Complex Arithmetic in Polar Form Given Z1 = 10∠60° and Z2 = 5∠45°, find Z1Z2, Z1/Z2, and Z1 + Z2 in polar form.

Question

Complex Arithmetic in Polar Form
Given [latex]Z_1 = 10\underline{/60^\circ} ext{ and } Z_2 = 5\underline{/45^\circ}, ext{ find } Z_1Z_2 , Z_1 /Z_2 , ext{ and } Z_1 + Z_2[/latex] in polar form.

Accepted Answer

For the product, we have

[latex]Z_1 imes Z_2=10 \underline{/60^\circ} imes 5 \underline{/45 ^\circ}=50 \underline{/105^\circ}[/latex]

Dividing the numbers, we have

[latex]\frac{Z_1}{Z_2} =\frac{10\underline{/60^\circ}}{5 \underline{/45^\circ}}=2 \underline{/15^\circ} [/latex]

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

[latex]\begin{matrix} A\underline{/ heta} &=&Ae^{j heta}&=&A\cos( heta)+jA\sin( heta) \quad \quad \quad \quad \quad ext{(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} [/latex]

Now, we can add the numbers. We denote the sum as [latex]Z_s[/latex]:

[latex]\begin{matrix} Z_s&=&Z_1+Z_2&=&5+j8.66+3.54+j3.54 \ &=&8.54+j12.2 \end{matrix} [/latex]

Next, we convert the sum to polar form:

[latex]|Z_s|=\sqrt{(8.54)^2+(12.2)^2}=14.9 \ an heta_s =\frac{12.2}{8.54} =1.43[/latex]

Taking the arctangent of both sides, we have

[latex] heta_s=\arctan(1.43)=55^\circ[/latex]

Because the real part of [latex]Z_s[/latex] is positive, the correct angle is the principal value of the arctangent (i.e., 55° is the correct angle). Thus, we obtain

[latex]Z_s=Z_1+Z_2=14.9 \underline{/55^\circ}[/latex]

Question A.5: Complex Arithmetic in Polar Form Given Z1 = 10∠60° and Z2 = ...

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