Cutoff frequency of arbitrary shape?

[olzanin]

Hi,

How can I calculate the cutoff frequency of some waveguide with arbitrary shape? It means no Rectangular or Round. All described examples in books are related to these shapes.

Best regards
olzanin

 

[djalli]

Olzanin read something for conformal mapping how to do it. You make your design to work with an initial rectangle.
You transform a rectangle to shape you want to produce the same output as your initial rectangle.

It is vast area of mathematics and you will love brilliant french,german and russian mathematicians. Can you say more on this so we can give you some more insights?

 

[eirp]

Conformal mapping is not able to calculate arbitrary shapes, just some kinds depending on angles of the polygone.

You have to employ some numerical technique to solve Helmholtz equation limited by given boundary (with approprite boundary condition - Neumann / Dirichlet).
Result will be eigenvectors (field shapes) and eigenvalues (related to cut-off frequency)

 

[djalli]

eirp hmm this is new to me. Any book you recommend to read in this area?

 

[eirp]

eirp hmm this is new to me. Any book you recommend to read in this area?

This is quite common technique,

check for example FEMLAB's manual/examples or simple example shown here
**broken link removed**

Also Numerical Techniques in EM by Sadiku (chapter 4) show some example of partially loaded waveguided solved by eigenmode approach (FEM)

Cheers
e.

 

[armsgroup]

you may start with wave equation and use finite difference method this will help to solve for any shape put you will need to reduse your step size and of course you need the boundary conditions then after get the equatios you will need to solve them numerically

 

[amicloud]

I once used a 2D-FDTD method to calculate the cutoff frequency.The cutoff frequency happens when the β is 0

 

[djalli]

and β is?

 

[eirp]

Generally the resonance condition (or cutoff when talking about WG) happens when

\[k^2=k^2_n\]

\[k^2=\omega^2\mu\epsilon\]

and \[k^2_n\] is calculated eigennumber.

If you imagine rectangular waveguide a x b with known eigennumbers:

\[k^2_{mn}=\big(\frac{m\pi}{a}\big)^2+\big(\frac{n\pi}{b}\big)^2\]

then the condition above gives

\[f_{m,n}=\frac{k_{m,n}}{2\pi\sqrt{\mu\epsilon}}\]

This is the familiar equation for calculating cut-offs of rect. waveguide.
The eigennumbers are however known analytically only in cases of separable geometries