Question

(i) Define the terms (I) resonant frequency, and (II) power factor of a series LCR circuit. For what value of the power factor will the power dissipated in the circuit be maximum?

(ii) An inductor of $\dfrac{5}{\pi}$ H, a capacitor of $\dfrac{50}{\pi}\ \mu\text{F}$ and a resistor of $400\ \Omega$ are connected in series across an ac voltage $v = 140 \sin(100\pi t)$ V. Calculate:

(I) impedance of the circuit, and

(II) rms value of current that flows in the circuit.

(Take $\sqrt{2} = 1.4$)

Hint:

Define the two terms, then use X_L = wL, X_C = 1/(wC), Z = sqrt(R^2 + (X_L - X_C)^2) and I_rms = V_rms/Z with V_rms = 140/1.4.

Answer 1

Step 1: Understanding the Question:
Part (i) asks for two definitions and the condition for maximum power. Part (ii) is a numerical on a series LCR circuit with \(L = \dfrac{5}{\pi}\) H, \(C = \dfrac{50}{\pi}\ \mu\)F, \(R = 400\ \Omega\) and \(v = 140\sin(100\pi t)\) V.

Step 2: Definition (I): resonant frequency:
In a series LCR circuit, the inductive reactance \(X_L = \omega L\) and the capacitive reactance \(X_C = \dfrac{1}{\omega C}\) cancel each other at one particular frequency.
The frequency at which \(X_L = X_C\) is called the resonant frequency. At this frequency the impedance is minimum and equal to \(R\), and the current is maximum.
\[ \omega_0 L = \frac{1}{\omega_0 C} \Rightarrow f_0 = \frac{1}{2\pi\sqrt{LC}} \]

Step 3: Definition (II): power factor:
The power factor is the cosine of the phase angle \(\phi\) between the applied voltage and the current. It is also the ratio of true (average) power to apparent power.
\[ \cos\phi = \frac{R}{Z} = \frac{P_{avg}}{V_{rms} I_{rms}} \]

Step 4: Condition for maximum power:
The average power is \(P = V_{rms} I_{rms}\cos\phi\). For given \(V_{rms}\) and \(I_{rms}\), power is largest when \(\cos\phi = 1\).
That means \(\phi = 0\), that is, the circuit behaves as a pure resistor (as at resonance). The maximum power is dissipated when the power factor is 1.

Step 5: Part (ii): reactances:
From \(v = 140\sin(100\pi t)\), the peak voltage is \(V_0 = 140\) V and \(\omega = 100\pi\) rad/s.
\[ X_L = \omega L = 100\pi \times \frac{5}{\pi} = 500\ \Omega \]
\[ X_C = \frac{1}{\omega C} = \frac{1}{100\pi \times \dfrac{50}{\pi}\times 10^{-6}} = \frac{1}{5\times 10^{-3}} = 200\ \Omega \]
\[ X_L - X_C = 500 - 200 = 300\ \Omega \]

Step 6: Part (ii) (I): impedance:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{(400)^2 + (300)^2} = \sqrt{250000} = 500\ \Omega \]

Step 7: Part (ii) (II): rms current:
The rms voltage is \(V_{rms} = \dfrac{V_0}{\sqrt{2}} = \dfrac{140}{1.4} = 100\) V.
\[ I_{rms} = \frac{V_{rms}}{Z} = \frac{100}{500} = 0.2\ \text{A} \]

Final Answer:
Resonant frequency is the frequency at which \(X_L = X_C\). Power factor is \(\cos\phi = R/Z\). Power is maximum when the power factor is 1. For the given circuit, the impedance is 500 ohm and the rms current is 0.2 A. \[ \boxed{Z = 500\ \Omega,\quad I_{rms} = 0.2\ \text{A}} \]