How accurately can you in principle measure temperature with an RTD or thermistor?
If we push a current through the resistor we'll get a voltage of across it. Now as the temperature changes the resistance will change by where is the temperature coefficient of the sensor. This will give us a signal
On the other hand the Johnson noise across the resistor will be where B is the bandwidth, and we get a signal-to-noise ratio of
The noise-quivalent-temperature (NET) can be defined as or
Here we can identify as the power dissipated in the resistor and simplify to
Here's a table with some common values for pt100 and 10k NTC thermistors. The sensitivity is determined by the sensor type. What limits is self-heating of the sensor which probably should be kept to a few milli-Kelvins in most precision applications. Thermistors with their higher sensitivity are an obvious choice for high-resolution applications, but the lower of a pt100 sensor can be compensated with a larger since most pt100 sensors are physically larger and thus have lower self-heating. pt100 sensors require 4-wire sensing, slightly more complex than a 2-wire measurement which is OK for a thermistor.
Sensor
Resistance
Sensitivity (divide by R to get alpha!)
Dissipated Power P
Noise-Equivalent-Temperature (1Hz bandwidth)
pt100
100 Ohms
0.391 Ohms/C
100 uW (I=1mA)
3 uK
NTC Thermistor
10 kOhms
-500 Ohms/C
9 uW (I=30uA)
0.9 uK
I conclude that it is not entirely obvious how to choose between a pt100 and a 10k thermistor. The thermistor is intrinsically more sensitive, but with good thermal contact to its surroundings self-heating in a pt100 sensor can be minimized and the same noise-requivalent-temperature achieved. In any case it looks like Johnson noise limits resolution to 1 uK or so in a 1 Hz bandwidth. If we AD-convert the voltage at 24-bit resolution (16M states) we can get a reasonable measurement range of ~32 K by matching 1 LSB = 2 uK.
Does anyone know of similar back-of-the-envelope calculations for other sensors (Thermocouples, AD590)?
(Time-lapse of ca 18 hour experiment. Bottom left is a spectrum-analyzer view of the beat-note signal. Top left is a frequency counter reading of the beat-note. Bottom right is a screen showing the a camera-view of the output-beam from the resonator)
This is a measurement of the thermal expansion of a fancy optical resonator made from Corning "Ultra Low Expansion" (ULE) glass. This material has a specified thermal expansion of 0.03 ppm/K around room temperature. This thermal expansion is roughly 800-times smaller than Aluminium, around 400-times smaller than Steel, and 40-times better than Invar - a steel grade specifically designed for low thermal expansion.
Can we do even better? Yes! Because ULE glass has a coefficient of thermal expansion (CTE) that crosses zero. Below a certain temperature it shrinks when heated, and above the zero-crossing temperature it expands when heated (like most materials do). This kind of behavior sounds exotic, but is found is something as common as water! (water is heaviest at around 4 C). If we can use the ULE resonator at or very close to this magic zero-crossing temperature it will be very very insensitive to small temperature fluctuations.
So in the experiment I am changing the temperature of the ULE glass and looking for the temperature where the CTE crosses zero (let's call this temperature T_ZCTE). The effect is fairly small: if we are 1 degree C off from T_ZCTE we would expect the 300 mm long piece of ULE glass to be 200 pm (picometers) longer than at T_ZCTE. That's about the size of a single water-molecule, so this length change isn't exactly something you can go and measure with your digital calipers!
Here's how it's done (this drawing is simplified, but shows the essential parts of the experiment):
We take a tuneable HeNe laser and lock the frequency of the laser to the ULE-cavity. The optical cavity/resonator is formed between mirrors that are bonded to the ends of the piece of ULE glass. We can lock the laser to one of the modes of the cavity, corresponding to a situation where (twice) the length of the cavity is an integer number of wavelenghts. Now as we change the temperature of the ULE-glass the laser will stay locked, and as the glass shrinks/expands the wavelength (or frequency/color) of the laser will change slightly.
Directly measuring the frequency of laser light isn't possible. Instead we take second HeNe laser, which is stabilized to have a fixed frequency, and detect a beat-note between the stabilized laser and the tuneable laser. The beat-note will have a frequency corresponding to the (absolute value of the) difference in frequency between the two lasers. Now measuring a length-change corresponding to the size of a single water-molecule (200 pm) shouldn't be that hard anymore!
Let's say the stabilized laser has a wavelength of (red light). Its frequency will be (that's around 474 THz). When the tuneable laser is locked to the cavity we force its wavelength to agree with where is an integer and is the length of the cavity. I've drawn only a small number of wavelengths in the figure, but a realistic integer is . We get and , very nearly but not quite the same wavelength/frequency as the stabilized laser. Now our photodiode which measures the beat-note will measure a frequency of .
How does this change when the ULE glass expands by 200 pm? When we heat or cool the cavity by 1 degree C the length changes to 300 mm + 200 pm, and the wavelength of the tuneable laser will change to . Now our beat-note detector will show . That's a change in the beat-note of more than 300 kHz - easily measurable!
That's how you measure a length-change corresponding to the diameter of a water molecule!
Why do this? Some of the best ultra-stable lasers known are made by locking the laser to this kind of ULE-resonator. Narrow linewidth ultra-stable lasers are interesting for a host of atomic physics and other fundamental physics experiments.
Update 2013 August: I made a drawing in inkscape of the experimental setup.
This figure shows most (if not all?) of the important components of this experiment. The AOM is not strictly required but I found it useful to shift the tuneable HeNe laser by +80 MHz to reach a TEM-00 mode of the ULE resonator. Not shown is a resonance-circuit (LC-tank) between the 2.24MHz sinewave-generator and the EOM. The EOM was temperature controlled by a TEC with an NTC thermistor giving temperature feedback.
A revised version of the circuit and PCB for a photodiode amplifier, to be used in PDH-locking (Pound-Drever-Hall) as well as RAM-nulling (residual amplitude modulation) in a laser experiment I am doing. The changes compared to the first prototype are:
The required bandwidth and gain is not easy to achieve in one stage, so there's a second stage of amplification after the transimpedance amplifier.
I'm suspicious of the noise caused by the switched-mode powersupply, as well as the DC2DC converter, of the previous design. So this circuit has just +/-5 V regulators and can be driven from a regular (known good) +/-12 V lab powersupply (or even two 9 V batteries).
Here is a schematic and simulation results produced with the free version of NI Multisim from Analog Devices. The design is for roughly 1 MOhm of transimpedance gain in total, here split between 7 kV/A transimpedance gain, and 144 V/V for the non-inverting second op-amp. At 1 kV/A of transimpedance gain a 5 uW optical signal at 633 nm (HeNe laser!) that produces a 2 uA photocurrent will result in a 2 V output signal. The AC analysis shows very slight gain-peaking for the transimpedance-stage (red trace) and a -3 dB bandwidth of >3 MHz overall (green trace).
The first op-amp used in the transimpedance stage only needs to have a bandwidth slightly exceeding the transimpedance gain bandwidth (the feedback resistor R1 together with the compensating cap C1, the capacitance of the photodiode C2, and the input-capacitance (not shown) of the op-amp form an RC low-pass filter). The AD8597 is marketed as "ultralow distortion/noise" and is fast enough (10 MHz). The second non-inverting op-amp needs a high gain-bandwidth-product (GBP) since we are amplifying ~100-fold here. The ADA4817 has a small-signal bandwidth of 1 GHz and GBP~400 MHz, so should work OK here.
A voltage of only 14 mV over the transimpedance-resistor is not ideal. The Johnson noise (which in principle a good designer can control/minimize) in the resistor will dominate over the shot noise (which we cannot avoid) in the optical signal. For shot-noise limited performance the rule of thumb is to make the voltage drop at least 51 mV (which will make Johnson and shot noise equal). Without tricks however that is not possible as here we have both a weak signal (2 uA of photocurrent), we want a high gain (1 kV/A in total), and we want to go fast (~3 MHz bandwidth)! If you relax any of those requirements (more power, less gain, slower response) it is straightforward to build a shot-noise limited amplifier in one or two stages.
The PCB, fresh from the mill:
Far right is a 3-pin TO-18 socket for the photodiode. Right-middle are the two op-amps with their feedback-resistors/caps, as well as two de-coupling caps for both +5V and -5V. Left-middle are 7805 and 7905 voltage regulators, and the BNC output-connector is far left. All the surface mount components are mounted on the top layer of the board, while the through-hole components are bottom-mounted. Resistors and caps are 1206-size. This PCB should fit the earlier enclosures I turned on the lathe.
Hopefully I will have time to assemble and test one or two of these next week. I should measure the actual frequency-response and compare it with the simulated one.
I've been measuring the beat-note (wikipedia talks about sound-waves, but it works for light-waves too) between two HeNe lasers. It jumps around maybe +/- 5 MHz quite rapidly which is not nice at all:
One laser is a commercial stabilized laser (I've tried both a HP5501A and a Mark-Tech 7900), and the other laser is a tunable one which I want to use for my experiment. But with this much jumping around the tunable laser is no good for the experiment I want to do 🙁
Some assembly of the PCB and enclosure for the first photodiode amplifier has happened today. Soldering the surface-mount components under a microscope was mostly easy - but trying to solder larger parts that require significant heating with the same tiny soldering iron used for small SMD parts was a mistake. The big parts were easy once I switched to a bigger soldering iron, but the BNC-connector was already a mess by then.
On the left a BNC connector. The black box in the middle is a DC-2-DC converter that produces +/-12VDC outputs from a single +9...18VDC input. The transimpedance amplifier on the right is based on an ADA4817 op-amp and is housed inside an RF screening can. On the far right is a white TO-18 socket for the photodiode.
The underside of the board has 7805 and 7905 voltage regulators that produce stable (hopefully!) +/-5V supply voltages for the op-amp.
Here two holes have been drilled in to the back-plate for the BNC-commector (ca 12.2 mm diameter) and a DC-input jack (5.5mm diameter with a 2.5mm pin). The PCB is attached to the back-plate and slides into the body, while the photodiode looks out through the hole of the face-plate.
The only setback was a disagreement between the 1-2-3 pin-sequence in the datasheet vs. my PCB-software for the SOT-89 packaged 7905 negative voltage regulator. If you look closely you can see it is soldered up-side-down on this board because the PCB footprint is wrong.
I'm making photodiode (transmipedance) amplifiers, and here is the first PCB being milled today. In the foreground a test-run where the cutter-height was too low resulting in too thin or vanishing PCB-traces. Note how the PCB material is not held in place along the Z-axis at all. The PCB-blank is just located in X/Y on the table using two locating pins/holes. In the Z-direction the idea is that the pneumatic cylinder pushes the lower flange of the spindle into contact with the PCB-material, and the exact cutter-height is adjusted relative to this flange only.
The toolchain is (old!) commercial software: PADS PowerLogic for schematic design, PADS PowerPCB for PCB-design, CircuitCam for converting the gerbers to HPGL, which BoardMaster uses to drive the mill (over RS232).
For general purpose 3D CAD at work we have Vertex (a Finnish Inventor/SolidWorks clone) and I used it to draw a model of the amplifier:
The size of the PCB and enclosure is mostly limited by how much of the powersupply one wants on-board, and how big connectors one wants to use. I'm using a standard BNC connector (SMA would have been smaller). The board is powered by a +9...18VDC supply which is DC2DC converted into +/-12 V and then regulated to +/- 5 V for the op-amp circuit. The box at the front is an RF shield for the amplifier itself. Light enters through an 8 mm hole in the face-plate and hits a TO-18 mounted photodiode. More on the circuit later.
The enclosure is 48 mm in diameter with a 16 mm thick face-plate, a 4 mm thick back-plate, and the body (55 mm length) bored out to an inner-diameter of 34 mm. The body should fit a 25x54mm PCB. The end-plates are attached to the body with five M3 screws on a 40 mm diameter bolt-circle. There is an M6 thread on the bottom of the face-plate, for attaching the amplifier to an optical-table or other instrumentation. I made two of these from 50 mm aluminium round-bar on a manual lathe and mill (using a rotary table for the holes/threads).
Note: for manual machining five evenly spaced holes the angle-sequence is: 0 - 72 - 144 - 216 - 288 - 0.
I'm thinking about polishing these a bit and then anodizing them. But for RF-shielding the contact-surfaces of all three parts would then have to be sanded/milled-down after andoizing. to ensure good electrical conductivity between the parts.
I made this ca 78x48x31 mm mount for a Faraday Isolator (Model IO-7-633 Optics For Research, now sold by Thorlabs) from 50x50 Aluminium bar on the manual mill at work. It raises the isolator up from the table by 31 mm. The isolator is attached to the mount with two M6 screws, 28 mm apart. The cap-head screws are countersunk so they don't protrude from the bottom. This mount is clamped to the optical table using the 8x5 mm slots in the sides.
We use Norland Optical Adhesive (NOA-81) for gluing bits and pieces together in the lab. The glue cures in UV-light. So far we've used a fluorescent solarium lamp for this, but we broke one of the two remaining lamps and can't seem to source new ones.
So I decided to try an LED solution. These are LEDEngin LZ1-10U600 LEDs with a center wavelength of 365 nm, 28 euros each from Mouser. They are glued to an aluminium plate using a heat-conducting glue, Loctite Output 384:
For simplicity I used a LighTech 18 W 700 mA constant-current powersupply that runs directly off AC mains. The powersupply has a maximum output voltage of 24 V which is enough to drive the five UV-LEDs in series (the UV-LED has a voltage-drop of 4.1 V)
Here's how the lamp looks like:
Most of the blue in this picture is fluorescence from the white paper underneath the lamp. A quick test shows that the NOA-81 glue cures very quickly indeed (seconds) with this lamp. Much faster than with the old fluorescent lamp (minutes) anyway. This kind of lamp may be useful for PCB-making also? I didn't keep the lamp on for very long yet, so I don't know how adequate the alu-plate is as a heat-sink.
Warning: The UV-light from these LEDs may be more or less harmful for your eyes and/or skin. Don't try this at home unless you know what you are doing!
Update: here is the plate. Let's hope it is accurate enough (it is not face-milled on the underside...)
A plate for holding 76 mm x 26 mm glass slides in the microscope. My first ever 'real' drawing with LibreCAD (that website has been down for two days now, so try also librecad on sourceforge).
The fourth paper from my thesis, entitled "Dual-trap optical tweezers with real-time force clamp control", has just been published online by Review of Scientific Instruments: http://link.aip.org/link/doi/10.1063/1.3615309
Here's a video from the paper. We are holding on to two micron sized plastic spheres with laser-beams (shown in the video as green/cyan cross-hairs). The lower beam/trap is stationary while the upper one is steerable. A ca 16um long DNA-molecule (invisible) is tethered between the beads.The experiment is performed in the presence of lambda exonuclease, an enzyme that "eats up" one strand of the DNA leaving just a single-stranded DNA-tether between the beads.
In the first part of the video a force-extension curve (bottom panel) is obtained using manual control. We stretch out the molecule by moving the upper trap upwards and check that the force-signal looks like it should when we have a single DNA-molecule of the right length between the beads.
In the second part, after t = 20 s, the tether is held force clamped at 3.4 pN (force shown in top panel). We're keeping the force constant with a PI-controller implemented on an FPGA that reads the force-signal from the lower bead and updates the position of the upper trap at around 200 kHz. As the molecule shortens the controller needs to move the upper trap/bead lower in order to maintain a 3.4 pN tension in the molecule. The video is at normal speed (1X) while the force extension curve is measured. During 13 min of force-clamp control the video is sped up 25-fold. During this time the exonuclease digests one strand of the double-stranded DNA molecule. When held at 3.4 pN of tension, single-stranded DNA is significantly shorter than double-stranded DNA. So the gradual conversion from a double-stranded tether to a single-stranded tether is seen as a decrease in the extension, i.e. a shortening of the distance between the plastic beads (middle panel). The tether broke at t = 880 s. Scale-bar 5 ?m.
through the resistor we'll get a voltage of 
across it. Now as the temperature changes the resistance will change by 
where 
is the temperature coefficient of the sensor. This will give us a signal
where B is the bandwidth, and we get a signal-to-noise ratio of
or
as the power dissipated in the resistor and simplify to
is self-heating of the sensor which probably should be kept to a few milli-Kelvins in most precision applications. Thermistors with their higher sensitivity are an obvious choice for high-resolution applications, but the lower 
