I would like to know if what is written in the attached diagram is correct or not.

Gain = Vs/E
S21 = Vs/Ve

--
F1AMM
François

Hi François,

Your S21 equation isn't quite correct. Using the terminology in your picture, it should be S21 = V2/E.

As to why, first consider that in a transmission line system, your V1 and V2 each consist of two waves, a forward wave (Vf) and a reflected wave (Vr).

So V1 = Vf1 + Vr1, and V2 = Vf2 + Vr2.

S21, by definition, equals Vf2/Vf1.

Now, assuming the terminating R equals the system Zo, then Vr2 = 0. From the equation, above, Vf2 = V2 - Vr2 = V2 - 0 = V2.

As for Vf1, imagine the source (Vs) and its source impedance (Zs) are connected to an infinitely long coax of characteristic impedance Zo, which is also equal to the source impedance. That is, your R = Zo, and Vs = your 2*E.

Vf is simple the voltage division of Vs by the source impedance and the coax impedance, and is therefore Vf = Vs*(Zo/(2*Zo)) = (2*E)*0.5 = E.

And so, S21 = Vf2/Vf1 = V2/E.

(I haven't thought about your gain equation yet).

Best regards,

- Jeff, k6jca

François,

Attached is a schematic showing an "ideal" VNA that only measures S11 and S21 (similar to the NanoVNA). Hopefully it is useful in understanding how a VNA that measures just S11 and S21 works.

Note that it assumes VNA circuitry resistances are exactly equal to Zo (e.g. 50 ohms) and that voltages are measured with high-impedance voltmeters that don't load the nodes they are measuring.

In actuality, nothing is ever ideal, and thus the directional-bridge in the NanoVNA looks different than this circuit (although similar in basic form).

Also, note that if Port 2's impedance is not exactly equal to Zo, then there will be a reflected wave, Vr2. The presence of this reflection will affect both the S21 measurement and the S11 measurement when the latter is made with a DUT (Device-Under-Test) connected between Ports 1 & 2.

VNAs that measure all four 2-port s-parameters (S11, S21, S12, and S22) can correct for this error. The NanoVNA cannot, and so Port 2's impedance needs to be as close to Zo as possible.

(On the other hand, the NanoVNA can correct its S11 and S21 measurements for errors in Port 1's impedance, although this requires using "Enhanced Response" error correction, which is not done in some versions of NanoVNA software).

Best regards,

- Jeff, k6jca

Hello

Thank you very much for your interest in my diagram. On the various websites, we only find very complicated formulas that I don't understand.

So, from what you wrote to me, S21 is indeed the expression of the amplification: V2/E which is equal to 1 when the DUT is a simple connection (potentiometric divider). I have always done my filter calculations with this type of diagram using an electromotive force generator 2E in series with a resistor R

Your diagram of the NanoVNA is very educational. I propose to integrate it, with your agreement, into my instructions (in French) concerning the NanoVNA. On this subject, too, we generally only find convoluted explanations that complicate everything.

--
F1AMM
François

Hi François,

Typically, I consider voltage gain to be S21. But I found one website stating that it is S21/(1+S11) (https://home.sandiego.edu/~ekim/e194rfs01/sparam.pdf -- see last page).

This latter equation equals Vf2/(Vf1+Vr1), or, per your original drawing, V2/V1 rather than V2/E.

Why this difference in voltage gain definitions? I don't know. Hopefully others here can comment.

And yes, please feel free to use my schematic, although I'd wait first to see if anyone here disagrees with it.

Best,

- Jeff, k6jca

Jeff didn’t mention it, so I will remind all reading (some of whom may not make the connection) that the S21 and its predecessor variables are all complex numbers.

Ed McCann
AG6CX

Jeff, Francois and others (and others):

Please look at the screenshot from page 22 0f Harvey Mudd pdf on
"S-Parameters Don't Map Clearly to Circuits."

S21 = b2/a2 = skip a term for now = V2/V1 * (1+S11)

The whole presentation attached for reference.

Also attaching Amakawa paper as a basic foundational document.

Best to all from Sausalito.

A Bientot!

Ed McCann
AG6CX







On Sun, Sep 15, 2024 at 9:48 AM Jeff Anderson via groups.io <jca1955=
sbcglobal.net@groups.io> wrote:

Hi François,

Typically, I consider voltage gain to be S21. But I found one website
stating that it is S21/(1+S11) (
https://home.sandiego.edu/~ekim/e194rfs01/sparam.pdf -- see last page).

This latter equation equals Vf2/(Vf1+Vr1), or, per your original drawing,
V2/V1 rather than V2/E.

Why this difference in voltage gain definitions? I don't know. Hopefully
others here can comment.

And yes, please feel free to use my schematic, although I'd wait first to
see if anyone here disagrees with it.

Best,

- Jeff, k6jca

Thanks, Ed.

Correct me if I am wrong. In other words, there are two ways to look at gain.

The first definition of gain is simply S21, which is the gain of the Forward Voltage *out* of the DUT (into a matched termination) divided by the Forward Voltage *into* the DUT. In other words:

S21 = Vf2/Vf1

The second definition is the familiar equation for Voltage Gain used for any lumped-element circuit:

Voltage Gain (i.e. Av) = Vout/Vin

Where Vin is the voltage at the input of the DUT (i.e. François' V1, which equals Vf1+Vr1), and Vout is François' V2, which equals Vf2, assuming the output termination is matched to Zo so that Vr2 = 0.

Therefore, Av = Vout/Vin = Vf2/(Vf1+Vr1) = (Vf2/Vf1)/(1+Vr1/Vf1) = S21/(1+S11)

Best,

- Jeff, k6jca

Hello

I offer you this document, page 15 and 16 (of the file).
https://web.ece.ucsb.edu/~long/ece145a/Notes4_Sparams.pdf

The conclusion is in the attached file S21_02.png

With the conventions of my first diagram in the attached file S21_AMM.png

Then Amplification = |V2|/E = S21

I'm not trying to be right; what matters to me is that we converge on the same conclusion.

--
F1AMM
François

I checked with a DUT consisting of a simple series resistor of 217.4 Ω

- Calculated S21 = |V2|/E = 0.315 (base 50Ω / 50Ω)
- Measured with my NanoVNA S21 = 0.32

S21 = |V2|/E is good
--
F1AMM
François

De la part de AG6CX
Envoyé : dimanche 15 septembre 2024 19:32

François,

what Ed has reminded everyone of—before my blood pressure went up to much—is that S-parameters are all, by their very nature, *complex* numbers. In other words, reactance may be accurately represented, whether capacitive or inductive. A simple number can not do that.

s_21 is *not* a direct measure of gain. Rather it is a *vector*, indicating both a magnitude and a phase angle, usually expressed as a complex number for ease of computation.

When we're using a VNA it is capable of so much more than the over-simplification of a VSWR number or a dB gain number. All of this became important 60 and more years ago when HP and others produced readily available—for industry anyhow—VNAs. That and the switch from valve (tube for left-pondians) to transistor (semiconductor) technology made accurate measurement of complex impedances essential. (Although semiconductors are generally less robust than valve technology, it's still possible to damage valves, especially power valves.)

But a big joy of a VNA is that as well as measuring S-parameters, the VNA can also compute alternative representations, such as:

• VSWR (for rough assistance in tuning aerials/antennas)

• series and parallel equivalent resistance and signed reactance
(for more useful help in tuning aerials/antennas)

• forward gain of an amplifier, or transfer function of a filter

• reverse gain of an amplifier, or “reverse” transfer function of a
filter

and quite a few more, for various purposes, at multiple frequencies of the user's choosing.

73, HTH,

Robin, G8DQX (who still has his original paper copy of HP AN-95, three-hole punched, hidden away somewhere!)

P.S. The answer to your question in message https://groups.io/g/nanovna-users/message/37933 depends on whether V1 and V2 are scalar or vector quantities. V2 *must* be a complex quantity, a phasor if you like, for the equation to hold. (And equally V1 must be a complex number.) Some of the confusion with scattering parameters is because the usual variables a and b are *complex numbers*. Thus old-style forward gain is |s_21 |, which gives the gain independent of any change of phase.

I share. In the early 1970s, when I worked in a CNES laboratory, we bought a Hewlett Packard Vector Network Analyzer (a fortune). Importation into France, by the French government, was prohibited but CNES benefited from an exemption. This device was used mainly in microwaves because the oscillators were tuned by YIG beads.

In our NanoVNAs, based on the diagram posted by Jeff, what do we measure in physical reality?

https://groups.io/g/nanovna-users/attachment/37936/0/290915%20ideal%20VNA%20only%20S11%20and%20S21.png

amplitudes? and phases? certainly not native S parameters. The VNA calculates the S parameters but from what type of data.

I got used to manipulating S11* (complex) to do my calculations and find Z. I want to do the same thing with S21.
--
F1AMM
François

De la part de G8DQX list
Envoyé : lundi 16 septembre 2024 12:08

On Mon, Sep 16, 2024 at 03:47 AM, François wrote:

In our NanoVNAs, based on the diagram posted by Jeff, what do we measure in
physical reality?

The NanoVNA is a "lumped-element" circuit (at least at low frequencies), meaning the distances between its circuit components are so short that we don't need to worry about wave propagation effects on circuit voltages and currents.

Physically, both the Port 1 voltage measurement (V1) and Port 2's voltage measurement (V2) are made at single points.

Now, on a transmission-line with reflections, there will be both a forward wave traveling from source to load and a reflected wave traveling from load to source. At any point on the transmission line, the voltage at that point and the current through that point have these relationships:

V @ point = Vf + Vr
I through point = If - Ir

Note, too, that the amplitude and phase of V and I at a single point on the transmission line constantly change. However, they are related due to the fact that If = Vf/Zo and Ir = Vr/Zo; Zo being the characteristic impedance of the transmission line.

The Directional Bridge in the VNA's "Port 1" is measuring one voltage and creating a second voltage. The voltage it creates is simply the source voltage divided by 2 and it represents Vf's value if the VNA were driving a transmission line of impedance Zo attached to Port 1.

The second voltage, measured at a single point (Port 2), equals Vf + Vr at this point *if* Port 1 were attached to a transmission line.

But are we really measuring forward and reflected waves? Sure, if we are attached to a transmission line of characteristic impedance Zo, Zo being the impedance the VNA was designed to work with.

But as I mentioned above, the directional bridge is simply looking at the voltage at a single point. An assumption is made that it is attached to a transmission line of impedance Zo, but it has no way to determine if that is indeed true. In fact, it could be attached to a resistor connected directly across its Port 1 -- another lumped element -- with no transmission line and thus no waves, yet the VNA calculates Vf and Vr as though forward and reflected waves actually existed.

Or it could be attached to a transmission line with a different characteristic impedance than the Zo it was designed for, for example, 75 ohms rather than 50. In this case, the Vf and Vr it calculates at Port 1 will not represent the actual Vf and Vr on this transmission line. For example, the 75 ohm line might be properly terminated and thus has no reflections, but the VNA, referenced to 50 ohms, will generate an S11 value as if there were an SWR of 1.5:1 on the line. And this value would accurately represent the impedance that the VNA measures at that point (i.e. 75 ohms), but the VNA-created Vf and Vr values are incorrect.

So the Vf and Vr values determined by a VNA are, in a sense, fictitious. Never the less, these two quantities can be used to accurately determine the impedance connected to a VNA's Port 1 (calculated as S11), irrespective of coax Zo, or even the existence of coax.

Hopefully this helps. When I was in college my professor mentioned that the way to measure SWR was to find the standing-wave voltage maximum on a transmission line and then move away by 1/4 wavelength, find the voltage minimum, and use these two values to calculate SWR. I always wondered how a "lumped-element" circuit could determine SWR until finally, one day years later, I decided to work it out for myself.

Best regards,

- Jeff, k6jca

P.S. François, I did a bit more editing of my diagram. It is attached. Don't know if it will be of more help.

On Mon, Sep 16, 2024 at 07:40 AM, Jeff Anderson wrote:

The second voltage, measured at a single point (Port 2), equals Vf + Vr at
this point *if* Port 1 were attached to a transmission line.

Typo correction!

This line should read: "The second voltage, measured at a single point (Port 1), equals Vf + Vr at this point *if* Port 1 were attached to a transmission line."

In other words, change "Port 2" to "Port 1".

- Jeff, k6jca

Francois, I've attached a 3rd iteration of my diagram. Because my source voltage was labeled V1, I changed its reference designator to "Vs" to differentiate it from your V1.

I also changed R3's reference designator to "Rs", as this resistor represents Port 1's source impedance.

Best regards,

- Jeff, k6jca

This diagram suggests that Vf2 and subsequently Vr2 originate from a source
at port2 but there is no signal source on Port2 ...

On Mon, 16 Sept 2024 at 20:21, Jeff Anderson via groups.io
<jca1955@...> wrote:

Francois, I've attached a 3rd iteration of my diagram. Because my source voltage was labeled V1, I changed its reference designator to "Vs" to differentiate it from your V1.

I also changed R3's reference designator to "Rs", as this resistor represents Port 1's source impedance.

Best regards,

- Jeff, k6jca

On Mon, Sep 16, 2024 at 11:54 AM, Dragan Milivojevic wrote:

This diagram suggests that Vf2 and subsequently Vr2 originate from a source
at port2 but there is no signal source on Port2 ...

Hence the Vf2 and Vr2 are the result of whatever is connected to Port2, presumably the DUT.
Hence also the note that IF R4=50ohms=Z0, then the equation holds. If Z0 is other than 50ohms, it is more complicated.

On Mon, Sep 16, 2024 at 12:06 PM, Stan Dye wrote:

If Z0 is other than 50ohms, it is more complicated.

Yes. In a "normal" VNA error-correction would correct for errors in Port 2's impedance. Unfortunately, this correction first requires measurement of all four 2-port s-parameters (S11, S21, S12, and S22), two of which the NanoVNA cannot measure (S12 and S22), and thus it cannot correct for Port 2 imperfections.

When performing S21 measurements, imperfections in a NanoVNA's Port 1 impedance can be corrected-for, but it requires the use of "Enhanced Response" error correction (a more complicated equation than what is found in some NanoVNA apps).

Regarding Dragan's point, I should add a DUT between Ports 1 & 2 to clarify the source of Vf2 (and thus Vr2).

Best regards,

- Jeff, k6jca

----- S21 -----

I am quite familiar with classical circuit calculations (jlω, 1/jCω etc) but I have trouble following the reasoning with the waves in one direction and the other.

Would you be so kind as to write me again, with the correct variables of the new diagram, the explanatory text that I will attach to your last diagram in my notice on NaonoVNA. Others will read it and will be more comfortable.

The important conclusion of this discussion is that the definition of S21 is identical to that of amplification, including in complex form.

----- S11 -----

For the readers of this thread, without wanting to show my knowledge, you will find attached the three magic formulas relative, this time to S11.

Z* is the impedance expressed in the form of a complex number. The asterisk (*) character indicates that this is a complex number. Excel is a great simplifying help to do these calculations because Excel knows complex numbers. Exemple avec un Excel en français :

=COMPLEXE(B30;C30) .
=COMPLEXE.DIV(COMPLEXE.SOMME(1;D30);COMPLEXE.DIFFERENCE(1;D30))

----- ToucheStone / Eznec -----

Be careful though that in TouchStone files (.s1p and .s2p) the S11 and S21 can be of the form A, jB (nanaovna-saver) or of the form Module, Argument (Eznec) with the argument expressed in degrees.

--
F1AMM
François

On Mon, Sep 16, 2024 at 10:28 PM, François wrote:

I have trouble following the reasoning with the waves in one direction and the
other.

Would you be so kind as to write me again, with the correct variables of the
new diagram

François,

If you are addressing this request to me, may I recommend that you ask specific questions regarding any points you do not understand.

Many thanks,

- Jeff, k6jca