082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

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336 Chapter 9 Complex numbers
6 Expressz = 1 ,where ω andC are real constants,
jωC
in the forma +bj.Plotzonan Argand diagram.
7 Ifz 1 = 4(cos40 ◦ +jsin40 ◦ )andz 2 = 3(cos70 ◦ +
jsin70 ◦ ),expressz 1 z 2 andz 1 /z 2 in polar form.
8 Simplify
( √ 2̸ (5π/4)) 2 (2̸ (−π/3)) 2
2̸ (−π/6)
Solutions
1 |z 1 | = √ 13,arg(z 1 ) = −0.5880
|z 2 | = 1,arg(z 2 ) = −π/2
|z 3 | = 1,arg(z 3 ) = π
|z 4 | = √ 20,arg(z 4 ) = −2.0344
|z 5 | = 3,arg(z 5 ) = 0
2 (a) √ 10̸ −0.3218 (b) 2̸ 0
(c)1̸ −π/2 (d) 13̸ 1.9656
3 (a) 2,5π/6
(b) 4 √ 2,π/4
z 1 z 2 =8 √ 2̸ 13π/12
√
2
z 1 /z 2 =
4 ̸ 7π/12
4 1+j, √ 3+j, √ 3 −j
6 − j
ωC
7 z 1 z 2 = 12(cos110 ◦ +jsin110 ◦ )
8 4
z 1 /z 2 = 4 3 (cos30◦ −jsin30 ◦ )
9.6 VECTORSANDCOMPLEXNUMBERS
It is often convenient to represent complex numbers by vectors in the x--y plane. Figure
9.6(a) shows the complex numberz = a + jb. Figure 9.6(b) shows the equivalent
vector. Figure 9.7 shows the complex numbersz 1
= 2 +j andz 2
= 1 +3j.
If we now evaluate z 3
= z 1
+z 2
we find z 3
= 3 + 4j which is also shown. If we
form a parallelogram, two sides of which are the representations ofz 1
andz 2
, we find
thatz 3
isthediagonaloftheparallelogram.Ifweregardz 1
andz 2
asvectorsintheplane
we see that there is a direct analogy between the triangle law of vector addition (see
Section7.2.3)and the addition ofcomplex numbers.
y
4
y
b
z = a + jb
y
b
(
a
b
3
2
z 2
z 3
a x a x
1
z 1
(a)
Figure9.6
Thecomplex numberz =a +jbandits equivalent vector.
(b)
0 1 2 3 4
Figure9.7
Vector addition in the complex plane.
x