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Second Order Systems

RCLParallelSeries.png

Consider the two circuits show above, the one on the left is an RCL series circuit with input $v(t)$ and output $i(t)$, while the one on the right is an RCL parallel circuit with input $i(t)$ and output $v(t)$. These two electrical systems are not equivalent, as they are described by different governing equations:

\begin{displaymath}
v_R(t)+v_L(t)+v_C(t)=R i(t)+L\frac{di(t)}{dt}+\frac{1}{C}\i...
...{R}{L} \dot{i}(t)+\frac{1}{LC} i(t)=\frac{1}{L}\dot{v}_0(t)
\end{displaymath}

and

\begin{displaymath}
i_R(t)+i_C(t)+i_L(t)=\frac{v(t)}{R}+C\frac{dv(t)}{dt}+\frac{...
...1}{RC} \dot{v}(t)+\frac{1}{LC} v(t)=\frac{1}{C}\dot{i}_0(t)
\end{displaymath}

In general, the canonical form of a general 2nd order system is

\begin{displaymath}\ddot{y}(t)+2\zeta\omega_n \dot{y}(t)+\omega_n^2  y(t)= x(t) \end{displaymath}

where $x(t)$ and $y(t)$ are the input and output, respectively, and $\zeta$ and $\omega_n$ are the two parameters that describe any 2nd order system. Comparing the canoncal form with two equations above we see that:

\begin{displaymath}
\omega_n=\frac{1}{\sqrt{LC}}
\end{displaymath}

and

\begin{displaymath}
\frac{R}{L}=2\zeta\omega_n=2\zeta_s\frac{1}{\sqrt{LC}},\;\;\;\;\mbox{i.e.}\;\;\;\;\;
\zeta_s=\frac{R}{2}\sqrt{\frac{C}{L}}
\end{displaymath}


\begin{displaymath}
\frac{1}{RC}=2\zeta_p\omega_n=2\zeta_p\frac{1}{\sqrt{LC}},\;...
...\;\mbox{i.e.}\;\;\;\;\;
\zeta_p=\frac{1}{2R}\sqrt{\frac{L}{C}}
\end{displaymath}

Note that we also have:

\begin{displaymath}
\zeta_p \zeta_s=\frac{1}{4},\;\;\;\;\;\zeta_p=\frac{1}{4\zeta_s},
\;\;\;\;\;\zeta_s=\frac{1}{4\zeta_p}
\end{displaymath}



Subsections

Ruye Wang 2014-03-23