Electrical Engineering is a free introductory textbook to the basics of electrical engineering. See the editorial for more information....

A More Convenient Notation

Author: E.E. Kimberly

The terms horizontal component and vertical component become burdensome. Also in some cases the axis of reference is neither vertical nor horizontal as seen by the observer. In mathematics, vectors are frequently expressed in what is called complex notation and the vector is called a complex quantity or complex number.

The quantity is not really complex in the usual sense, but it is called complex because it is composed of two parts corresponding to the horizontal component and the vertical component or, as they are sometimes called, the in-phase component and the quadrature component. The general mathematical form used is a+jb, in which a is the in-phase component and b is the quadrature component. These in-phase and quadrature components by definition correspond to the horizontal and vertical components, respectively, as previously used. The prefix j indicates that b is a component to be added to a at an angle of 90° counter-clockwise. The significance of j is illustrated in Figs. 4-5 and 4-6.

Fig. 4-5. Vector Addition a+b
Fig. 4-6. Vector Addition a+jb

The prefix j is usually defined as an operator which rotates counterclockwise through 90° any vector by which it is multiplied. The operator j is the same as the mathematician's imaginary number i which is . A function expressed in complex form is usually designated by a dot above it; thus, E = Ea+jEb. However, a generally accepted negligence has resulted in omission of the dot where such omission will not result in confusion.

Example 4-3. - Solve Example 4-2 by using complex notation.

Solution. - The given vectors are expressed as follows:


Last Update: 2010-10-06