Download presentation

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

Similar presentations

Presentation is loading. Please wait....

OK

© 2010 Pearson Education, Inc. All rights reserved

© 2010 Pearson Education, Inc. All rights reserved

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on team building activities Raster scan display and random scan display ppt on tv Ppt on bluetooth free download Ppt on business cycle phases in order Ppt on online banking system One act play ppt on dvd Ppt on obesity prevention initiative Ppt on difference between product and service marketing Ppt on types of houses around the world Ppt on web browser security