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.
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| 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] | |
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Resonansi « MAGATRIKA
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