Given: Z1= -j2.5 ohms, Z2=j4; Z3=5 ohms; Z4= (1+j5) ohms. If the four impedances are connected in parallel, find the equivalent impedance in ohms.
Given Z1=-j2.5, Z2=j4, Z3=j5, Z4=1+j5 ohms. If the four impedances are connected in parallel, find the equivalent impedance in ohms.
To find the equivalent impedance of the four impedances connected in parallel, we need to apply the formula for calculating the total impedance in a parallel combination.
In a parallel circuit, the reciprocal of the total impedance is equal to the sum of the reciprocals of the individual impedances.
So, let's calculate the reciprocal of each impedance:
Reciprocal of Z1 = 1 / (-j2.5) = 1 / (-j2.5) * (j2.5) / (j2.5) = -j2.5 / (j^2 + 2.5^2) = j2.5 / 6.25 = j0.4
Reciprocal of Z2 = 1 / (j4) = -j0.25
Reciprocal of Z3 = 1 / 5 = 0.2
Reciprocal of Z4 = 1 / (1 + j5) = (1 - j5) / (1 + j5) * (1 - j5) / (1 - j5) = (1 - j5) / (1^2 + 5^2) = (1 - j5) / 26 = 0.038 - j0.192
Now, we can sum up the reciprocals of these impedances:
Reciprocal of total impedance = Reciprocal of Z1 + Reciprocal of Z2 + Reciprocal of Z3 + Reciprocal of Z4
= j0.4 + (-j0.25) + 0.2 + (0.038 - j0.192)
= j0.162 - j0.192
To find the equivalent impedance, we need to calculate the reciprocal of the sum above:
Reciprocal of total impedance = 1 / (j0.162 - j0.192)
= (j0.162 + j0.192) / ((j0.162 - j0.192)(j0.162 + j0.192))
= (j0.162 + j0.192) / (j0.026 - j0.06152)
= (j0.162 + j0.192) / (j0.026 + j0.06152)
= (0.162 + 0.192j) / (0.026 + 0.06152j)
Finally, we can calculate the equivalent impedance by simplifying the complex division above.