Electrical circuits lecture notes b. Tech

Vector Rotation 
 
 
So by multiplying an imaginary number by j
2
will rotate the vector by 180
o
anticlockwise, 
multiplying by j
3
rotates it 270
o
and by j
4
rotates it 360
o
or back to its original position. 
Multiplication by j
10
or by j
30
will cause the vector to rotate anticlockwise by the appropriate 
amount. In each successive rotation, the magnitude of the vector always remains the same. 
Complex and Polar forms of Representation: 
 
In Electrical Engineering there are different ways to represent a complex number either 
graphically or mathematically. One such way that uses the cosine and sine rule is called the 
Cartesian or Rectangular Form
. 
A complex number is represented by a real part and an imaginary part that takes the 
generalised form of: 
Z=x+jy 
Where 
Z - is the Complex Number representing the Vector 
x - is the Real part or the Active component 
y - is the Imaginary part or the Reactive component 
j - is defined by √-1 
 

---

In the rectangular form, a complex number can be represented as a point on a two-
dimensional plane called the complex or s-plane. So for example, Z = 6 + j4 represents a 
single point whose coordinates represent 6 on the horizontal real axis and 4 on the vertical 
imaginary axis as shown. 
Complex Numbers using the Complex or s-plane: 
Complex Numbers using Polar Form: 
 
Unlike rectangular form which plots points in the complex plane, the Polar Form of a 
complex number is written in terms of its magnitude and angle. Thus, a polar form vector is 
presented as: Z = A 
∠
±θ, where: Z is the complex number in polar form, A is the magnitude 
or modulo of the vector and θ is its angle or argument of A which can be either positive or 
negative. The magnitude and angle of the point still remains the same as for the rectangular 
form above, this time in polar form the location of the point is represented in a “triangular 
form” as shown below.