Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

More General Differential Equations

A more general linear constant coefficient differential equation can be written as:

\begin{eqnarray*}
\sum_{k=0}^{N}a_{k}\frac{d^{k}v}{dt^{k}}=\sum_{k=0}^{M}b_{k}\frac{d^{k}i}{dt^{k}}
\end{eqnarray*}

or, in the frequency domain, assuming zero initial conditions,

$\displaystyle \sum_{k=0}^{N}a_{k}s^k V(s)=\sum_{k=0}^{M}b_{k}s^k I(s)
$

We can define a transfer-function relationship as follows:

\begin{eqnarray*}
Z(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{V(s)}{I(s)} =
\frac
{b_0 + b_1 s + b_2 s^2 + \cdots + b_M s^M}
{1 + a_1 s + a_2 s^2 + \cdots + a_N s^N}
\end{eqnarray*}

where we have normalized $ a_0\neq 0$ to $ 1$ . Note that $ Z(s)$ is a rational function of $ s$ of order $ \max(N,M)$ .

If $ i(t)$ and $ v(t)$ are measured at the same point, then $ Z(s)$ is a driving point impedance, as depicted below:

\epsfig{file=eps/1port.eps,width=2in}

If the circuit (or mechanical system) is physically passive, then $ Z(s)$ must be positive real.


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download NumericalInt.pdf
Download NumericalInt_2up.pdf
Download NumericalInt_4up.pdf

``Discrete-Time Lumped Models'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]