LC Oscillations, Quantitatively — Question 5

Question 5

Component tolerances combine through the square root of the LC product

A nominal LC oscillator uses positive L0 and C0. Actual values are $L=L_0(1+\epsilon_L)$ and $C=C_0(1+\epsilon_C)$, with independent bounds $|\epsilon_L|\le.02$ and $|\epsilon_C|\le.03$.

1. Derive the first-order fractional frequency error.

2. Find exact worst-case high and low frequency ratios.

Question 5 - Solution

Component tolerances combine through the square root of the LC product

(a) Method and calculation

$$\omega/\omega_0=[(1+\epsilon_L)(1+\epsilon_C)]^{-1/2}.$$$$\boxed{\Delta\omega/\omega_0\simeq-\tfrac12(\epsilon_L+\epsilon_C)}.$$

(b) Interpretation and further calculation

$$\omega_{\max}/\omega_0=(.98\cdot.97)^{-1/2},\quad\omega_{\min}/\omega_0=(1.02\cdot1.03)^{-1/2}.$$$$\boxed{\omega_{\max}/\omega_0=1.02565,\quad\omega_{\min}/\omega_0=.975621}.$$

Physical check

Worst-case bounded errors add; a root-sum-square rule would require a statistical error model.