Alternating Current Set-2

Q1. A sinusoidal source $v(t)=200\sqrt{2}\sin(\omega t)\ \text{V}$ (so $V_{\rm rms}=200\ \text{V}$) is applied to a series circuit containing $R=50\ \Omega$ and an inductor whose reactance at that frequency is $X_L=50\ \Omega$. Calculate (i) the RMS current in the circuit and (ii) the average power dissipated in the resistor.

Q2. In a series RLC circuit driven at its resonance frequency with $R\ll X_L,X_C$, it is often observed that the magnitudes of voltages across the inductor and capacitor are much larger than the source voltage though they cancel in phasor sum. Which statement best explains this observation?

Q3. The instantaneous voltage and current for a load are given by $v(t)=220\sqrt{2}\sin(\omega t)\ \text{V}$ and $i(t)=10\sqrt{2}\sin(\omega t+\pi/3)\ \text{A}$. From these waveforms (phase lead of current by $\pi/3$), the average real power absorbed by the load is

Q4. Assertion (A): The average power absorbed by an ideal capacitor connected to a sinusoidal source over one complete cycle is zero.
Reason (R): In an ideal capacitor the current leads the voltage by $\pi/2$, so the instantaneous power $p(t)$ is symmetric with equal positive and negative areas over a cycle.

Q5. A plant draws $P=1200\ \text{W}$ from a $200\ \text{V}$ (rms) AC supply with lagging power factor $\cos\phi_1=0.60$. A capacitor bank is added in parallel to improve the power factor to $\cos\phi_2=0.90$ (lagging). Determine approximately (i) the reactive power $Q_C$ (in VAR) that the capacitor must supply and (ii) the new supply current $I_{\rm new}$ (rms, in A).

Q6. A series $R$–$L$ circuit is connected to $v(t)=120\sqrt{2}\sin(\omega t)\ \text{V}$. The measured RMS current is $I_{\rm rms}=2.0\ \text{A}$ and the current lags the applied voltage by $30^\circ$. Determine the numerical values of $R$ and $X_L$ (in $\Omega$).

Q7. A series RLC circuit shows resonance at $f_0=1.00\ \text{kHz}$. The amplitude of current falls to $I_0/\sqrt{2}$ at $f_1=950\ \text{Hz}$ and $f_2=1050\ \text{Hz}$. If the inductance is $L=0.10\ \text{H}$, estimate (i) the quality factor $Q$ and (ii) the series resistance $R$ of the circuit (use $Q=\dfrac{f_0}{f_2-f_1}$ and $Q=\dfrac{\omega_0 L}{R}$).

Q8. An inductive load has impedance $Z=10+j100\ \Omega$ (i.e. $R=10\ \Omega,\ X_L=100\ \Omega$) across a $220\ \text{V}$ (rms) supply. A capacitor with reactance $X_C=100\ \Omega$ is inserted in series with the load (same frequency). Estimate the RMS supply current before and after adding the capacitor, and the RMS voltage across the inductor after adding the capacitor.

Q9. Assertion (A): The power factor of a purely inductive AC circuit is zero.
Reason (R): The inductive reactance $X_L=\omega L$ increases with frequency $\omega$.

Q10. A series circuit has $R=2\ \Omega$, $L=0.10\ \text{H}$ and $C=10\ \mu\text{F}$. The source is $v(t)=100\sqrt{2}\sin(\omega t)\ \text{V}$ and the circuit is driven at its resonance angular frequency $\omega_0$. Calculate (i) the quality factor $Q$ and (ii) the peak (amplitude) voltage across the inductor at resonance.