264 COMPLEX NUMBERS- (− 2 +j)
1
4
⎡⎣Moduli 1.223, arguments
38 ◦ 22 ′, 128◦ 22 ′,
218 ◦ 22 ′and 308◦ 22 ′⎤⎦- (− 6 −j5)
1(^2) [
Moduli 2.795, arguments
109 ◦ 54 ′, 289◦ 54 ′
]
- (4−j3)
− 2
[^3
Moduli 0.3420, arguments 24◦ 35 ′,
144 ◦ 35 ′and 264◦ 35 ′]- 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 polarform,Z 0 andγ.[
Z 0 = 390. 2 ∠− 10. 43 ◦,
γ= 0. 1029 ∠ 61. 92 ◦]24.4 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 example 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), hencej^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=rejθWhen a complex number is written in this way, it is
said to be expressed inexponential form.
There are therefore three ways of expressing a
complex number:1.z=(a+jb), called Cartesian or rectangu-
lar form,2.z=r(cosθ+jsinθ)orr∠θ, calledpolar form,
and3.z=rejθcalledexponential form.The exponential form is obtained from the polar
form. For example, 4∠ 30 ◦becomes 4ejπ(^6) in expo-
nential form. (Note that inrejθ,θmust be in radians.)