Complex Operations Assume Z1 = 3∠45°, Z2 = 4∠45°, calculate Z1 + Z2, Z1Z2, and Z1/Z2.

Question

Complex Operations

Assume [latex]Z_1=3 \angle 45^{\circ}, Z_2=4 \angle 45^{\circ}[/latex], calculate [latex]Z_1+Z_2, Z_1 Z_2[/latex], and [latex]Z_1 / Z_2[/latex].

Accepted Answer

First, convert [latex]Z_1[/latex] and [latex]Z_2[/latex] to rectangular form, then add them.

Using Equation (B.1):

[latex]L(x(t))=X(s)=\int_0^{\infty} x(t) e^{-s t} dt[/latex] (B.1)

The real part a of the [latex]Z_1[/latex] is:

[latex]a=3 imes \cos \left(45^{\circ} ight)=2.1213[/latex]

The imaginary part b of the [latex]Z_1[/latex] is:

[latex]b=3 imes \sin \left(45^{\circ} ight)=2.1213[/latex]

The real part c of the [latex]Z_2[/latex] is:

[latex]c=4 imes \cos \left(45^{\circ} ight)=2.8284[/latex]

The imaginary part d of the [latex]Z_2[/latex] is:

[latex]d=4 imes \sin \left(45^{\circ} ight)=2.8284[/latex]

Therefore:

[latex]Z_1+Z_2=2.1213+j 2.1213+2.8284+j 2.8284=4.9497+j 4.9497[/latex]

Convert the result to polar form:

[latex]\begin{aligned}Z_1+Z_2 &=7 \angle 45^{\circ} \Z_1 imes Z_2 &=(3 imes 4) \angle\left(45^{\circ}+45^{\circ} ight)=12 \angle 90^{\circ} \\frac{Z_1}{Z_2} &=\frac{3}{4} \angle\left(45^{\circ}-45^{\circ} ight)=0.75 \angle 0^{\circ}\end{aligned}[/latex]

Question C.6: Complex Operations Assume Z1 = 3∠45°, Z2 = 4∠45°, calculate ...

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