SINGLE PHASE AC CIRCUIT THEORY: PHASOR (CONTD.)

     Addition

To add two or more complex numbers, simply add the real and imaginary parts separately. For example, if

C₁ = ± A₁ ± j B₁      and C₂ = ± A₂ ± j B₂

Then            C₁ + C₂ = (± A₁ ± A₂) + j (±B₁ ± B₂)                        (3.1)

There is really no need to memorize the equation. Simply set one above the other and consider the real and imagi­nary parts separately, as shown in Solution (b) of Example 3.1.

Example 3.1:

a.      Add C₁ = 2 + j 4 and C₂ = 3 + j 1.

b.     Add C₁ = 3 + j 6 and C₂ = -6 + j 3.

Solutions:

a.      By Eq. (3.1),

C₁ + C₂ = (2 + 3) + j (4 + 1) = 5 + j 5

An alternate method is

2 + j 4

3 + j 1

5 +j 5

a.      By Eq. (3.1),

          C₁ + C₂ = (3 - 6) + j (6 + 3) = -3 + j 9

An alternate method is

3 + j 6

         -6 + j 3

  

        -3 +j 9

       Subtraction

In subtraction, the real and imaginary parts are again con­sidered separately. For example, if

C₁ = ± A₁ ± j B₁      and C₂ = ± A₂ ± j B₂

Then            C₁ - C₂ = [± A₁ – (± A₂)] + j [±B₁ – (± B₂)]     

Again, there is no need to memorize the equation if the alternate method of solution in Example 3.2 is em­ployed.

Example 3.2:

a.      Subtract C₂ = 1 + j 4 from C₁ = 4 + j 6.

b.     Subtract C₂ = -2 + j 5 from C₁ = 3 + j 3.

Solutions:

a.      By Eq. (3.2),

C₁ - C₂ = (4 - 1) + j (6 - 4) = 3 + j 2  

An alternate method is

4 + j 6

      - (1 + j 4)

3 +j 2

a.      By Eq. (3.2),

          C₁ - C₂ = [3 (- 2)] + j (3 - 5) = 5 - j 2

An alternate method is

3 + j 3

     - (-2 + j 5)

        

         5 - j 2