Resonance
- Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedances or admittances of the circuit elements cancel each other.
- In series RLC circuit, resonance occurs when the inductive and capacitive reactanecs are equal in magnitude but cancel each other beause they are 180$^\circ$ apart in phase. \begin{align*} X_L &= X_C\\ 2 \pi f L &= \dfrac{1}{2 \pi fC}\\ f^2 &= \dfrac{1}{4 \pi^2 LC}\\ f &= \sqrt{\dfrac{1}{4 \pi^2 LC}}\\ f &= \dfrac{1}{2 \pi \sqrt{LC}}\\ \omega &= \dfrac{1}{\sqrt{LC}} \end{align*}
- Quality factor, Q is a quantitative measure of sharpness of the peak, which relates to the maximum or peak energy stored in the circuit (the reactance) to the energy dissipated (the resistance) during each cycle of oscillation. In other words, it is the ratio of resonant frequency to bandwidth.
- Higher the quality factor, the smaller is the bandwidth. \[Q = \dfrac{f_r}{BW}\]
- In parallel resonance, the total admittance is given by, \begin{align*} Y_{total} &= Y_1 + Y_2 + Y_3\\ &= \dfrac{1}{R} + \dfrac{1}{j \omega L} + \dfrac{1}{\dfrac{-j}{\omega C}}\\ &= \dfrac{1}{R} + \dfrac{-j}{\omega L} + j\omega C\\ &= \dfrac{1}{R} + j(\omega C - \dfrac{1}{\omega L}) \end{align*} Resonance occurs when, \[\omega C = \dfrac{1}{\omega L}\]