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ALTERNATING CURRENT
(AC) CIRCUITS
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Time-variant Voltage
Time-varying voltage that is commercially
available in large quantities and is commonly
called the ac voltage. (The letters ac are an
abbreviation for alternating current.)
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Alternating waveform available from
commercial supplies.
The term alternating indicates only that thewaveform alternates between two prescribed
levels in a set time sequence
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A sinusoid is a signal that has the form of thesine or cosine function.
A sinusoidal current is usually referred to as
alternating current (ac). Such a current reverses at regular time
intervals and has alternately positive and
negative values. Circuits driven by sinusoidalcurrent or voltage sources are called accircuits.
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Sinusoidal ac voltages are available from a
variety of sources.
The most common source is the typical home
outlet, which provides an ac voltage that
originates at a power plant; such a power
plant is most commonly fueled by water
power, oil, gas, or nuclear fusion in each casean ac generator (also called an alternator).
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Sinusoidal Voltage
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A sketch of Vmsin tas a function of t.
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A sketch of Vmsin tas a function of t.
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It is evident that the sinusoid repeats itselfevery T seconds; thus, T is called theperiod ofthe sinusoid. Observe that T= 2.
While is in radians per second (rad/s), fis inHertz
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Periodic Function
A periodic function is one that satisfies
f (t) =f (t + nT), for all t and for all integers n.
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The period T of the periodic function is the
time of one complete cycle or the number of
seconds per cycle.
The reciprocal of this quantity is the number
of cycles per second, known as the cyclic
frequency f of the sinusoid. Thus,
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Instantaneous value: The magnitude of a
waveform at any instant of time; denoted by
lowercase letters
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Peak amplitude: The maximum value of awaveform as measured denoted by uppercaseletters (such as Emfor sources of voltage and Vmforthe voltage drop across a load).
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Peak-to-peak value: Denoted by Ep-p or Vp-p, thefull voltage between positive and negative peaks ofthe waveform, that is, the sum of the magnitude ofthe positive and negative peaks.
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Period (T ): The time interval between successiverepetitions of a periodic waveform (the period T1,T2, and T3), as long as successive similar points ofthe periodic waveform are used in determining T.
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General expression for the sinusoid
Where (t + ) is the argument and is the
phase. Both argument and phase can be in
radians or degrees.
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Check Your Understanding
Determine the amplitude, phase, period, and
frequency of the sinusoid:
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Check Your Understanding
Given the sinusoid 5 sin(4t600), calculate its
amplitude, phase, angular frequency, period,
and frequency.
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Check Your Understanding
Determine the angular velocity of a sine wave
having a frequency of 60 Hz.
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Effective (rms) value of a sinusoid
The equivalent dc value is called the effective
value of the sinusoidal quantity.
Root-mean-square (rms) value is the root-
mean-square or effective value of a waveform.
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Check Your Understanding
Determine the effective or rms values of the
sinusoidal waveform.
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Check Your Understanding
Determine the effective or rms values of the
sinusoidal waveform.
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PHASORS
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Sinusoids are easily expressed in terms of
phasors, which are more convenient to work
with than sine and cosine functions.
A phasor is a complex number that represents
the amplitude and phase of a sinusoid.
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Phasors provide a simple means of analyzinglinear circuits excited by sinusoidal sources;solutions of such circuits would be intractableotherwise.
The notion of solving ac circuits using phasorswas first introduced by Charles Steinmetz in1893.
Before we completely define phasors and applythem to circuit analysis, we need to bethoroughly familiar with complex numbers.
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Complex Numbers
A complex number z can be written in
rectangular form as
where j = 1; x is the real part of z; y is the
imaginary part of z.
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BASIC OPERATIONS OFCOMPLEX NUMBERS
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Check Your Understanding
Evaluate
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Check Your Understanding
Evaluate
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Check Your Understanding
Evaluate
ANSWER:
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The idea of phasor representation is based on
Eulers identity. In general,
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Given a sinusoid v(t) = Vmcos(t + )
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V is the phasor representation of the sinusoid
v(t). In other words, a phasor is a complexrepresentation of the magnitude and phase of a
sinusoid.
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Time-domain and Phasor-domain
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Sinusoid-Phasor Transformation
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Check Your Understanding
Transform the sinusoid to phasor:
i = 6 cos(50t 400
) A
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Check Your Understanding
Transform the sinusoid to phasor:
v = 4 sin(30t + 500
) V
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Check Your Understanding
Express the sinusoid represented by this phasor.
V =j8e-j/6
V
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Check Your Understanding
Express the sinusoid represented by this phasor.
I= 3j4 A
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Check Your Understanding
Given i1(t) = 4 cos(t + 300 ) A and
i2(t) = 5 sin(t200) A, determine their sum.
ANSWER:
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PHASOR RELATIONSHIPS FORCIRCUIT ELEMENTS
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Voltage-Current Relations for a Resistor
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Voltage-Current Relations for an Inductor
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Voltage-Current Relations for a Capacitor
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Voltage-Current Relationships
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Check Your Understanding
Determine the current that flows through an 8
resistor connected to a voltage source
vs= 110 cos 377t V.
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Check Your Understanding
The voltage v = 12 cos(60t + 45o) Vis applied to
a 0.1-H inductor. Determine the steady-state
current through the inductor.
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The voltage v = 12 cos(60t + 450)V is applied to
a 0.1-H inductor. Determine the steady-statecurrent through the inductor.
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Check Your Understanding
If voltage v = 6 cos(100t30o)V is applied to a
50F capacitor, calculate the current through
the capacitor.
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IMPEDANCE AND ADMITTANCE
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IMPEDANCE
The impedance Z of a circuit is the ratio of the
phasor voltage V to the phasor current I,
measured in ohms ().
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The impedance represents the opposition which
the circuit exhibits to the flow of sinusoidalcurrent. Although the impedance is the ratio of
two phasors, it is not a phasor, because it does
not correspond to a sinusoidally varyingquantity.
Impedances and
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Impedances and
Admittances of Passive Elements
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As a complex quantity, the impedance may be
expressed in rectangular form as
where R = Re Z is the resistanceandX = Im Z is
the reactance.
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The reactance X may be positive or negative.We say that the impedance is inductive whenX is positive or capacitive whenX is negative.
Thus, impedance Z = R + jX is said to beinductive or lagging since current lags voltage,while impedance Z = R jX is capacitive orleadingbecause current leads voltage.
The impedance, resistance, and reactance areall measured in ohms.
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The impedance may also be expressed in polar
form as
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Check Your Understanding
Determine the impedance of the circuit.
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Check Your Understanding
Determine the impedance of the circuit.
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Check Your Understanding
Determine the impedance of the circuit. Assume
that the circuit operates at = 50 rad/s.
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Admittance
The admittance Y is the reciprocal of
impedance, measured in siemens (S).
The admittance Y of an element (or a circuit)
is the ratio of the phasor current through it tothe phasor voltage across it, or
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As a complex quantity, we may write Y as
Where G =Re Y is called the conductanceand B
=Im Y is called the susceptance.
Admittance, conductance, and susceptance areall expressed in the unit of siemens (or mhos).
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Admittance
h k d d
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Check Your Understanding
Determine the admittance of the circuit.
h k d d
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Check Your Understanding
Determine the admittance of the circuit.
Ch k d di
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Check Your Understanding
Determine the admittance of the circuit. Assume
that the circuit operates at = 50 rad/s.
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REVIEW
#1
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#1
In a linear circuit, the voltage source is
(a) What is the angular frequency of the voltage?(b) What is the frequency of the source?
(c) Determine the period of the voltage.
#2
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#2
In a linear circuit, the current source is
Determine isat t = 2 ms.
#3
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#3
Express the function in cosine form:
#4
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#4
Express the function in cosine form:
#5
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#5
Express v = 8 cos(7t + 15o) in sine form.
#6
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#6
Evaluate:
Express your results in rectangular form.
#7
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#7
Evaluate:
Express your results in rectangular form.
#8
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#8
Evaluate the determinant
Express your results in polar form.
#9
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#9
Transform the sinusoid to phasor:
#10
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#10
Transform the sinusoid to phasor:
#11
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#11
Express the sum of the following sinusoidalsignals in the form ofAcos(t+ ) withA > 0
and 0 < < 360.
#11
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#11
Express the sum of the following sinusoidalsignals in the form ofAcos(t+ ) withA > 0
and 0 < < 360.
#12
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#12
Determine a single sinusoid corresponding to:
#13
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#13
A series RCL circuit has R = 30 ,XC = j50 ,and XL = j90 . Determine the impedance of
the circuit.
#14
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#14
Determine the impedance of the circuit.
#15
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#15
Two elements are connected in series as shown.If i = 12 cos(2t 30o) A, determine the element
values.
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IMPEDANCE COMBINATIONS
SERIES IMPEDANCES
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SERIES IMPEDANCES
TWO IMPEDANCES IN SERIES
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TWO IMPEDANCES IN SERIES
Check Your Understanding
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Check Your Understanding
The impedances Z1= 10 + j12 and Z2= 6j9 are connected in series. Determine the total
impedance.
Check Your Understanding
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Check Your Understanding
Determine the total impedance.
PARALLEL IMPEDANCES
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PARALLEL IMPEDANCES
TWO IMPEDANCES IN PARALLEL
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TWO IMPEDANCES IN PARALLEL
Check Your Understanding
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Check Your Understanding
Determine the total impedance of the circuit.
Check Your Understanding
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Check Your Understanding
Determine the total impedance of the circuit at2 kHz.
Check Your Understanding
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Check Your Understanding
Determine the total impedance of the circuit.
Check Your Understanding
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Check Your Understanding
At = 50 rad/s, determine Zinof the circuit.
VOLTAGE DIVISION PRINCIPLE
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VOLTAGE DIVISION PRINCIPLE
Check Your Understanding
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Check Your Understanding
Determine v.
Check Your Understanding
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Check Your Understanding
If vs= 5 cos 2t V in the circuit, determine Vo.
CURRENT DIVISION PRINCIPLE
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CURRENT DIVISION PRINCIPLE
Check Your Understanding
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Check Your Understanding
Determine i.
Check Your Understanding
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Check Your Understanding
Determine i.
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AC CIRCUITS ANALYSIS
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The techniques of voltage/current division,series/parallel combination of
impedance/admittance, circuit reduction, and
Y - transformation all apply to ac circuitanalysis.
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Basic circuit laws (Ohms and Kirchhoffs)apply to ac circuits in the same manner as
they do for dc circuits; that is,
Check Your Understanding
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Check Your Understanding
Determine v(t) and i(t) in the circuit.
Check Your Understanding
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Check Your Understanding
What is the instantaneous voltage across a 2-Fcapacitor when the current through it is
i = 4 cos(106t + 25o) A?
Check Your Understanding
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Check Your Understanding
The voltage across a 4-mH inductor is
v = 60 cos(500t 65o) V. Determine the
instantaneous current through it.
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