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}} $ |
| Mechanical Oscillation | Mechanics and Oscillation | Types of Oscillation |