What's the relation between antenna length and resonating frequency?

[project.email]

dear friends
I want to know what's the relation between the antenna length and resonating frequency
how can we determine the resonance frequency without any simulation or measurement just by looking at antenna dimension?
i.e in this antenna with this dimensions, do you know what is the resonating frequency?
Is there any equation for this?
thanks all.

 

[RCinFLA]

There are several factors. The most important is the effective dielectric constant within which the resonator exists. The velocity factor is related to 1/sqrt(dielectric constant).

If it is a radiating antenna then by default all the field is not confined within the high k dielectric so the effective dielectric constant will be somewhat less then the er of 10.2 listed.

Beside the effective dielectric constant shortening the resonator, the amount of stray capacitance at the end of resonator will provide some additional shortening.

 

[project.email]

Dear friend thanks for your reply
I read in a book that I found this antenna :

"The length of the spiral radiator is determined for the antenna to resonate at the desired frequency (402–405 MHz) in the FDTD geometry and the coaxial feed is located for a good 50  match."

this antenna must be resonating in about 400MHz but what's the relation between the length of radiator and the resonating frequency?!
I'm so confused!

 

[FvM]

how can we determine the resonance frequency without any simulation or measurement just by looking at antenna dimension?

As a first step, you would want to "see" the oscillation mode of the resonator. Do you? If you have seen several similar designs, you can probably guess about it, but you should better perform an EM simulation to be sure. It's not a simple thing like a λ/4 or λ/2 resonator.

You can also assume, that the geometry has been selected on purpose, most likely to achieve a particular radiation pattern. I don't believe that except for some antenna experts many people will recognize the expectable radiation pattern.

P.S.: The center line length of the microstrip is very near to a λ/2 resonator. In so far, we can say, that you can easily determine the resonance frequency. But this neither explains the specific shape nor the feed point position and why we achieve 50 ohm input impedance for it.

 

[project.email]

Dear FvM thanks for your reply
I performed a simulation and found the Mode of feeding is TEM.but I'm not interested in it.
an important question here is that by this configuration and feeding I get the different result with the mentioned result in the book.
by the above feeding I got the following S_parameter and by changing the feeding position I found that this antenna dose not resonate at 402-405MHz but it resonate at 450MHz.dose something wrong in my analysis or it may the writers of that book has make a mistake in the drawing the above picture?!

this is the result with the above feeding position mentioned in the book:


and it's the better resonating at 450MHZ by changing the feeding position


furthermore I found some equation about PIFA antenna design to determine resonate frequency:
PIFA - Planar Inverted-F Antennas
but I think with this equations we can not also find the resonate frequency correctly!
using equation[4] for this antenna we have:
32+24=3e8÷4f√10.2--->f=51.09MHz:roll: