Series RLC resonance, half power, bandwidth

Resonance in a series RLC circuit is the condition in which the inductive reactance equals the capacitive reactance (X_L = X_C), so the net reactance is zero, the impedance is purely resistive (Z = R) and the supply voltage and current are in phase.

Resonance condition

Z = R + j(ωL − 1/ωC)
Resonance: ωL = 1/ωC
ω0 = 1/√(LC) rad/s      f0 = 1/(2π√(LC)) Hz

At resonance: Z is minimum (= R), current is maximum I0 = V/R, power factor = 1, and V_L = V_C in magnitude (equal and opposite), each equal to Q·V.

 I
 I0 |          ***
    |         *   *
I0/√2|- - - -*- - - *
    |       *|     |*
    |     **|     | **
    |_____*_|_____|_*____ ω
          ω1  ω0  ω2
          |<-BW->|

Half power frequencies

The frequencies at which the power delivered to the circuit falls to half of its maximum value (P = I0²R/2) are called half power frequencies (ω1 lower, ω2 upper). At these points the current is I0/√2 = 0.707 I0, so |Z| = √2 R. This needs the net reactance to equal the resistance:

|X| = |ωL − 1/ωC| = R
Above resonance:  ωL − 1/ωC = +R  →  ω²LC − ωRC − 1 = 0
Below resonance:  ωL − 1/ωC = −R  →  ω²LC + ωRC − 1 = 0

Taking the positive roots:

ω2 = R/2L + √[(R/2L)² + 1/LC]
ω1 = −R/2L + √[(R/2L)² + 1/LC]

Also ω1·ω2 = 1/LC = ω0², so ω0 is the geometric mean of ω1 and ω2.

Bandwidth

Bandwidth is the width of the band of frequencies between the half power points.

BW = ω2 − ω1 = R/L  rad/s
BW = f2 − f1 = R/(2πL)  Hz

Using the quality factor Q = ω0L/R = 1/(ω0CR):

BW = ω0/Q     (or f0/Q in Hz)

For high Q (Q > 10): ω1 ≈ ω0 − BW/2 and ω2 ≈ ω0 + BW/2.

Answer: resonance at ω0 = 1/√(LC); half power frequencies ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC]; bandwidth BW = R/L rad/s = ω0/Q.