[SI-LIST] Re: Copper Fill --- correction
- From: Scott McMorrow <scott@xxxxxxxxxxxxx>
- Date: Fri, 14 Jun 2002 11:07:39 -0700
A final correction to the resonance formula.
I often transpose velocity of propagation with delay per unit length.
Vp = 1/(delay per unit length)
Steve Corey pointed out that my units did not work out, which is due my
internal processing error. (I'd like to say this was age, but since
I've been doing it all my life, it's more of a "feature not a bug" that
I learn to live with.
The correct equation for resonance of a square cavity is:
resonant frequency = Vp/(2 * w)
or 1/ (2 * w * delay per unit length)
Vp = C/sqrt(Er)
C = speed of light
delay per unit length = sqrt (Er)/ C.
Going back to the books this comes from the general equation for the
resonant wavelength of a rectangular closed cavity:
lambda = 2 /sqrt [ (l/a)2 + (m/b)2 + (n/c)2]
where a,b,c are the dimensions of the cavity
and l,m,n are the resonant modes.
For my case, I have set the modes to 1,0,0, and reduced the equation to:
lambda = 2/sqrt(l/a)2 = 2a, which is the same as my 2w.
resonant frequency = Vp/lambda = Vp/2a (or Vp/2w)
resonant frequency = Vp/2a or Vp/(2 * w) from my example.
resonant frequency = 1/ (2 * w * delay per unit length)
regards,
scott
--
Scott McMorrow
Teraspeed Consulting Group LLC
2926 SE Yamhill St.
Portland, OR 97214
(503) 239-5536
http://www.teraspeed.com
Scott McMorrow wrote:
>Okay, now I really need some coffee ... and it's 5:30 in the evening.
>Here's yet another correction and more elaboration.
>
>A square patch of copper on top of a dielectric and plane forms a crude
>cavity resonator which will have a number of natural resonant
>eigenmodes. The first resonant point is a half-wave resonance. The
>resonance frequency is equal to the round trip of a wave propagating
>back and forth across the region, much like what would happen in a
>square pool of water between the two walls. For a one square inch patch
>of copper, the resonant frequency on FR-4 is somewhere between 2.7 and
>3.3 GHz
>
>A very crude square patch
>
> w
>--------------
>| | resonant frequency = 1/( 2 * w * Vp)
>| | Vp = velocity of propagation of wave in
>dielectric medium = c/sqrt(Er)
>| | w = width and height of square
>| |
>| |
>---------------
>
>
>Adding vias to the patch effectively pins the structure down and raises
>it's resonant frequency. The spacing of the vias control the resonance,
>with a half-wave resonance being the lowest generally supported by the
>structure. Stitching the patch at all four corners actually does nothing
>to change the resonant frequency of the cavity. However, placing a
>fifth via at the center will push the resonance up by a factor of 2.
>
> w
>@------------@
>| | resonant frequency = 1/( 4 * w * Vp)
>| | Vp = velocity of propagation of wave in
>dielectric medium
>| @ | w = width and height of square
>| | @ = Via stitch
>| |
>@------------@
>
>In essence, the via spacing has changed from w to w/2. These formulas
>are quite approximate, but will place us in the general playing field. A
>full-wave field solver such as PowerSI from Sigrity, or SIwave and HFSS
>from Ansoft, will provide much more exact resonance frequency analysis.
>
>So, the "rough" formula I gave for via spacing should be:
>
>via spacing =1/(2 * (f * Vp)) not 2/(f * Vp) as I had indicated.
>
>In my previous example, this would amount to a via spacing of 0.59
>inches, not 1.18.
>
>
>Sorry about my confusion.
>
>
>Scott
>
>
>
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