458 AC POWER SYSTEMS
in whichθPis the phase angle between the voltage and the current of any particular phase, and
cosθPis the power factor.
Under most normal operating conditions, the various components of the three-phase system
are characterized by completephase symmetry. If such phase symmetry is assured throughout
the power system, it is desirable to simplify the analytical efforts to a great extent by the use of
per-phase analysis. Also recall that three-phase systems are most often represented bysingle-line
(one-line) diagrams.
Theenergy Eassociated with the instantaneous power over a period of timeTseconds is
given by
E=PT (10.2.7)
whereEis the energy in joules that is transferred during the interval, andPis the average value
of the real-power component in joules per second. Note that the reactive-power component does
not contribute to the energy that is dissipated in the load. The energy associated with the reactive
power component is transferred between the electric fields (which result from the application of
the sinusoidal voltage between the phase conductors and ground) and the magnetic fields (which
result from the flow of sinusoidal current through the phase conductors).
Many industrial loads have lagging power factors. Electric utilities may assess penalties for
the delivery of reactive power when the power factor of the customer’s load is below a minimum
level, such as 90%. Capacitors are often used in conjunction with such loads for the purpose
ofpower factor correction or improvement. An appropriate capacitor connected in parallel with
an inductive load cancels out the reactive power and the combined load may have unity power
factor, thereby minimizing the current drawn from the source. For problems and examples on this
subject, one may also refer to Sections 3.1 and 4.2.EXAMPLE 10.2.1A 60-Hz, three-phase motor draws 25 kVA at 0.707 lagging power factor from a 220-V source.
It is desired to improve the power factor to 0.9 lagging by connecting a capacitor bank across the
terminals of the motor.(a) Determine the line current before and after the addition of the capacitor bank.
(b) Specify the required kVA (kVAR) rating of the capacitor bank. Also sketch a power
triangle depicting the power factor correction by using capacitors.
(c) If the motor and the capacitor bank are wye-connected in parallel, find the capacitance
per phase of the capacitor bank assuming that it is balanced.
(d) How would the result of part (c) change if the motor and the capacitor bank were to be
delta-connected in parallel?SolutionThe real and reactive powers of the load (motor) are:
PM= 25 × 0. 707 = 17 .68 kW
QM=25 sin(cos−^10. 707 )= 17 .68 kVAR