Question A.3: Rectangular-to-Polar Conversion Convert Z5 = 10 + j5 and Z6 ...

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 \quad \quad \quad \quad \quad (A.1) \\ |Z_5|=\sqrt{x_5^2+y_5^2}=\sqrt{10^2+5^2}=11.18

and

|Z_6|=\sqrt{x_6^2+y_6^2}=\sqrt{(-10)^2+5^2}=11.18

To find the angles, we use Equation A.2.

\tan(\theta_5)=\frac{y_5}{x_5}=\frac{5}{10}=0.5

Taking the arctangent of both sides, we have

\theta_5=\arctan(0.5)=26.57^\circ

Thus, we can write

Z_5=10+j5=11.18 \underline{/26.57^\circ}

This is illustrated in Figure A.4.

Evaluating Equation A.2 for Z_6 , we have

\tan{(\theta_6)}=\frac{y_6}{x_6} =\frac{5}{-10} =-0.5

Now if we take the arctan of both sides, we obtain

\theta_6=-26.57^\circ

However,Z_6 = -10 + j5 is shown in Figure A.4. Clearly, the value that we have found for u6 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

\theta_6=180+\arctan\left(\frac{y_6}{x_6} \right) =180-26.57=153.43^\circ

Finally, we can write

Z_6=-10+j5=11.18\underline{/153.43^\circ}