How is the impedance of an RL circuit affected when a capacitor is added in parallel?

Impedance of the RL Circuit:

The impedance of an RL circuit is a complex quantity that includes both resistance (R) and reactance (XL) due to the inductor. The reactance of an inductor is given by XL = 2πfL, where f is the frequency of the AC signal, and L is the inductance of the inductor.

The impedance of the RL circuit (Z_RL) is given by:

Z_RL = R + jXL

where "j" represents the imaginary unit (√(-1)).

Impedance of the Capacitor:

The impedance of a capacitor (Z_C) is also a complex quantity and depends on the capacitance (C) and the frequency of the AC signal. The reactance of a capacitor is given by XC = 1 / (2πfC).

The impedance of the capacitor is given by:

Z_C = 1 / (jXC)

Z_C = -j / (2πfC)

Now, when the capacitor is added in parallel to the RL circuit, the total impedance (Z_total) is given by the reciprocal of the sum of the reciprocals of the individual impedances:

1 / Z_total = 1 / Z_RL + 1 / Z_C

To find the total impedance (Z_total), you can simply invert both sides of the equation:

Z_total = 1 / (1 / Z_RL + 1 / Z_C)

Substitute the expressions for Z_RL and Z_C:

Z_total = 1 / (1 / (R + jXL) + 1 / (-j / (2πfC)))

To simplify further, we can find a common denominator:

Z_total = 1 / (1 / (R + jXL) - j / (2πfC))

Next, to remove the complex fraction, we multiply the numerator and denominator by the complex conjugate of the denominator:

Z_total = (R + jXL) / ((R + jXL) - j / (2πfC)) * ((R - jXL) / (R - jXL))

Now, we can expand and simplify the expression:

Z_total = (R^2 + jRXL - jRXL + j^2X^2L^2) / (R - jXL - jR + j^2XL)

Since j^2 = -1, we get:

Z_total = (R^2 - X^2L) / (R - j(2πfL - XL))

The real part of the impedance (resistance) is R^2 - X^2L, and the imaginary part (reactance) is -(2πfL - XL).

In summary, when a capacitor is added in parallel to an RL circuit, the total impedance becomes a complex quantity with both resistance and reactance components. The presence of the capacitor affects the overall impedance, and the reactance of the capacitor (XC) opposes the reactance of the inductor (XL) in the circuit. The impedance at any particular frequency will depend on the values of R, L, C, and the frequency of the AC signal.