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Rashid, M. H. Sinusoidal Excitation and Phasors The Engineering Handbook. Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

1998 by CRC PRESS LLC

105Sinusoidal Excitation and Phasors

105.1 Sinusoidal Source105.2 Phasor105.3 Passive Circuit Elements in Phasors Domain

Resistor Inductor Capacitor Sinusoidal Responses

Muhammad H. RashidPurdue University

A sinusoidal forcing function known as a sinusoid is one of the most important excitations. Inelectrical engineering, the carrier signals for communications are sinusoids, and the sinusoid is alsothe dominant signal in the power industry. Sinusoids abound in nature, as, for example, in themotion of a pendulum, in the bouncing of a ball, and in the vibrations of strings andmembranes.

Because of the importance of sinusoids, the output response of a circuit due to a sinusoidal inputsignal is an important criterion in determining the performance of a circuit. In this section, we willdefine a sinusoidal function, represent it in a phasor form, and then illustrate the techniques fordetermining the sinusoidal responses.

105.1 Sinusoidal SourceA sinusoidal source (independent or dependent) produces a signal that varies sinusoidally withtime. In electrical engineering, it is normally a voltage or a current. A sinusoid can be expressed asa sine function or a cosine function. There is no clear-cut choice for the use of either function.However, the sine function is more commonly used. Using a sine function, the instantaneousvoltage, which is shown in Fig. 105.1, can be represented by

v(t) = Vm sin !t (105:1)

where Vm is the maximum amplitude, and ! is the angular or radian frequency in rad/s.


A sinusoid is a periodic function defined generally by the property

v(t+ T ) = v(t) (105:2)

where T is the period, the time for one complete cycle. That is, the function goes through onecomplete cycle every T seconds and is then repeated. The period T is related to the angularfrequency ! by

T =2

!(105:3)

In 1 s, the function goes through 1/T cycles, or periods. The frequency is then

f =1

T=

!

2(105:4)

The number of cycles per second, or hertz (abbreviated Hz), named for the German physicistHeinrich R. Hertz (1857-1894), is the standard unit of frequency. The frequency and the angularfrequency are related by

! = 2f (105:5)

So far, we have assumed that a sinusoid starts at t = 0 . However, it could be phase shifted by anangle and has the more general expression given by

v(t) = Vm sin (!t + ) (105:6)

where is the phase angle, or simply phase. Since !t is in radians, should also be expressed inradians. However, it is often convenient to specify in degrees. That is,

v(t) = Vm sin!t +

3

(105:7)

Figure 105.1 Three sinusoids with different phase angles.


or

v(t) = Vm sin (!t + 60) (105:8)

is acceptable, although Eq. (105.8) is mathematically inconsistent. While computing the value ofsin(!t + ) , one should use the same units (radians or degrees) for both !t and . If has apositive value, the sinusoid is said to have a leading phase angle. If has a negative value, it issaid to have a lagging phase angle.

105.2 PhasorA phasor is a complex number that represents the amplitude and phase angle of a sinusoidalfunction. It transforms a sinusoidal function from time domain to the complex number domain. Thephasor concept, which is generally credited to electrical engineer Charles Proteus Steinmetz(1865-1923), is based on the Euler's exponential function of the trigonometricfunction

ej = cos + j sin (105:9)

Thus, a cosine function may be regarded as the real part of the exponential function and a sinefunction as the imaginary part of the exponential function. That is,

cos = Reej

(105:10)

sin = Imej

(105:11)

Therefore, we can write the sinusoidal voltage function of Eq. (105.6) as

v(t) = Vm sin(!t+ ) (105:12)

= Vm Imnej(!t+)

o= Im

Vme

jej!t

(105:13)

which indicates that the coefficient of the exponential ej!t is a complex number. It is the phasorrepresentation, or phasor transform, of the sinusoidal function and is represented by a boldfaceletter. Thus, the phasor V becomes

V = Vmej (exponential form) (105:14)

= Vm cos + jVm sin (polar form) (105:15)

Since phasors are used extensively in analysis of electrical engineering circuits, the exponentialfunction ej is abbreviated in a shorthand notation for the sake of simplicity


as

ej = 1 6 (105:16)

A graphical relationship between a phasor and sinusoid is shown in Fig. 105.2. A unit phasor maybe regarded as a unit vector having an initial phase displacement of and rotating in thecounterclockwise direction at an angular speed of ! .

Figure 105.2 Rotating phasor.

105.3 Passive Circuit Elements in Phasors DomainIf we want to find the response of an electrical circuit due to a sinusoidal forcing function, first weneed to establish the relationship between the phasor voltage across and the phasor current throughpassive elements such as the resistor, inductor, and capacitor.

ResistorLet us assume that a sinusoidal current of

i(t) = Im sin(!t+ ) (105:17)

where Im is the maximum amplitude of the current in amperes, and is the phase angle of thecurrent.

Using Ohm's law for the resistor in Fig. 105.3(a), the instantaneous voltage across the terminalsof the resistor is related to its instantaneous current by

v(t) = Ri(t) = R fIm sin(!t + )g = RIm sin(!t + ) (105:18)

which can be represented as a phasor by


V = RImej = RIm 6 (105:19)

Since Im 6 = I is the phasor representation of the current, we can write

V = RI (105:20)

That is, the phasor voltage across the terminals of a resistor is the resistor times the phasor current.Figure 105.3(b) shows the V and I relationship of a resistor. A resistor has no phase shift betweenthe voltage and its current. The current is said to be in time phase with the voltage, as shown inFig. 105.3(c).

Figure 105.3 Voltage-current relationships of a resistor.

InductorAssuming a sinusoidal current of i = Im sin (!t+ ) , the voltage across an inductor shown in Fig.105.4(a) is related to its current by

v = Ldi

dt= !LIm cos (!t+ ) (105:21)

Using the trigonometric identity of cos A = sin(A + =2) , Eq. (105.21) can be rewrittenas

v = !LIm sin!t + +

2

(105:22)

The phasor representation of the voltage given by Eq. (105.22) is


V = !LImej(+=2) = !LIm 6 ( + =2)

= !LImej ej=2

= !LImej (cos =2 + j sin =2)

= j!LImej = j!LI

(105:23)

Thus, the phasor voltage across the terminals of an inductor equals j!L times the phasor current.Since the operator j gives a phase shift of +90, then jI = jI 6 = Im 6 ( + =2) . That is, thecurrent lags behind the voltage by 90, or the voltage leads the current by 90. The voltage andcurrent relationship in the phasor domain is shown in Fig. 105.4(b) and in the time domain in Fig.105.4(c).

Figure 105.4 Voltage-current relationships of an inductor.

CapacitorAssuming a sinusoidal voltage of v = Vm sin (!t + ) , the current through a capacitor shown inFig. 105.5(a) is related to its voltage by

i = Cdv

dt= j!CVm cos (!t+ ) (105:24)

Using the trigonometric identity, Eq. (105.24) can be rewritten as

i = !CVm sin!t + +

2

(105:25)


I = j!CV (105:26)

Thus, the phasor current through a capacitor equals j!C times the phasor voltage. That is, thevoltage lags behind the current by 90, or the current leads the voltage by 90. The voltage andcurrent relationship in the phasor domain is shown in Fig. 105.5(b) and in the time domain in Fig.105.5(c).

Figure 105.5 Voltage-current relationships of a capacitor.

Sinusoidal ResponsesWe can conclude from the previous discussions that the phasor relationship between the voltageand the current of an element takes the general form of

V = ZI (105:27)

where Z is the impedance of the circuit element. That is, the impedance of a resistor is R, theimpedance of an inductor is j!L , and the impedance of a capacitor is 1=j!C . Thus, for a circuithaving L and/or C, the impedance Z will be a complex number with a real part R and an imaginary

The phasor representation of the current given by Eq. (105.25) is


part X such that

Z = R + jX = Z 6 (105:28)

where

Z = [R2 +X2 ]1=2 (105:29)

and

= tan1 (X=R) (105:30)

If the Kirchhoff's voltage law for a set of n sinusoidal voltages in the time domain is givenby

v1 + v2 + v3 + v4 + + vn = 0 (105:31)

then the equivalent statement in the phasor domain can be written as

V1 + V2 + V3 + V4 + + Vn = 0 (105:32)

Similarly, if the Kirchhoff's current law for a set of n sinusoidal currents in the time domain isgiven by

i1 + i2 + i3 + i4 + + in = 0 (105:33)

then the equivalent statement in the phasor domain can be written as

I1 + I2 + I3 + I4 + + In = 0 (105:34)

Let us consider the RLC circuit shown in Fig. 105.6. We will use the phasor concept in findingits current in response to an input source voltage of vs = Vm sin(!t + ) . We use the phasornotation

Vs = Vm 6 (105:35)

Applying the Kirchhoff's voltage law, we get

Vs = (R + j!L + 1=j!C)I = ZI (105:36)

where Z is the total impedance of the series loop formed by R, L, and C. That is,


Z = R + j(!L 1=!C) = [R2 + (!L 1=!C)2]1=2 6 = Z 6 (105:37)

where Z is the impedance magnitude and is given by

Z = [R2 + (!L 1=!C)2]1=2 (105:38)

and is the impedance angle given by

= tan1!L 1=!C

R

(105:39)

Dividing the phasor voltage by the phasor impedance gives the phasor currentas

I =VsZ

=Vs 6 Z 6

=Vs 6

Z(105:40)

which indicates that the current lags the input voltage by an angle of . Converting to the timedomain, the current is given by

i =Vmp

R2 + (!L 1=!C )2sin(!t+ ) (105:41)

Figure 105.6 A series RLC circuit.

Let us consider the parallel RLC circuit shown in Fig. 105.7. Applying the Kirchhoff's current


law at the node a, we get

I = I1 + I2 + I3 (105:42)

Since the three impedances are: ZR = R;Z1 = R1 + j!L , and ZC = 1=j!C , we can substitutefor I by using Ohm's law. That is,

I =Vs 6 R

+Vs 6

R1 + j!L+

Vs 6 1=j!C

I = Vs 6

"1

R+

1pR21 + (!L)

26 1 + j!C

#(105:43)

where 1 is the impedance angle of R1 and L, and it is given by

1 = tan1 !L

R1(105:44)

Equation (105.42) can be written as

I = Vs 6 Y 6 = VsY 6 ( ) (105:45)

where Y 6 is the equivalent admittance of R, R1 , and L such that it is related to the equivalentimpedance Z by

Y 6 = 1Z 6

=1

R+

1

R1 + j!L+ j(!C) (105:46)

Equation (105.43) can be written in the time domain as

i(t) = VmY sin(!t + ) (105:47)

where is the impedance angle of (R1 + j!L) , which is in parallel with R and 1=j!C . Inconverting from the phasor domain to the time domain, we have assumed that the maximumamplitude of a sinusoid is the magnitude of its phasor representation. However, in practice, themagnitude of a sinusoid is normally quoted in its root-mean-square (rms) value, which is definedby

Vrms =

s1

T

ZV 2m cos

2(!t + )dt (105:48)

which, after completing the integration and simplification, becomes


Vrms =Vmp2

(105:49)

Thus, v = Vm sin (!t + ) =p2Vrms sin(!t + ) , and it will be represented in a phasor form by

Vrms 6 . For example, 170 sin(!t+ ) (170=p2) 6 = 120 6 .

Figure 105.7 A parallel RLC circuit.


Patented May 1, 1888#381,968

Tesla was an electrical genius from Serbo-Croatia. He is credited with over 700 inventions; thisone perhaps his most important. He invented what is now known as the induction motor. It ranfrom alternating current (AC) and had no commutating brushes to burn up like the direct current(DC) motors used up to that time. An AC induction motor was inherently synchronous andtherefore ran at a constant speed regardless of load; a great advantage in industrialapplications.

Westinghouse bought the rights to this invention in July 1888. Through the 1890's, the inductionmotor's application to industry helped force the eventual world-wide domination by AC powersystems which up to that time had been DC systems like those promoted by Edison. ( 1995,DewRay Products, Inc. Used with permission.)

Defining TermsImpedance: An impedance is a measure of the opposition to a current flow due to a sinusoidal

voltage source. It is a complex number and has a real and an imaginarypart.

Phasor: A phasor is the vector representation of a sinusoid in the complex domain. It representsthe amplitude and the phase angle of a sinusoid.

Sinusoid: A sinusoid is a sinusoidal time-dependent periodic forcing function which has amaximum amplitude and can also have a phase delay.

ReferencesBalabania, N. 1994. Electric Circuits. McGraw-Hill, New York.Dorf, R. C. 1993. Introduction to Electric Circuits. John Wiley & Sons, New York, NY.Hayt, W. H., Jr., and Kemmerley, J. E. 1993. Engineering Circuit Analysis. McGraw-Hill, New

York.Irwin, D. J. 1989. Basic Engineering Circuit Analysis. Macmillan, New York.Jackson, H. W. 1986. Introduction to Electric Circuits. Prentice Hall, Englewood Cliffs,

NJ.Johnson, D. E., Hilbert, J. L., and Johnson, J. R. 1989. Basic Electric Circuit Analysis. Prentice

Hall, Englewood Cliffs, NJ.Nilson, J. W. 1993. Electric Circuits. Addison-Wesley, Reading, MA.

Further InformationA general review of linear circuits is presented in Linear Bircuits by M. E. Van Valkenburh,

Prentice Hall (Chapters 9-11).The applications of phasors in determining frequency responses are presented in Basic Network

Theory by J. Vlach, Van Nostrand Reinhold (Chapter 8).

ELECTRO-MAGNETIC MOTORNikola Tesla


Many examples of computer-aided simulations by the Industry Standard Circuit Simulator SPICEare given in SPICE for Circuits and Electronics Using PSpice by M. H. Rashid, PrenticeHall.

The monthly journal IEEE Transactions on Circuits and SystemsFundamental Theory andApplications reports advances in the techniques for analyzing electrical and electroniccircuits. For subscription information contact IEEE Service Center, 445 Hoes Lane, P.O. Box1331, Piscataway, NJ 08855-1331. Phone (800) 678-IEEE.


The Engineering HandbookContentsSinusoidal Excitation and Phasors105.1 Sinusoidal Source105.2 Phasor105.3 Passive Circuit Elements in Phasors DomainResistorInductorCapacitorSinusoidal Responses

Defining TermsReferencesFurther Information

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