Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

This is an AMP page - Open full page! for all features.

Complex Numbers

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

Sponsored Links

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 + j b                         (1)

where

Z = complex number

a = real part

j b = 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:

Example - Complex numbers on the Cartesian form

The complex numbers

ZA = 3 + j 2                              (2a)

ZB = -3 + j 3                             (2b)

ZC = -2 - j 2                             (2c)

can be represented in the Argand diagram:

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 + j 2    

ZB = -3 + j 3  

Z(A+B) = (3 + (-3)) + (j 2 + j 3)

         = j 5

Complex numbers on the Polar form

A complex number on the polar form can be expressed as

Z = r (cosθ + j sinθ)                   (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"

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 ej θ                    (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 + j 2    

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 + j b

Zb = c + j d

Za + Zb = (a + j b) + (c + j d)   

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

or alternative

Za = ra (cosθa + j sinθa)

Zb = rb (cosθb + j sinθb)

Za + Zbra (cosθa + j sinθa) + rb (cosθb + j sinθb)

   = (ra cosθa + rb cosθb) + j (ra sinθarb sinθb)                            (6b)

or alternatively

Za = ra ejθa     

Zb rb ejθb  

Za + Zb = ra ejθa + rb ejθb  

   = (ra cosθa + rb cosθb) + j (ra sinθarb sinθb)                            (6c)

Example - Adding Complex Numbers

Za = 3 + j 2

Zb = 5 - j 4

Za + Zb = (3 + j 2) + (5 - j 4)

  = (3 + 5) + j(2 + (-4))

  = 8 - j 2

Example - Adding Complex Numbers

Za = 3 (cos 35 + j sin 35)

Zb = 2 (cos 15 + j sin 15)

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

  = 4.38 - j 2.2 

Subtracting Complex Numbers

Za = a + j b

Zb = c + j d

Za - Zb = (a + j b) - (c + j d)

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

Example - Subtracting Complex Numbers

Za = 3 (cos 35 + j sin 35)

Zb = 2 (cos 15 + j sin 15)

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

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

     =  0.52 + j 1.2

Multiplication of Complex Numbers

Za = a + j b

Zb = c + j d

Za Zb = (a + j b) (c + j d)

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

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

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

Za Zb = (a + j b) (c + j d)

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

Example - Multiplying Complex Numbers

Za3 + j 2

Zb5 - j 4

Za Zb = (3 + j 2) (5 - j 4)

  = (3 5 - 2 (-4)) + j(3 (-4) + 2 5)

  = 23 - j 2

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 - j2 b2

   = a2 - (- b2)

   = a2 + b2                                 (9)

Example - Multiplying a Complex Number with its Conjugate

Za3 + j 2

Za*3 - j 2

Za Za* = (3 + j 2) (3 - j 2)

  = 32 + 22

  = 13

Division of Complex Numbers

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

Za = a + jb

Zbc + j d

Za / Zb = (a + j b) / (c + j d)

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

= (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.

Sponsored Links

Related Topics

Related Documents

Sponsored Links

Share

Search Engineering ToolBox

  • the most efficient way to navigate the Engineering ToolBox!

SketchUp Extension - Online 3D modeling!

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the Sketchup Extension Warehouse!

Privacy

We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience.

Some of our calculators and applications let you save application data to your local computer. These applications will - due to browser restrictions - send data between your browser and our server. We don't save this data.

Google use cookies for serving our ads and handling visitor statistics. Please read Google Privacy & Terms for more information about how you can control adserving and the information collected.

AddThis use cookies for handling links to social media. Please read AddThis Privacy for more information.

Topics

Unit Converters

Temperature

oC
oF


Load Calculator!

Length

m
km
in
ft
yards
miles
naut miles


Load Calculator!

Area

m2
km2
in2
ft2
miles2
acres


Load Calculator!

Volume

m3
liters
in3
ft3
us gal


Load Calculator!

Weight

kgf
N
lbf


Load Calculator!

Velocity

m/s
km/h
ft/min
ft/s
mph
knots


Load Calculator!

Pressure

Pa (N/m2)
bar
mm H2O
kg/cm2
psi
inches H2O


Load Calculator!

Flow

m3/s
m3/h
US gpm
cfm


Load Calculator!

12 13

This website use cookies. By continuing to browse you are agreeing to our use of cookies! Learn more