Complex Numbers

1 August 2006 Slide 2 Definition A complex number z is a number of the form where x is the real part and y the imaginary part, written as x = Re z, y = Im z. 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 equal and their imaginary parts are equal.

1 August 2006 Slide 3 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 Represent z = x + jy geometrically as the point P(x,y) in the x-y plane, or as the vector from the origin to P(x,y). The complex plane x-y plane is also known as the complex plane.

1 August 2006 Slide 4 Polar Coordinates With z takes the polar form: r is called the absolute value or modulus or magnitude of z and is denoted by | z |. Note that :

1 August 2006 Slide 5 Complex plane, polar form of a complex number Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x- axis to OP in the above figure. From the figure,

1 August 2006 Slide 6 θ is called the argument of z and is denoted by arg z. Thus, For z = 0, θ is undefined. A complex number z ≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually The value of θ that lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.

1 August 2006 Slide 7 Euler Formula – an alternate polar form The polar form of a complex number can be rewritten as : This leads to the complex exponential function : Further leads to :

1 August 2006 Slide 8 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 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 9 x +1+1 x z1z1 z2z2 Im Re -2-2 r1r1 r2r2

1 August 2006 Slide 10 A complex number, z = 1 + j, has a magnitude Example 1 and argument : Hence its principal argument is : rad Hence in polar form :

1 August 2006 Slide 11 A complex number, z = 1 - j, has a magnitude Example 2 and argument : Hence its principal argument is : rad Hence in polar form : In what way does the polar form help in manipulating complex numbers?

1 August 2006 Slide 12 What about z 1 =0+j, z 2 =0-j, z 3 =2+j0, z 4 =-2? Other Examples

1 August 2006 Slide 13 ● ● ● Im Re z 1 = + j z 2 = - j z 3 = 2z 4 = -2 ●

1 August 2006 Slide 14 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 15 Suppose we have 2 complex numbers, z 1 and z 2 given by : Easier with normal form than polar form Easier with polar form than normal form magnitudes multiply!phases add!

1 August 2006 Slide 16 For a complex number z 2 ≠ 0, magnitudes divide! phases subtract!

1 August 2006 Slide 17 Given the transfer function model : Generally, this is a frequency response model if s is taken to be. A common engineering problem involving complex numbers In Engineering, you are often required to plot the frequency response with respect to the frequency, .

1 August 2006 Slide 18 Let’s calculate H(s) at s=j10. x Re Im 2 For a start :

1 August 2006 Slide 19 Let’s calculate H(s) at s=j1. x Re Im 2 x

1 August 2006 Slide 20 What happens when the frequency tends to infinity? When the frequency tends to infinity, H(s) tends to zero in magnitude and the phase tends to !

1 August 2006 Slide 21 Polar Plot of H(s) showing the magnitude and phase of H(s)

1 August 2006 Slide 22 Frequency response of the system Alternate view of the magnitude and phase of H(s)