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Complex Numbers

Complex numbers are used in alternating current theory and mechanical vector analysis.

There are two main forms of complex numbers

  • Cartesian
  • Polar

Complex numbers on the Cartesian form

A complex number consists of a real part and an imaginary part and can be expressed on the Cartesian form as

Z = a + jb                         (1)

where

Z = complex number

a = real part

jb = imaginary part (it is common to use i instead of j)

A complex number can be represented in a Cartesian axis diagram with an real and an imaginary axis - also called the Argand diagram:

Complex numbers - cartesian argand form example

Example - Complex numbers on the Cartesian form

The complex numbers

ZA = 3 + j2                              (2a)

ZB = -3 + j3                             (2b)

ZC = -2 - j2                             (2c)

can be represented in the Argand diagram:

Complex numbers -  cartesian argand form example

Addition and Subtraction of Complex numbers

Complex numbers are added/subtracted by adding/subtracting the separately the real parts and the imaginary parts of the number.

Example - Adding two Complex numbers

ZA = 3 + j2    

ZB = -3 + j3  

Z(A+B) = (3 + (-3)) + (j2 + j3)

         = j5

complex number cartesian agrand form adding

Complex numbers on the Polar form

A complex number on the polar form can be expressed as

Z = r (cos(θ) + jsin(θ))                   (3)

where

r = modulus (or magnitude) of Z - and is written as "mod Z" or |Z|

θ = argument (or amplitude) of Z - and is written as "arg Z"

Complex numbers - polar form

r can be determined using Pythagoras' theorem

r = (a2 + b2)1/2                          (4)

θ can be determined by trigonometry

θ = tan-1(b / a)                          (5) 

(3) can also be expressed as

Z = r e                    (6)

As we can se from (1), (3) and (6) - a complex number can be written in three different ways.

Example - Complex number on the Polar form

The complex number

Za = 3 + j2    

can be expressed on the polar form by calculating the modulus and the argument.

The "modulus" can be calculated by using eq. (4):

r = (32 + 22)1/2  

  =  3.606

The "argument" can be calculated by using eq. (5):

θ = tan-1(2 / 3)

   = 33.69o

The complex number on polar form (3):

Za = 3.606 (cos (33.69) + j sin( 33.69)) 

Or alternatively (6)

Za = 3.606 ej 33.69 

Adding or Subtraction of Complex Numbers

Adding Complex Numbers

Za = a + jb

Zb = c + jd

Za + Zb = (a + jb) + (c + jd)   

   = (a + c) + j(b + d)                            (6)

or alternative

Za = ra (cos(θa) + jsin(θa))

Zb = rb (cos(θb) + jsin(θb))

Za + Zb = ra (cos(θa) + jsin(θa) + rb (cos(θb) + jsin(θb))

   = (ra cos(θa) + rb cos(θb)) + j(ra sin(θa) +  rb sin(θb))                            (6b)

or alternatively

Za = ra ejθa     

Zb =  rb ejθb  

Za + Zb = ra ejθa + rb ejθb  

   = (ra cos(θa) + rb cos(θb)) + j(ra sin(θa) +  rb sin(θb))                            (6c)

Example - Adding Complex Numbers

Za = 3 + j2

Zb = 5 - j4

Za + Zb = (3 + j2) + (5 - j4)

  = (3 + 5) + j(2 + (-4))

  = 8 - j2

Example - Adding Complex Numbers

Za = 3 (cos(35) + jsin(35))

Zb = 2 (cos(15) + jsin(15))

Za + Zb = (3 cos(35)+ 2 cos(15)) + j(3 sin(35) +  2 sin(15))

  = 4.38 - j2.2 

Subtracting Complex Numbers

Za = a + jb

Zb = c + jd

Za - Zb = (a + jb) - (c + jd)

    = (a - c) + j(b - d)                           (7)

Example - Subtracting Complex Numbers

Za = 3 (cos(35) + jsin(35))

Zb = 2 (cos(15) + jsin(15))

Za - Zb = 3 (cos(35) + jsin(35)) - 2 (cos(15) + jsin(15))

     = (3 cos(35) - 2 cos(15)) + j(3 sin(35) - 2 sin(15))

     =  0.52 + j1.2

Multiplication of Complex Numbers

Za = a + jb

Zb = c + jd

Za Zb = (a + jb) (c + jd)

   = a c + a (jd) + (jb) c + (jb) (jd)

   = a c + j(a d) + j(b c) + j2(b d)                           (8)

Since j2 = -1  - (8) can be transformed to 

Za Zb = (a + jb) (c + jd)

    = (a c - b d) + j(a d + b c)                          (8b)

Example - Multiplying Complex Numbers

Za = 3 + j2

Zb = 5 - j4

Za Zb = (3 + j2) (5 - j4)

  = (3 5 - 2 (-4)) + j(3 (-4) + 2 5)

  = 23 - j2

Complex Conjugate

The complex conjugate of (a + jb) is (a - jb).

Multiplying a complex number with its complex conjugate results in a real number like

Za = a + jb

Za* = a - jb

Za Za* = (a + jb) (a - jb)

   = a2 - j(a b) + j(a b) - j2b2

   = a2 - (- b2)

   = a2 + b2                                 (9)

Example - Multiplying a Complex Number with its Conjugate

Za = 3 + j2

Za* = 3 - j2

Za Za* = (3 + j2) (3 - j2)

  = 32 + 22

  = 13

Division of Complex Numbers

Division of complex numbers can be done with the help of the denominators conjugate:

Za = a + jb

Zb = c + jd

Za / Zb = (a + jb) / (c + jd)

= ((a + jb) / (c + jd)) ((c - jd) / (c - jd))

= (a c +  j(a d) + j(b c) + j2(b d)) / (c2 + d2)                                  (10)

Multiplying both the nominator and the denominator with the conjugate of the denominator is called rationalizing.

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