Editing LC circuit (section)

==<span class="anchor" id="LC_series_anchor">Series circuit</span>==
[[File:Series LC Circuit.svg|thumb|right|Series LC circuit]]

In the series configuration of the LC circuit, the inductor (L) and capacitor (C) are connected in series, as shown here. The total voltage {{mvar|V}} across the open terminals is simply the sum of the voltage across the inductor and the voltage across the capacitor. The current {{mvar|I}} into the positive terminal of the circuit is equal to the current through both the capacitor and the inductor.

: <math>\begin{align}
 V &= V_L + V_C, \\
 I &= I_L = I_C.
\end{align}</math>

=== Resonance ===
[[Inductive reactance]] <math>\ X_\mathsf{L} = \omega L\ </math> increases as frequency increases, while [[reactance (electronics)#Capacitive reactance|capacitive reactance]] <math>\ X_\mathsf{C} = \frac{1}{\ \omega C\ }\ </math> decreases with increase in frequency (defined here as a positive number). At one particular frequency, these two reactances are equal and the voltages across them are equal and opposite in sign; that frequency is called the resonant frequency {{math| ''f''<sub>0</sub> }} for the given circuit.

Hence, at resonance,

: <math>\begin{align}
 X_\mathsf{L} &= X_\mathsf{C}\ , \\
 \omega L &= \frac{ 1 }{\ \omega C\ } ~.
\end{align}</math>

Solving for {{mvar|ω}}, we have

: <math>\omega = \omega_0 = \frac{ 1 }{\ \sqrt{ L C\;}\ }\ ,</math>

which is defined as the resonant angular frequency of the circuit. Converting angular frequency (in radians per second) into frequency (in [[Hertz (unit)|Hertz]]), one has

: <math>f_0 = \frac{ \omega_0 }{\ 2 \pi\ } = \frac{ 1 }{\ 2 \pi \sqrt{ L C\;}\ }\ ,</math>
and
: <math>X_{\mathsf{L} 0} = X_{\mathsf{C} 0} = \sqrt{\frac{\ L\ }{C}\;}</math>
at <math>\omega_0</math>.

In a series configuration, {{mvar|X}}{{sub|C}} and {{mvar|X}}{{sub|L}} cancel each other out. In real, rather than idealised, components, the current is opposed, mostly by the resistance of the coil windings. Thus, the current supplied to a series resonant circuit is maximal at resonance.
* In the limit as {{math| ''f'' → ''f''<sub>0</sub> }} current is maximal. Circuit impedance is minimal. In this state, a circuit is called an ''acceptor circuit''<ref>{{cite web |title=What is an acceptor circuit? |website=qsstudy.com |series=Physics |url=http://www.qsstudy.com/physics/what-is-acceptor-circuit.html}} ].</ref>
* For &nbsp; {{math|''f''  <  ''f''<sub>0</sub>}} , &emsp; {{mvar|X}}{{sub|L}} {{math|≪ ''X''}}{{sub|C}}&nbsp;; &nbsp; hence, the circuit is capacitive.
* For &nbsp; {{math|''f''  >  ''f''<sub>0</sub>}} , &emsp; {{mvar|X}}{{sub|L}} {{math| ≫ ''X''}}{{sub|C}}&nbsp;; &nbsp; hence, the circuit is inductive.

===Impedance===
In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.

First consider the [[Electrical impedance|impedance]] of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:

: <math> Z = Z_\mathsf{L} + Z_\mathsf{C} ~.</math>

Writing the inductive impedance as {{mvar| Z}}{{sub|L}} {{math| {{=}} ''jωL'' }} and capacitive impedance as {{mvar| Z}}{{sub|C}} {{math| {{=}} {{sfrac|1| ''j ω C'' }} }} and substituting gives

:<math> Z(\omega) = j \omega L + \frac{ 1 }{\ j\omega C\ } ~.</math>

Writing this expression under a common denominator gives

:<math> Z(\omega) = j \left( \frac{\ \omega^2 L C - 1\ }{\omega C} \right) ~.</math>

Finally, defining the natural angular frequency as

:<math> \omega_0 = \frac{ 1 }{\ \sqrt{ L C \;}\ }\ ,</math>

the impedance becomes

:<math> Z(\omega) = j\ L\ \left( \frac{\ \omega^2 - \omega_0^2\ }{ \omega } \right) =  j\ \omega_0 L\ \left( \frac{ \omega }{\ \omega_0\ } - \frac{\ \omega_0\ }{ \omega } \right) =  j\ \frac{ 1 }{\ \omega_0 C\ } \left( \frac{ \omega }{\ \omega_0\ } - \frac{\ \omega_0\ }{ \omega } \right)\ ,</math>

where <math>\, \omega_0 L\ \,</math> gives the reactance of the inductor at resonance.

The numerator implies that in the limit as {{math| ''ω'' → ±''ω''<sub>0</sub> }}, the total impedance {{mvar| Z }} will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as a [[band-pass filter]] having zero impedance at the resonant frequency of the LC circuit.