Capacitive network and resonance

[dmorris619]

I am designing a loop antenna with a switchable resonance point from 500Hz to 99500Hz in increments of 1000. I am trying to switch resonance by changing the value of the capacitor by making a network of caps in parallel that are turned on with a mosfet. My problem is that the resonant frequency causes the values of the capacitor to follow a 1/x^2 trend meaning the standard C,C/2,C/4 binary combination will not work.

To get to my question, does anyone have an idea on how i can run through all 100 resonance points using a switch capacitive network? If that isn't possible are there other methods I can consider?

 

[berkeman]

I am designing a loop antenna with a switchable resonance point from 500Hz to 99500Hz in increments of 1000. I am trying to switch resonance by changing the value of the capacitor by making a network of caps in parallel that are turned on with a mosfet. My problem is that the resonant frequency causes the values of the capacitor to follow a 1/x^2 trend meaning the standard C,C/2,C/4 binary combination will not work.

To get to my question, does anyone have an idea on how i can run through all 100 resonance points using a switch capacitive network? If that isn't possible are there other methods I can consider?

Can you use a topology like an R/2R ladder DAC uses? Where you connect various combinations of caps to step through a linear progression?

 

[dmorris619]

Thats what I am trying to do but i can't figure out the progression. Its not linear i believe.

 

[berkeman]

Thats what I am trying to do but i can't figure out the progression. Its not linear i believe.

It's linear if you make the steps 500Hz. 200 steps of 500Hz let you hit the values you want. And 200 steps fits into an 8-bit R/2R ladder DAC topology (or C/2C ladder in your case...).

 

[dmorris619]

why does 500 hz make it linear but not 1000? Do you think you could provide me with an explanation?

 

[berkeman]

why does 500 hz make it linear but not 1000? Do you think you could provide me with an explanation?

I am designing a loop antenna with a switchable resonance point from 500Hz to 99500Hz in increments of 1000.

My point was that it would be difficult to make a circuit that has jumps of 1000Hz, with a base of 500Hz and a final target of 99,500Hz. But it seems very do-able to make a circuit that has uniform steps of 500Hz, which allows you to hit all the frequencies that you want. The C/2C ladder DAC codes that you use to get the frequencies will start off at 1, and increment by 2:

1      500Hz
3     1500Hz
5     2500Hz
etc.

And the C/2C ladder DAC has FETs to ground for each of the 8 capacitor taps on the ladder.

Keep in mind that capacitors don't combine in series like resistors do, so there may need to be adjustments to the C/2C ladder versus the traditional R/2R ladder. I haven't looked into what those differences might be, but hopefully you can figure out a reasonable combination of caps for the ladder.

 

[dmorris619]

Thanks, I'll give that a shot and see

 

[berkeman]

BTW, although capacitors do not combine in series/parallel like resistors do, inductors do combine like resistors. So it might be easier to use an L/2L ladder DAC arrangement, with a single resonant capacitor. You can keep that option in mind as you look at the C/2C options...

 

[dmorris619]

I am struggling to figure out each value for the capacitive ladder network. I do not believe that the network is based purely on two caps C and 2C repeated, however it is very likely that I am missing something. If it isn't two values of capacitors repeated then are there any suggestions for how i go about determining the values beyond just brute force substitution. I do have MATLAB available to me if that opens up possibilities.

 

[dmorris619]

Keep in mind that I am trying to get 100 diferent combinations using 8 capacitors so the standard dac ladder network methodology will not work because i'll end with 200 caps

 

[berkeman]

Keep in mind that I am trying to get 100 diferent combinations using 8 capacitors so the standard dac ladder network methodology will not work because i'll end with 200 caps

No, with R-2R topology, you get a binary weighted set of values. So 8 bits is plenty for making 100 values.

I'll have a look at the ladder values for the capacitor base ladder, hopefully later today.

 

[berkeman]

Hmm, wait. I think I missed that the resonances go as 1/SQRT(LC) -- the SQRT part is probably what you referring to as being a problem.

How accurate do you need these resonances to be?

 

[Phrak]

Perhaps there may be a way to get the 1,1/4,1/9,... series using a gyrator. Just a thought.

 

[berkeman]

Yeah, or if we could vary both L and C at the same time, we could get rid of the SQRT part...

Or just use a frequency divider and PLL arrangement, synthesizer style...

 

[Phrak]

Sorry, I'm not familiar with PLL designs. But both the reciprocal and the square are a problem.

To approach an ideal gyrator with Rs and Ls and Cs requires a judicious selection in the sizes of its discrete components, so it probably wouldn't be accurate over a large range of frequencies. $j\omega RC >> 1$ in the following.

The approximate equation for a gyrator is given in Wikipedia as

$$Z_{in} = R_L + j \omega R_L RC$$

R_L should be small so that the second term is dominant.

But we don't have to use just Rs and Cs if the op amp can handle it. All the stuff on the right hand side can be replaced with complex impedances, Z_xx, including the capacitance.

$$Z_{in} = Z_L + Z_L Z_2/Z_1 \ ,$$

where Z_L is small.

We can address the squaring problem by simultaneously switching in different Z₂/Z₁ components together. But, I see, we don't get the squared reciprocal, so the idea is a bust as far as I can see.

Edit: Wait, it might work. Z₂ should be a capacitor, and Z₁ a resistor. But still fraught with practical problems, as the op amp becomes an integrator, on top of other things.

 

[sophiecentaur]

Have you considered the likely Q of your resonant circuit? I can't remember what sort of Q is feasible from 'audio frequency' em radiators but it may well be that you just don't need to get the C values very accurate at the low frequency end (big C values) . That may make things much easier for you. The actual design of the antenna would be very relevant and you'd probably need to do some Q measurements of a real antenna before getting sucked into the details of how to tune it for best reception.

 

[dmorris619]

From all of your help I think the best approach to my loop antenna is to use a low pass filter. However my main concern is that the gain across the entire spectrum is now not even(I understand that the gain from the capacitors wasn't even either but it was linear and the bandwidths of interest were my bin size, which means post processing could reduce the gain.) and this will affect my representation across the entire spectrum. Does anyone know how I can create a sharper corner or essentially make the drop off sharper than 20db/dec, or could I possibly apply a filter after Fourier Transforming it digitally to account for the gain loss.

Also I am concerned about the noise generated by a low pass filter. I calculated that I need a 483 ohm resistor to put the cut off at 100K and my loop antenna has a resistance of 3.2 ohms. Can I just use a white noise calculation on this or is there a better approximation of the noise that I can use to calculate it.

 

[sophiecentaur]

I feel that, if you are concerned about noise, then it would be in your interest to resonate the coil. The impedance of small loops at low frequency is very low and you need to try hard to get useful signal power out. An RC low pass filter would not, I think, be best for the same reason and LC combinations can give sharper skirts.
There is, of course, the other aspect of receiving small signals in the presence of high interference levels. You can expect a number of high level interfering sources in that region of the spectrum. Your front end stages will have higher requirements for linearity if you don't give yourself some selectivity before amplification.
BTW, did you ever consider a ferrite rod antenna? Virtually all MF /LF receivers use them and i think the situation could be even better at your very lowest frequencies.

 

[dmorris619]

I've considered ferrite rod antennas and to be honest they are not off the table, I just started with an air core for testing.

If I resonate the circuit though, I don't get a large Q factor for my low frequency signals over the bandwidth I want. The only thing I can do to increase the Q factor is decrease the bandwidth and I'm not sure I can do that for the sake of my data logging on the dsp side.

One thing I think that should be mentioned is that this antenna is not meant to receive a specific signal, its just meant to read the magnetic field levels in my frequency region, so there really isn't interference(other than electrical radiation, since I only want the magnetic).

My analog is a little rusty in terms of amplification, so can you explain what you mean by my linear region. I'm assuming it has to do with the voltage output from my antenna and that it may have to cover 30microvolts to 5 volts.

So if I must use resonance would a network of capacitors AND inductors allow me to get all the combinations I want? What kind of affect would this have on the noise in my circuit.

 

[sophiecentaur]

If I resonate the circuit though, I don't get a large Q factor for my low frequency signals over the bandwidth I want. The only thing I can do to increase the Q factor is decrease the bandwidth and I'm not sure I can do that for the sake of my data logging on the dsp side.

...

My analog is a little rusty in terms of amplification, so can you explain what you mean by my linear region. I'm assuming it has to do with the voltage output from my antenna and that it may have to cover 30microvolts to 5 volts.

So if I must use resonance would a network of capacitors AND inductors allow me to get all the combinations I want? What kind of affect would this have on the noise in my circuit.

You need to tailor your input filter response to match the spectrum of the signal you want - for best performance. This may require something better than a 'single humped' response - like a typical IF filter'

Linearity may be relevant if there are high level of interfering signals present. A big signal can cause cross modulation onto your wanted signal and it can also produce intermodulation products which can lie right on top of the wanted frequency. Once they're introduced, you can't easily eliminate non-linear products so linearity is often a problem - which is why 'passive' selectivity can be such good value. If you are looking for very low level signals and you're concerned about SNR (low current in the pre amplifier) then this will conflict with the requirement for linearity (high current in the pre amplifier). Is the 'noise' you refer to, front end noise or is it received noise?
There are 1001 designs for LC filters, for band /high /low pass and no one with a need for good noise performance would introduce unnecessary 'hot' resistive elements before the first stage of amplification.

 

[dmorris619]

But can i have interefering signals if i just want to read the strength of the magnetic field and not a specific signal.

How do i make a resonance curve with more than one response? Maybe I am missing your point but are you saying i need to shift the resonance point or what?

The noise I am referring to is the noise in the circuit because as i said aside from electrical noise the antenna shouldn't be picking up noise in that range because its all signal.

 

[sophiecentaur]

If you are just looking at the magnetic field and are sure there is no interference then you needn't concern yourself with selectivity. All you need is a wide band, low noise amp with a very crude input filter (LC and no resistors, of course, if you are serious about receiver noise). I don't know what you are actually using it for but I should be very surprised if there were no unwanted signals in your experiment. When you are looking down in the receiver noise levels are you not sure to be getting something man-made as well?

There is the possible issue of matching the loop antenna to the amplifier (for best SNR), which could require a transformer or some alternative network.

btw, did you consider a screened loop antenna? It is good at rejecting E fields. It's basically a loop wound within a split torus (loop of tube) and still picks up the H field but rejects the E field by many dB as it's balanced out by the electric screen. I read about this, as it happens, in another thread, recently.

 

[dmorris619]

I am going to shield my loop antenna by encasing it in a screen but i am unsure if i need to use a ferrous material like iron or if aluminum will be okay. I think aluminum will be fine but want to double check.

The only signal noise i can think of is frequencies outside the spectrum of interest and signals that have an intensity out of my range 1mg to 100mg.

By filtering it with a low pass filter(using just l and c) i can get rid of the noise outside the frequency range but the intensity will be hard to figure out.

I still am interested in using resonance but am unsure how to be able to shift the resonance point with an inductive and capacitive ladder network.

How would using a transformer help impedance match the antenna?

 

[berkeman]

How would using a transformer help impedance match the antenna?

You use the turns ratio of the coupling/matching transformer to match the two impedances (antenna impedance to receiver input impedance). Do you know the equation for how an impedance transforms through a transformer, in terms of the number of turns on each side of the transformer?

 

[dmorris619]

Yes power is held contant and either current is decreased or voltage is stepped up. Its how wall outlets convert 120vac to 12vdc(also using a rectifier). So does stepping up the turns increase the impedance to the amplifier?

 

[sophiecentaur]

You can use an autotransformer configuration - i.e. more of a 'Tapped Inductor' but there are many designs using ferrite cores and a balun transformer can ensure that your antenna is balanced whilst the receiver input is unbalanced - another good way to eliminate unwanted fields.
Google with terms such as antenna matching transformer and you'll see what I mean. Impedance ratio is turns ratio squared (ideally)
I have a feeling that your Q will not be very high so all you may need is a small number of fairly wide band resonances which should put all your wanted signals near enough to a peak.

For electric screening you just need a conductor. Aluminium is fine and cheap as long as you can form it into a circle of suitable diameter. I seem to remember seeing one made of copper plumbing pipe; much more well behaved for forming shapes with. To get the circle, the pipe was stuffed very hard, full of sand and bent around a barrel. There were minimal kinks and it looked quite good. You could even talk nicely to a plumber and use a pipe bender. The screen can always be made by wrapping copper foil around the outside of the loop, if looks are not important.

This may be of interest - he's matched various loops in this link.
http://www.qsl.net/wa1ion/bev/bb_antenna_matching.pdf"