Lab Manual: Limits of Detection

Overview

Resolution limit

Second order system

Second order circuit with alternating current source and parallel resistor, capacitor, and inductor.

$ \dfrac{1}{Z_{eq}}=\dfrac{1}{Z_R}+\dfrac{1}{Z_L}+\dfrac{1}{Z_C} $

$ Z_{eq}=\frac{Z_R Z_L Z_C}{Z_R Z_L + Z_R Z_C + Z_L Z_C} $

$ Z_{eq}=\frac{\hat{V}_o(s)}{\hat{I}_{in}(s)}=\frac{RL/C}{RLs+\dfrac{R}{Cs}+\dfrac{L}{C}}=\frac{Ls}{LCs^2+\dfrac{L}{R}s+1} $

Mechanical circuit analogy

Ideal mechanical and electronic lumped elements.

$ \frac{\hat{V}_o(s)}{\hat{F}_{in}(s)}=\frac{\dfrac{1}{m}s}{s^2+ \dfrac{b}{m} s+\dfrac{k}{m} } $

Underdamped system: atomic force microscope

Overdamped system: laser tweezers

Optical trap procedure

AFM procedure