EE 188 Extra Credit Homework Solutions:
Complex Numbers and Phasors
1. Convert 3 + j4 to polar form; write the angle first in radians, then in degrees.
Solution: 3 + j4 = 5ej0.9273 = 56 0.9273 rads = 56 53.13o
2. Convert 5ejπ/3 to rectangular form; please note that the angle is given in radians.
Solution: 5ejπ/3 = 2.5 + j4.33
3. Convert 25ej170 to rectangular form.
o
Solution: 25ej170 = −24.62 + j4.34
o
4. Which form is better for adding complex numbers, rectangular or polar?
Solution: Rectangular form is better for adding complex numbers together.
Add Z1 = 2ej30 and Z2 = 5ej100 together. Write your result in both rectangular and polar form.
Solution: Z1 = 1.732 + j1, Z2 = −0.868 + j4.924;
Z1 + Z2 = 0.864 + j5.924 = 5.9866 81.7o .
o
o
5. Which form is better for multiplying complex numbers, rectangular or polar?
Solution: Polar form is better for multiplying complex numbers together.
Multiply Z1 = 2 + j3 and Z2 = −4 − j12 together. Write your result in both rectangular and polar
form.
Solution: Z1 = 3.6066 56.31o, Z2 = 12.6496 − 108.43o = 12.6496 + 251.57o.
Z1 Z2 = 45.6126 − 52.12o = 45.6126 307.88o
Note: If you solved for the angle of Z2 as tan−1 (−12/ − 4) = tan−1 (3), you’ll get the wrong answer
because tan−1 (3) = 71.57o . Using that method, you’re solving for the inverse tangent of a positive
number, which results in an angle in the first quadrant. The correct answer will be an answer in the
third quadrant, because both real and imaginary components have negative values. So you must add
180o to get the correct angle in the third quadrant: 71.57o + 180o = 251.57o (or subtract by 180o
which gives the equivalent answer of −108.43o ).
If your calculator allows you to input both real and imaginary components when it converts to polar
form (i.e. TI-89), then you should get the correct answer. The problem occurs when you can’t
directly convert from rectangular to polar, and instead must find the angle φ as = tan−1 (y/x). In
that case, you must recognize that the answer should be in the third quadrant, and add 180o to the
result for φ.
The same problem occurs when the angle should be in the second quadrant (negative real part,
positive imaginary part). y/x will be negative, and the result for φ = tan−1 (y/x) will be in the
fourth quadrant. You must again add or subtract 180o to your answer to get the correct result.
Again, this is only a problem when your calculator does not convert from rectangular to polar
directly, and you must do the conversion to phi as φ = tan−1 (y/x).
6. Given v(t) = 12 cos(2000t − 25o ), write its corresponding phasor V.
Solution: V = 12 6 −25o .
, write its corresponding sinusoidal voltage v(t) when ω = 1000 rad/s. Write
7. If the phasor V = 5 6 2π
3
v(t) as a cosine wave.
Solution: v(t) = 5 cos(1000t + 2π/3).
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