1. Complex numbers
i or j

2. An Imaginary Number, when squared, gives a negative result.
Complex numbers
An Imaginary Number, when squared, gives a negative result.
imaginary2 = negative

3. i = √-1 Complex numbers i is used in maths But
j is used in electronics and engineering (because i is already used as a symbol for current)

4. Complex numbers
i = √-1 i2 = -1 i3 = -√-1 i4 = 1 i5 = √-1

5. Complex numbers
Example What is i6 ?
i6 = i4 × i2
= 1 × -1
= -1

6. Adding complex numbers
(4 +j3) + (3 + j5)
4 +j j5
7 + j8

7. Adding complex numbers
(3 +j6) + (2 – j3)
3 +j6 + 2 – j3
5 + j3

8. Subtracting complex numbers
(6 +j8) - (2 + j3)
6 +j8 - 2 – j3
4 + j5
Note the change of sign

9. Multiplying complex numbers
Example 1,
6(3 +j4) =
18 + j24
Example 2
j8 + 3(3 – j2) =
j8 + 9 – j6
j2 + 9

10. Multiplying complex numbers
(3 + j2)(4 + j)
Use F.O.I.L.
(3x4) + (3xj) + (j2 x4) + (j2 x j)
12 + j3 + j8 + j22
j2 = -1
12 +j11 – 2
10 + j11

11. Multiplying complex numbers
(5 - j2)(2 + j2)
Use F.O.I.L.
(5 x 2) + (5 x j2) - (j2 x2) - (-j2 x j2)
10 + j10 – j4 - j24
j2 = -1
10 – j6 + 4
14 - j10

12. Multiplying complex numbers
(4 - j2)(3 - j)
Use F.O.I.L.
(4 x 3) - (4 x j) - (j2 x3) + (j2 x j)
12 – j4 – j6 + j22
j2 = -1
12 - j10 – 2
10 - j10

13. Multiplying a conjugate pair
(4 - j2)(4 + j2)
Use F.O.I.L.
(4 x 4) + (4 x j2) - (j2 x 4) - (j2 x j2)
16 + j8 – j8 - j24
j2 = -1
16 + 4 20

14. Dividing complex numbers
(2 +6j)/2j = 2/2j + 6j/2j = 1/j +3 J-1 + 3

15. Dividing complex numbers
(6 + j3)/ (3+j2)
Multiply by the conjugate of the denominator
(6 + j3) x (3 – j2)
(3 + j2) (3 - j2)
18 – j12 +j9 –j26
9 – j6 + j6 –j24

16. Dividing complex numbers
18 - j – j24= 24 – j

17. Argand Diagrams
Imaginary axis y
Real axis x
Z = x +yj

18. Argand Diagrams r
r = √(x2 + y2)

19. Argand Diagrams
tanΦ = y/x Φ

20. Argand Diagrams x = r cosΦ r yj = r sinΦ Z = x +yj Φ Imaginary axis y
Real axis x
Z = x +yj
x = r cosΦ r
yj = r sinΦ Φ

21. Argand diagrams are used to calculate impedance in RLC circuits
Example
Argand diagrams are used to calculate impedance in RLC circuits

22. Example
The impedance of a circuit is given by the complex number 3 +j4 Construct the Argand diagram for 3 +j4

23. Example
Imaginary axis y
Real axis x
Z = 3 +j4 3 r j4

24. Example
From the Argand diagram derive the expression for the impedance in polar form

25. Example r = √(32 + 42) = √(9 + 16) √(25) = 5 r Z = 3 +j4 3
Imaginary axis y
Real axis x
Z = 3 +j4 3 r j4

26. Example r tanΦ = 4/3 Φ = 53.13 Z = 3 +j4 3 Imaginary axis y j4
Real axis x
Z = 3 +j4 3 r j4

27. Example r Answer Z = 5 53.13 Z = 3 +j4 3 Imaginary axis y j4
Real axis x
Z = 3 +j4 3 r j4

28. Multiplying and dividing polar form
6∟20° x 4∟30°
Multiply the length (modulus) and add the argument (angle)
= 24∟50°
9∟10° / 3∟40° = 9/3 ∟(10°-40°)
divide the length (modulus) and subtract the argument (angle)
= 3∟-30°

29. Argand diagrams as phasor diagrams
The voltage of a circuit is given as V = 3 + j3 and the current drawn is given as I = 8 + j2 Find the phase difference between V and I Find the power (VI.cosФ)

30. Argand diagrams as phasor diagrams
Voltage = √ ( ) = √18 = 4.24 Volts
Current = √( ) = √ 68
= amps

31. Argand diagrams as phasor diagrams
Voltage phase angle tanΦ = 3/3 =1,
Φ = 45o
Current phase angle tanΦ = 2/8 =0.25,
Φ = 14.0o

32. Argand diagrams as phasor diagrams
Phase difference between V and I = 45o o = 31o power = VIcosΦ 4.24 x 8.25 cos31o 4.24 x 8.25 x .86 = 30 watts