What are the salient differences between Lumped and Distributed systems? In what contexts are distributed systems the appropriate model and in what are lumped systems the appropriate model?

Also, Lumped systems are said to be described by ordinary differential equations while the latter is said to be described by partial differential equations. Can someone explain why?

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    $\begingroup$ While this is an interesting and valid question, it has really nothing to do with signal processing and may be better off in physics or a related discipline (electronics, acoustics, etc.) $\endgroup$
    – Hilmar
    May 24 '13 at 10:38
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    $\begingroup$ @Hilmar Same comment when I posted it on Electronics.stackexchange.com $\endgroup$
    – Gowtham
    May 24 '13 at 15:51

The elements building a lumped system are thought of being concentrated at singular points in space. The classical example is an electrical circuit with passive elements like resistor, inductance and capacitor. The physical quantities current and voltage are functions of time (only). E. g. the current at a capacitor with capacity $C$ is given by $$ i(t) = C\frac{\mathrm d v(t)}{\mathrm d t} $$ Where $C$ is a constant (and so are $R$ and $L$). This leads to ordinary differential equations.

In contrast, the elements in distributed systems are thought of being distributed in space, so that physical quantities depend on both time and space. The classical example is the electrical line where inductance, capacity and resistance are not constant but functions of length $x$. This leads to partial derivatives of $i(t,x)$ and $v(t,x)$ in $t$ and $x$.

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    $\begingroup$ Looked at it from an elementary stand point: apply a current to a capacitor and the voltage is there with no complications. Now apply the same current to the end of a transmission line: the current generates a wavefront traveling along the line. To talk about the voltage you have to describe where you are measuring the voltage leading to two variables; position and time. The meaning of partial differential equations is that you have two or more variables determining the resulting function. $\endgroup$
    – rrogers
    Apr 15 '15 at 13:48

It is important to realize that the terms lumped or distributed are not properties of the system itself. These properties are related to the size of the system compared to the wavelength of the voltages and currents passing through it. So a resistor is or isn't a lumped element (even though it is usually meant to be one), depending on the frequency of the applied signals.

An element can be considered as lumped if its size is much smaller than the wavelength of the applied voltages and currents. In this case wave propagation effects may be neglected and a lumped element can be treated as a black box which is completely described by the behavior at its terminals.

EDIT: I forgot to mention the differential equations. Lumped systems are described by ordinary differential equations because due to the small size of the system (compared to the wavelength), the spatial derivatives can be neglected and we only need to consider time derivatives. On the other hand, for distributed systems we need to take wave propagation phenomena into account and we get non-zero spatial as well as time derivatives, which leads to partial differential equations.

  • $\begingroup$ Very interesting answer! Could you tell me what "wave propagation phenomena" are you referring to? many thanks! $\endgroup$
    – sunrise
    Jun 20 '17 at 10:14

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