Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

See Also

Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Introduction to Physical Signal Models
       Physical Models
          Impedance Networks

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Impedance Networks

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

$\displaystyle 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

$\displaystyle 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 [449]2.15(or Fourier transform [451]), 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 [449],

$\displaystyle 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

$\displaystyle 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

$\displaystyle 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: Impedance diagram for the force-driven, series arrangement of mass and spring shown in Fig.1.9.
\includegraphics{eps/lseriesidCopy}

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:

$\displaystyle 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.

Previous: Equivalent Circuits
Next: Wave Digital Filters

Order a Hardcopy of Physical Audio Signal Processing

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )