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Complex Arithmetic
Introduction
Each problem in this problem set presents an equation involving complex numbers and variables.
Use complex arithmetic to determine the values of the variables.
Appendix B of Introduction to Electric Circuits by R.C. Dorf and J.A Svoboda provides a reviewof complex arithmetic.
Worked Examples
Example 1
Given
4565
15 8
j je Ae j
θ °=
− +
Find the values of A and θ .
Solution:
( )45 15245 45 107
152
6 6 305 5 1.76
15 8 17 17
j j j
je e e
j e
°− °° °= = =
− +
je− °
Example 2:
Given:
45102.36 je
a j b
°=
+
Find the values of a and b.
Solution:
45
45
104.24 3 3
2.36
j
je j a
e
− °
°= = − = j b+
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Example 3
Given:
( )3 8 3 j 2 Ae jθ − + = j
Find the values of A and θ .
Solution:
( )90
90 111 21
111
32 32 323.75
3 8 8.54 8.54
j j j j
j
j e Ae e e
j e
θ
°
°− ° − °
°= = = =
− +
Example 4:
Given:
( ) 1352 3 5 j j Ae j eθ − °= + +
Find the values of A and θ .
Solution:
( ) ( ) ( )
( ) (
135
161
2 3 5 2 3 3.54 3.54
2 3.54 3 3.54
1.54 0.54
1.63
j j
j
Ae j e j j
j j
j
e
θ − °
− °
= + + = + + − −
= − + −
= − −
=
)
Example 5:
Given:
15
4 3
2
j
j
j Ae
e
θ
°
−=
Find the values of A and θ .
Solution:
( )37
37 15 52
15 15
4 3 5 52.5
2 2 2
j j j j
j j
j e Ae e e
e e
θ
− °
− °− ° − °
° °
−= = = =
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Example 6:
Given:
135
5 13
6 j
ja j b
e °
− += +
Find the values of a and b.
Solution:
( )111
111 135
135 135
24
5 13 13.9 13.9
6 6 6
2.32 2.12 0.94
j j
j j
j
j ea j b e
e e
e j
°
°− °
° °
− °
− ++ = = =
= = −
Example 7:
Given:6
4 3 j
ja j b
−= − −
+
Find the values of a and b.
Solution:
( )90
90 143
143
53
6 6 6
4 3 5 5
1.2 0.722 0.958
j j
j
j
j ea j b e
j e
e j
− °
− °+ °
− °
°
−+ = = =
− −
= = +
Example 8
Given:
( )120 156 4 3 2 j je j e a° °
− + + = + j b
Find the values of a and b.
Solution:
( ) ( )
( )
( ) ( )
120 120
120 120
120 120 240
6 4 3 1.93 0.52 6 2.07 3.52
6 4.08
6 4.08 24.48
12.2 21.2
j j
j j
j j
a j b e j j e j
e e
e e
j
° °
° °
°+ ° °
+ = − + + + = − +
=
= × =
= − −
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Example 9:
Given:120
4 3 j
Ae j b°
j+ = − +
Find the values of A and b.
Solution:120 4 3
0.5 0.866 4 3
j Ae j b j
j A j b
°
j
+ = − +
− + + = − +
Equating real and imaginary parts:
0.5 4 8 A A− = − ⇒ =
and
0.866 3 3 0.866 8 3.93 A b b+ = ⇒ = − × = −
Example 10:
Given:
( )1206 4 8 j je j b e θ °18− + + =
Find the values of A and θ .
Solution:
( )120
120
120
6 4 8 18
184 8cos 8sin 3 1.5
6
j j
j
j
e j b e
j b j e je
θ
θ θ
°
− °
°
− + + =
− + + + = = = − − 2.6
Equating real and imaginary parts:
8 cos 2.5 71.8θ θ = ⇒ = ° and
8 sin 2.6 2.6 8 cos(71.8 ) 10.2b bθ + = − ⇒ = − − ° = −
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Example 11:
Given:
( ) 604 2 2 ja j j Ae+ = −
Find the values of a and A.
Solution:
( )
( )
604 2 2
8 2 2 0.5 0.866
10 2 0.5 0.866
ja j j Ae
j a A j A
j a A j A
°+ = −
− + = − +
− = +
Equating real and imaginary parts:
10 0.5 20 A A= ⇒ = and
( )2 0.866 20 8.66a a− = ⇒ = −
