Complex Numbers

1. Complex Numbers

2. Definition • A complex numberz 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.

3. 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-yplane, or as the vector from the origin to P(x,y). Complex Plane The complex plane x-y plane is also known as the complex plane.

4. Note that : 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|.

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,

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 θthat lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.

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 :

8. In mathematics terms, q is referred to as the argument of z and it can be positive or negative. In engineering terms, q 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.

9. Im z1 x r1 +q1 -q2 Re r2 x z2

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

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

12. Other Examples What about z1=0+j, z2=0-j, z3=2+j0, z4=-2?

13. Im z1 = + j ● z4 = -2 z3 = 2 ● ● Re ● z2 = - j

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.

15. Suppose we have 2 complex numbers, z1 and z2 given by : Easier with normal form than polar form Easier with polar form than normal form magnitudes multiply! phases add!

16. For a complex number z2 ≠ 0, phases subtract! magnitudes divide!

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

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

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

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 -900!

21. Polar Plot of H(s) showing the magnitude and phase of H(s)

22. Frequency response of the system Alternate view of the magnitude and phase of H(s)