Impedance Networks

The concept of impedance is central in classical electrical engineering. The simplest case is Ohm's Law for a resistor $R$ :

$$V(t) \eqsp R\, I(t)$$

where $V(t)$ denotes the voltage across the resistor at time $t$ , and $I(t)$ is the current through the resistor. For the corresponding mechanical element, the dashpot, Ohm's law becomes

$$f(t) \eqsp \mu\, v(t)$$

where $f(t)$ is the force across the dashpot at time $t$ , and $v(t)$ is its compression velocity. The dashpot value $\mu$ is thus a mechanical resistance.^2.14

Thanks to the Laplace transform [452]^2.15(or Fourier transform [454]), 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, using the differentiation theorem for Laplace transforms [452],

$$F(s) \eqsp m\, A(s) \eqsp m\, sV(s) \eqsp m\, s^2X(s)$$

where $F(s)$ denotes the Laplace transform of $f(t)$ (`` $F(s) = {\cal L}_s\{f\}$ ''), and similarly for displacement, velocity, and acceleration Laplace transforms. (It is assumed that all initial conditions are zero, i.e., $f(0)=x(0)=v(0)=a(0)=0$ .) The mass impedance is therefore

$$R_m(s) \isdefs \frac{F(s)}{V(s)} \eqsp ms.$$

Specializing the Laplace transform to the Fourier transform by setting $s=j\omega$ gives

$$R_m(j\omega) \eqsp jm\omega.$$

Similarly, the impedance of a spring having spring-constant $k$ is given by

$$\begin{eqnarray*} R_k(s) &=& \frac{k}{s}\\ [5pt] R_k(j\omega) &=& \frac{k}{j\omega}. \end{eqnarray*}$$

The important benefit of this frequency-domain formulation of impedance is that it allows every interconnection of masses, springs, and dashpots (every RLC equivalent circuit) to be treated as a simple resistor network, parametrized by frequency.

Figure 1.11: Impedance diagram for the force-driven, series arrangement of mass and spring shown in Fig.1.9.

As an example, Fig.1.11 gives the impedance diagram corresponding to the equivalent circuit in Fig.1.10. Viewing the circuit as a (frequency-dependent) resistor network, it is easy to write down, say, the Laplace transform of the force across the spring using the voltage divider formula:

$$F_k(s) \eqsp F_{\mbox{ext}}(s) \frac{R_k(s)}{R_m(s)+R_k(s)} \eqsp F_{\mbox{ext}}(s)\frac{k/m}{s^2+k/m}$$

These sorts of equivalent-circuit and impedance-network models of mechanical systems, and their digitization to digital-filter form, are discussed further in Chapter 7.