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# Resonance

### Formulas

#### 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 = wL 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 = wrL / 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.

 NOTATION The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. B C E f G h I j L P Q susceptance capacitance voltage source frequency conductance h-operator current j-operator inductance active power reactive power [siemens, S] [farads, F] [volts, V] [hertz, Hz] [siemens, S] [1Ð120°] [amps, A] [1Ð90°] [henrys, H] [watts, W] [VAreactive, VArs] Q R S t V W X Y Z f w quality factor resistance apparent power time voltage drop energy reactance admittance impedance phase angle angular frequency [number] [ohms, W] [volt-amps, VA] [seconds, s] [volts, V] [joules, J] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [rad/sec]