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