alternating current )

8/11/2019 Alternating Current )

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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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Given the sinusoid 5 sin(4t600), calculate its

amplitude, phase, angular frequency, period,

and frequency.


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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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Determine the effective or rms values of the

sinusoidal waveform.


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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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Evaluate


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Evaluate


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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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Transform the sinusoid to phasor:

i = 6 cos(50t 400

) A


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v = 4 sin(30t + 500

) V


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Express the sinusoid represented by this phasor.

V =j8e-j/6

V


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Express the sinusoid represented by this phasor.

I= 3j4 A


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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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Determine the current that flows through an 8

resistor connected to a voltage source

vs= 110 cos 377t V.


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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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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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Determine the impedance of the circuit.


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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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Determine the admittance of the circuit.

h k d d


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Determine the admittance of the circuit.

Ch k d di


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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


#10


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#10


#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


#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



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The impedances Z1= 10 + j12 and Z2= 6j9 are connected in series. Determine the total

impedance.



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Determine the total impedance.

PARALLEL IMPEDANCES


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PARALLEL IMPEDANCES

TWO IMPEDANCES IN PARALLEL


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TWO IMPEDANCES IN PARALLEL



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Determine the total impedance of the circuit.



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Determine the total impedance of the circuit at2 kHz.



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Determine the total impedance of the circuit.



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At = 50 rad/s, determine Zinof the circuit.

VOLTAGE DIVISION PRINCIPLE


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VOLTAGE DIVISION PRINCIPLE



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Determine v.



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If vs= 5 cos 2t V in the circuit, determine Vo.

CURRENT DIVISION PRINCIPLE


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CURRENT DIVISION PRINCIPLE



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Determine i.



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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,



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Determine v(t) and i(t) in the circuit.



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What is the instantaneous voltage across a 2-Fcapacitor when the current through it is

i = 4 cos(106t + 25o) A?



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The voltage across a 4-mH inductor is

v = 60 cos(500t 65o) V. Determine the

instantaneous current through it.


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