FORMS OF COMPLEX NUMBERS Machinery's Handbook, 31st Edition
60
Fig. 1. Complex coordinate system Fig. 2. Examples of standard (rectangular) form, z = a + bi z = 5 + 3 i lies in the first quadrant, with a and b both positive; z = –1 – i is in the third quad- rant. Complex numbers can be converted from the standard form to any of three vector forms : polar, trigonometric, and exponential. Polar form of a complex number: Vectors are objects that have both magnitude ( r ) and direction ( θ ). Vectors are essential in electrical engineering and other fields for repre - senting many processes such as alternating current and voltage. They are represented graphically by an arrow, as seen in Fig. 3.
z = r θ
Fig. 3. (a) Polar form of a complex number; (b) Magnitude r = | z | relationship to a and b ; (c) Angle θ to a, b .
The polar form of a complex number is z = r θ . z is a vector in the sense of its magnitude (length) r and direction angle θ from the horizontal (Fig. 3a). By Pythagorean theorem, r 2 = a 2 + b 2 ; thus, magnitude r a b = + 2 2 , which is called the modulus , is denoted | z |. From trigonometry, tan θ = b / a ; hence, θ = tan –1 ( b / a ). Trigonometric form of a complex number: Another form that shows the directional na- ture of complex numbers is the trigonometric form of z . From trigonometry, cos θ = a / r and sin θ = b / r ; hence, a = r cos θ , b = r sin θ , so, a + bi = r cos θ + ir sin θ = r ( a cos θ + i sin θ ) Exponential form of a complex number: Recall that the number e is the base of the natural logarithm (see page 37). A complex number is represented in exponential form through Euler’s formulas, which are used widely in electrical engineering applications:
i
i
i
i
θ
θ
θ
θ
−
−
e e +
e e i − 2
i
θ
,
,
e
i
= + θ cos
θ
θ
θ
=
=
sin
cos
sin
2
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online