Impedance

Step 1: Formula for parallel impedance.
The total impedance

Z_%7B%5Ctext%7Btotal%7D%7D

for two components in parallel is given by:

%5Cfrac%7B1%7D%7BZ_%7B%5Ctext%7Btotal%7D%7D%7D%20%3D%20%5Cfrac%7B1%7D%7BZ_1%7D%20%2B%20%5Cfrac%7B1%7D%7BZ_2%7D.

Step 2: Substitute the given impedances.
Given that

Z_1%20%3D%203%20%2B%20j4

ohms and

Z_2%20%3D%203%20-%20j4

ohms, we substitute these into the formula:

%5Cfrac%7B1%7D%7BZ_%7B%5Ctext%7Btotal%7D%7D%7D%20%3D%20%5Cfrac%7B1%7D%7B3%20%2B%20j4%7D%20%2B%20%5Cfrac%7B1%7D%7B3%20-%20j4%7D.

Simplifying:

%5Cfrac%7B1%7D%7BZ_%7B%5Ctext%7Btotal%7D%7D%7D%20%3D%20%5Cfrac%7B(3%20-%20j4)%20%2B%20(3%20%2B%20j4)%7D%7B(3%20%2B%20j4)(3%20-%20j4)%7D%20%3D%20%5Cfrac%7B6%7D%7B9%20%2B%2016%7D%20%3D%20%5Cfrac%7B6%7D%7B25%7D.

Thus:

Z_%7B%5Ctext%7Btotal%7D%7D%20%3D%206%20%5C%2C%20%5Ctext%7Bohms%7D.

Step 3: Conclusion.
Thus, the overall impedance is 6 ohms, corresponding to option (D).

Chapter Questions
1 MCQs