Question A.3: Rectangular-to-Polar Conversion Convert Z5 = 10 + j5 and Z6 ......
Rectangular-to-Polar Conversion
Convert Z_5 = 10 + j5 and Z_6 = −10 + j5 to polar form.
The complex numbers are illustrated in Figure A.4. First, we use Equation A.1 to find the magnitudes of each of the numbers. Thus,
|Z|^2 = x^2 + y^2 (A.1)
y = |Z| \sin(θ) (A.4)
|Z_5| = \sqrt{x_5^2 + y_5^2} = \sqrt{10^2 + 5^2}=11.18and
|Z_6| = \sqrt{x_6^2 + 6_5^2} = \sqrt{(-10)^2 + 5^2}=11.18To find the angles, we use Equation A.2.
\tan (\theta) = \frac{y}{x} (A.2)
\tan (θ_5) = \frac{y_5}{x_5}=\frac{5}{10}=0.5
Taking the arctangent of both sides, we have
θ_5 = \arctan(0.5) = 26.57°
Thus, we can write
Z_5 = 10 + j5 = 11.18\angle 26.57°
This is illustrated in Figure A.4.
Evaluating Equation A.2 for Z_6, we have
Now if we take the arctan of both sides, we obtain
θ_6 = −26.57°
However, Z_6 = −10+j5 is shown in Figure A.4. Clearly,the value that we have found for θ_6 is incorrect. The reason for this is that the arctangent function is multivalued. The value actually given by most calculators or computer programs is the principal value. If the number falls to the left of the imaginary axis (i.e.,if the real part is negative), we must add (or subtract) 180° to arctan(y/x) to obtain the correct angle. Thus, the true angle for Z_6 is
Finally, we can write
Z_6 = −10 + j5 = 11.18 \angle 153.43°