Frequency Domain Interpretation

The equation for the inductor, assuming zero initial conditions, transforms to

$$\begin{eqnarray*} V(s)=LsI(s) \end{eqnarray*}$$

where $s$ is the complex frequency variable. Taking $z$ transforms of the sequences $v$ and $i$ in the backward-difference scheme yields:

$$\begin{eqnarray*} V(z^{-1})=L\frac{1-z^{-1}}{T}I(z^{-1}) \end{eqnarray*}$$

Thus we can think of our discretized scheme as one obtained under the mapping $s\to \frac{1-z^{-1}}{T}$. So here we are mapping from the $s$ plane to the $z$ plane. The following figure illustrates where real continuous time frequencies (the $j\omega$ axis) are mapped:

dc ($s=0$) mapped to dc ($z=1$)

infinite frequency mapped to ($z=0$)

We can write the scheme for the trapezoid rule as follows:

$$\begin{eqnarray*} V(z^{-1})=\frac{2L}{T}\frac{1-z^{-1}}{1+z^{-1}}I(z^{-1}) \end{eqnarray*}$$

• the mapping $s\to\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$ maps all real frequencies to real frequencies uniquely. In partcular dc$\to$dc, and infinite frequency maps to $z=-1$.

  One way of examining the frequency mapping more closely is by looking at $\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$ on the unit circle, i.e., where $z=e^{j\omega T}$. This yields:

  $$\begin{eqnarray*} \frac{2}{T}\frac{1-e^{-j\omega T}}{1+e^{-j\omega T}} &=& \frac{2j}{T}\tan(\omega T/2) \end{eqnarray*}$$
  

• notice in particular the behaviour of the mapping near dc ($\omega$ = 0):
  $$\begin{eqnarray*} \frac{2j}{T}\tan(\omega T/2) \simeq j\omega +O(T^{3}) \end{eqnarray*}$$
  

  where, since $\tan(\theta)$ is odd, there are no even-order terms in its series expansion. Thus, the trapezoid rule is a second-order accurate approximation to a derivative, in the limit of small $T$ (i.e., near dc).