## Resonance

**
Series
Resonance
A series circuit comprising an inductance L, a resistance R and a
capacitance C has an impedance Z_{S} of:
Z_{S} = R + j(X_{L} - X_{C})
where X_{L} = wL
and X_{C} = 1 / wC **

At resonance, the imaginary part of Z_{S}
is zero:

X_{C} = X_{L}

Z_{Sr} = R

w_{r}
= (1 / LC)^{½} = 2pf_{r}

The quality factor at resonance Q_{r} is:

Q_{r} =
w_{r}L
/ R = (L / CR^{2})^{½} = (1 / R )(L / C)^{½} = 1 /
w_{r}CR

Parallel resonance

A parallel circuit comprising an inductance L with a series resistance R,
connected in parallel with a capacitance C, has an admittance Y_{P}
of:

Y_{P} = 1 / (R + jX_{L}) + 1 / (- jX_{C}) = (R / (R^{2}
+ X_{L}^{2})) - j(X_{L} / (R^{2} + X_{L}^{2})
- 1 / X_{C})

where X_{L} =
wL
and X_{C} = 1 /
wC

**
At resonance, the imaginary part of Y _{P}
is zero:
X_{C} = (R^{2} + X_{L}^{2}) / X_{L}
= X_{L} + R^{2} / X_{L} = X_{L}(1 + R^{2}
/ X_{L}^{2})
Z_{Pr} = Y_{Pr}^{-1} = (R^{2} + X_{L}^{2})
/ R = X_{L}X_{C} / R = L / CR
w_{r}
= (1 / LC - R^{2} / L^{2})^{½} = 2pf_{r}
The quality factor at resonance Q_{r} is:
Q_{r} =
w_{r}L
/ R = (L / CR^{2} - 1)^{½} = (1 / R )(L / C - R^{2})^{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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