Electromagnetic Oscillation

$U_C$ $\mathrm{V}$ voltage at the capacitor
$U_L$ $\mathrm{V}$ voltage at the inductor
$I$ $\mathrm{A}$ current
$L$ $\mathrm{H, Vs/A}$ inductance
$C$ $\mathrm{F, As/V}$ capacitance
$Q$ $\mathrm{As}$ electric charge
$q$ $\mathrm{As}$ momentary charge
$u$ $\mathrm{V}$ momentary voltage
$i$ $\mathrm{A}$ momentary current
$\varphi$ $\mathrm{rad}$ phase angle
$\omega_0$ $\mathrm{s^{-1}}$ angular eigenfrequency
$f$ $\mathrm{Hz}$ eigenfrequency
$T$ $\mathrm{s}$ period duration
$R$ $\mathrm{\Omega}$ ohmic resistance
$\delta$ $\mathrm{s^{-1}}$ decay coefficient
$\omega$ $\mathrm{s^{-1}}$ eigenfrequency of damped oscillation
$G$   quality
$D$   damping ratio

Undamped Electromagnetic Oscillation

$$ U_C = \frac{Q}{C} \qquad
U_L = – L \, \frac{\mathrm{d}I}{\mathrm{d}t} $$

$$ – \frac{Q}{C} – L \, \frac{\mathrm{d}I}{\mathrm{d}t} = 0 $$

Differential equation:

$$ \ddot{q} + \frac{q}{L \, C} = 0 $$

$$ q(t) = \hat{q} \, \sin\left( \omega_0 \, t + \varphi_0 \right) \qquad
q(t) = \hat{q} \, \sin \varphi $$

$$ u(t) = \hat{u} \, \sin\left( \omega_0 \, t + \varphi_0 \right) \qquad
u(t) = \hat{u} \, \sin \varphi $$

$$ i(t) = \hat{i} \, \sin\left( \omega_0 \, t + \varphi_0 – \frac{\pi}{2} \right) \qquad
i(t) = \hat{i} \, \sin\left( \varphi – \frac{\pi}{2} \right) $$

$$ T = 2 \, \pi \, \sqrt{L \, C} $$

$$ I = \frac{\mathrm{d}Q}{\mathrm{d}t} \qquad
L = \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\mathrm{d}Q}{\mathrm{d}t} \right) = L \, \ddot{Q} $$

Damped Electromagnetic Oscillation

Differential equation:

$$ \ddot{q} + \frac{R}{L} \, \dot{q} + \frac{1}{L \, C} \, q = 0 $$

$$ q(t) = \hat{q} \, \mathrm{e}^{-\delta\,t} \, \sin\left( \omega_0 \, t + \varphi_0 \right) $$

$$ \delta = \frac{R}{2 \, L} \qquad
\omega = \sqrt{ \omega_0^2 + \delta_0^2 } \qquad
\omega = \sqrt{ \frac{1}{L \, C} – \left( \frac{R}{2 \, L} \right)^2 } $$

$$ G = \frac{1}{R} \, \sqrt{ \frac{L}{C} } \qquad
G = \frac{1}{2 \, D} $$

$$ D = \frac{R}{2} \, \sqrt{ \frac{C}{L} } \qquad
D = \frac{\delta}{\omega_0} $$

Relationship Mechanical – Electromagnetic Oscillation

Mechan.: deflection, elongation $x$
Electr.: charge of capacitor $q$
     
Mechan.: velocity $v = \dot{x}$
Electr.: current $i = \dot{q}$
     
Mechan.: mass $m$
Electr.: inductance $L$
     
Mechan.: spring constant $k$
Electr.: 1 / capacitance $1/C$
     
Mechan.: damping ratio $b$
Electr.: resistance $R$

Potential Energy

Mechan.: $ E_{POT} = \displaystyle{\frac{k}{2} \, x^2} $   $E_{POT} = \displaystyle{\frac{k \, x^2}{2}} \, \sin^2 \varphi $
Electr.: $ E_{EL} = \displaystyle{\frac{q^2}{2 \, C}} $    

Kinetic Energy

Mechan.: $ E_{KIN} = \displaystyle{\frac{m}{2} \, v^2} $   $ E_{KIN} = \displaystyle{\frac{m \, v^2}{2} \, \cos^2 \varphi} $
Electr.: $ E_{MAG} = \displaystyle{\frac{L \, i^2}{2}} $    

Undamped Oscillation

Mechan.: $ x = \hat{x} \, \sin \left( \omega_0 \, t + \varphi_0 \right) $   $ \omega_0 = \displaystyle{\sqrt{\frac{k}{m}}} $
Electr.: $ q = \hat{q} \, \sin \left( \omega_0 \, t + \varphi_0 \right) $   $ \omega_0 = \displaystyle{\sqrt{\frac{1}{L \, C}}} $

Damped Oscillation

Mechan.: $ x = \hat{x} \, \mathrm{e}^{-\delta\,t} \, \sin \left( \omega_0 \, t + \varphi_0 \right) $   $ \omega_0 = \displaystyle{\sqrt{\omega_0^2 + \delta_0^2}} $   $ \delta = \displaystyle{\frac{b}{2 \, m}} $
Electr.: $ q = \hat{q} \, \mathrm{e}^{-\delta\,t} \, \sin \left( \omega_0 \, t + \varphi_0 \right) $   $ \omega_0 = \displaystyle{\sqrt{\omega_0^2 + \delta_0^2}} $   $ \delta = \displaystyle{\frac{R}{2 \, L}} $