ZB = -3 + j 3
Z(A+B) = (3 + (-3)) + (j 2 + j 3)
= j 5
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.
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
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 + Zb = ra (cosθa + j sinθa) + rb (cosθb + j sinθb)
= (ra cosθa + rb cosθb) + j (ra sinθa + rb sinθb) (6b)
or alternatively
Za = ra ejθa
Zb = rb ejθb
Za + Zb = ra ejθa + rb ejθb
= (ra cosθa + rb cosθb) + j (ra sinθa + rb sinθb) (6c)
Za = 3 + j 2
Zb = 5 - j 4
Za + Zb = (3 + j 2) + (5 - j 4)
= (3 + 5) + j(2 + (-4))
= 8 - j 2
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
Za = a + j b
Zb = c + j d
Za - Zb = (a + j b) - (c + j d)
= (a - c) + j(b - d) (7)
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
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)
Za = 3 + j 2
Zb = 5 - j 4
Za Zb = (3 + j 2) (5 - j 4)
= (3 5 - 2 (-4)) + j(3 (-4) + 2 5)
= 23 - j 2
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)
Za = 3 + j 2
Za* = 3 - j 2
Za Za* = (3 + j 2) (3 - j 2)
= 32 + 22
= 13
Division of complex numbers can be done with the help of the denominators conjugate:
Za = a + jb
Zb = c + 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.
Mathematical rules and laws - numbers, areas, volumes, exponents, trigonometric functions and more.
Real, imaginary and apparent power in AC circuits.
Principal algebraic expressions formulas.
Prefix names used for multiples and submultiples units.
Designation of large number in US vs. other countries.
Derivatives and differentiation expressions.
Law of fractions
Exponential functions related to the hyperbola.
Calculate square, cube, square root and cubic root. Values tabulated for numbers ranging 1 to 100.
Convert between Cartesian and Polar coordinates.
Radian is the SI unit of angle. Convert between degrees and radians. Calculate angular velocity.
Roman numerals are a combinations of seven letters.
Triangle analytical geometry.
Sine, cosine and tangent - the natural trigonometric functions.
Online vector calculator - add vectors with different magnitude and direction - like forces, velocities and more.
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!
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.
If you want to promote your products or services in the Engineering ToolBox - please use Google Adwords. You can target the Engineering ToolBox by using AdWords Managed Placements.