Complex Numbers Definition A complex number z is a number of the form x jy where j 1 x is the real part and y the imaginary part, written as x = Re z, = Im z. y j is called the imaginary unit If x = 0, then z = jy is a pure imaginary number. The complex conjugate of a complex number, z = x + jy, denoted by z* , is given by z* = x jy. Two complex numbers are equal if and only if their real parts are e qual and their imaginary parts are equal. 1 August 2006 Slide 2 Complex Plane

A complex number can be plotted on a plane with two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis y P z = x + iy O The complex plane 1 August 2006 Represent z = x + jy geometrica lly as the point P(x,y) in the x-y plane, or fro as the vector OP to P(x,y). m the origin x

x-y plane is also known as the complex plane. Slide 3 Polar Coordinates With x r cos , y r sin z takes the polar form: z r (cos j sin ) r is called the absolute value or modulus or magnitude of z and is denoted by |z|. z r x 2 y 2 zz* Note that : * zz ( x jy )( x jy ) 2 x y 1 August 2006

2 Slide 4 Im P y z = x + iy r | = z | x O Re

Complex plane, polar form of a complex number Geometrically, |z| is the distance of the point z from the origin while is the directed angle from the positive xaxis to OP in the above figure. From the figure, 1 August 2006 tan 1 y x Slide 5 is called the argument of z and is denoted by arg z. Thus, y arg z tan x 1

z 0 For z = 0, is undefined. A complex number z 0 has infinitely many possible argum ents, each one differing from the rest by some multiple of 2 . In fact, arg z is actually tan 1 y 2n , n 0,1,2,... x The value of that lies in the interval (-, ] is called the pr inciple argument of z ( 0) and is denoted by Arg z. 1 August 2006 Slide 6 Euler Formula an alternate polar form The polar form of a complex number can be rewritten as :

z r (cos j sin ) x jy re j This leads to the complex exponential function : e z e x jy e x e jy x e cos y j sin y 1 j cos e e j 2 1 j sin e e j 2j Further leads to : 1 August 2006

Slide 7 In mathematics terms, is referred to as the argument of z and it can be positive or negative. In engineering terms, is generally referred to as phase of z and it can be positive or negative. It is denoted as z The magnitude of z is the same both in Mathematics and engineering, although in engineering, there are also different interpretations depending on what physical system one is referring to. Magnitudes are always > 0. The application of complex numbers in engineering will be dealt with later. 1 August 2006 Slide 8 Im z1 x r1 z1 r1e j1

+ 1 -2 Re r2 z 2 r2 e j 2 x z2 1 August 2006 r1 , r2 , 1 , 2 0 Slide 9 Example 1 A complex number, z = 1 + j , has a magnitude | z | (12 12 ) 2 and argument :

1 z tan 2n 2n rad 1 4 1 Hence its principal argument is : Arg z / 4 rad Hence in polar form : j z 2 cos j sin 2e 4 4 4

1 August 2006 Slide 10 Example 2 A complex number, z = 1 - j , has a magnitude | z | (12 12 ) 2 and argument : 1 z tan 2n 2n rad 1 4 1 rad Hence its principal argument is : Arg z 4

Hence in polar form : z 2e j 4 2 cos j sin 4 4 In what way does the polar form help in manipulating complex numbers? 1 August 2006 Slide 11 Other Examples What about z1=0+j, z2=0-j, z3=2+j0, z4=-2?

z1 0 j1 1e j 0.5 10.5 z 3 2 j 0 j0 2e 20 1 August 2006 z 2 0 j1 1e j 0.5 1 0.5 z 4 2 j 0 2e j 2 Slide 12 Im z4 = -2

z1 = + j 0.5 1 August 2006 z3 = 2 Re z2 = - j Slide 13 Arithmetic Operations in Polar Form The representation of z by its real and imaginary parts is useful for addition and subtraction. For multiplication and division, representation by the polar form has apparent geometric

meaning. 1 August 2006 Slide 14 Suppose we have 2 complex numbers, z1 and z2 given by : z1 x1 jy1 r1e j1 z 2 x 2 jy 2 r2 e j 2 z1 z 2 x1 jy1 x 2 jy 2 x1 x 2 j y1 y 2 z1 z 2 r1e j1

r1 r2 e magnitudes multiply! 1 August 2006 r e 2 j 2 j (1 ( 2 )) Easier with normal form than polar form Easier with polar form than normal form phases add! Slide 15 For a complex number z2 0, j1

z1 r1e r1 j (1 ( 2 )) r1 j (1 2 ) j 2 e e z 2 r2 e r2 r2 magnitudes divide! z1 r1 z2 r2 1 August 2006 phases subtract! z 1 ( 2 ) 1 2 Slide 16 A common engineering problem involving complex numbers

Given the transfer function model : 20 H (s) s 1 Generally, this is a frequency response model if s is taken to be s j . In Engineering, you are often required to plot the frequency response with respect to the frequency, . 1 August 2006 Slide 17 20e j 0 For a start : H ( s 0) j 0 200 1e Lets calculate H(s) at s=j10. 20 H ( j10) j10 1 H ( j10)

20 20e j 0 101e 10 j tan 1 1 2 20 log10 2 dB 5.98 dB 101 H ( j10) 0 tan 1 (10) 1.47 rad 84.30 Im 84.30 Re 2 x 2e i1.47 1 August 2006 Slide 18

Lets calculate H(s) at s=j1. 20 H ( j1) j1 1 H ( j1) 20 20e j 0 2e 1 j tan 1 1 14.142 20 log 10 14.142dB 23 dB 2 H ( j1) 0 tan 1 (1) 0.7854 rad 45 0 Im 84.30 45 0

Re 2 x H (i10) H (i1) x 1 August 2006 Slide 19 What happens when the frequency tends to infinity? H ( s ) s j 20 0 ? j 1 H ( s ) s j 20 20

j tan 1 j 1 e 0 90 0 When the frequency tends to infinity, H(s) tends to zero in magnitude and the phase tends to -900! 1 August 2006 Slide 20 Polar Plot of H(s) showing the magnitude and phase of H(s) 1 August 2006 Slide 21 Frequency response of the system Alternate view of the magnitude and phase of H(s) 1 August 2006 Slide 22