ABSTRACT
A complex number is of the form (a+ jb) where a is a real number and jb is an imaginary number. Hence (1+ j2) and (5− j7) are examples of complex numbers.
By definition, j = √−1 and j2 = −1
Complex numbers are widely used in the analysis of series, parallel and series-parallel electrical networks supplied by alternating voltages (see Chapters 24 to 26), in deriving balance equations with a.c. bridges (see Chapter 27), in analysing a.c. circuits using Kirchhoff’s laws (Chapter 30), mesh and nodal analysis (Chapter 31), the superposition theorem (Chapter 32), with Thévenin’s and Norton’s theorems (Chapter 33) and with delta-star and star-delta transforms (Chapter 34) and in many other aspects of higher electrical engineering. The advantage of the use of complex numbers is that the manipulative processes become simply algebraic processes. A complex number can be represented pictorially on an
Argand diagram. In Figure 23.1, the line 0A represents the complex number (2+ j3), 0B represents (3− j), 0C represents (−2+ j2) and 0D represents (−4− j3). A complex number of the form a+ jb is called a
Cartesian or rectangular complex number. The significance of the j operator is shown in Figure 23.2. In Figure 23.2(a) the number 4 (i.e. 4+ j0) is shown drawn as a phasor horizontally to the right of the origin on the real axis. (Such a phasor could represent, for example, an alternating current, i= 4 sinωt amperes, when time t is zero.)
The number j4 (i.e. 0+ j4) is shown in Figure 23.2(b) drawn vertically upwards from the origin on the imaginary axis. Hence multiplying the number 4 by the operator j results in an anticlockwise phase-shift of 90◦ without
Figure 23.1 The Argand diagram
Multiplying j4 by j gives j24, i.e. −4, and is shown in Figure 23.2(c) as a phasor four units long on the horizontal real axis to the left of the origin — an anticlockwise phase-shift of 90◦ compared with the position shown in Figure 23.2(b). Thus multiplying by j2 reverses the original direction of a phasor.
