Transmission along an LC network
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Determine the dispersion relation for a periodic wave crossing the line.
What is the cutoff frequency ?
Is the voltage across capacitor of the cell number and the current in the coil follows (see figure), we can write :
The last equation is derived with respect to time :
And used two loop laws :
We are interested in the propagation of plane waves that we write in the form :
where is the "width" of a cell.
The previous differential equation then becomes :
Or, finally :
It is requested dispersion relation.
The cut-off frequency is obtained when is equal to :
Determine the propagation equation satisfied by the voltage across the capacitors within the approximation of continuous medium.
It is assumed that the distance between the cells is small compared to the wavelength ( .
We can then define a function such as (Taylor-Young Development of the second order) :
And the propagation equation becomes :
We find a d'Alembert equation, characteristic of a propagation without dissipation at the speed .
We take the dispersion relation obtained in the first question :
And assume that (assumption of continuous medium).
The phase velocity (equal here also to the group velocity) is then :
We find the speed of the wave obtained in the second question.