Alternating Current Set-3

Q1. A sinusoidal voltage $v(t)=311\sin(100\pi t)\ \text{V}$ is applied across a resistor $R=50\ \Omega$. What is the average power dissipated in the resistor?

Q2. A series circuit contains $R=10\ \Omega$, $L=50\ \text{mH}$ and $C=50\ \mu\text{F}$. The supply has peak voltage $V_0=100\ \text{V}$ at angular frequency $\omega=1000\ \text{rad/s}$. At this $\omega$ the inductive reactance $X_L$ and capacitive reactance $X_C$ give a net reactance $X=X_L-X_C$. What fraction of the average power dissipated in the resistor at resonance is dissipated at $\omega=1000\ \text{rad/s}$?

Q3. Assertion (A): For a series RLC circuit at resonance driven by a sinusoidal source of fixed amplitude $V_0$, the average power absorbed by the circuit is independent of the resistance $R$.
Reason (R): At resonance the current amplitude is $I_{\text{res}}=V_0/R$, so the average power $P=V_0^2/(2R)$ which depends on $R$.

Q4. An AC source of rms voltage $100\ \text{V}$ at $50\ \text{Hz}$ is connected to a coil. Measured rms current is $2.0\ \text{A}$ and measured average power dissipated in the coil is $120\ \text{W}$. Treating the coil as a series combination of $R$ and $X_L$, determine $R$ and $X_L$ (at $50\ \text{Hz}$).

Q5. The instantaneous waveforms are given by $v(t)=100\sqrt{2}\sin(\omega t)\ \text{V}$ and $i(t)=5\sqrt{2}\sin(\omega t-\tfrac{\pi}{3})\ \text{A}$. Using the waveforms, what is the average power absorbed by the circuit?

Q6. A periodic voltage $v(t)=100\sin\omega t+60\sin3\omega t$ (peak values shown) is applied across a resistor $R=10\ \Omega$. Assuming no DC component and that different harmonics are orthogonal over a period, the average power dissipated in the resistor is:

Q7. A single-phase load consumes $5.0\ \text{kW}$ at a power factor of $0.80$ (lagging) from a $230\ \text{V}$, $50\ \text{Hz}$ supply. What capacitance (in $\mu\text{F}$) must be connected in parallel with the load to improve the power factor to unity?

Q8. Assertion (A): If a periodic voltage and current are expressed as Fourier series (only sinusoidal harmonics, no DC), the average power delivered to a linear circuit equals the sum of average powers contributed by each harmonic (no cross terms).
Reason (R): Different sinusoidal harmonics are orthogonal over a period, so the time average of the product of two different-frequency sinusoids is zero, eliminating cross terms.

Q9. A sinusoidal voltage crosses zero with positive slope at $t=0$. The sinusoidal current crosses zero with positive slope at $t=2.0\ \text{ms}$. If the frequency is $50\ \text{Hz}$, the current

Q10. A series RLC circuit has $R=10\ \Omega$, $L=0.20\ \text{H}$ and $C=50\ \mu\text{F}$. The bandwidth $\Delta\omega$ is defined as the difference between the angular frequencies where the power falls to half its maximum (half-power points). The bandwidth $\Delta\omega$ (in rad/s) equals: