An LRC oscillator circuit consists of a 50-$\varOmega$ resistor and a 100 nF capacitor.What inductance is necessary for the decay rate of the circuit (the 1/e value)to be equal to the oscillation period?

Question

Quizplus · Accepted Answer

For an LRC circuit, the decay rate (1/e value) being equal to the oscillation period implies that the damping is critical. The formula for the oscillation period $T$ in an underdamped LRC circuit is given by $T = 2\pi\sqrt{LC}$, where $L$ is the inductance and $C$ is the capacitance. The decay rate $\alpha$ is given by $\alpha = R/(2L)$, where $R$ is the resistance. Setting $T = 1/\alpha$, we get $2\pi\sqrt{LC} = 2L/R$. Rearranging for $L$ and substituting $R = 50\ \Omega$ and $C = 100\ nF = 100 \times 10^{-9}\ F$, we solve for $L$. Solving this equation gives us the correct inductance value necessary for the decay rate to equal the oscillation period, which matches the value for option C, 9.93 mH.

An LRC Oscillator Circuit Consists of a 50- Ω\varOmegaΩ Resistor and a 100 NF Capacitor

An LRC Oscillator Circuit Consists of a 50- $\varOmega$ Resistor and a 100 NF Capacitor