Re: Fast TR

J. Eric Bracken (bracken@ansoft.com)
Thu, 05 Sep 1996 17:12:36 -0400

Ok, here's the promised example.

Structure:
---------

The structure is not a connector, like Fabrizio wanted, but for simplicity
a 90 degree bend in a PCB trace. Here is the geometry:

- +-----------------------------------------+
^ | |
0.5mm | | |
| | |
v | |
- +------------------------------+ |
^ | |
|<----------- 1.5 mm -------|->| |
| | |
| | |
| | |
| | |
1.5mm | | |
| | |
| | |
| | |
| | |
| | |
v | |
- +----------+
0.5mm
|<-------->|


Not drawn to scale, unless you're using the Lucidatypewriter font!

Other details: the trace is a stripline, sitting on top of a substrate
with relative epsilon = 4.0. The substrate is 250um thick; beneath it
is a ground plane. Above the trace is air (rel. eps. = 1.0).

The characteristic impedance of this line is about 50 ohms, and the
propagation velocity is about 1.7 * c.

Theory:
------

Consider the problem of two short transmission line pieces, each with
characteristic impedance Z0 and a delay of T/2. Assume that they meet
at a point where there is a capacitance C to ground. We can do some
circuit-theory math and compute the scattering parameters for this
simple circuit, with the result that

2 -sT
S12 = S21 = ------------ e
2 + s C Z0


- s C Z0 -sT
S11 = S22 = ------------ e
2 + s C Z0

One important point is that these scattering parameters are normalized
to reference impedances of Z0 at each port.

Points to Remember:
-------------------

The S12, S21 parameters are "low-pass" functions. Below the frequency
omega = 2/(C*Z0), the magnitude will be approximately 1, and the phase
will be linearly decreasing with frequency, with a slope of -T.

The S11, S22 parameters (the "reflections") are "high-pass." There's
no reflection at DC. At low frequency, the phase is -90 degrees, and
this is linearly decreasing with slope -T (until we hit the pole, at
very high frequency.) The magnitude will be increasing linearly
with frequency until we hit the pole.

In the high-frequency limit, S12, S21 go to zero--the capacitor "shorts
out" the signal transmission path. The magnitudes of S11 and S22 go to
1 (total reflections), with the phase still linearly changing.

Field Simulation Results:
------------------------

After running the full-wave solver (a new integral-equation code called
Strata), the computed S-parameters were:

S11, S22:

f(Ghz) magnitude phase (deg)
----------------------------------------------------------------
0.5 0.00629838843382856 -93.3732317606879
0.6 0.00756469882186052 -94.1015741942106
0.7 0.00883645917070845 -94.6497780023877
0.8 0.0100859798457592 -95.4125895351485
0.9 0.0113501137028483 -96.1151512598425
1 0.0126040062920698 -96.7824570098957
1.1 0.0138559068090599 -97.4588402571775
1.2 0.0151056281359217 -98.1354261354934
1.3 0.0163530380694489 -98.8139995536391
1.4 0.017596165593154 -99.4963181396051
1.5 0.0188375539840172 -100.169769685603
1.6 0.0200753227600295 -100.847509119433
1.7 0.021309560982257 -101.524304110675
1.8 0.0225398519089814 -102.201644837998
1.9 0.0237660863610423 -102.878799678111
2 0.0249878764245314 -103.55577674792
2.1 0.0262051690804417 -104.232953834727
2.2 0.0274175249091954 -104.910720408465
2.3 0.0286248586300752 -105.58641589322
2.4 0.0298268936635684 -106.263628137272
2.5 0.0310236022493272 -106.940656969319
2.6 0.0322108071902056 -107.617898360109
2.7 0.0333993397738608 -108.294478171557
2.8 0.0345778969381823 -108.970381870114
2.9 0.0357506954065757 -109.648019918542
3 0.0369168624119478 -110.325070935431
3.1 0.0380763444358136 -111.001605544961
3.2 0.0392291064418385 -111.679253574184
3.3 0.0403746095636276 -112.355686739942
3.4 0.0415128891513402 -113.032206457171
3.5 0.0426444343622587 -113.709836502195
3.6 0.0437694144485022 -114.385870439913
3.7 0.0448846509718373 -115.063337759818
3.8 0.0459933115030725 -115.740282173917
3.9 0.0470944815582409 -116.417100943914
4 0.0481863428443381 -117.094712133267
4.1 0.0492718801084387 -117.771856244246
4.2 0.0503487686081146 -118.449425091717
4.3 0.051417308647525 -119.126635343505
4.4 0.0524784162278128 -119.804614534177
4.5 0.0535309691220054 -120.482134819185
4.6 0.0545736834903432 -121.16010968651
4.7 0.0556086776287063 -121.838564687749
4.8 0.0566373689145524 -122.516031415445
4.9 0.057655376658151 -123.194472213203
5 0.0586657255237102 -123.873186865888

S12, S21

f(Ghz) magnitude phase (deg)
----------------------------------------------------------------
0.5 0.999977279976794 -3.47236337980649
0.6 0.999973539409345 -4.16625371069462
0.7 0.999948785284645 -4.85983469011781
0.8 0.999946389143216 -5.5545933789936
0.9 0.999936401491237 -6.2485094773938
1 0.999919916207834 -6.94286346187756
1.1 0.999903136796343 -7.63722728533762
1.2 0.999884807817131 -8.331601242563
1.3 0.999865287785009 -9.02598336589715
1.4 0.999845025068813 -9.7204468802595
1.5 0.999821464112314 -10.4148881377847
1.6 0.999797230967763 -11.1093708800811
1.7 0.999771271715513 -11.8038879584318
1.8 0.999743936807286 -12.4984489405392
1.9 0.99971507143912 -13.1930526139606
2 0.999684675107323 -13.8877086561972
2.1 0.999652866664669 -14.5824135883738
2.2 0.999620054152801 -15.2771901916798
2.3 0.999584630706598 -15.9720083748332
2.4 0.99954845929186 -16.6668986018081
2.5 0.999510894650072 -17.3618514985107
2.6 0.999472059368003 -18.0571470776796
2.7 0.99943136045185 -18.7520073862102
2.8 0.9993886883078 -19.4472185947995
2.9 0.999346044295399 -20.1424798458104
3 0.999301582542841 -20.8378474661754
3.1 0.999255468223036 -21.5333100851578
3.2 0.999208411320521 -22.2288649922803
3.3 0.999159451626275 -22.9245348953736
3.4 0.999109146073435 -23.6203126363656
3.5 0.999058127507417 -24.3161819481978
3.6 0.999005004725641 -25.012139244453
3.7 0.998951085454251 -25.7082874713764
3.8 0.99889564097471 -26.4045280468092
3.9 0.998838752580002 -27.1008852590106
4 0.998781177446802 -27.7974129610608
4.1 0.998721510667334 -28.4940375915803
4.2 0.998661299329911 -29.1908105050006
4.3 0.998598911658435 -29.8877435665324
4.4 0.998536381538866 -30.5847927356922
4.5 0.998471225822516 -31.2820155061099
4.6 0.998405612296481 -31.9794303779462
4.7 0.998339255021106 -32.676987921886
4.8 0.998268960111797 -33.374678945558
4.9 0.998198923816041 -34.072582128191
5 0.998127865959929 -34.7705681390661

Analysis:
---------

Notice that the S-parameters conform pretty well to the expected
analytic forms for S12 and S11. You can see this if you plot them.

>From the slope of the phase of S12, we can compute the effective delay
of the bend:

T = (-6.94deg - (-27.80deg))/(4GHz - 1GHz) * 1/360deg = 19.3ps

This is pretty close to what we would expect if we just measured the
lengths of the transmission line stubs. Each piece is 1.5mm long on
the "inside" edge and 2mm long on the outside, so the "average" total
length is 3.5mm. From this and the propagation velocity, we'd get a
delay of 20.6ps.

The more interesting calculation is for the effective capacitance of
the bend. Assuming that the s*C*Z0 in the denominator of S11 is small
compared to 2, the magnitude of S11 is just s*C*Z0/2. So, we get

2 |S11| 2 * ( 0.031 )
C = ------------- = ------------------------- = 79 fF
omega * Z0 2*Pi*(2.5GHz)*(50 Ohms)

(we could have used any frequency, S11 pair here.) You can substitute
this back into s*C*Z0 and verify that it's indeed fairly small, and
so the procedure works.

With all respects to Ed Sayre, this "extra" capacitance is NOT predicted
by simple transmission line models of the bend.

__________________________________________________________________
________ ________ __ ________
/\_______\ /\_______\ /\_\ /\_______\ J. Eric Bracken, Ph.D.
/ / ______// / ____ // / // / ______/ Ansoft Corporation
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\/_______/ \/_/ \/_/ \/_/ \/_______/ bracken@ansoft.com

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