91Ó°ÊÓ

Given: \( Z_{1} = 2 + i \) \( Z_{2} = 4 - 3i \)

The formula for the total impedance in parallel circuits is: \[ \frac{1}{Z} = \frac{1}{Z_{1}} + \frac{1}{Z_{2}} \]

First, find the reciprocal of each given impedance individually:\[ \frac{1}{Z_{1}} = \frac{1}{2+i} \]To simplify, multiply numerator and denominator by the complex conjugate of the denominator:\[ \frac{1}{2+i} \times \frac{2-i}{2-i} = \frac{2-i}{(2+i)(2-i)} = \frac{2-i}{4+1} = \frac{2-i}{5} = \frac{2}{5} - \frac{i}{5} \]So, \( \frac{1}{Z_{1}} = \frac{2}{5} - \frac{i}{5} \)Next, find the reciprocal of \( Z_{2} \):\[ \frac{1}{Z_{2}} = \frac{1}{4-3i} \]To simplify, multiply numerator and denominator by the complex conjugate of the denominator:\[ \frac{1}{4-3i} \times \frac{4+3i}{4+3i} = \frac{4+3i}{(4-3i)(4+3i)} = \frac{4+3i}{16+9} = \frac{4+3i}{25} = \frac{4}{25} + \frac{3i}{25} \]So, \( \frac{1}{Z_{2}} = \frac{4}{25} + \frac{3i}{25} \)

Now, add the reciprocals:\[ \frac{1}{Z} = \frac{2}{5} - \frac{i}{5} + \frac{4}{25} + \frac{3i}{25} \]Combine the real parts and the imaginary parts separately:\[ \frac{1}{Z} = \left( \frac{2}{5} + \frac{4}{25} \right) + \left( -\frac{1}{5} + \frac{3}{25} \right)i \]Convert fractions to have a common denominator:\[ \frac{2}{5} = \frac{10}{25} \]So, the real part is:\[ \frac{10}{25} + \frac{4}{25} = \frac{14}{25} \]And the imaginary part is:\[ -\frac{1}{5} = -\frac{5}{25} \]\[ -\frac{5}{25} + \frac{3}{25} = -\frac{2}{25} \]Therefore,\[ \frac{1}{Z} = \frac{14}{25} - \frac{2}{25}i \]

Now take the reciprocal of the sum to find the total impedance:\[ Z = \left( \frac{14}{25} - \frac{2}{25}i \right)^{-1} \]To find the inverse of a complex number in the form \( a + bi \), multiply by the complex conjugate over itself:\[ Z = \frac{(\frac{14}{25} - \frac{2}{25}i)}{(\frac{14}{25} - \frac{2}{25}i)} \times \frac{\frac{14}{25} + \frac{2}{25}i}{\frac{14}{25} + \frac{2}{25}i} \]Simplify the expression by considering the magnitudes:\[ Z = \frac{\frac{14}{25} + \frac{2}{25}i}{(\frac{14}{25})^2 + (\frac{-2}{25})^2} \]\[ = \frac{\frac{14}{25} + \frac{2}{25}i}{\frac{196}{625} + \frac{4}{625}} \]\[ = \frac{\frac{14}{25} + \frac{2}{25}i}{\frac{200}{625}} \]\[ = \frac{\frac{14}{25} + \frac{2}{25}i}{\frac{8}{25}} \]Multiply numerator and denominator by 25/8 (reciprocal of 8/25):\[ Z = \left( \frac{14}{25} + \frac{2}{25}i \right) \times \frac{25}{8} \]\[ = \frac{14}{8} + \frac{2}{8}i \]\[ = \frac{7}{4} + \frac{1}{4}i \]So, the total impedance is:\[ Z = \frac{7}{4} + \frac{1}{4}i \]