Determine the impedance of a replacement three branch parallel circuit in which R=1.0k ohms, Xc= 500 ohms, Xl= 1.2k ohms
To determine the impedance of a three-branch parallel circuit, you need to calculate the total impedance (Z) by using the parallel impedance formula:
1/Z = 1/Z1 + 1/Z2 + 1/Z3
Where Z1, Z2, and Z3 are the impedances of each branch.
Given the following values:
R = 1.0k ohms
Xc = 500 ohms
Xl = 1.2k ohms
Convert all the given values to their respective complex impedances.
The impedance of a resistor (R) is purely resistive, so it can be represented as:
Zr = R = 1.0k ohms
The impedance of a capacitor (Xc) is purely imaginary and is given by:
Zc = 1 / (j * Xc) = -j * (1 / Xc) = -j * (1 / 500)
The impedance of an inductor (Xl) is purely imaginary and is given by:
Zl = j * Xl = j * 1.2k
Now, substitute the values of Zr, Zc, and Zl into the parallel impedance formula and calculate the total impedance (Z).
1/Z = 1/Zr + 1/Zc + 1/Zl
1 / Z = 1 / (1.0k) + 1 / (-j * 1/500) + 1 / (j * 1.2k)
To find the inverse of a complex number, you need to multiply both the numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of -j * 1/500 is j * 1/500.
1 / Z = 1 / (1.0k) + (1 / (-j * 1/500)) * (j * 1/500) + 1 / (j * 1.2k) * (j * 1.2k)
1 / Z = 1 / (1.0k) - j / 500 * j / 500 + 1 / (1.2k * j) * 1.2k * j
Continue simplifying:
1 / Z = 1 / (1.0k) + 1 / (500 * 500) + 1 / (-1.44m)
Now invert both sides:
Z = 1 / (1 / (1.0k) + 1 / (500 * 500) + 1 / (-1.44m))
Evaluate this equation using a calculator or computer to obtain the final impedance value.