the sinusoidal
waveform which is available at household electrical sockets as
well as in power lines connected to residential and industrial
area.
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1.1 Radian
• The radian is the standard unit
of angular measure
• An angle's measurement in
radians is numerically equal to
the length of a corresponding
arc of a unit circle, so one
radian is just under 57.3 degrees
(when the arc length is equal to
the radius)
• This is SI unit of angel
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1.2 Time Period
• It follows that the magnitude in radians of one complete
revolution (360 degrees) is the length of the entire
circumference divided by the radius, 2π. Thus 2π radians is
equal to 360 degrees, meaning that one radian is equal to 180/π
degrees.
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1.3 Degree
A degree (in full, a degree
of arc, arc degree, or
arcdegree), usually denoted
by ° (the degree symbol), is
a measurement of plane
angle, representing 1⁄360 of
a full rotation; one degree
is equivalent to π/180
radians. This is not SI unit
of angel
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1.4 Degree and Radian
2π rad = 360o
1 rad = 180o/π
1.5 Chart to convert between
Degrees and Radians
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1.6 Angular Frequency
• Angular frequency ω (also referred to by the
terms angular speed, radial frequency, circular
frequency,
orbital
frequency,
radian
frequency, and pulsatance) is a measure of
rotation rate
• Angular frequency ω (in radians per second),
is larger than frequency ν (in cycles per
second, also called Hz), by a factor of 2π
• This figure uses the symbol ν, rather than f to
denote frequency
• One revolution is equal to 2π radians, hence
ω= 2πf = 2π/T
ω is the angular frequency or angular speed
(measured in radians per second),
T is the period (measured in seconds),
f is the ordinary frequency (measured in
hertz)
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2. Characteristics of Sinusoids
• Consider a sinusoidally varying voltage
v(t)=Vm sin(ωt)
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
ωt = the argument of the sinusoid
• The function repeats itself every 2π radians, and its period is therefore
2π radians.
• A sine wave having a period T must execute 1/T periods each second;
its frequency f is 1/T hertz, abbreviated Hz.
The sinusoidal function v(t) = Vm sin ωt is plotted (a) versus ωt and (b) versus t.
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2.1 Lagging and Leading (1/4)
• v1(t) = Vm sin ωt
• v2(t) = Vm sin(ωt + θ) includes a phase angle θ in its argument
• Since corresponding points on the sinusoid Vm sin(ωt + θ) occur θ rad, or
θ/ω seconds, earlier, we say Vm sin(ωt + θ) leads Vm sinωt by θ rad
• It is also correct to say that sin ωt
• is lagging sin(ωt + θ) by θ rad,
• is leading sin(ωt + θ) by −θ rad, or
• is leading sin(ωt − θ) by θ rad
Two sinusoidal waves whose phases are
to be compared must:
1. Both be written as sine waves, or both
as cosine waves
2. Both be written with positive
amplitudes
3. Each have the same frequency
• In either case, leading or lagging, we
say that the sinusoids are out of phase
• If the phase angles are equal, the
sinusoids are said to be in phase
Engr Habeel Ahmad
2.1 Lagging and Leading (2/4)
𝒗𝟏 = 𝟓𝒔𝒊𝒏𝟏𝟎𝟎𝝅𝒕;
𝒗𝟐 = 𝟓𝐬𝐢𝐧(𝟏𝟎𝟎𝝅𝒕 + 𝟑𝟎𝒐 )
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Engr Habeel Ahmad
2.1 Lagging and Leading (3/4)
𝒗𝟏 = 𝟓𝒔𝒊𝒏𝟏𝟎𝟎𝝅𝒕;
𝒗𝟐 = 𝟓𝐬𝐢𝐧(𝟏𝟎𝟎𝝅𝒕 − 𝟗𝟎𝒐 )
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2.1 Lagging and Leading (4/4)
• In electrical engineering, the phase angle is commonly given in
degrees, rather than radians; to avoid confusion we should be
sure to always use the degree symbol. Thus, instead of writing:
• We customarily use:
• In evaluating this expression at a specific instant of time, e.g., t
= 10−4 s, 2π1000t becomes 0.2π radian, and this should be
expressed as 36° before 30° is subtracted from it
• Don’t confuse your apples with your oranges
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2.2 Converting Sines to Cosines (1/2)
• The
sine and cosine are
essentially the same function, but
with a 90o phase difference
• Thus,
sin ωt = cos(ωt − 90o)
• Multiples of 360o may be added
to or subtracted from the
argument of any sinusoidal
function without changing the
value of the function
• Hence, we may say that:
Leads
or v1 lags v2 as
by 130o
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2.2 Converting Sines to Cosines (2/2)
• A graphical representation of the two
sinusoids v1 and v2
• The magnitude of each sine function
is represented by the length of the
corresponding arrow, and the phase
angle by the orientation with respect
to the positive x axis
• In this diagram, v1 leads v2 by 100° +
30° = 130°, although it could also be
argued that v2 leads v1 by 230°
• It is customary, however, to express
the phase difference by an angle less
than or equal to 180° in magnitude
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2.3 Practice