# Measuring the transfer function of a "black box"

Discussion in 'General Electronics Chat' started by zvir, Jun 25, 2014.

1. ### zvir Thread Starter New Member

Oct 22, 2012
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Hi, people.

I'm a physics student, and currently I'm measuring some stuff in the lab, and I have hit a wall, and would kindly ask for help. So, my problem is as follows: I am using a certain measurement device (Agilent RLC meter, not sure about the exact model; it measures R and X components of the complex impedance [or |Z|-θ, whatever]) to measure the electrical properties of certain samples. However, there is a cryostat between the sample and the measurement device, and it's influence on the signal can't be neglected. I am attaching a crude 'schematic' of my measurement system.

So, I need to find a way to get the cryostat out of my measurements. It's impedance is too complex to approximate with some replacement circuit. I am not at all interested in it's properties or internal structure. What I have is a few sets of complex impedance measurements where the DUT is a known resistor, in [20Hz, 30kHz] range.

What I need now is a way to process the data raw data I get off my measurement device, in order to remove the 'black box' in the middle. I believe the best way to do it is to construct a transfer function of the box, but I don't know how to do it from the measurements I have. Any help would be appreciated.

P.S. please forgive my limited drawing skills.

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2. ### alfacliff Well-Known Member

Dec 13, 2013
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use a non reactive load on the measured side of the cryostat, if you model the cryostat, you should be able to subrtact its effects from your measurements.

3. ### MrAl Distinguished Member

Jun 17, 2014
2,676
550
Hi,

What do you mean that "it's impedance is too complex to approximate with some replacement circuit". That's the whole idea of getting to the point where you can write out some equations that will reduce to the character of the device under test.
Do you have any data or model number of the device between your measurement unit and your device under test?
Do you know if that in between device is linear?

4. ### zvir Thread Starter New Member

Oct 22, 2012
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Isn't that exactly what I said i want to do? Read my original post again: "What I have is a few sets of complex impedance measurements where the DUT is a known resistor, in [20Hz, 30kHz] range". I just don't know how to extract the cryostat model from that.

I tried to approximate the cryostat+sample system with a not-so-simple RLC array. The results were not really great.

And that's exactly what I need, an equation to process my measurements with, in order to get only the character of the sample, and not sample+cryostat.

I do not have any model numbers. The device in between is made out of lead capacitances and inductances, some low-pass filters etc., and contains no active components, it should be linear. Also, everything is measured in small-signal mode (10mV order of magnitude), so it's safe to assume that everything behaves linearly. Some of those components are known, but due to a lot of unwanted properties of the system as a whole, the device is still difficult to model. E.g., the sample is inside a box made of several layers of metal, and the leads connecting the sample and the measurement unit are thin cryo-resistant coaxial cables with relatively high resistance.

5. ### KeepItSimpleStupid Well-Known Member

Mar 4, 2014
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I think your missing the whole point and/or not reading the directions.

The meter has to have the ability to do short and open test at all the frequencies of interest. This calibrates the extraneous stuff like the leads. You need to watch carefully how the leads and shields are connected, You may have to use isolated BNC's. You may find that the capacitance of the cables are too high.

I had good luck using twisted teflon wires becase I was unable to get high temp coax. I used the twisted teflon as a guard. The chamber provided the shield. some of the DC measurements were in the pA range. The hall effect holder had the ability to use a potentiometer for nulling because we couln't afford the right instrumentation.

I've set up LCR measurements for both cryostats and environmental chambers. I also designed a couple of nice holders for hall effect measurements for thin and bulk films.

6. ### zvir Thread Starter New Member

Oct 22, 2012
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No, I think you're missing the point here. If I only have a set of data, which was measured earlier, and without the meter's corrections, how would I do those corrections manually?

7. ### MrAl Distinguished Member

Jun 17, 2014
2,676
550
Hi again,

Well one way would be to use admittance parameters or maybe impedance parameters using the theory of two-port networks.
This requires some testing of the device in question of course, but if it's linear it should not be too difficult.

For example, using admittance parameters we get the two, two-port network equations:
I1=y11*V1+y12*V2
I2=y21*V1+y22*V2

These are just two algebraic equations, and we want to solve for the admittance parameters y11, y12, y21, and y22.
The voltages are V1 the input voltage and V2 the output voltage, and I1 the input current and I2 the output current looking into the output of the network.

To solve for the parameters, we apply one voltage (V1 or V2) and short out the other port. So if we apply V1 we short V2, and measure both I1 and I2. We then apply V2 and short V1, then measure I1 and I2 again.
When we short V2 for example the two equations reduce to:
I1=y11*V1
I2=y21*V1

and note these are easy to solve now because we know I1, I2, and V1.

I am pretty sure that for a linear network all the parameters should admit a single complex solution for all frequencies, but you should be able to find out more about this by looking up some two-port network theory.

Of course you'd have to test it to make sure everything was right.
You'd also have to test it for accuracy related to measurement resolution, because obviously if you have a 1H inductor in series with a 1uH (unknown) inductor the ratio is a million to 1 so there will be significant error unless the measurements are impractically super accurate.

If there are inductors inside the device then one or more of them might act in a non linear fashion simply because the core acts differently depending on excitation and even frequency. This would means you'd have to do a lot more tests over some frequency range and compile the results and determine either how it changes, or else just use the data to interpolate between measured points. This could get pretty complicated though.

Last edited: Jun 28, 2014
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8. ### zvir Thread Starter New Member

Oct 22, 2012
8
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Kudos to you, sir! This is exactly what I was looking for!

After spending a couple of days reading about two-port network theory, I derived some equations to calculate the z-parameters of a two-port network from the measurements of input impedance with the second port of the network open, shorted, and connected to a known load.

In case anyone else has the same problem, I'll just put my results here, for everyone to see.

So, a two-port network is described with a matrix equation:

$\begin{bmatrix}V_1\\V_2 \end{bmatrix}=\begin{bmatrix} z_{11}&&z_{12}\\z_{21}&&z_{22} \end{bmatrix}\cdot \begin{bmatrix} I_1\\I_2 \end{bmatrix}$

This matrix equation can be rewritten as:

$V_1=z_{11}I_1 + z_{12}I_2
V_2=z_{21}I_1+z_{22}I_2$

If a load $Z_L$ is connected to port 2 of the network, there is an additional constraint to the system:
$V_2=Z_L\cdot I_2$

Port 1 voltage and current are related as:
$V_1=Z_{input}\cdot I_1$

Also, all networks that consist only of linear passive components exhibit a property called reciprocity, which introduces another constraint to the system: $z_{12}=z_{21}=z_{off}$ (off = offdiagonal).

Using this system of equations, we can solve for $Z_{input}$:
$Z_{input}=z_{11}-\frac{z_{off}^2}{Z_L+Z_{22}}$

If port 2 is open, $Z_L=\inf$, which gives us the first parameter:
$Z_{In,Open}=z_{11}$

Setting $Z_L$ to zero (shorted output port), and to some known value (resistor is probably the best, since it only exhibits real part of the impedance), gives us two more equations, with two more unknown parameters.

Solving for the impedance parameters:

$Z_{11}=Z_{In,Open}$

$Z_{22}=\frac{Z_{In,Open}-Z_{In,Load}}{Z_{In,Load}-Z_{In,Short}}\cdot Z_L$

$Z_{12}=Z_{21}=\sqrt{\left(Z_{In,Open}-Z_{In,Short}\right)\cdot z_{22}}$

where:
$Z_{In,Open}$: input impedance of the network with port 2 open,
$Z_{In,Short}$: input impedance of the network with port 2 shorted,
$Z_{In,Load}$: input impedance of the network with $Z_L$ connected across port 2.

Please note, all the z-parameters are complex numbers.

With all these parameters calculated, the impedance of the load can now be extracted:
$Z_L=\frac{Z_{off}^2+\left(Z_{In,Unknown}-z_{11}\right)\cdot z_{22}}{z_{11}-Z_{In,Unknown}}$
where $Z_{In,Unknown}$ is input impedance of the system with the unknown load connected across port 2.

Once again, I wish to give my thanks, and I hope that someone else will find these equations useful.

9. ### The Electrician AAC Fanatic!

Oct 9, 2007
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This is the easy part.

Now you have to find a representation of all four Z parameters as complex functions of a complex variable over the 20 Hz to 30 kHz frequency range.

This problem is known as "System Identification", and you can find some discussion and a lot of references here:

http://en.wikipedia.org/wiki/System_identification

10. ### zvir Thread Starter New Member

Oct 22, 2012
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So, I checked my calculations by connecting a known capacitor (483nF, measured with an ordinary multimeter) across the output port, measuring the impedance of the entire system and reconstructing the capacitor impedance using previously calculated impedance parameters. The results were quite good, I calculated capacitance of 481nF.

Interesting fact: I calculated all four impedance parameters, and they all seem to have the same value (well, not exactly the same, but within 1% from one another). I have no explanation for that, and I'd really like to know what's going on here. Any ideas?

Frequency dependence is not really an issue here. I can just measure the sample at the same frequencies I used to measure the parameters, or I can interpolate between points. Interpolation across the frequency spectrum should work, because the impedance is a smooth, monotonic function of frequency.

However, there is one problem. I will have my measurements spread across a large temperature range (will be going as low as 10mK), resistive parts of the cryostat will change properties (total resistance will change by more than an order of magnitude). I read something about a theorem by Sidney Darlington, which states that any RLC network can be represented as an LC network terminated with a certain resistive element (http://en.wikipedia.org/wiki/Equivalent_impedance_transforms#Eliminating_resistors). This would mean that i can use a cascade of two networks (LC and R) and their transmission parameters to calculate the load, which is a very elegant solution.
That makes sense if the resistive elements are constant, however, if they are prone to change, the response of the entire network may change. E.g., even a simple RC low-pass filter changes it's properties with varying R. As I see it, there is no justification to consider the Darlington's theorem valid in this case. But still, I might be wrong. Does anyone have some more experience with this?

11. ### MrAl Distinguished Member

Jun 17, 2014
2,676
550
Hi,

You could check that theorem for a correlation between the original network resistor variations and the programmed resistor variation. You might find a simple correlation between the two. I might try this myself too just to see if it turns out to be simple.
That way you would only have to vary the programmed resistance in accordance with the temperature.

Wow, 10mK that's quite low. Trying to reach that new state of matter?

It should be pretty easy to calculate the transformed network though if you do decide to go with it, or so it seems anyway.

What network did you use to do the test you talked about with the capacitor calculation?

Oh yeah, is your in between device a true two port or are one input lead and one output lead common (that makes three leads instead of four)?

12. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,308
343
What are the Z parameters, measured at what frequency?

13. ### zvir Thread Starter New Member

Oct 22, 2012
8
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I'm not sure I understand this part. Care to elaborate a bit? Or could you recommend some literature where I could find more information on this theorem?

You never know, maybe there's a Nobel somewhere out there, just waiting for me.

Yes, the network is a true two-port. It's actually only two lines going from top of the cryostat to the sample holder. All interaction between those two lines is only due to capacitive and inductive effects, there are no electrical connections of any kind between the two. Also, every line passes through some weird stuff (copper powder filters, these are probably the most problematic element of all), which additionally complicate the frequency response.

I just connected the capacitor to the sample holder in the cryostat and measured from the connectors on top. So, I was actually using the same network to be used in proper measurements. If you like, you can find some graphs to check if the method is good here: http://borovnica.sok.hr/~zvir/

I did three sets of measurements, all of them with my measurement instrument connected to port 1; with port 2 open, then with port 2 shorted, and then with a known load (1k resistor, in my case) connected to port 2. Then I calculated the Z-parameters of the network, connected the capacitor to port 2, mesured the impedance of the system again, and applied corrections.
These graphs show the measured impedance of the system (blue), impedance of the load (calculated using previously calculated parameters, red), and the theoretical values of impedance for an ideal capacitor (green).

I'd really like to be able to modify this model to account for temperature changes. Tomorrow (or the day after), I'll be cooling the cryostat down, so I plan on doing a bunch of measurements on different temperatures, using two other lines, just to check how the response of that two-port behaves. Using that data, I will probably be able to extrapolate how each individual pair of lines changes. I just can't decide whether to use open or short measurements...

Here they are: http://borovnica.sok.hr/~zvir/zparams.txt
First column is the frequency, then Re[z11], Im[z11], Re[z12] etc. Please note that these parameters are valid only for room temperature, at lower temperatures the response probably changes. I will probably have more data on that in a few days.