228 Higher Engineering Mathematics
- (− 2 +j)
1(^4) ⎡
⎣
Modulus 1. 223 ,arguments
38. 36 ◦, 128. 36 ◦,
218. 36 ◦and 308. 36 ◦
⎤
⎦
- (− 6 −j 5 )
1(^2) [
Modulus 2. 795 ,arguments
109. 90 ◦, 289. 90 ◦
]
- ( 4 −j 3 )
− 2
[^3
Modulus 0. 3420 ,arguments 24. 58 ◦,
144. 58 ◦and 264. 58 ◦]- For a transmission line, the characteristic
impedanceZ 0 and the propagation coefficient
γare given by:
Z 0 =√(
R+jωL
G+jωC)
andγ=√
[(R+jωL)(G+jωC)]GivenR= 25 ,L= 5 × 10 −^3 H,
G= 80 × 10 −^6 siemens, C= 0. 04 × 10 −^6 F
and ω= 2000 πrad/s, determine, in polar
form,Z 0 andγ.[
Z 0 = 390. 2 ∠− 10. 43 ◦,
γ= 0. 1029 ∠ 61. 92 ◦]21.4 The exponential form of a
complex number
Certain mathematical functions may be expressed as
power series (for example, by Maclaurin’s series—see
Chapter 8), three examples being:(i) ex= 1 +x+x^2
2!+x^3
3!+x^4
4!+x^5
5!+··· (1)(ii) sinx=x−x^3
3!+x^5
5!−x^7
7!+··· (2)(iii) cosx= 1 −x^2
2!+x^4
4!−x^6
6!+··· (3)Replacingxin equation (1) by the imaginary number
jθgives:ejθ= 1 +jθ+(jθ)^2
2!+(jθ)^3
3!+(jθ)^4
4!+(jθ)^5
5!+···= 1 +jθ+j^2 θ^2
2!+j^3 θ^3
3!+j^4 θ^4
4!+j^5 θ^5
5!+···By definition, j=√
(− 1 ), hence j^2 =−1, j^3 =−j,
j^4 =1,j^5 =j, and so on.Thus ejθ= 1 +jθ−θ^2
2!−jθ^3
3!+θ^4
4!+jθ^5
5!−···
Grouping real and imaginary terms gives:ejθ=(
1 −θ^2
2!+θ^4
4!−···)+j(
θ−θ^3
3!+θ^5
5!−···)However, from equations (2) and (3):
(
1 −θ^2
2!+θ^4
4!−···)
=cosθand(
θ−θ^3
3!+θ^5
5!−···)
=sinθThus ejθ=cosθ+jsinθ (4)Writing−θforθin equation (4), gives:ej(−θ)=cos(−θ)+jsin(−θ)However, cos(−θ)=cosθand sin(−θ)=−sinθThus e−jθ=cosθ−jsinθ (5)The polar form of a complex number z is:
z=r(cosθ+jsinθ). But, from equation (4),
cosθ+jsinθ=ejθ.
Therefore z=rejθ
When a complex number is written in this way, it is said
to be expressed inexponential form.
There are therefore three ways of expressing a com-
plex number:- z=(a+jb), called Cartesian or rectangu-
lar form, - z=r(cosθ+jsinθ)orr∠θ,calledpolarform,and
- z=rejθcalledexponential form.
The exponential form is obtained from the polar form.
For example, 4∠ 30 ◦becomes 4ej
π(^6) in exponential
form. (Note that inrejθ,θmust be in radians.)