Question A.5: Complex Arithmetic in Polar Form Given Z1 = 10∠60° and Z2 = ...
Complex Arithmetic in Polar Form
Given Z_1 = 10\underline{/60^\circ} \text{ and } Z_2 = 5\underline{/45^\circ}, \text{ find } Z_1Z_2 , Z_1 /Z_2 , \text{ and } Z_1 + Z_2 in polar form.
For the product, we have
Z_1 \times Z_2=10 \underline{/60^\circ} \times 5 \underline{/45 ^\circ}=50 \underline{/105^\circ}
Dividing the numbers, we have
\frac{Z_1}{Z_2} =\frac{10\underline{/60^\circ}}{5 \underline{/45^\circ}}=2 \underline{/15^\circ}
Before we can add (or subtract) the numbers, we must convert them to rectangular form. Using Equation A.14 to convert the polar numbers to rectangular, we get
\begin{matrix} A\underline{/\theta} &=&Ae^{j \theta}&=&A\cos(\theta)+jA\sin(\theta) \quad \quad \quad \quad \quad \text{(A.14)} \\ Z_1&=&10 \underline{/60^\circ}&=&10\cos(60^\circ)+j10 \sin(60^\circ) \\ &=&5+j8.66 \\ Z_2&=&5 \underline{/45^\circ}&=&5 \cos(45^\circ)+j5\sin(45^\circ) \\ &=&3.54+j3.54 \end{matrix}
Now, we can add the numbers. We denote the sum as Z_s:
\begin{matrix} Z_s&=&Z_1+Z_2&=&5+j8.66+3.54+j3.54 \\ &=&8.54+j12.2 \end{matrix}
Next, we convert the sum to polar form:
|Z_s|=\sqrt{(8.54)^2+(12.2)^2}=14.9 \\ \tan \theta_s =\frac{12.2}{8.54} =1.43
Taking the arctangent of both sides, we have
\theta_s=\arctan(1.43)=55^\circ
Because the real part of Z_s is positive, the correct angle is the principal value of the arctangent (i.e., 55° is the correct angle). Thus, we obtain
Z_s=Z_1+Z_2=14.9 \underline{/55^\circ}