Distributed-element model
In
The distributed model is used where the wavelength becomes comparable to the physical dimensions of the circuit, making the lumped model inaccurate. This occurs at high frequencies, where the wavelength is very short, or on low-frequency, but very long, transmission lines such as overhead power lines.
Applications
The distributed-element model is more accurate but more complex than the lumped-element model. The use of infinitesimals will often require the application of calculus, whereas circuits analysed by the lumped-element model can be solved with linear algebra. The distributed model is consequently usually only applied when accuracy calls for its use. The location of this point is dependent on the accuracy required in a specific application, but essentially, it needs to be used in circuits where the wavelengths of the signals have become comparable to the physical dimensions of the components. An often-quoted engineering rule of thumb (not to be taken too literally because there are many exceptions) is that parts larger than one-tenth of a wavelength will usually need to be analysed as distributed elements.[1]
Transmission lines
The primary line constants are normally taken to be constant with position along the line leading to a particularly simple analysis and model. However, this is not always the case, variations in physical dimensions along the line will cause variations in the primary constants, that is, they have now to be described as functions of distance. Most often, such a situation represents an unwanted deviation from the ideal, such as a manufacturing error, however, there are a number of components where such longitudinal variations are deliberately introduced as part of the function of the component. A well-known example of this is the
Where
. This can be a wildly different impedance.High-frequency transistors
Another example of the use of distributed elements is in the modelling of the base region of a bipolar junction transistor at high frequencies. The analysis of charge carriers crossing the base region is inaccurate when the base region is simply treated as a lumped element. A more successful model is a simplified transmission line model, which includes the base material's distributed bulk resistance and the substrate's distributed capacitance. This model is represented in figure 2.
Resistivity measurements
In many situations, it is desired to measure
The model used here needs to be truly 3-dimensional (transmission line models are usually described by elements of a one-dimensional line). It is also possible that the resistances of the elements will be functions of the coordinates, indeed, in the geophysical application, it may well be that regions of changed resistivity are the very things that it is desired to detect.[4]
Inductor windings
Another example where a simple one-dimensional model will not suffice is the windings of an inductor. Coils of wire have capacitance between adjacent turns (and more remote turns as well, but the effect progressively diminishes). For a single-layer solenoid, the distributed capacitance will mostly lie between adjacent turns, as shown in figure 4, between turns T1 and T2, but for multiple-layer windings and more accurate models distributed capacitance to other turns must also be considered. This model is fairly difficult to deal with in simple calculations and, for the most part, is avoided. The most common approach is to roll up all the distributed capacitance into one lumped element in parallel with the inductance and resistance of the coil. This lumped model works successfully at low frequencies but falls apart at high frequencies where the usual practice is to simply measure (or specify) an overall Q for the inductor without associating a specific equivalent circuit.[5]
See also
References
Bibliography
- Kenneth L. Kaiser, Electromagnetic compatibility handbook, CRC Press, 2004 ISBN 0-8493-2087-9.
- Karl Lark-Horovitz, ISBN 0-12-475946-7.
- Robert B. Northrop, Introduction to instrumentation and measurements, CRC Press, 1997 ISBN 0-8493-7898-2.
- P. Vallabh Sharma, Environmental and engineering geophysics, Cambridge University Press, 1997 ISBN 0-521-57632-6.