### Formulas

#### 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 / wCAt 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 / wCAt 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.

NOTATION |
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BCEfGhIjLPQ |
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] |
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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] |