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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Introduction to Lumped Models
       One-Port Network Theory
          Series Combination of One-Ports

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Series Combination of One-Ports

Figure L.6 shows the series combination of two one-ports.

Figure L.6: Two one-port networks combined in series. The impedance of the series combination is $ R(s) = F(s)/V(s) = R_1(s) + R_2(s)$.
\includegraphics[scale=0.9]{eps/lseries}

Impedances add in series, so the aggregate impedance is $ R(s) = R_1(s) + R_2(s)$, and the admittance is

$\displaystyle \Gamma(s) = \frac{1}{\frac{1}{\Gamma_1(s)} + \frac{1}{\Gamma_2(s)}} = \frac{\Gamma_1(s) \Gamma_2(s) }{\Gamma_1(s) + \Gamma_2(s)} $

The latter expression is the handy product-over-sum rule for combining admittances in series.

In a physical situation, if two elements are connected in such a way that they share a common velocity, then they are in series. An example is a mass connected to one end of a spring where the other end is attached to a rigid support and the force is applied to the mass, as shown in Fig. L.7.

Figure L.7: A mass and spring combined as one-ports in series.
\includegraphics[scale=0.9]{eps/lseriesExample}

Figure L.8 shows the electrical equivalent circuit corresponding to Fig.L.7.

Figure: Electrical equivalent circuit of the series mass-spring driven by an external force diagrammed in Fig.L.7.
\begin{figure}\input fig/lseriesec.pstex_t \end{figure}

Figure: Impedance diagram for the force-driven, series arrangement of mass and spring shown in Fig.L.7.
\begin{figure}\input fig/lseriesid.pstex_t \end{figure}

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Previous: One-Port Network Theory
Next: Parallel Combination of One-Ports
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.


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