Resonators, general susceptibility, and AC effects

General resonator

Assume the driving force have the form $F(t) = F_0 e^{-i \omega t}$

Equation of motion

$\ddot{x}(t)+\eta \dot{x}(t)+\omega_{0}^{2} x(t)=\frac{F_{0}}{m} e^{-i \omega t}$

with a response of form $x(t) = \tilde x_0 e^{-i\omega t}$ , we get

$\tilde{x}*{0}(\omega)=\frac{F*{0} / m}{\omega_{0}^{2}-\omega^{2}-i \eta \omega}$

General susceptibility is defined as

$\chi_{n}(\omega) = \frac{\tilde x_0(\omega_0)}{\tilde x_0(\omega)}=\frac{\omega_{0}^{2}}{\left(\omega_{0}^{2}-\omega^{2}\right)-i \eta \omega}$

Imaginary part ( $\chi''$ )
- Symmetric, gaussian function
- FWHM
Real part ( $\chi'$ )
- Antisymmetric
- $\chi_n \to 1$ when $\omega \to 0$
- $\chi_n \to 0$ when $\omega \to \infty$

Q-factor relate to the imaginary part, or the $\eta$ term=

The phase of the response is deﬁned by

$\tilde{\chi}*{n}=\left|\chi*{n}\right| e^{i \delta} \quad \tan \delta=\frac{\chi^{\prime \prime}}{\chi^{\prime}}$

$\tilde{R}=\tilde{\chi} \tilde{S}$

$P(t)=S(t) \frac{d R(t)}{d t}$

$P(t)$ measures the power absorbed per unit space.

$\langle P\rangle=\frac{S_{0}^{2} \omega}{2} \chi^{\prime \prime}$