PRACTICE EXAM 4 PROBLEM 2 SOLUTION
1250
F 14
EX:
Express the value of the following sum in rectangular form:
(e− j 30° ) + j e j 30° +
*
SOL'N:
3e− j60°
The asterisk mean "conjugate", so we change the j to –j inside the
parentheses. The | | means magnitude, but the magnitude of this polarform number is unity, so the term inside the | | equals unity. ( | e jx |= 1 for
real number x.) Thus, we have the following equivalent expression:
e j 30° + j + 3e− j60°
We now convert to rectangular form for the addition. We use Euler's
formula:
Ae jφ = A cos φ + jAsin φ
Applying Euler's formula to the first and third terms in the sum yields the
following:
cos(30°) + j sin(30°) + j + 3 cos(−60°) + j 3 sin(−60°)
Noting that cos -x = cos x and sin -x = -sin x, and using exact values for
the 30°, 60°, 90° triangles, we have the following result:
3
1
1
3
+ j + j+ 3 − j 3
2
2
2
2
We rearrange to put real values together and imaginary values together for
the final addition:
⎛ 3
⎛1
1⎞
3⎞
⎜ 2 + 3 2⎟ + j⎜ 2 +1− 3 2 ⎟
⎝
⎠
⎝
⎠
or
3 + j0 = 3
We check this answer with a graphical addition of phasors:
The answer does appear to be 3 (red arrow is sum of vectors in blue that
represent complex numbers being summed).