0195136047.pdf

110 TIME-DEPENDENT CIRCUIT ANALYSIS

EXAMPLE 3.1.4

Letω= 2 π×60 rad/s corresponding to a frequency of 60 Hz.

(a) Considerv(t)= 100

√

2 cos(ωt+30°)V andi(t)= 10

√

2 sin(ωt+30°)A. Find the

corresponding phasorsV ̄andI ̄by choosing the rms values for the phasor magnitudes.

(b) Consider the phasorsV ̄andI ̄as obtained in part (a), and show how to obtain the time-

domain functions.

Solution

(a) v(t)= 100

√

2 cos(ωt+30°)V=Re

[

100

√

2 ej30°ejωt

]

Suppressing the explicit time variation and choosing the rms value for the phasor

magnitude, the phasor representation in the frequency domain then becomes

V ̄= 100 30°= 100 ej30°= 100 ejπ/^6 = 100 (cos30°+jsin30°)V

Similarly, for the current:

I ̄= 10  −60°= 10 e−j60°= 10 e−jπ/^3 = 10 (cos60°−jsin60°)A

since sin(ωt+30°)=cos(ωt+30°−90°)=cos(ωt−60°).

(b) Now thatV ̄ = 100 30°, the corresponding time-domain functionv(t) is obtained as

v(t)=Re [

√

2 Ve ̄ jωt]=Re [

√

2 100ej30°ejωt], orv(t)=Re [100

√

2 ej(ωt+30°)]=

100

√

2cos(ωt+30°)V. Note that the multiplicative factor of

√

2 is used to obtain the

peak amplitude, since for sinusoids the peak value is

√

2 times the rms value.

Similarly,i(t)=Re [10

√

2 ej(ωt−60°)] = 10

√

2cos(ωt−60°) = 10

√

2sin

(ωt+30°)A, whereω= 2 πf= 2 π×60 rad/s in this case.

For the three linear time-invariant passive elementsR(pure resistance),L(pure inductance),

andC(pure capacitance), the relationships between voltage and current in the time domain as

well as in the frequency domain are shown in Figure 3.1.1.

Phasors, being complex numbers, can be represented in the complex plane in the conventional

polar form as an arrow, having a length corresponding to the magnitude of the phasor, and an angle

(with respect to the positive real axis) that is the phase of the phasor. In aphasor diagram, the

various phasor quantities corresponding to a given network may be combined in such a way that

one or both of Kirchhoff’s laws are satisfied. In general the phasor method of analyzing circuits

is credited to Charles Proteus Steinmetz (1865–1923), a well-known electrical engineer with the

General Electric Company in the early part of the 20th century.

The phasor diagram may be drawn in a number of ways, such as in a polar (or ray) form,

with all phasors originating at the origin, or in a polygonal form, with one phasor located at the

end of another, or in a combination of these, depending on the convenience and the point to be

made. Such diagrams provide geometrical insight into the voltage and current relationships in a

network. They are particularly helpful in visualizing steady-state phenomena in the analysis of

networks with sinusoidal signals.