Sharpness of resonance, forced e-m oscillation

Sharpness of resonance

It describes how rapidly the current amplitude (or any response) drops when the driving frequency moves away from the resonant frequency. Small damping (small R) gives a sharp resonance. It is measured by the quality factor Q = ω0L/R = ω0/Δω, where Δω is the half-power bandwidth.

Current amplitude of forced e.m. oscillation

A series L, C, R circuit with applied emf E = E0 cos ωt.

 ~E0 cos ωt ---- L ---- R ---- C ---+
   |_________________________________|

1. Equation: L d²q/dt² + R dq/dt + q/C = E0 cos ωt.
2. In the steady state the charge oscillates at the driving frequency: q = q0 cos(ωt − φ).
3. Substitute, write cos ωt = cos(ωt − φ)cosφ − sin(ωt − φ)sinφ, and equate coefficients:

q0 (1/C − Lω²) = E0 cosφ
q0 R ω         = E0 sinφ

4. Square and add:

q0 = E0 / {ω √[R² + (ωL − 1/ωC)²]}

5. The current is i = dq/dt = −ωq0 sin(ωt − φ), so the current amplitude is I0 = ωq0:

I0 = E0 / √[R² + (ωL − 1/ωC)²] = E0/Z

The current lags the emf by ψ = φ − π/2, where tanψ = (ωL − 1/ωC)/R. 6. I0 is maximum (E0/R) when ωL = 1/ωC, i.e. ω = 1/√(LC). The smaller R is, the taller and sharper the peak of I0 against ω.