RLC Circuit Analysis at Half-Power Frequencies

Let's analyze each statement regarding RLC circuits at half-power frequencies.

Statement 1: RLC Series Circuit Half-Power Frequency

In a series RLC circuit, the impedance is given by \(Z = R + jX\), where \(R\) is the resistance and \(X = \omega L - \frac{1}{\omega C}\) is the reactance. The magnitude of impedance is \(|Z| = \sqrt{R^2 + X^2}\).

The power dissipated in the circuit is \(P = \frac{V^2 R}{|Z|^2}\), where \(V\) is the RMS voltage across the circuit. At resonance (\(\omega_0\)), the reactance \(X=0\), so \(|Z| = R\), and the power is maximum, \(P_{max} = \frac{V^2}{R}\).

Half-power frequencies (\(\omega_l\) and \(\omega_h\)) are the frequencies where the power is half of the maximum power: \(P = \frac{P_{max}}{2}\).

So, \(\frac{V^2 R}{|Z|^2} = \frac{1}{2} \frac{V^2}{R}\).

This simplifies to \(|Z|^2 = 2R^2\), which means \(|Z| = \sqrt{2}R\).

Substituting the expression for \(|Z|\):

\(\sqrt{R^2 + X^2} = \sqrt{2}R\)

\(R^2 + X^2 = 2R^2\)

\(X^2 = R^2\)

\(|X| = R\)

Thus, at half-power frequencies in a series RLC circuit, the magnitude of the total reactance is equal to the resistance. Statement 1 is TRUE.

Statement 2: RLC Parallel Circuit Half-Power Frequency

In a parallel RLC circuit driven by a voltage source, it's convenient to use admittance, \(Y = G + jB\), where \(G = \frac{1}{R}\) is the conductance and \(B = \omega C - \frac{1}{\omega L}\) is the total susceptance. The magnitude of admittance is \(|Y| = \sqrt{G^2 + B^2}\).

For a voltage source V across the parallel combination, the current drawn is \(I = VY\). The power dissipated in the resistor is \(P = \frac{V^2}{R} = V^2 G\). Note that this power is constant if V is constant.

The concept of half-power frequencies in a parallel RLC circuit (driven by a voltage source) usually refers to the frequencies where the magnitude of the current drawn from the source is \(\sqrt{2}\) times the minimum current drawn at resonance.

At resonance (\(\omega_0\)), the susceptance \(B=0\), so \(|Y| = G\), and the current magnitude is minimum, \(|I_{min}| = V|Y| = VG\).

At half-power frequencies, \(|I| = \sqrt{2} |I_{min}|\).

So, \(V|Y| = \sqrt{2} VG\), which means \(|Y| = \sqrt{2}G\).

Substituting the expression for \(|Y|\):

\(\sqrt{G^2 + B^2} = \sqrt{2}G\)

\(G^2 + B^2 = 2G^2\)

\(B^2 = G^2\)

\(|B| = G\)

Thus, at half-power frequencies in a parallel RLC circuit, the magnitude of the total susceptance is equal to the conductance. Statement 2 is TRUE.

Conclusion

Based on the analysis of both statements, Statement 1 is TRUE and Statement 2 is TRUE.

Therefore, the correct answer is that both Statement 1 and Statement 2 are TRUE.