Category: Complex Numbers

As with series circuits, parallel networks may be analyzed by using phasor diagrams. However, with parallel networks containing more than two branches, this can become very complicated. It is with parallel AC network analysis in particular that the full benefit of using complex numbers may be appreciated. The theory for parallel AC networks introduced previously…

Simple AC circuits may be analyzed by using phasor diagrams. However, when circuits become more complicated, analysis is considerably simplified by using complex numbers. It is essential that the basic operations used with complex numbers, as outlined in this chapter thus far, are thoroughly understood before proceeding with AC circuit analysis. 7.5.1 Series AC Circuits…

Figure 7.3 Polar form of complex numbers This latter form is usually abbreviated to Z=r∠θ, and is called the polar form of a complex number. r is called the modulus (or magnitude of Z) and is written as mod Z or |Z|. r is determined from Pythagoras’s theorem on triangle OAZ: The modulus is represented on the Argand diagram by the distance OZ. θ is…

If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. Hence, if a+jb=c+jd, then a=c and b=d. This is a useful property, since equations having two unknown quantities can be solved from one equation. Complex equations are used when deriving balance equations with AC bridges. Example 7.3 Solve the following complex…

(a) Addition and subtraction and  Thus,  and  (b) Multiplication But j2=–1, thus, For example, (c) Complex conjugate The complex conjugate of (a+jb) is (a –jb). For example, the conjugate of (3 –j2) is (3+j2). The product of a complex number and its complex conjugate is always a real number, and this is an important property used when dividing complex numbers. Thus, For…

A complex number is of the form (a+jb) where a is a real number and jb is an imaginary number. Therefore, (1+j2) and (5 –j7) are examples of complex numbers. By definition,  and j2=–1 (Note: In electrical engineering, the letter j is used to represent  instead of the letter i, as commonly used in pure mathematics, because i is reserved for current.) Complex numbers are widely used in the analysis…