Higher Engineering Mathematics

254 COMPLEX NUMBERS

Multiplying equation (1) by 2 gives:

2 x+ 2 y= 4 (3)

Adding equations (2) and (3) gives:

−x=7, i.e.,x=− 7

From equation (1), y= 9 , which may be

checked in equation (2).

Now try the following exercise.

Exercise 102 Further problems on complex

equations

In Problems 1 to 4 solve the complex equations.

1. (2+j)(3−j2)=a+jb [a=8,b=−1]

2.

2 +j

1 −j

=j(x+jy)

[

x=

3

2

,y=−

1

2

]

3. (2−j3)=

√

(a+jb)[a=−5,b=−12]

4. (x−j 2 y)−(y−jx)= 2 +j [x=3,y=1]


5. If Z=R+jωL+ 1 /jωC, express Z in
   (a+jb) form whenR=10,L=5,C= 0. 04
   andω=4. [Z= 10 +j 13 .75]

23.6 The polar form of a complex

number

(i) Let a complex numberzbex+jyas shown

in the Argand diagram of Fig. 23.4. Let dis-

tanceOZberand the angleOZmakes with the

positive real axis beθ.

From trigonometry, x=rcosθand

y=rsinθ

Hence Z=x+jy =rcosθ+jrsinθ

=r(cosθ+jsinθ)

Z=r(cosθ+jsinθ) is usually abbreviated to

Z=r∠θwhich is known as thepolar formof

a complex number.

(ii)ris called themodulus(or magnitude) ofZand

is written as modZor|Z|.

ris determined using Pythagoras’ theorem on

triangleOAZin Fig. 23.4,

Z

r jy

θ

O

x

A Real axis

Imaginary

axis

Figure 23.4

i.e. r=

√

(x^2 +y^2 )

(iii)θis called theargument(or amplitude) ofZ

and is written as argZ.

By trigonometry on triangleOAZ,

argZ= θ=tan−^1

y

x

(iv) Whenever changing from cartesian form to

polar form, or vice-versa, a sketch is invalu-

able for determining the quadrant in which the

complex number occurs.

Problem 9. Determine the modulus and argu-

ment of the complex numberZ= 2 +j3, and

expressZin polar form.

Z= 2 +j3 lies in the first quadrant as shown in

Fig. 23.5.

r

0 2 Real axis

j 3

Imaginary

axis

θ

Figure 23.5