Addition and Subtraction of Polar form
Addition
or subtraction cannot be performed in polar form unless the complex numbers
have the same angle q or differ
only by multiples of 1800.
Example 3.3:
or 
Multiplication
To
multiply two complex numbers in rectangular form, multiply the real and
imaginary parts of one in turn by the real and imaginary parts of the other.
For example, if
C1 = A1 + j B1 and C2 = A2
+ j B2
Then C1.
C2 = A1 + j B1
A2
+ j B2
A1A2 + j B1 A2
+ j
A1B2 + j 2B1 B2
A1A2 + j (B1 A2 + A1B2)
+ B1 B2 (-1)
And C1. C2 = A1A2
- B1 B2 + j (B1 A2 + A1B2)
(3.3)
In Example 3.4(b), we obtain a
solution without resorting to memorizing Eq. (3.3). Simply carry along the j factor when multiplying each part of
one vector with the real and imaginary parts of the other.
Example 3.4:
a.
Find C1.
C2 if C1 = 2 + j 3 and C2 = 5 + j 10.
b.
Find C1.
C2 if C1 = -2 - j 3 and C2 = +4 - j 6.
Solutions:
a. Using the
format above (Eq. (3.3)), we have
C1. C2
= [(2)(5) – (3)(10)]+ j [(3)(5) + (2)(10)]
= -20 + j 35
b. Without
using the format, we obtain
-2 - j
3
+4 - j
6
-8 - j
12
+j 12 + j 2 18
-8 -18 + j
(-12 + 12)
C1. C2
=
-26 = 26
In polar form, the
magnitudes are multiplied and the angles added algebraically. For example, for
C1 = C1
and C2
= C 2 
and C2
= C 2
C1. C2
= C1 C2 
(3.4)
(3.4)
Example 3.5:
a.
Find C1.
C2 if C1 = 5
and C2 = 10
.
and C2 = 10.
b.
Find C1.
C2 if C1 = 5
and C2 = 10
.
and C2 = 10.
Solutions:
a. From Eq.
(3.4), we have
C1. C2
= (5)(10)
=50 
=50
b. From Eq.
(3.4), we have
C1. C2 = (2)(7)
=14 
=14
To multiply a complex number in
rectangular form by a real number requires that both the real part and the
imaginary part be multiplied by the real number. For example,
(10)(2 + j 3) = 20 + j 30
and (50 
) (0 + j 6)
= j 300 = 300
) (0 + j 6)
= j 300 = 300
3.3
Division
To
divide two complex numbers in rectangular form, multiply the numerator and
denominator by the conjugate of the denominator and the resulting real and
imaginary parts collected. That is, for
C1 = A1 + j B1 and C2 = A2
+ j B2
|
Then C1 (A1 + j B1)( A2 - j B2)
C2 (A2 + j B2)( A2 - j B2)
=
|
|
|
C1 A1A2 + B1B2 A2B1 - A1B2
C2 
The
equation does not have to be memorized if the steps above used to obtain Eq. (3.5)
are employed. That is, first multiply the numerator by the complex conjugate of
the denominator and separate the real and imaginary terms. Then divide each
term by the real number obtained by multiplying the denominator by its
conjugate.
Example 3.6:
a.
Find C1/
C2 if C1 = 1 + j 4 and C2 = 4 + j 5.
b.
Find C1/
C2 if C1 = -4 - j 8 and C2 = +6 - j 1.
Solutions:
a. By Eq. (3.5),
C1/C2
= 
0.585 + j 0.268
0.585 + j 0.268
b. Using an
alternate method, we have
-4 - j
8
+6 - j
1
-24 - j
48
+j 4 + j 2 8
-24 + j
(-48 + 4)+8 = -16 – j 52
+6 - j 1
+6 - j
1
36 - j
6
-j 6 + j 2 1
36 + 0 +1 = 37
C1/C2
= 
= - 0.432 - j
1.405
= - 0.432 - j
1.405
To
divide a complex number in rectangular form by a real number, both the
real part and the imaginary part must be divided by the real number. For
example,
and 
=3.4
=3.4
In
polar form, division
is accomplished by simply dividing the magnitude of the
numerator by the magnitude of the denominator and subtracting the
angle of the denominator from that of the numerator. That is,
for
C1 = C1
and C2
= C 2
and C2
= C 2
C1/C2 = C1/C2
(3.6)
(3.6)
Example 3.7:
a.
Find C1/
C2 if C1 = 15
and C2 = 2
.
and C2 = 2.
b.
Find C1/
C2 if C1 = 8 and C2 = 16
.
and C2 = 16.
Solutions:
a. From Eq.
(3.6), we have
C1/ C2
= (15)/(2) 
=7.5 
=7.5
b. From Eq.
(3.6), we have
C1/ C2 = (8)/(16) 
=0.5 
=0.5
We
obtain the reciprocal in the
rectangular form by multiplying the numerator and denominator by the complex
conjugate of the denominator.
and 
(3.7)
(3.7)
In
the polar form the reciprocal is
(3.8)
Some
concluding examples using the four basic operations follow.
Example 3.8: Perform the following operations, leaving the
answer in polar or rectangular form.
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