The following is a LCR series circuit (L stands for inductor, C for capacitor, and R for resistor).

The inductive reactance is

` ````
X
```_{L} = ωL

The capacitive reactance is

` ````
X
```_{C} = 1/(ωC)

The impedance is

` ````
Z = R +j(X
```_{L}-X_{C})

The impedance is at a minimum when `X`

which happens when
_{L}=X_{C}

` ````
ωL = 1/(ωC)
ω
```^{2}L = 1/C
ω^{2} = 1/(LC)
ω = 1/√(LC)

This is known as the *resonant frequency*.
At this point the instantaneous current
is at its maximum and in phase with the instantaneous voltage.

The bandwidth — the range of frequencies either side of `ω`

where the instantaneous current is above the
cut-off frequency — is

` ````
Δω = R/L
```

The *Q factor* gives us an indication of the sharpness of the current peak.
A high Q factor is a sharp peak, and a low Q factor is a broad peak.
This can be calculated as follows:

` ````
Q = ωL/R
= 1/(RωC) [ωL = 1/(ωC)]
= 1/R×√(L/C)
```

The *selectivity* of the circuit is given by the Q factor.
Also, at the resonant frequency:

` ````
V
```_{C} = V_{L} = QV

Fischer-Cripps. A.C., *The Electronics Companion.* Institute of Physics, 2005.

Copyright © 2014 Barry Watson. All rights reserved.