Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Ads

Chapters

Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Elements
          A Physical Derivation of Wave Digital Elements
             Reflectances of Elementary Impedances

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?

  

Reflectances of Elementary Impedances


Capacitor Reflectance.

For a capacitor of $ C$ Farads, the driving-point impedance is (see §L.1.3)

$\displaystyle R_C(s)=\frac{1}{Cs} $

(or $ k/s$ for a spring with constant $ k$). Substituting into Eq.$ \,$(Q.1) gives the reflectance

$\displaystyle S_C(s) = \frac{R_C(s)-R_0}{R_C(s)+R_0} = \frac{1 - R_0 C s}{1 + R_0 C s} \protect$ (Q.5)


Inductor Reflectance.

For an inductor of $ L$ Henrys, we have

$\displaystyle R_L(s)$ $\displaystyle =$ $\displaystyle Ls$  
$\displaystyle \,\,\Rightarrow\,\,S_L(s)$ $\displaystyle =$ $\displaystyle \frac{Ls-R_0}{Ls+R_0} = \frac{ s - R_0/L }{ s + R_0/L} \protect$ (Q.6)


Resistor Reflectance.

Finally, for a resistor of $ R$ Ohms, we get

$\displaystyle S_R(s) = \frac{R-R_0}{R+R_0} = \frac{1 - R_0/R }{ 1 + R_0/R } \protect$ (Q.7)

Note that both the capacitor and inductor reflectances are stable allpass filters, as they must be. Also, the resistor reflectance is always less than 1, no matter what waveguide impedance $ R_0>0$ we choose.

Order a Hardcopy of Physical Audio Signal Processing

Previous: Reflectance of a General Lumped Waveguide Termination
Next: Choosing Waveguide Impedance to Simplify Element Reflectance
written by 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.