Derivation of the Wave Digital Inductor

It is instructive to see what happens when this change of variables in applied to the inductor.

1. We have the following difference equation for the inductor (using the trapezoidal rule of numerical integration, or bilinear transform, as you prefer):
   $$i_n = i_{n-1} + \frac{T}{2L} (v_n + v_{n-1})$$

2. Perform the wave-variable substitution
   $$v_n = \frac{a_n+b_n}{2} \qquad i_n = \frac{a_n-b_n}{2R_0}$$
   to get
   $$\frac{a_n - b_n}{2R_0} = \frac{a_{n-1} - b_{n-1}}{2R_0} + \frac{T}{2L}\left(\frac{a_n + b_n}{2} + \frac{a_{n-1} + b_{n-1}}{2}\right)$$

3. Now choose $R_0 = 2L/T$ to obtain
   $$\begin{eqnarray*} a_n - b_n &=& a_{n-1} - b_{n-1} + (a_n + b_n + a_{n-1} + b_{n-1}) \\ &=& 2 a_{n-1} + a_n + b_n \end{eqnarray*}$$
   

   which further simplifies to

   $$\zbox{b_n = - a_{n-1}} \qquad \hbox{(Wave Digital Inductor)}$$

Now there is no direct path from input to output.

In terms of wave variables, with simplest choices of the port resistances, we obtain the following wave digital filter elements (the elementary one-ports):

Elementary wave-digital one-ports. The port impedances for the wave-digital inductor, capacitor, and resistor (on the right) are defined as $2L/T$, $T/(2C)$, and $R$, respectively.