228 Higher Engineering Mathematics
5. (−2 + j )
1
4
⎡
⎣
Modulus 1.223, arguments
38.36 ◦ , 128.36 ◦ ,
218.36 ◦ and 308.36 ◦
⎤
⎦
6. (−6 − j 5)
1
2
Modulus 2.795, arguments
109.90 ◦ , 289.90 ◦
7. (4 − j 3)
−2
3
Modulus 0.3420, arguments 24.58
◦
,
144.58 ◦ and 264.58 ◦
8. For a transmission line, the characteristic
impedance Z 0 and the propagation coefficient
γ are given by:
Z 0 =
R + j ωL
G + j ωC
and
γ =
[(R + j ωL)(G + j ωC)]
Given R = 25 , L =5 × 10 −3 H,
G = 80 × 10 −6 siemens, C = 0.04 × 10 −6 F
and ω = 2000 π rad/s, determine, in polar
form, Z 0 and γ .
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) e
x
= 1 + x +
x 2
2!
+
x 3
3!
+
x 4
4!
+
x 5
5!
+ · · ·
(1)
(ii) sin x = x −
x 3
3!
+
x 5
5!
−
x 7
7!
+ · · ·
(2)
(iii) cos x = 1 −
x 2
2!
+
x 4
4!
−
x 6
6!
+ · · ·
(3)
Replacing x in equation (1) by the imaginary number
j θ gives:
e
j θ
= 1+ j θ +
( j θ) 2
2!
+
( j θ) 3
3!
+
( j θ) 4
4!
+
( j θ) 5
5!
+· · ·
= 1 + j θ +
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 e j θ = 1 + j θ −
θ 2
2!
− j
θ 3
3!
+
θ 4
4!
+ j
θ 5
5!
− · · ·
Grouping real and imaginary terms gives:
e
j θ
=
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!
− · · ·
= cos θ
and
θ −
θ
3
3!
+
θ
5
5!
− · · ·
= sin θ
Thus
e
jθ
= cos θ + j sin θ
(4)
Writing −θ for θ in equation (4), gives:
e
j (−θ)
= cos(−θ) + j sin(−θ)
However, cos(−θ)= cos θ and sin(−θ)= −sin θ
Thus
e
−jθ
= cos θ − j sin θ
(5)
The polar form of a complex number z is:
z =r(cos θ + j sin θ). But, from equation (4),
cos θ + j sin θ = e jθ .
Therefore
z = re jθ
When a complex number is written in this way, it is said
to be expressed in exponential form.
There are therefore three ways of expressing a complex number:
1. z =(a + j b), called Cartesian or rectangular form,
2. z =r(cos θ + j sin θ) or r∠θ, called polar form, and
3. z =re j θ called exponential form.
The exponential form is obtained from the polar form.
For example, 4∠30 ◦ becomes 4e
j
π
6 in exponential
form. (Note that in re j θ , θ must be in radians.)
5. (−2 + j )
1
4
⎡
⎣
Modulus 1.223, arguments
38.36 ◦ , 128.36 ◦ ,
218.36 ◦ and 308.36 ◦
⎤
⎦
6. (−6 − j 5)
1
2
Modulus 2.795, arguments
109.90 ◦ , 289.90 ◦
7. (4 − j 3)
−2
3
Modulus 0.3420, arguments 24.58
◦
,
144.58 ◦ and 264.58 ◦
8. For a transmission line, the characteristic
impedance Z 0 and the propagation coefficient
γ are given by:
Z 0 =
R + j ωL
G + j ωC
and
γ =
[(R + j ωL)(G + j ωC)]
Given R = 25 , L =5 × 10 −3 H,
G = 80 × 10 −6 siemens, C = 0.04 × 10 −6 F
and ω = 2000 π rad/s, determine, in polar
form, Z 0 and γ .
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) e
x
= 1 + x +
x 2
2!
+
x 3
3!
+
x 4
4!
+
x 5
5!
+ · · ·
(1)
(ii) sin x = x −
x 3
3!
+
x 5
5!
−
x 7
7!
+ · · ·
(2)
(iii) cos x = 1 −
x 2
2!
+
x 4
4!
−
x 6
6!
+ · · ·
(3)
Replacing x in equation (1) by the imaginary number
j θ gives:
e
j θ
= 1+ j θ +
( j θ) 2
2!
+
( j θ) 3
3!
+
( j θ) 4
4!
+
( j θ) 5
5!
+· · ·
= 1 + j θ +
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 e j θ = 1 + j θ −
θ 2
2!
− j
θ 3
3!
+
θ 4
4!
+ j
θ 5
5!
− · · ·
Grouping real and imaginary terms gives:
e
j θ
=
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!
− · · ·
= cos θ
and
θ −
θ
3
3!
+
θ
5
5!
− · · ·
= sin θ
Thus
e
jθ
= cos θ + j sin θ
(4)
Writing −θ for θ in equation (4), gives:
e
j (−θ)
= cos(−θ) + j sin(−θ)
However, cos(−θ)= cos θ and sin(−θ)= −sin θ
Thus
e
−jθ
= cos θ − j sin θ
(5)
The polar form of a complex number z is:
z =r(cos θ + j sin θ). But, from equation (4),
cos θ + j sin θ = e jθ .
Therefore
z = re jθ
When a complex number is written in this way, it is said
to be expressed in exponential form.
There are therefore three ways of expressing a complex number:
1. z =(a + j b), called Cartesian or rectangular form,
2. z =r(cos θ + j sin θ) or r∠θ, called polar form, and
3. z =re j θ called exponential form.
The exponential form is obtained from the polar form.
For example, 4∠30 ◦ becomes 4e
j
π
6 in exponential
form. (Note that in re j θ , θ must be in radians.)