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Electric Circuits, 9th Edition P76

Electric Circuits, 9th Edition P76. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 726 Complex Numbers A complex number in polar form can be put in rectangular form by writing ceje c cos0 jsin0 c cos 0 jc sin 0 a jb. B.5 The transition from rectangular to polar form makes use of the geometry of the right triangle namely. ce B.6 where tan0 b a. B.7 It is not obvious from Eq. B.7 in which quadrant the angle 6 lies. The ambiguity can be resolved by a graphical representation of the complex number. Figure B.l The graphical representation of jb when a and b are both positive. 4 3 5 36.87 a 4 3 5 143.13 b 5 21 l87o -4- 3 5 216X7 c d Figure B.2 The graphical representation of four complex numbers. B.2 The Graphical Representation of a Complex Number A complex number is represented graphically on a complex-number plane which uses the horizontal axis for plotting the real component and the vertical axis for plotting the imaginary component. The angle of the complex number is measured counterclockwise from the positive real axis. The graphical plot of the complex number n a jb c 0 if we assume that a and b are both positive is shown in Fig. B.l. This plot makes very clear the relationship between the rectangular and polar forms. Any point in the complex-number plane is uniquely defined by giving either its distance from each axis that is a and b or its radial distance from the origin c and the angle of the radial measurement 6. It follows from Fig. B.l that 0 is in the first quadrant when a and b are both positive in the second quadrant when a is negative and b is positive in the third quadrant when a and b are both negative and in the fourth quadrant when a is positive and b is negative. These observations are illustrated in Fig. B.2 where we have plotted 4 3 -4 3 -4 - 3 and 4 - 3. Note that we can also specify 6 as a clockwise angle from the positive real axis. Thus in Fig. B.2 c we could also designate 4 - 3 as 5 -143.13 . In Fig. B.2 d we observe that 5 323 13 5 -36.87 . It is customary to express 0 in terms of negative values when 6 lies in the third