112 TIME-DEPENDENT CIRCUIT ANALYSIS
analysis when a given voltage or current phasor is taken as the reference. Since a phasor diagram is
a frozen picture at one instant of time, giving the relative locations of various phasors involved, and
the whole diagram of phasors is assumed to be rotating counterclockwise at a constant frequency,
different phasors may be made to be the reference simply by rotating the entire phasor diagram
in either the clockwise or the counterclockwise direction.
From the viewpoints of ease and convenience, it would be a matter of common sense to
choose the current phasor as the reference in the case of series circuits, in which all the elements
are connected in series, and the common quantity for all the elements involved is the current.
Similarly for the case of parallel circuits, in which all the elements are connected in parallel
and the common quantity for all the elements involved is the voltage, a good choice for the
reference is the voltage phasor. As for the series–parallel circuits, no firm rule applies to all
situations.
Referring to Figure 3.1.1(a), in a purely resistive circuit, notice that the current is in phase
with the voltage. Observe the waveforms ofiRandvRin the time domain and their relative
locations in the phasor domain, notice the phasorsV ̄Rchosen here as a reference, andI ̄R, which
is in phase withV ̄R. The phase angle between voltage and current is zero degrees, and the cosine
of that angle, namely, unity or 1 in this case, is known as thepower factor. Thus a purely resistive
circuit is said to have unity power factor.
In the case of a pure inductor, as shown in Figure 3.1.1(b), the currentlags(behind) the
voltage by 90°, or one can also say that the voltageleadsthe current by 90°. Observe the current
and voltage waveforms in the time domain along with their relative positions, as well as the
relative phasor locations ofV ̄LandI ̄Lin the phasor domain, withV ̄Lchosen here as a reference.
The phase angle between voltage and current is 90°, and the cosine of that angle being zero, the
power factor for the case of a pure inductor is said to be zero power factor lagging, since the
current lags the voltage by 90°.
For the case of a pure capacitor, as illustrated in Figure 3.1.1(c), the current leads the voltage
by 90°, or one can also say that the voltage lags the curent by 90°. Notice the current and voltage
waveforms in the time domain along with their relative positions, as well as the relative phasor
locations ofV ̄CandI ̄Cin the phasor domain, withV ̄Cchosen here as a reference. The phase angle
between voltage and current is 90°, and the cosine of that angle being zero, the power factor for
the case of a pure capacitor is said to be zero power factor leading, since the current leads the
voltage by 90°.
The circuit analysis techniques presented in Chapter 2 (where only resistive networks are
considered for the sake of simplicity) apply to ac circuits using the phasor method. However,
the constant voltages and currents in dc circuits are replaced by phasor voltages and currents
in ac circuits. Similarly, resistances and conductances are replaced by the complex quantities
for impedance and admittance. Nodal and mesh analyses, being well-organized and systematic
methods, are applied to ac circuits along with the concepts of equivalent circuits, superposition,
and wye–delta transformation.Power and Power Factor in ac Circuits
Power is the rate of change of energy with respect to time. The unit of power is a watt (W),
which is a joule per second (J/s). The use of rms or effective values of voltage and current allows
the average power to be found from phasor quantities. Let us consider a circuit consisting of
an impedanceZ φ=R+jXexcited by an applied voltage ofv(t)=√
2 Vrmscos(ωt+φ),
producing a current ofi(t)=√
2 Irmscosωt. The corresponding voltage and current phasors are
then given byV φandI0°, which satisfy the Ohm’s-law relationV/ ̄ I ̄=V φ/I 0°=Z φ.