Complex Impedances

• Models of damping in practical physical systems are rarely completely independent of frequency, like the ideal dashpot

• Thanks to the Laplace transform (or Fourier transform), the concept of impedance easily extends to masses and springs as well

• We need only allow impedances to be frequency-dependent

• For example, the Laplace transform of Newton's $f=ma$ yields, by the differentiation theorem,
  $$F(s) \eqsp m\, X(s) \eqsp m\, sV(s) \eqsp m\, s^2A(s)$$
  where
  • $F(s) = {\cal L}_s\{f\} =$ Laplace transform of $f(t)$ (initial conditions assumed zero)
  • Impedance of a point-mass is
    $$R_m(s) \isdefs \frac{F(s)}{V(s)} \eqsp ms$$

  • Specializing the Laplace transform to the Fourier transform by setting $s=j\omega$ gives
    $$R_m(j\omega) \eqsp jm\omega$$

  • Impedance of a spring with spring-constant $k$ is
    $$\begin{eqnarray*} R_k(s) &=& \frac{k}{s}\\ [5pt] R_k(j\omega) &=& \frac{k}{j\omega} \end{eqnarray*}$$