The Series RLC Circuit — Question 10

Question 10

High quality factor makes the current sensitive to small frequency errors

For a series RLC circuit let $\mathcal Q=\omega_0L/R$, $\omega=\omega_0(1+\epsilon)$, and $|\epsilon|\ll1$. The drive voltage amplitude is fixed.

1. Derive the leading detuning dependence of current relative to its resonant value.

2. For quality factor 25, estimate the allowable fractional frequency error if current must remain at least .900 of its resonant value.

Question 10 - Solution

High quality factor makes the current sensitive to small frequency errors

(a) Method and calculation

$$\frac XR=\mathcal Q\left[(1+\epsilon)-\frac1{1+\epsilon}\right]\simeq2\mathcal Q\epsilon.$$$$\boxed{I/I_0\simeq[1+4\mathcal Q^2\epsilon^2]^{-1/2}}.$$

(b) Interpretation and further calculation

$$I/I_0\ge.900\Rightarrow|\epsilon|\lesssim\frac{\sqrt{.900^{-2}-1}}{2\mathcal Q}.$$$$\boxed{|\epsilon|\lesssim.00968644\quad(\mathcal Q=25)}.$$

Physical check

The approximation retains a possibly order-one product of quality factor and a small fractional detuning.