RMS Values of AC
For sinusoidal AC: $I = I_0 \sin(\omega t)$:
$$I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0, \quad V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0$$
Series LCR Circuit & Resonance
- Impedance ($Z$):
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
Where $X_L = \omega L$ and $X_C = \frac{1}{\omega C}$.
Resonant Frequency ($\omega_0$):
Occurs when $X_L = X_C \implies Z = R$ (minimum impedance, maximum current):
$$\omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi\sqrt{LC}}$$
Power in AC Circuit:
$$P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi \quad (\cos\phi = \frac{R}{Z} \text{ is Power Factor})$$
