Introduction to Electric Circuits

84 Single-phase a.c. circuits

The phasor V3 has the same length as V2 but is 90 ~ ahead of it so that V3 = j V2.

But from equation (4.19) V2 - jV1, so

V3 -j(jV~) = jZv~ (4.20)

Now V3 has the same length as V1 and is in the opposite direction to it, which

means that V3 = -V~. It follows that

j .2 - -1 and j= V-1 (4.21)

Since it is impossible to take the square root of -1, it (V-l) is said to be
imaginary and the vertical or j-axis is often referred to as the imaginary axis. It
is also known as the quadrature axis. The horizontal or reference axis is also
called the real axis.
Now V 4 has the same length as V3 and is 90 ~ ahead of it so that V 4 = j V 3.
From equation (4.20) 1/3 = J 9 2 V 1, so

V 4 --j(j2V1) : j(-1)V, - -jV 1

and lies along the negative imaginary axis. Finally, applying the j operator to V4

shifts it through 90 ~ in an anticlockwise direction bringing it to the positive real

axis as Va. This is verified by noting that

jV4 -j(-jV1) - -j2V1 - -(-1)V1 - V1

Note also that

jV4 = j(jV3) -jj(jV2) -jjj(jV~) -j4v 1 = v 1

It follows that

j .4 = 1 (4.22)

The Argand diagram consists of four quadrants.

9 In the first quadrant, the real and imaginary axes are both positive and, as

shown in Fig. 4.22(b), the angle 4'1 takes values between 0 and 90 ~ from the

real positive direction.

9 In the second quadrant, the real axis is negative and the imaginary axis is

positive and phasors lying in this quadrant are between 90 ~ and 180 ~ from

the reference direction.

9 In the third quadrant, both the real and the imaginary axes are negative and

phasors are between 180 ~ and 270 ~ from the reference.

9 Finally, in the fourth quadrant the real axis is positive and the imaginary

axis is negative, the angles from the reference direction being between 270 ~

and 360 ~.

Note that in all quadrants the angles ((~)1, (])2, (~3, and 4}4) are obtained from

tan -I (imaginary component/real component).