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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]

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