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:

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:

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.