Power system fundamentals - Wind Power Integration: Connection and System Operational Aspects - page 31

components - resistance, inductance and capacitance - respond to AC. The
instantaneous voltage v resulting from an instantaneous cosinusoidal current i is
considered in each case, based on the reference directions shown in Figure 2.6.
Resistance (R)
We know from Ohm's Law that v ¼ Ri . Taking i as reference, we have

i ¼ I m cos w t
∴ v ¼ RI m cos w t

The voltage drop across the resistor is in phase with the current through it. The
corresponding phasor diagram is shown in Figure 2.7. Note that the complex phasor
quantities are shown as bold letters.
The active and reactive powers for the resistance are given by (2.14):

P ¼ VI cos f ¼ VI ¼ RI 2
Q ¼ VI sin f ¼ 0

Inductance (L)
We may obtain inductor voltage for a cosinusoidal current from (2.11):

v ¼ L d i
d t
i ¼ I m cos w t
∴ v ¼ w LI m cos ðw t þ p= 2 Þ

The corresponding phasor diagram is shown in Figure 2.8. The voltage phasor
leads the current phasor by 90 ( p /2 rad). This phase shift is achieved by use of the j
operator. j is itself a phasor, with a magnitude of 1 and an angle of 90 measured
anti-clockwise. If j is multiplied by itself, the product will be 1 with an angle of
180 p :

The impedance of the inductance is complex, and is given by Z ¼ j w L and the
magnitude of the inductive impedance is referred to as its reactance, given by
X ¼w L .

anti-clockwise, or 1. From this it is seen that j ¼

i

v

Figure 2.6 Reference directions for v and i

V = R I

I

Figure 2.7 Phasor diagram for resistance