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Delay-Free Loops

We might think, at this point, that we are done, because we can simply ``replace'' each RLC type circuit element by the derived signal flow path. Consider what happens with the discretization of the following simple LC filter:

\epsfbox{eps/LCfilter.eps}

Here, $ u(t)$ is an input voltage, $ y(t)$ the output voltage, $ i(t)$ the current, and in addition, we will define $ v(t)$ to be the voltage across the inductor. We thus have the three differential equations:

\begin{eqnarray*}
v&=&L\frac{di}{dt}\\
i&=&C\frac{dy}{dt}\\
v&=&-u-y
\end{eqnarray*}

The first two come from the definitions of the inductor and capacitor respectively, the third from Kirchoff's voltage law.

Discretizing these equations according to the trapezoid rule yields:

\begin{eqnarray*}
i_{n} &=& i_{n-1}+\frac{T}{2L}(v_{n}+v_{n-1}) \\
y_{n} &=& y_{n-1}+\frac{T}{2C}(i_{n}+i_{n-1}) \\
v_{n} &=& -(u_{n}+y_{n})
\end{eqnarray*}

and this yields the following signal flow diagram:

\epsfbox{eps/LCdiscrete.eps}

Note however, the delay-free loop which prohibits this implementation from being realizable.

In other terms, connecting two explicit finite difference models has resulted in an implicit finite difference model.


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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-25 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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