Stray/Parasitic Capacitance - Impacts phase velocity?

[Strides]

Hey everyone,

I'm trying to measure the stray capacitance in a circuit comprised of an rf signal generator, an oscilloscope, a coaxial cable (short-circuited) and some capacitors. I measured the resonant frequencies of the coaxil cable for varying values of capacity (0pF to 850pF) between 4MHz and 100MHz. I then used the following equation to plot X vs capacity and find the corresponding characteristic impedance and stray capacitance:

$$X = ZC_{x} + ZC_{s}$$

Where X is:

$$X = 1/(2πf tan(2πfl/v_p))$$

(I was only given the function for X, so not entirely sure what it describes, I'll be grateful to anyone who is able to offer some insight or background theory)

l = length of test cable, 2m
f = frequency
v = phase velocity

Results from graph:

Z = 55.167
Cs = 29.5pF

Does this value seem sensible for the given frequency/capacitance range?

I then calculated the phase velocity of the coaxil cable to be around 1.94*10^(-8), and that the stray capacitance increases the phase velocity by around a mere 7m/s. Again is this expected for the given range?

Thanks for all the help, in advance.

 

[berkeman]

Can you upload a picture of your setup, and a schematic diagram?

29pF is a little high for parasitic capacitance, but it's hard to say without understanding your setup better.

 

[Strides]

At the moment, all I've got is a simple diagram, I hope that suffices.

 

[berkeman]

And you want to measure which parasitic capacitance? On the far side of the resistor where the cable is (minus the expected cable capacitance)? What is the capacitance of your 'scope probes?

 

[Strides]

The scope probes are just coaxil cables connected to the oscilloscope, which should be around 50Ω & 90pF per meter. With a length of a couple cm, I assume roughly 2pF per channel?

I was hoping to measure any parasitic capacitance which may effect the phase velocity. I was only given the above equation in the experiment to measure it by, so I'm unsure of primarily where that would be based in the circuit. I would be grateful for any explanation that you're willing to give regarding that.

 

[sophiecentaur]

X=ZCx+ZCsX=ZCx+ZCs​

That would apply for a series stray C. For parallel C, you would need the Admittance equivalent.

 

[Strides]

I've been specifically told to use the equation above for the serial stray capacitance, so I assume that in this experiment the parallel stray capacitance that you've mentioned must be taken as negligible.

 

[Strides]

I mainly just want to know, if the increase in the phase velocity for the given stray capacitance is an acceptable value for the given range of frequency?

 

[sophiecentaur]

I've been specifically told to use the equation above for the serial stray capacitance, so I assume that in this experiment the parallel stray capacitance that you've mentioned must be taken as negligible.

The C's in your diagram are in parallel. I wonder what serial capacitance is suggested - a capacitative coupling perhaps. In any case that is not mentioned on the diagram. Perhaps you should check or ask for advice from the guy who set the question?
I'm not sure of the context of the question. Are you finding the phase velocity from the electrical length of the transmission line and the Impedance? The additional (parallel) C 's will modify the electrical length.
I don't imagine you would be using a Smith Chart to solve the problem these days but the graphical method is great for getting to understand what is going on.

 

[Strides]

Here's the lab guide that I've been using, the question for stray capacitance is at 7.4. I've already found the phase velocity from the length of the transmission line without taking the stray capacitance into consideration.

 

[sophiecentaur]

Here's the lab guide that I've been using, the question for stray capacitance is at 7.4. I've already found the phase velocity from the length of the transmission line without taking the stray capacitance into consideration.

OK. Now I have read the instruction sheet, I am a bit clearer. The circuit diagram is a bit sloppy. Why is the signal source ground not shown and what is its source resistance? Is that series R high - making a current source? It enables you to detect when there's a resonance, using your two scope probes.
To answer your question about the "χ". That is just a substitution to get rid of that mess of symbols in (20). That's a typical 'book work' trick which saves writing and shows the pattern of things - and just manages to confuse people like you and me! But it is worth getting used to that sort of thing.
As a reality check, you have 2m of coax, which will be about equivalent to 1.5m of freespace. That would resonate for a signal of 6m λ in free space ( = around 20MHz Edit.Sorry =50MHz ) with no parasitics. Is that something like what you found? What does your graph look like?
The parasitic C that you inferred (30pF) may be right for a cheapo probe arrangement. How 'lively' was your system? By that I mean how much did things change if you put your hands on the probes? Normally, for a reliable measurement, the setup in the box in the diagram would involve the probes being held in a connector with a good ground. I could believe the answer you got but, like Berkeman, I would need a bit more information about the details.

I've been specifically told to use the equation above for the serial stray capacitance,

The suffix "s" stands for "stray" and not, I think for 'series'. All the capacitances are basically in parallel with the measurement end of the line.

I am not sure that saying the added C increases the Phase Velocity is right. It just alters the calculated value (as stated in 7.4); it's an added measurement error.