Calculate Parameter Alpha in Series RLC Circuit Power Factor

To find the value of $\alpha$, we analyze the given parameters of the series RLC circuit.

The power factor of a series RLC circuit is defined as: $\cos\phi = \frac{R}{Z}$

where:

• $R$ is the resistance of the circuit ($R = 100\text{ }\Omega$)
• $Z$ is the total impedance of the circuit, given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$
• $\cos\phi$ is the power factor ($\cos\phi = 0.5$)

Substitute the given value of the power factor into the expression: $0.5 = \frac{R}{Z} \implies Z = 2R$

Using the impedance formula $Z = \sqrt{R^2 + (X_L - X_C)^2}$: $2R = \sqrt{R^2 + (X_L - X_C)^2}$

Squaring both sides: $4R^2 = R^2 + (X_L - X_C)^2$

$3R^2 = (X_L - X_C)^2$

Taking the square root on both sides, the absolute difference between inductive reactance ($X_L$) and capacitive reactance ($X_C$) is: $|X_L - X_C| = \sqrt{3}R$

Given that the difference between the inductive reactance and capacitive reactance is $\sqrt{3}\alpha\text{ }\Omega$: $|X_L - X_C| = \sqrt{3}\alpha$

Equating the two expressions for $|X_L - X_C|$: $\sqrt{3}\alpha = \sqrt{3}R$

$\alpha = R$

Since $R = 100\text{ }\Omega$, we get: $\alpha = 100$