With the recent discussion about high return loss loads, I made the point
that it is a bit pointless worrying about it unless there’s firmware
support for calibration kits in NanoVNA, which as far as I am aware there’s
not.

The following is the the phase of a *female N calibration standard* from an
HP 85054B calibration kit, costing more than $20,000

https://www.keysight.com/en/pd-1000002019%3Aepsg%3Apro-pn-85054B/standard-mechanical-calibration-kit-dc-to-18-ghz-type-n-50-ohm

The photographs show the VNA calibrated to 6 GHz - I happened to VNA warmed
up and calibrated up to 6 GHz.

On the far left of the Smith chart is the theoretically ideal short. On the
right is the theoretical ideal open.

This VNA covers 50 MHz to 20 MHz, so with a minimum frequency of 50 MHz,
the phase would not be the idealised 180 degrees, but it would be quite
close to 180 degrees. Hence the trace starts not quite on the far left.

At 6 GHz the phase has changed from 180 to 59.34 degrees. For simplicity,
assume that the phase is 60 degrees. So the phase has changed 180-60=120
degrees over 6 GHz. Assuming a linear phase change with frequency, which is
approximately true, that’s a phase shift of 120/6= 20 degrees per GHz.

So if this $20,000 kit was used to calibrate a NanoVNA using the NanoVNA
firmware, a 20 degree phase error would exist at 1 GHz. With such a large
phase error, it would be pointless worrying about the loads.

I would add it is possible to make an N female short with virtually no
delay, but HP chose not to do it.

When it comes to a male N calibration standard, it is technically
impossible to make it with zero offset.

This link might help an understanding of what is happening on those plots.

https://www.kirkbymicrowave.co.uk/Support/FAQ/Why-do-I-not-see-a-dot-on-the-Smith-Chart-for-the-opens-and-shorts/


Dave



--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

David,

Good stuff, thanks for posting.

Assuming the connectors physically stout enough to give repeatable results,
would it be possible to calibrate out most of the errors encountered in a cheap or homebrew calibration set?
Is this something that could be done through post-processing in nanovna-saver?

FYI, the two images in your "not-see-a-dot" FAQ don't show up
in either my Chrome browser or in Firefox.

Jerry, KE7ER

Well, I think for low-reflection DUTs the error might still be dominated by the residual directivity (and hence the return loss of the calibration load). But I agree that while the discussion about the loads is interesting, it will generally not be the limiting factor with the NanoVNA.

For the NanoVNA-V2 in particular, with frequencies of 3GHz and beyond, the firmware calibration routine and the included open and short standards are somewhat inadequate. By comparison, the load is fine... Apparently, there are plans for a 6GHz version, but unless calibration is sorted out, I'm not sure there is much of a point.

Not all is lost, with a PC connected you have a lot more options (of course for portable use that will not do you much good). NanoVNA-QT can use arbitrary touchstone files for the standards and NanoVNA-Saver can use a polynomial model. I usually use scikit-rf with Python instead, because it is more flexible for 2-port calibration. QT only does simple normalization for transmission and I suspect 2-port calibration in Saver is essentially broken.

Even if this is fixed, there doesn't seem to be a suitable low-cost option for a calkit at the moment (low-cost relative to the NanoVNA, I am aware of your products). At a minimum, it would need characterized offset delays. As you illustrate so nicely, the phase error can become signifcant and at some point you might be better off without a full SOL calibration.

The SDR-Kits calkits try to fill this gap, but beyond 1.5GHz the performance doesn't seem to be too great (at least for the female one). This is maybe not surprising, as they are aimed primarily at the 1.5 GHz VNWA. I suspect that the issue is mainly with the open and could be solved if the fringing capacitance were included in the model. If anyone has measured the SDR-Kits parts with a properly calibrated analyzer up to at least 3GHz, I would be very interested in the touchstone files.

Concerning zero-delay/flush shorts, I think this is not really the issue. For 3.5mm it is quite possible to make male and female flush shorts (at least you can order them from Maury Microwave). However, I expect it will be much harder to make a flush open with low capacitance. And I suspect this is the reason why you usually don't find a flush short in a calkit. If the difference in delay between the short and open is too large, the phase will eventually cross-over and the calibration equation becomes singular.

For flush open etc you might find this interesting:
https://www.youtube.com/watch?v=-PK9Bn7Ixnw


Concerning zero-delay/flush shorts, I think this is not really the issue.

On Tue, 18 Aug 2020 at 03:06, Dragan Milivojevic <d.milivojevic@...>
wrote:

For flush open etc you might find this interesting:

https://www.youtube.com/watch?v=-PK9Bn7Ixnw

It is technically flawed in several ways as are several other videos from

the same author. *Every* video I have seen from him are wrong, although
this is is not as bad as others.

It’s not that important that the phase of the open and short are exactly
180 degrees apart. The $11,000 18 GHz Keysight 85050B APC7 calkit is one
example of a kit where no attempt is made to achieve 180 degrees, although
it is technically possible to approach it. I don’t know the figures off
hand, and I can’t be bothered to measure or calculate it, but I would
expect that the phase probably varies from 170 to 180 degrees on the 85050B
yet that kit is very accurate, with an error on the worst standard not
exceeding 0.4 degrees to 8 GHz and 0.6 degrees up to 18 GHz

He makes no attempt to explain why he is aiming to get 180 degrees, other
than to say that his HP kit has 180 degrees. I doubt he has a clue why, as
he never says why.

For waveguide calibration kits you don’t use opens for calibration. Leaving
a waveguide open just makes an antenna with a return loss of about 13 dB.

Instead two shorts of different delays are used. Since wavelength in
waveguide is not inversely proportional to frequency, there’s a massive
difference between the phases in waveguide as the frequency is swept.

2) He has a totally flawed video on reference planes. His misunderstanding
of that can be seen in this video too, but it more subtle.

I find YouTube to be awash with people creating videos about things they
know
a little about. He is one such example.

Dave.
--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

On Mon, 17 Aug 2020 at 22:50, Jerry Gaffke via groups.io <jgaffke=
yahoo.com@groups.io> wrote:

David,

Good stuff, thanks for posting.

Assuming the connectors physically stout enough to give repeatable results,

That particular short is very repeatable, as the female centre conductor
has no slots in it, so it doesn’t expand in diameter when the male pin
enters.

would it be possible to calibrate out most of the errors encountered in a
cheap or homebrew calibration set?

Is this something that could be done through post-processing in
nanovna-saver?

If a low-cost kit is designed properly, (and I am yet to see one that is),
then yes, errors can be largely corrected in software, but it doesn’t solve
the current problems with the NanoVNA if you want to use it standalone,
without a PC.

You will not get the accuracy of a Keysight kit, as the female centre pin
will always expand in diameter on SMA. It would be very expensive to make
an SMA connector that avoids that problem

FYI, the two images in your "not-see-a-dot" FAQ don't show up

Yes, I noticed that after I posted it. I will resolve that today. It is
weird what caused that problem,

Jerry, KE7ER

G8WRB
--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

On Tue, 18 Aug 2020 at 00:01, <switchabl@...> wrote:

The SDR-Kits calkits try to fill this gap, but beyond 1.5GHz the
performance doesn't seem to be too great (at least for the female one).
This is maybe not surprising, as they are aimed primarily at the 1.5 GHz
VNWA. I suspect that the issue is mainly with the open and could be solved
if the fringing capacitance were included in the model. If anyone has
measured the SDR-Kits parts with a properly calibrated analyzer up to at
least 3GHz, I would be very interested in the touchstone files.

There’s too much variation from one to the other. You really need the
Touchstone files of the parts you have, which pushes up production costs
significantly.

However, I expect it will be much harder to make a flush open with low

capacitance.

And it is unnecessary to do so.

And I suspect this is the reason why you usually don't find a flush short

in a calkit.

Indeed so, which is why I laugh when I see all these papers and videos
explaining why the delay on the short needs to be so close to zero.



If the difference in delay between the short and open is too large, the

phase will eventually cross-over and the calibration equation becomes
singular.

I think you mean if the difference in phase is too *small*. Yes, indeed
they would, but in practice the kit would become unusable before the
different in phase became zero. I believe about 20 phase degrees difference
is the minimum needed. Below that the calibration would be unstable. Noise
in the instrument would become more significant, as it tried to measure two
devices very similar to each other.

Dave
--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

Dr David Kirkby Ph.D C.Eng MIET
Email: drkirkby@... Web:
https://www.kirkbymicrowave.co.uk/
Kirkby Microwave Ltd (Tel 01621-680100 / +44 1621-680100)
Stokes Hall Lodge, Burnham Rd, Chelmsford, Essex, CM3 6DT.





On Mon, 17 Aug 2020 at 22:04, Dr. David Kirkby <
drkirkby@...> wrote:

With the recent discussion about high return loss loads, I made the point
that it is a bit pointless worrying about it unless there’s firmware
support for calibration kits in NanoVNA, which as far as I am aware there’s
not.

The following is the the phase of a *female N calibration standard* from
an HP 85054B calibration kit, costing more than $20,000

<snip>

This link might help an understanding of what is happening on those plots.

https://www.kirkbymicrowave.co.uk/Support/FAQ/Why-do-I-not-see-a-dot-on-the-Smith-Chart-for-the-opens-and-shorts/

Dave

As someone pointed out, the link was broken. I know what caused it to
break. I recently changed the server and have not enabled the .htaccess
file which was performing some redirects. /FAQ has now been moved to
/Support/FAQ, so the link broke. I will find any more and fix them, as the
"permanent redirects have been in place for almost a year now.". I should
fix the remove the redirects and sort out whats missing.

If anyone can find any more broken links on
https://www.kirkbymicrowave.co.uk/ please let me know.

Dave

On Tue, Aug 18, 2020 at 07:36 AM, Dr. David Kirkby, Kirkby Microwave Ltd wrote:

There’s too much variation from one to the other. You really need the
Touchstone files of the parts you have, which pushes up production costs
significantly.

For now I'm just hoping it will be close enough to at least somewhat improve the results. I wouldn't use them for a data-based calibration either, the idea is to maybe just fit delay and a single capacitance parameter (for the open). Ideally someone could supply data for a whole batch, so the variation could be quantified as well. But that seems rather unlikely (and maybe not worth thte effort).
I am very much aware of the limitations of this approach.

Compared to the kit that comes with the NanoVNA-V2 it is at least usable. The supplied short has a good chance of eventually damaging the test port and it is probably better to just leave the test port open rather than use the supplied dust cap. Ironically, the load is mostly fine...

If the difference in delay between the short and open is too large, the

phase will eventually cross-over and the calibration equation becomes
singular.

I think you mean if the difference in phase is too *small*. Yes, indeed
they would, but in practice the kit would become unusable before the
different in phase became zero. I believe about 20 phase degrees difference
is the minimum needed. Below that the calibration would be unstable. Noise
in the instrument would become more significant, as it tried to measure two
devices very similar to each other.

Indeed, I apologize for the somewhat ambiguous wording. What I meant to say is that when difference in offset delay is too large, there will be frequencies where the phase difference becomes zero. And of course anything even close to zero is a problem, since even if the calibration equation is not technically singular, it will still be quite ill-conditioned and any uncertainties (including noise) will be greatly amplified.

In this respect, 180° is theoretically ideal from a mathematical point of view. But as you remarked, as long as the deviations don't become too large, the effect will be very small and 170° should be just as good from a practical perspective. It certainly seems ill-advised to trade off any other properties in order to maintain an exact 180° phase difference.
I wonder if this is a relic from scalar network analyzers where phase deviations cannot be corrected.

David,

Having O-S-L correction factors only available in host software like nanovna-saver
seems a fine solution. Though there are larger ARM processors that should have
no trouble doing this sort of thing in a standalone unit.

In what ways are current low cost calibration kits not "designed properly"?
Is it mostly this issue of the SMA female center pin?
I'm guessing that variations in capacitance, inductance, and resistance
with frequency can be compensated for in software.

Are there reasonably cheap connectors that would be better than our SMA's
with regard to this center pin issue?

Would it be possible to determine parameters for this compensation
without access to more accurate equipment?
Seems doubtful.
Though once somebody with a $20k VNA creates a few O-S-L kits
with compensation parameters, perhaps those could be used with
a nanoVNA to create more of the same with somewhat less accuracy.

For me, this is mostly academic, but fun to think about.
I'd be quite happy piddling around down at 30mhz, learning about
Smith charts and how to design a matching network,
I have no real need for better accuracy.
The nanoVNA is a fantastic tool right out of the box.

Jerry, KE7ER

Links within the FAQ now work. thanks!

I could understand the results shown in that FAQ
if the VNA is calibrated using one set of O-S-L standards
and the VNA is then used to measure the open and short standard
from some other calibration kit.

But if the VNA is measuring the standards to which it was calibrated,
should it not show the Smith chart at the top of the FAQ, with three nice clean dots?

Jerry, KE7ER

Every video on the topic or all of them?
What is wrong with it? As I see it he is just trying to make
his homemade set behave similar to the HP one.
Could you point to a better article/video about
the topic (making your own cal standards).

On Tue, 18 Aug 2020 at 06:59, Dr. David Kirkby, Kirkby Microwave Ltd <
drkirkby@...> wrote:

On Tue, 18 Aug 2020 at 03:06, Dragan Milivojevic <d.milivojevic@...>
wrote:

For flush open etc you might find this interesting:

https://www.youtube.com/watch?v=-PK9Bn7Ixnw

It is technically flawed in several ways as are several other videos from

the same author. *Every* video I have seen from him are wrong, although
this is is not as bad as others.

It’s not that important that the phase of the open and short are exactly
180 degrees apart. The $11,000 18 GHz Keysight 85050B APC7 calkit is one
example of a kit where no attempt is made to achieve 180 degrees, although
it is technically possible to approach it. I don’t know the figures off
hand, and I can’t be bothered to measure or calculate it, but I would
expect that the phase probably varies from 170 to 180 degrees on the 85050B
yet that kit is very accurate, with an error on the worst standard not
exceeding 0.4 degrees to 8 GHz and 0.6 degrees up to 18 GHz

He makes no attempt to explain why he is aiming to get 180 degrees, other
than to say that his HP kit has 180 degrees. I doubt he has a clue why, as
he never says why.

For waveguide calibration kits you don’t use opens for calibration. Leaving
a waveguide open just makes an antenna with a return loss of about 13 dB.

Instead two shorts of different delays are used. Since wavelength in
waveguide is not inversely proportional to frequency, there’s a massive
difference between the phases in waveguide as the frequency is swept.

2) He has a totally flawed video on reference planes. His misunderstanding
of that can be seen in this video too, but it more subtle.

I find YouTube to be awash with people creating videos about things they
know
a little about. He is one such example.

Dave.
--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

On Tue, 18 Aug 2020 at 18:38, Jerry Gaffke via groups.io <jgaffke=
yahoo.com@groups.io> wrote:

Links within the FAQ now work. thanks!

I could understand the results shown in that FAQ

if the VNA is calibrated using one set of O-S-L standards

and the VNA is then used to measure the open and short standard

from some other calibration kit.

But if the VNA is measuring the standards to which it was calibrated,

should it not show the Smith chart at the top of the FAQ, with three nice
clean dots?

Jerry, KE7ER

If, after calibration, there were perfect opens and shorts* at the
reference plane*, there would be dots. But the standards apply opens and
shorts some distance from the reference plane, so the arcs exist.

The delay on that female short is 27.99 ps. The delay on the male short is
significantly longer (63.078 ps), so had I shown the male, the arc would
have been more than twice the length.

Dave


--
Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom

To do a proper cal on most modern VNA or PNA first the kit used it identified to
the machine. The -032kit (SMA) from HP is know the the Agilent E8357 PNA
by the kit number and will correct for any flaws in the actual physical standards
by adding the correction factors already stored. The end result is three dots.

Substitute a different kit and you get a different result without calibrating.
I used kits for N, BNC, SMA, SMB and TNC and also terminated wires
as a part of the nominal work done (antenna measurements).

However whats going on requires understanding transmission lines, and
PNA internal operation. Agilent has a few good presentations on it.

However the NanoVNA is good and simple but nowhere near the level of
accuracy, and dynamic range. Before you get into the load and cal you
have to also recognize the error level and internal math limitations.

In the end its a plane of calibration that has to be understood, not
how long it is to the short (phase distance).

Allison
-----------------
No direct email, it goes to bit bucket due address harvesting in groups.IO

OK, I think I understand now.
When calibrating a VNA to the standards, an adjustment is made by the software
as to exactly where the reference plane is.

The calibration could have been structured such that we see the three dots
when measuring the calibration standards.
But then the reference plane would not be at a convenient position
for performing measurements.

Jerry, KE7ER

On Tue, 18 Aug 2020 at 16:28, Jerry Gaffke via groups.io <jgaffke=
yahoo.com@groups.io> wrote:

David,

Having O-S-L correction factors only available in host software like
nanovna-saver

Not if you want to take the unit mobile and use it without a computer,
which for me at least is the big attraction of a NanoVNA.

Jerry, KE7ER

If I want a laboratory, I own two HP ones, have access to a PNA-X. (I also
have a VNWA which I have never used) The big attraction to me for the
NanoVNA is its portability, and that gets lost if you have to hook it up to
external software.

Everyones uses are different. But IMHO, the thing the NanoVNA lacks most is
any sensible form of calibration, in the firmware. That would require one
to enter calibration kit parameters.

If you are only working at HF, it is not an issue.

Dave

The internal software doe not have not have that cal standard set in it.
The side effect is the user then has to "do the math" and adjust as needed.
Tedious but still better than not having anything.

The down side is you have to understand what you have and how best
to exploit it.

At HF its less significant and even at 400mhz its not so bad.

Allison
-----------------
No direct email, it goes to bit bucket due address harvesting in groups.IO

On 8/18/20 1:15 PM, Dr. David Kirkby, Kirkby Microwave Ltd wrote:

On Tue, 18 Aug 2020 at 16:28, Jerry Gaffke via groups.io <jgaffke=
yahoo.com@groups.io> wrote:

David,

Having O-S-L correction factors only available in host software like
nanovna-saver

Not if you want to take the unit mobile and use it without a computer,
which for me at least is the big attraction of a NanoVNA.
Jerry, KE7ER

If I want a laboratory, I own two HP ones, have access to a PNA-X. (I also
have a VNWA which I have never used) The big attraction to me for the
NanoVNA is its portability, and that gets lost if you have to hook it up to
external software.
Everyones uses are different. But IMHO, the thing the NanoVNA lacks most is
any sensible form of calibration, in the firmware. That would require one
to enter calibration kit parameters.

And is probably a non trivial modification - you'd need a UI for entering and editing the parameters, a UI to select them (not so tough), and a way to preserve them across power cycles.




In main.c (working from hugen's repo), here's probably where you need to fool with it..


static void apply_error_term_at(int i)
{
// S11m' = S11m - Ed
// S11a = S11m' / (Er + Es S11m')
float s11mr = measured[0][i][0] - cal_data[ETERM_ED][i][0];
float s11mi = measured[0][i][1] - cal_data[ETERM_ED][i][1];
float err = cal_data[ETERM_ER][i][0] + s11mr * cal_data[ETERM_ES][i][0] - s11mi * cal_data[ETERM_ES][i][1];
float eri = cal_data[ETERM_ER][i][1] + s11mr * cal_data[ETERM_ES][i][1] + s11mi * cal_data[ETERM_ES][i][0];
float sq = err*err + eri*eri;
float s11ar = (s11mr * err + s11mi * eri) / sq;
float s11ai = (s11mi * err - s11mr * eri) / sq;
measured[0][i][0] = s11ar;
measured[0][i][1] = s11ai;

// CAUTION: Et is inversed for efficiency
// S21m' = S21m - Ex
// S21a = S21m' (1-EsS11a)Et
float s21mr = measured[1][i][0] - cal_data[ETERM_EX][i][0];
float s21mi = measured[1][i][1] - cal_data[ETERM_EX][i][1];
float esr = 1 - (cal_data[ETERM_ES][i][0] * s11ar - cal_data[ETERM_ES][i][1] * s11ai);
float esi = - (cal_data[ETERM_ES][i][1] * s11ar + cal_data[ETERM_ES][i][0] * s11ai);
float etr = esr * cal_data[ETERM_ET][i][0] - esi * cal_data[ETERM_ET][i][1];
float eti = esr * cal_data[ETERM_ET][i][1] + esi * cal_data[ETERM_ET][i][0];
float s21ar = s21mr * etr - s21mi * eti;
float s21ai = s21mi * etr + s21mr * eti;
measured[1][i][0] = s21ar;
measured[1][i][1] = s21ai;
}


and here, after the measurements are made, where the measurements are turned into cal parameters.



void cal_done(void)
{
chMtxLock(&mutex_sweep);
ensure_edit_config();
if (!(cal_status & CALSTAT_LOAD))
eterm_set(ETERM_ED, 0.0, 0.0);
//adjust_ed();
if ((cal_status & CALSTAT_SHORT) && (cal_status & CALSTAT_OPEN)) {
eterm_calc_es();
eterm_calc_er(-1);
} else if (cal_status & CALSTAT_OPEN) {
eterm_copy(CAL_SHORT, CAL_OPEN);
eterm_set(ETERM_ES, 0.0, 0.0);
eterm_calc_er(1);
} else if (cal_status & CALSTAT_SHORT) {
eterm_set(ETERM_ES, 0.0, 0.0);
cal_status &= ~CALSTAT_SHORT;
eterm_calc_er(-1);
} else {
eterm_set(ETERM_ER, 1.0, 0.0);
eterm_set(ETERM_ES, 0.0, 0.0);
}

if (!(cal_status & CALSTAT_ISOLN))
eterm_set(ETERM_EX, 0.0, 0.0);
if (cal_status & CALSTAT_THRU) {
eterm_calc_et();
} else {
eterm_set(ETERM_ET, 1.0, 0.0);
}

cal_status |= CALSTAT_APPLY;
redraw_request |= REDRAW_CAL_STATUS;
chMtxUnlock(&mutex_sweep);
}

I couldn't figure out who from the morass of comments,
but somebody wrote
"It’s not that important that the phase of the open and short are exactly
180 degrees apart".

This is true only if the calibration routines know about this! Otherwise
all bets are off. And the only way for the routines to know about this is
through the cal stds definitions table for that physical set of standards.

Dana

On Wed, 19 Aug 2020 at 00:04, Dana Whitlow <k8yumdoober@...> wrote:

I couldn't figure out who from the morass of comments,

but somebody wrote

"It’s not that important that the phase of the open and short are exactly

180 degrees apart".



This is true only if the calibration routines know about this! Otherwise

all bets are off. And the only way for the routines to know about this is

through the cal stds definitions table for that physical set of standards.



Dana

Even if the standards are 180 degrees apart in phase, the calibration
routines need to know the exact phases at each frequency. +100 degrees and
-80 are significantly different from +110 degrees and -70 degrees, yet both
differ by exactly 180 degrees.

In any case, it is impossible to keep the phases 180 degrees apart at all
frequencies. A polynomial is used to express the fringing capacitance of
the open and sometimes the inductance of a short too, although the
inductance of the short is a significantly smaller effect.

--

Dr. David Kirkby,
Kirkby Microwave Ltd,
drkirkby@...
https://www.kirkbymicrowave.co.uk/
Telephone 01621-680100./ +44 1621 680100

Registered in England & Wales, company number 08914892.
Registered office:
Stokes Hall Lodge, Burnham Rd, Althorne, Chelmsford, Essex, CM3 6DT, United
Kingdom