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Resonance

 

Series Resonance
A series circuit comprising an inductance L, a resistance R and a capacitance C has an impedance ZS of:
ZS = R + j(XL - XC)
where XL =
w
L and XC = 1 / wC

At resonance, the imaginary part of ZS is zero:
XC = XL
ZSr = R
wr = (1 / LC)½ = 2pfr
The quality factor at resonance Qr is:
Qr =
wrL / R = (L / CR2)½ = (1 / R )(L / C)½ = 1 / wrCR

Parallel resonance
A parallel circuit comprising an inductance L with a series resistance R, connected in parallel with a capacitance C, has an admittance YP of:
YP = 1 / (R + jXL) + 1 / (- jXC) = (R / (R2 + XL2)) - j(XL / (R2 + XL2) - 1 / XC)
where XL =
wL and XC = 1 / wC

At resonance, the imaginary part of YP is zero:
XC = (R2 + XL2) / XL = XL + R2 / XL = XL(1 + R2 / XL2)
ZPr = YPr-1 = (R2 + XL2) / R = XLXC / R = L / CR
wr = (1 / LC - R2 / L2)½ = 2pfr
The quality factor at resonance Qr is:
Qr =
wr
L / R = (L / CR2 - 1)½ = (1 / R )(L / C - R2)½

Note that for the same values of L, R and C, the parallel resonance frequency is lower than the series resonance frequency, but if the ratio R / L is small then the parallel resonance frequency is close to the series resonance frequency.