ejercicios_numeros_complejos
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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 review
of complex arithmetic.
Worked Examples
Example 1
Given
4565
15 8
j je Aej
Find the values of A and .
Solution:
45 15245 45 107
152
6 6 305 5 1.76
15 8 17 17
jj 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
ej b
Example 3
Given:
3 8 3j 2Ae j j
Find the values of A and .
Solution:90
90 111 21
111
32 32 323.75
3 8 8.54 8.54
jjj j
j
j eAe e e
j e
Example 4:
Given:1352 3 5j jAe j e
Find the values of A and .
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
Example 5:
Given:
15
4 3
2
j
j
jAe
e
Find the values of A and .
Solution:37
37 15 52
15 15
4 3 5 52.5
2 2 2
jjj j
j j
j eAe e e
e e
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
jj
j j
j
j ea j b e
e e
e j
Example 7:
Given:
64 3
jj
a 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
jj
j
j
j ea j b e
j e
e j
Example 8
Given:120 156 4 3 2j 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
Example 9:
Given:120 4 3jAe j b j
Find the values of A and b.
Solution:120 4 3
0.5 0.866 4 3
jAe j b j
A j A j b j
Equating real and imaginary parts:
0.5 4 8A A
and
0.866 3 3 0.866 8 3.93A b b
Example 10:
Given:1206 4 8j je j b e 18
Find the values of A and .
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
2.6
Equating real and imaginary parts:
8 cos 2.5 71.8
and
8 sin 2.6 2.6 8 cos(71.8 ) 10.2b b
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 20A A
and
2 0.866 20 8.66a a
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