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604 ⏐⏐⏐ THE BASIC ELEMENTS AND PHASORS
In polar form, the magnitudes are multiplied and the angles added
algebraically. For example, for
C1 � Z1 �v1 and C2 � Z2 �v2 we write
EXAMPLE 14.23
a. Find C 1 ⋅ C 2 if
b. Find C 1 ⋅ C 2 if
C 1 � 5 �20° and C 2 � 10 �30°
C 1 � 2 ��40° and C 2 � 7 ��120°
(14.33)
Solutions:
a. C 1 ⋅ C 2 � (5 �20°)(10 �30°) � (5)(10) /20° � 30° � 50 �50°
b. C 1 ⋅ C 2 � (2 ��40°)(7 ��120°) � (2)(7) /�40° � 120°
� 14 ��80°
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°(0 � j 6) � j 300 � 300 �90°
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, if
C1 � X1 � jY1 and C2 � X2 � jY2 C1 �
C2
then �
C 1
�
C 1 ⋅ C 2 � Z 1Z 2 / v 1 � v 2
(X 1 � jY 1)(X 2 � jY 2)
���
(X 2 � jY 2)(X 2 � jY 2)
(X 1X 2 � Y 1Y 2) � j(X 2Y 1 � X 1Y 2)
����
X 2 2 � Y 2 2
X
and � � j
2Y1 � X1Y2 ��
2 (14.34)
X 2 � Y 2 X1X2 � Y1Y2 ��
2
X 2 � Y 2
2 �
C2
2
The equation does not have to be memorized if the steps above used
to obtain it 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 sum of each term of the
denominator squared.
EXAMPLE 14.24
a. Find C 1/C 2 if C 1 � 1 � j 4 and C 2 � 4 � j 5.
b. Find C 1/C 2 if C 1 ��4 � j 8 and C 2 ��6 � j 1.
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