Impedance, noise, and measurement

Factual Questions
1

There seem to be two contradictory conventional wisdoms about how to accurately measure weak electrical signals relative to their noise.

In the world of data acquisition, as I understand the conventional wisdom, the only design consideration in the measurement is to keep the impedance of the voltage measurement large relative to the impedance of the sensor.

In the world of communications, on the other hand, as I understand the conventional wisdom, we need to transfer as much of the sensor’s output power (not voltage) to whatever amplifier forms the front end of the voltage measurement circuit, so we want to have the same impedance in the amplifier that we do in the sensor. This point is often confused by the general public, but the most power transfer occurs when the source and destination impedances are the same, whereas other criteria such as efficiency might dictate other impedance matching.

Thus I think I understand two contradictory versions of how to do this, and so must be misunderstanding something somewhere. The misunderstanding might lie in whether transferring maximum power from the sensor is important for the sake of noise, or elsewhere.

I have to choose between three sensors. All the sensors have voltage outputs, but they have different voltage ranges and different impedances. The application bandwidth is trivial to measure, so all the candidate sensors are more than fast enough. But they are all limited by Johnson-Nyquist (or thermal) noise. The bandwidth extends practically to zero Hz, so impedance matching transformers aren’t an option. Under typical measurement conditions the Robust Sensor has an output of 1.5e-8 V and an impedance of 3.5e3 ohms. Its measurement is barely useable due to the noise. The Beautiful Sensor has an output of 7.0e-5 V and an impedance of 9.0e4 ohms. Its measurement is clear and beautiful, however its lifetime appears to be days or maybe weeks, which would be a little difficult to design around (but manageable). The New Sensor has an output of 1.7e-8 and an impedance of 1.0e-1 ohms. There aren’t any New Sensors to test yet and won’t be for a little while, and yet other things are moving along, so it would be valuable to correctly guess whether they will win or not. Of course, we must all make our own informed choices balancing risk with everything else, but there’s good reason to expect the New Sensor to have a long lifetime and other desireable qualities, so the prediction of which sensor wins out is pretty much a prediction of measurement quality.

The new sensor has a signal size like the unsatisfactory Robust Sensor, which is a bad sign. But its Johnson-Nyquist noise based on the signal size and the impedance should be like that of the Beautiful Sensor, which is a good sign.

Now my questions are, is the New Sensor going to yield a clear and beautiful measurement like the Beautiful Sensor? And, must the measurement system match its impedance to take advantage of its attractive Johnson-Nyquist noise?

2

Bloody hell, why can’t you just have a nuisance circuit breaker tripping like everyone else? [/bump]

3

My first reaction is surprise that a project where it’s important to predict stuff like this doesn’t have access to somebody who knows this kind of thing inside and out. You’re getting your answer from a general-purpose message board? Please tell me you don’t work for NASA.

I believe that impedance matching is irrelevant. You just, as you say, want your amplifiers input impedance to be large compared to the sensor’s impedance. That shouldn’t be hard given those numbers.

You can calculate something proportional to a signal-to-noise ratio, using the fact that Johnson noise is proportional to sqrt® (I’m equating sensor impedance with resistance; I’m not sure that’s appropriate). Conclusion: new sensor is a bit worse that really good one and a lot better than really bad one. However, its voltage is significantly lower. Can you really measure a voltage that low without introducing significant noise?

4

Colour me cycnical, but this reads like an exam or assignment question.

For instance, the bandwidth isn’t given, it is however “trivial to measure”. Just so happens that the bandwidth cancels out if you only consider Johnson noise in the comparison. A realistic desgin study to decide on the sensor would need to include current noise and 1/f noise in the amplifier to arrive at a final noise figure, and that may affect the choice of sensor. These are not available from first principles, unlike Johnson noise.

The Robust sensor tells you the noise figure that isn’t any good, the Beautiful the noise figure that is workable. So the question is simply whether the New sensor is good enough.

The other question is a curious one. Rather neatly phrased with a nice bit of deliberate confusion thrown in. Answer, no you are not transferring power, only voltage. But you need to consider the noise in the measuring device. Uncorreleted noise adds as the square root of the sum of squares, but the amplitude of the signal is reduced by the ratio of the sensors impedance to the input impedance (consider the sensor’s Thevenian equivalent here.) You want to maximise the signal to noise ratio. You have an expression for noise and an expression for the signal amplitude, both dependant upon the ratio of the impedances. Solve. Compare answer with initial question.

You finally may need to consider whether a 0.1 Ohm input impedance allows for a workable amplifier circuit. But this isn’t decidable from first principles.

5

What kind of sensors are you talking about, Napier?

6

An EE here…

You have two very different topics here; Measurement and Signal/Power transfer.

RE: “In the world of data acquisition, as I understand the conventional wisdom, the only design consideration in the measurement is to keep the impedance of the voltage measurement large relative to the impedance of the sensor.”

The objective here is to make a measurement without disturbing it (We’ll leave Heisenberg out of this.) Let’s say you have a resistor in a circuit and you want to know what the voltage is across it. You connect a voltmeter across the resistor. The input impedance of the voltmeter must be much much higher than the impedance of the resistor, or the voltage across the two of them, in parallel, will be less than it was originally across the resistor. Result: you’ll have an inaccurate reading.

RE: "In the world of communications, on the other hand, "

This isn’t just in communications, its any place where you wish to deliver a signal with maximum power from one place to another. Consider a weak microphone driving an amplifier input circuit . In this case it works out that the most signal power is tranferred to the amplifier when the input impedance of the amplifier equals the output impedance of the microphone. Also, when you get into things like driving a cable or long circuit board etches (especially with digital signals and at higher frequencies) matching the impedances reduces or eliminates the generation of spurious signals due to reflections at the point of connection.

You may need to break out Ohm’s Law and go play with a circuit of a voltage source with two resistors in series across it and do some voltage and power calculations to convince yourself of all this.

But I agree it is surprising you’re working on something that raises a question like that and don’t have access to someone else to whom this is all fundemental.

7

Apples and oranges, like Fred said, and neither one applies to your situation. Your “data aquisition” rule is only useful if you don’t care about noise and don’t want to calibrate out the voltage drop that occurs with a finite impedance voltmeter or amplifier. The power match case only matters if you are trying to maximize the power transfer. Someone else brought up yet another condition, i.e. impedance matching to the transmission lines to avoid reflections. This is also irrelevant to your situation, since I am assuming you are at frequencies low enough that you are not worried that the cable is a substantial fraction of a wavelength long.

Your real issue then, as pointed out by others, is signal-to-noise optimization. To do this, you need to understand the transfer characteristics of your sensor and all the sources of noise in your sensor and in the measurement circuit (usually just the first stage of amplification).

To begin with, let’s assume that your signal band is high enough that we can ignore the very important topic of 1/f noise. At any given frequency, you can model the sensor as some complex impedance in series with the signal voltage and the Nyquist noise voltage. The Nyquist noise power scales in proportion to the resistive component of the sensor impedance as you know. In comparing the sensors, the first thing you want to know is how the signal voltage compares to the Nyquist voltage. There is no way your signal-to-noise ratio can get any better than it is coming directly out of the sensor.

Next you need to consider the noise sources in the amplifier. For simplicity in this discussion, assume that the amplifier input impedance is very large. For a linear amplifier, the noise can be modeled as two sources of noise referred to the input of the amplifier. A source of voltage noise that will add directly to the signal voltage and a source of current noise that will induce a voltage in the sensor, proportional to the resistance of the sensor. If the sensor has a very large resistance, the sensor signal and its Nyquist noise will typically also be large and the dominant source of noise will be due to amplifier current noise. On the other hand, if the sensor resistance is small, the signal will typically be small also, and the noise will be dominated by the amplifier’s voltage noise.

Now you need to optimize. Without knowing all the details of your situation, I can’t do that here, but suffice to say that normally what you want to do is not to “power match” but to “noise match”. That is, the sensor and amplifier should be chosen so that the sensor resistance is equal to the “noise match resistance” of the amplifier. This resistance is given by the square root of the ratio of the spectral density of the voltage noise to the spectral density of the current noise of the amplifier. This assumes that you can either choose amplifiers with different noise match resistances (but the same noise temperature), you can choose sensors with different resistances but with signals that scale in proportion to the Nyquist noise (V^2 proportional to R), or that you can put a transformer between the sensor and the amplifier.

In your case, you have a small number of options, so you are not going to achieve the optimum, but you can still calculate which is better. All you need to do is look up the noise charactertics of your sensors and your amplifier, and as the British say, “do the sums”. Beware that at low frequencies, most amplifiers and sensors exhibit noise that gets higher and higher as the frequency drops, often in a power law. This is usually called 1/f noise. The analysis is done the same way, but the best performance can get nasty quickly if you insist on being able to handle very low frequencies.

8

Thanks, all. Well, HongKongFooey, maybe no thanks on that one, but I do have a landline phone that keeps loosing the dial tone at the network interface box. It takes weeks for the phone company to come, and the dial tone has mysteriously returned by then, though I check it every few days and show them the phone I keep unplugged sitting on the floor downstairs by the door where the box is, and they give me eyerolls. You got anything for that?

Uncertain, what can I tell you? It’s not like we’re in business to make these things. This is an unusual need for us. Besides, sometimes you get excellent advice, or at least leads, on the SDMB. Note the couple of posts above this answer. You say “You can calculate something proportional to a signal-to-noise ratio…”; that’s what I meant by my next to last paragraph.

Francis Vaughan, I take as some kind of weird compliment that this sounds as well worded as a test question, varied though they are. It isn’t. Your reply, though, deserves more study…

Crafter_Man, nice to hear from you - I can’t say what the sensor type is, as it’s somebody else’s secret. They are unusual. It isn’t very exciting, though.

Uncle Fred, I can believe maximum power transfer happens when impedances are matched, as I said in the OP. But with “You have two very different topics here; Measurement and Signal/Power transfer” you hit on something - which of the two are important, if I have a measurement troubled by noise? Of course I don’t want the measurement circuit to load the sensor down completely, but, do I need to try to use all the power coming from the sensor? Is that important for noise? I thought it was, using as an example the expense of antennae and the impedance matching between them and amplifiers, reasoning that they wouldn’t build the amps to match the antenna power if it wasn’t important to. But now I am wondering if the only reason was to avoid reflections, as JWT mentions? Maybe my original question should have been worded, “Is impedance matching important for minimizing noise?” Though my last words were practically there…

JWTKottekoe, there is a great deal here, which I am going to go study at some length. Thank you.

BTW, my interest is in frequencies no higher than 1 Hz, and maybe I don’t even care about 0.1 and higher. So perhaps what will trouble me most is what people call “drift” (though it isn’t clear to me why that is a different kind of noise and not just a different regime for measuring noise). I might wind up using reed switches to reverse the sensor polarity avery so often just to deal with that, or buying a chopper stabilized amp.

9

Sleeper cell

10

Antenna theory is related, but for more complex (bad pun there) reasons, to power transfer. Once you are moving electrical fields around where the wavelength is the same as the distances involved it gets wierd. You don’t have this problem.

1Hz or 0.1Hz - well that is very low. Low enough that it changes things. Johnson noise is proportional to the square root of the bandwidth. Your bandwidth is so low that it is very likely that performance of your system is dominated by other (and non-stochastic) noise sources. Also the very low frequency means that 1/f noise may be a big issue. I would venture to suggest, again, that you really have no hope of answering the question until you look at viable amplifier designs, and evaluate the system end to end.

In general, noise can be defined as anthing in the measured result that isn’t what you want. So drift is noise. Just very low frequency noise. However it might not be stochastic, but that doesn’t mean it isn’t noise. Non-stochastic noise adds linearly. A drift compensation system is nothing more than a low pass filter, just by a different name. However I feel nervous that any active drift compensation may introduce more noise than you are prepared to accept. Again, you really need to crunch the numbers with some real system designs.

11

The information is a bit too vague for me to comment on. Are we talking about periodic signals, or non-periodic? DC component in an AC signal? Can you provide a plot of example data?

Are you allowed to modulate the signal? Have you considered chopping techniques, lock-in amplifier techniques, etc.?

12

Thanks for giving this some thought.

My signal is not periodic. The interesting part of the signal is below 0.1 Hz or so, and I don’t expect anything else in particular at any frequency (that is, I am not trying to ignore some other well defined competing signal, just miscellaneous nondescript noise). The signal can be of either polarity. I can’t provide a plot, but it seems like my signal might look about like the acceleration of a car in the forward direction (that is, braking produces a negative signal) recorded over the length of a long trip, so typically a long term average would be zero, but there would be shorter term variations of interest. However, at least as I understand my application today, my interest is only in the averages over a time scale of a few seconds, or even perhaps 100 or 1000 seconds. Probably there is no penalty here for ignoring or losing the signal for seconds at a time, even 100 seconds or so. If there is a big blip while I am looking the other way, there is no special problem. It is more important to correctly know where zero is than it is to know the magnitude of the signal. That is, I care much more about offset error than I do about gain error. Electrically, my signal is isolated relative to ground and is balanced (that is its real or complex impedance relative to ground would be the same on either of its two legs), and I have been measuring it with a differential input DAQ device. In fact, for experiments, it happens that I have been using a battery powered datalogger whose chassis floats. Eventually this measurement will become more permanent, powered from the mains. I think I have a great degree of freedom in how I physically connect my signal; for example if there was reason to I could ground one side or I could create a center tap and connect it to some synthesized waveform WRT ground.

My sensor (whichever one I choose) is detecting a physical phenomenon that I cannot modulate. But, I could modulate the electrical output of the sensor before trying to measure it. I am considering chopping techniques. I could imagine alternating between the signal and a short circuit, and perhaps better I could imagine alternating between the signal and the reversed signal (that is, flipping back and forth a 2 pole 2 throw switch with crossed connections between the “on” and “off” contacts).

I don’t understand the distinction between lock-in amplifiers and chopper stabilized amplifiers. Is the distinction that chopper-stabilized amps do not use knowledge of the phase of the chopping, while lock-in amplifiers do? Are lock-in amplifiers only useful in situations where the signal of interest is itself a sine wave of known frequency? or any waveform of known frequency? Or could one effectively measure an inherently nonperiodic signal that was then periodically polarity-reversed? I figured if I kept reversing the measurement polarity, I would subtract groups of polarity A from groups of polarity B to get a stabilized measurement. Maybe I’d acquire the signal voltage at 10 Hz and reverse its polarity every 3 seconds and use blocks of around 25 raw measurements after waiting for some settling. There isn’t a resolution problem - the short term precision of all my candidate systems would allow detecting changes the size of my signal. Would a lock-in amplifier provide a better stabilized measurement?