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\begin{document}
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\begin{titlepage} \setcounter{page}{0} \vspace{2.5cm}
\begin{center}
\Large\bf{Inductor (L), Capacitor (C), Resistor (R)\\ 
          \vspace{0.5cm} Circuit (LCR) Notes}\\
\vspace{2.0cm} \large{by}\\ \vspace{2.0cm}
\large{\bf Bernd A. Berg }\\ \vspace{2.0cm}
{\bf Department of Physics}\\ \vspace{0.4cm}
{\bf Florida State University}\\ \vspace{0.4cm}
{\bf Tallahassee, FL~32306, USA.}\\ \vspace{0.4cm}
{(Version March 6, 2011)} \vspace{2.0cm} \vfill

{\bf Copyright \copyright\ by the author.}\\

\end{center}

\end{titlepage}

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\chapter{}\label{LCR}

\section{LCR circuit}

We consider a \index{Resistor}resistor $(R)$, a\index{Capacitor}
capacitor $(C)$ and an \index{Inductor}inductor $(L)$ in series, 
see \index{LCR circuit}\index{Circuit}figure~\ref{fig_LCR}. Charge 
conservation implies
\begin{equation} \label{currents}
 I_R\ =\ I_C\ =\ I_L\ = I
\end{equation}
where $I_R$, $I_C$ and $I_L$ are the currents at the resistor,
%
\begin{figure}[hb] \vspace{5pc}
\centerline{\hbox{ \psfig{figure=fig/LCR.eps,width=10cm} }}
\caption{Series LCR circuit with an ac generator.\label{fig_LCR}}
\end{figure}
%
capacitor and inductor, respectively. The voltages are related to
the currents by the following equations
\begin{equation} \label{V_RC}
V_R\ =\ R\,I ~~{\rm and}~~ V_C\ =\ {Q\over C} ~~{\rm with}~~
{d Q\over d t}\ =\ I
\end{equation}
where $Q(t)$ is the charge on the capacitor. As inductor we consider 
an idealized solenoid (i.e., we neglect boundary effects which 
distinguish a real solenoid from an infinitely long one). The 
induced (back) electromotive force (voltage over the inductor) is 
the derivative of the magnetic flux
\begin{equation} \label{V_L}
  \epsilon_{\rm back}\ =\ V_L\ =\ \frac{1}{c}{d\Phi_m\over dt}\ 
  =\ L\,{d I\over d t}
\end{equation}
where the constant $L>0$ is called \index{Self inductance}{\it self 
inductance}\index{Inductance} of the coil. In accordance with Lenz's 
law the induced electromotive 
$\epsilon_{\rm back}$ will work against the electromotive force 
$\epsilon_{\rm emf}$ which drives the circuit.  We use complex 
calculus and the physical quantities are given by the real parts. 
For the total voltage we have
\begin{equation} \label{emf}
  \epsilon_{\rm emf}\ =\ \epsilon_{\max}\, e^{-i\omega t}\
  =\ V\ =\ V_L+V_R+V_C 
\end{equation}
where $\omega$ is the \index{Driving frequency}driving frequency. 
We replace in this relation the voltages by their equations listed 
above and eliminate the current $I$ in favor of the time derivative
of the charge of the capacitor and to arrive at the second order 
differential equation ($-i\omega\,t$ is Jackson convention)
\begin{equation} \label{LCR_QDeqn}
L\, {d^2 Q\over d t^2} + R\, {d Q\over d t} + {Q\over C}\ 
=\ \epsilon_{\max}\, e^{ - i \omega t}\ . 
\end{equation}
The general solution of this equation is obtained as superposition 
of a special solution of the inhomogenous equation and the general 
solution of the homogeneous equation, $Q=Q_h+Q_i$. A special solution 
of the inhomogeneous equation is
\begin{equation} \label{Q_i}
 Q_i\ =\ Q_0\, e^{ - i \omega t} 
\end{equation}
where
\begin{equation} \label{Q_amplitude}
\left( -\omega^2\,L - i\,\omega\,R + {1\over C} \right)\, Q_0 
= \epsilon_{\max} ~~\Rightarrow~~ Q_0 =  { \epsilon_{\max}
   \over  -\omega^2\,L - i\,\omega\,R + 1/C }\ . 
\end{equation}
The general solution of the homogeneous equation follows from the
ansatz $Q_h\sim e^{i \alpha t}$, which implies the algebraic equation
$$   -\alpha^2\,L + i\,\alpha\,R + C^{-1}\ =\ 0\ . $$
The solution is
\begin{equation} \label{omega_h}
  \alpha\ =\ i\,\frac{R}{2L} \pm \omega_h ~~{\rm with}~~
  \omega_h = \sqrt{ {1\over LC} - {R^2\over 4L^2} }
\end{equation}
and we arrive at the general solution of the homogeneous equation
\begin{equation} \label{Q_h}
 Q_h\ =\ A_1\, e^{-t R / (2L)}\, e^{ +i\omega_h\,t} +
         A_2\, e^{-t R / (2L)}\, e^{- i\omega_h\,t}\ . 
\end{equation}
Due to the exponential damping the homogeneous solution dies out 
for large times and we are left with the inhomogeneous solution 
$Q=Q_i$~(\ref{Q_i}). The homogeneous solution has to be used to
get the initial values right, e.g.~for turning the circuit on or 
off. In the large time limit the essential features of the LCR 
circuit are those of the inhomogeneous solution.

For the inhomogeneous solution (\ref{Q_i}) $Q_0$ is a complex constant.
To separate the amplitude (magnitude) of $Q_0$ from its  phase we write
\begin{equation} \label{Q0}
 Q_0\ =\ Q_{\max}\, e^{-i\delta} ~~{\rm with}~~ Q_{\max}\ =\ |Q_0|\ 
 =\ {\epsilon_{\max}\over \sqrt{ \left( 1/ C-\omega^2\,L \right)^2 
 + \omega^2\,R^2 } }\ . 
\end{equation}
Following standard conventions, we rewrite the expression for
$Q_{\max}$ as
\begin{equation} \label{impedance_etc}
 Q_{\max}\ =\ {\epsilon_{\max}\over \omega\, \sqrt{ 
 \left[ 1/(\omega\,C) - \omega\,L \right]^2 + R^2 } }\ =\ 
 {\epsilon_{\max}\over \omega\, \sqrt{(X_C-X_L)^2 + R^2 } }\ =\
 {\epsilon_{\max} \over \omega\, Z}\ .
\end{equation}
Here\index{Capacitive reactance}
\begin{eqnarray} \label{X_C}
  X_C &=& \frac{1}{\omega\,C}~~{\rm is\ called}\ 
                   {\it capacitive\ reactance,} \\ \label{X_L}
  X_L &=& \omega\,L~~{\rm is\ called}\ {\it inductive\ reactance,}
\end{eqnarray}
and\index{Reactance}\index{Inductive reactance} the difference 
$X_L-X_C$ is called \index{Total reactance}{\it total reactance}. 
The quantity\index{Impedance}
\begin{equation} \label{Z_impedance}
 Z = \sqrt{(X_C-X_L)^2+R^2}~~{\rm is\ called}\ {\it impedance}.  
\end{equation} 
Using this notation the original equation (\ref{Q_amplitude}) for 
$Q_0$ becomes 
\begin{equation}
  Q_0\,=\,\frac{\epsilon_{\max}}{\omega\,\left(X_C-X_L-iR\right)}
  \,=\,{\epsilon_{\max}\,(X_C-X_L+i\,R)\over\omega\,Z^2}\ .
\end{equation}
It follows from $Q_0=Q_{\max}\,\cos(\delta)-i\,Q_{\max}\,\sin(\delta)$
that the phase $\delta$ is given by\footnote{Note that in Tipler,
Physics 1995 edition, the phase $\delta$ is defined with respect to 
the current instead of the charge, what leads to the equation 
$\delta=(X_L-X_C)/R$.}
\begin{equation}
  \tan {\delta}\ =\ - {\sin (\delta )\over \cos (\delta )}\ =\
  - {{\rm Im}\,Q_0\over {\rm Re}\,Q_0}\ =\ \frac{R}{X_L-X_C}\ . 
\end{equation}
From equation~(\ref{V_RC}) we see that $V_C$ is in phase with the
charge $Q$ 
\begin{equation}
  V_C = {Q_{\max}\over C}\, e^{-i\delta}\, e^{-i\omega t}
\end{equation}
and the phase of the current $I$ is shifted by $\pi/2$ with respect 
to $V_C$:
\begin{eqnarray} \nonumber
  I\,=\,{d Q\over d t}\,=\,-i\,\omega\,Q_{\max}\,e^{-i\delta}\,
  e^{-i\omega t} &=& \omega\,Q_{\max}\, e^{-i\delta-i\pi/2}\, 
  e^{-i\omega t} \\ \label{current_phase}
  &=& I_{\max}\, e^{-i\delta-i\pi/2}\, e^{-i\omega t}
\end{eqnarray}
where we used $-i=\exp(-i\pi/2)$. Note also that $I_{\max} = 
\omega\, Q_{\max}$ together with equation~(\ref{Q0}) implies
\begin{equation} \label{Imax}
  I_{\max}\,=\,\frac{\epsilon_{\max}}{Z}\,=\,{\epsilon_{\max}\over
  \sqrt{\left[ 1/(\omega\,C)-\omega\,L \right]^2 + R^2 } }\ .
\end{equation}
The voltage over the resistor is in phase with the current
\begin{equation}
V_R\ =\ {I_{\max}\over R}\,e^{-i\delta-i\pi/2}\,e^{-i\omega t}
\end{equation}
whereas the voltage over the inductor is shifted by another $\pi/2$ 
phase factor:
\begin{equation}
  V_L\ =\ L\, {d I\over d t}\ =\ \omega\,L\,I_{\max}\,
  e^{-i\delta t-i\pi}\,e^{i\omega t}\ .
\end{equation}
The current amplitude (\ref{Imax}) takes its maximum at the
\index{Resonance frequency}{\it resonance frequency} 
\begin{equation} \label{omega_resonance}
 \omega _0\ =\ {1 \over \sqrt{LC} }\ .
\end{equation}
Note that $Q_{\max}(\omega_0)=\epsilon_{\max}/(\omega_0 R)$ due
to equation~(\ref{impedance_etc}). For $R\ne 0$ the maximum of 
$Q_{\max}$ is shifted by an amount of order $(R/L)^2$ away from 
$\omega_0$. We find it by differentiation of $Q_{\max}$ with 
respect to $\omega^2$:
\begin{equation} \label{omega_max}
 - 2\,L\,( C^{-1}-\omega^2_{\max}\,L)+R^2=0 ~~\Longrightarrow~~
 \omega_{\max} = \sqrt{ {1\over LC} - {R^2\over 2\,L^2} }\ .
\end{equation}
Note, that this $\omega_{\max}$ value is lower than the frequency 
$\omega_h$ of the damped homogeneous solution~(\ref{Q_h}), while
the resonance frequency $\omega_0$ is larger.

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