Introduction to Electric Circuits

4.4 Complex notation 83

by the length of the phasor, while Z_~b~ indicates that it is ~b~ degrees ahead of
(leading) the reference direction. Similarly,

=

the minus sign indicating that V 2 is (])2 degrees behind (lagging) the reference
direction.
Note that V1 has two components at right angles as shown in Fig. 4.21(b).
These are V~r along the reference direction and Vlq at right angles to the
reference direction. Now Vlr -- V 1 COS (~1 and Vlq -- V 1 sin ~ba so that

V 1 -" Vlr -+- Vlq -- V 1 cos (])1 + V1 sin ~bl (a phasorial addition)

To indicate that V1 sin ~bl is at right angles to V1 cos 4h an operator j is
introduced which, when placed before a quantity simply indicates that that
quantity has been shifted through 90 ~ in an anticlockwise direction with respect
to any quantity which does not have a j in front of it. (Incidentally mathema-
ticians use the letter i placed after the quantity.)
Using this notation we have that
V~ - V1 cos 4~1 + jV1 sin ~1 -- VI[COS IJ)l + j sin ~bl]

and

V2 = V2 cos ~b2 - j V2 sin ~b2 = V2[cos ~b2 - j sin ~b2]

The minus sign in front of the j indicates that the quantity V2 sin ~b2 has been
shifted through 90 ~ in a clockwise direction.

The Argand diagram

In Fig. 4.22(a), which is called an Argand diagram, the phasor V~ is shown in the
positive reference direction. The phasor V2 has the same length as V1 but is 90 ~
ahead of it so, using the complex (or 'j') notation, we write

V2 = jV, (4.19)

Figure 4.22

V2 jVl

-- V l 0 V"--1

V4 -j%

(a)

V2

-,r....

_, (~2

V3

Vl

r

V4

(b)