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diff --git a/content/posts/multichannel-led-driver/index.rst b/content/posts/multichannel-led-driver/index.rst deleted file mode 100644 index 7de03d5..0000000 --- a/content/posts/multichannel-led-driver/index.rst +++ /dev/null @@ -1,456 +0,0 @@ ---- -title: "32-Channel LED tape driver" -date: 2018-05-02T11:31:14+02:00 ---- - -Theoretical basics -================== - -Together, a friend and I outfitted the small staircase at Berlin's Chaos Computer Club with nice, shiny RGB-WW LED tape -for ambient lighting. This tape is like regular RGB tape but with an additional warm white channel, which makes for much -more natural pastels and whites. There are several variants of RGBW tape. Cheap ones have separate RGB and white LEDs, -which is fine for indirect lighting but does not work for direct lighting. Since we wanted to mount our tape in channels -at the front of the steps, we had to use the slightly more expensive variant with integrated RGBW LEDs. These are LEDs -in the 5050 (5.0mm by 5.0mm) form factor common with RGB LEDs that have a small section divided off for the white -channel. The red, green and blue LED chips sit together in the larger section covered with clear epoxy and the white -channel is made up from the usual blue LED inside a yellow phosphor in the smaller section. - -Since we wanted to light up all of 15 steps, and for greatest visual effect we would have liked to be able to control -each step individually we had to find a way to control 60 channels of LED tape with a reasonable amount of hardware. - -LED tape has integrated series resistors and runs off a fixed 12V or 24V constant-voltage supply. This means you don't -need a complex constant-current driver as you'd need with high-power LEDs. You can just hook up a section of LED tape -to a beefy MOSFET to control it. Traditionally, you would do *Pulse Width Modulation* (PWM) on the MOSFET's input to -control the LED tape's brightness. - -Pulse Width Modulation ----------------------- - -`Pulse Width Modulation`_ is a technique of controlling the brightness of a load such as an LED with a digital signal. -The basic idea is that if you turn the LED on and off much too fast for anyone to notice, you can control its power by -changing how long you turn it on versus how long you leave it off. - -PWM divides each second into a large number of periods. At the beginning of each period, you turn the LED on. After -that, you wait a certain time until you turn it off. Then, you wait for the next period to begin. The periods are always -the same length but you can set when you turn off the LED. If you turn it off right away, it's off almost all the time -and it looks like it's off to your eye. If you turn it off right at the end, it's on almost all the time and it looks -super bright to your eye. Now, if you turn it off halfway into the cycle, it's on half the time and it will look to your -eye as half as bright as before. This means that you can control the LED's brightness with only a digital signal and -good timing. - -.. raw:: html - - <figure> - <img src="images/pwm_schema.jpg" alt="A visualization of PWM at different duty cycles."> - <figcaption>Waveforms of two PWM cycles at different duty cycles.</figcaption> - </figure> - -PWM works great if you have a dedicated PWM output on your microcontroller. It's extremely simple in both hardware and -software. Unfortunately for us, controlling 32 channels with PWM is not that easy. Cheap microcontrollers only have `a -handful of hardware PWM outputs`_, so we'd either have to do everything in software, bit-banging our LED modulation, or -we'd have to use a dedicated chip. - -Doing PWM in software is both error-prone and slow. Since the maximum dynamic range of a PWM signal is limited by the -shortest duty cycle it can do, software PWM being slow means it has poor PWM resolution at maybe 8 bits at most. Poor -color resolution is not a problem if all you're doing is to fade around the `HSV rainbow`_, but for ambient lighting -where you *really* want to control the brightness down to a faint shimmer you need all the color resolution you can get. - -If you rule out software PWM, what remains are dedicated `hardware PWM controllers`_. Most of these have either of three -issues: - -* They're expensive -* They don't have generous PWM resolution either (12 bits if you're lucky) -* They're meant to drive small LEDs such as a 7-segment display directly and you can't just hook up a MOSFET to their - output - -This means we're stuck in a dilemma between two poor solutions if we'd want to do PWM. Luckily for us, PWM is not the -only modulation in town. - -.. _`Pulse Width Modulation`: https://en.wikipedia.org/wiki/Pulse-width_modulation -.. _`a handful of hardware PWM outputs`: https://www.nxp.com/parametricSearch#/&c=c731_c380_c173_c161_c163&page=1 -.. _`HSV rainbow`: https://en.wikipedia.org/wiki/HSL_and_HSV -.. _`hardware PWM controllers`: http://www.ti.com/lit/ds/symlink/tlc5940.pdf - -Binary Code Modulation ----------------------- - -PWM is the bread-and-butter of the maker crowd. Everyone and their cat is doing it and it works really well most of the -time. Unbeknownst to most of the maker crowd, there is however another popular modulation method that's mostly used in -professional LED systems: Enter `*Binary Code Modulation* (BCM) <http://www.batsocks.co.uk/readme/art_bcm_1.htm>`_. - -BCM is to PWM sort of what barcodes are to handwriting. While PWM is easy to understand and simple to implement if all -you have is a counter and an IO pin, BCM is more complicated. On the other hand, computers can do complicated and BCM -really shines in multi-channel applications. - -Similar to PWM, BCM works by turning on and off the LED in short periods fast enough to make your eye perceive it as -partially on all the time. In PWM the channel's brightness is linearly dependent on its duty cycle, i.e. the percentage -it is turned on. In PWM the duty cycle D is the total period T divided by the on period T_on. The issue with doing PWM -on many channels at once is that you have to turn off each channel at the exact time to match its duty cycle. -Controlling many IO pins at once with precise timing is really hard to do in software. - -BCM avoids this by further dividing each period into smaller periods which we'll call *bit periods* and splitting each -channel's duty cycle into chunks the size of these bit periods. The amazingly elegant thing in BCM now is that as you -can guess from the name these bit periods are weighted in powers of two. Say the shortest bit period lasts 1 -microsecond. Then the second-shortest bit period is 2 microseconds and the third is 4, the fifth 8, the sixth 16 and so -on. - -.. raw:: html - - <figure> - <img src="images/bcm_schema.jpg" alt="A visualization of BCM at different duty cycles."> - <figcaption>Waveforms of a single 4-bit BCM cycle at different duty cycles. This BCM can produce 16 different - levels.</figcaption> - </figure> - -Staggered like this, you turn on the LED for integer value of microseconds by turning it on in the bit periods -corresponding to the binary bits of that value. If I want my LED to light for 19 microseconds every period, I turn it on -in the 16 microsecond bit period, the 2 microsecond bit period and the 1 microsecond bit period and leave it off for the -4 and 8 mircosecond bit periods. - -Now, how this is better instead of just more complicated than plain old PWM might not be clear yet. But consider this: -Turning on and off a large number of channels, each at its own arbitrary time is hard because doing the timing in -software is hard. We can't use hardware timers since we only have two or three of those, and we have 32 channels. -However, we can use one hardware timer to trigger a really cheap external latch to turn on or off the 32 channels all at -once. With this setup, we can only controll all channels at once, but we can do so with very precise timing. - -All we need to do is to set our timer to the durations of the BCM bit periods, and we can get the same result as we'd -get with PWM with only one hardware timer and a bit of code that is not timing-critical anymore. - -Applications of Binary Code Modulation -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -BCM is a truly wondrous technique, and outside of hobbyist circles it is in fact very widely known. Though we're using -it to control just 32 channels here, you can do much more channels without any problems. The most common application -where BCM is invariably used is *any* kind of LED screen. Controlling the thousands and thousands of LEDs in an LED -screen with PWM with a dedicated timer for each LED would not be feasible. With BCM, all you need to dedicate to a -single LED is a flipflop (or part of one if you're multiplexing). In fact, there is a whole range of `ICs with no other -purpose than to enable BCM on large LED matrices <http://www.vabolis.lt/stuff/MBI5026.pdf>`_. Basically, these are a -high-speed shift register with latched outputs much like the venerable 74HC595_, only their outputs are constant-current -sinks made so that you can directly connect an LED to them. - -.. _74HC595: http://www.ti.com/lit/ds/symlink/sn74hc595.pdf - -Running BCM on LED tape -~~~~~~~~~~~~~~~~~~~~~~~ - -In our case, we don't need any special driver chips to control our LED tape. We just connect the outputs of a 74HC595_ -shift register to one MOSFET_ each, and then we directly connect the LED tape to these MOSFETs. The MOSFETs allow us to -drive a couple of amps into the LED tape from the weak outputs of the shift register. - -The BCM timing is done by hooking up two timer channels of our microcontroller to the shift registers *strobe* and -*reset* inputs. We set the timer to PWM mode so we can generate pulses with precise timing. At the beginning of each -bit period, a pulse will strobe the data for this bit period that we shifted in previously. At the end of the bit -period, one pulse will reset the shift register and one will strobe the freshly-reset zeros into the outputs. - -.. raw:: html - - <figure> - <img src="images/olsndot_output_schematic.jpg" alt="From left to right, we see the STM32, one of the shift - registers, and the LEDs and MOSFETs. The LED tape is driven to ground by the MOSFETs, which are in turn directly - driven from the shift register outputs. The shift register is wired up to the STM32 with its clock and data - inputs on SCK and MOSI and its RESET and STROBE inputs on channel 2 and 3 of timer 1."> - <figcaption> - The schematic of a single output of this LED driver. Multiple shift register stages can be cascaded. - </figcaption> - </figure> - - -Our implementation of this system runs on an STM32F030F4P6_, the smallest, cheapest ARM microcontroller you can get from -ST. This microcontroller has only 16kB of flash and 1kB of RAM, but that's plenty for our use. We use its SPI controller -to feed the modulation data to the shift registers really fast, and we use two timer channels to control the shift -registers' reset and strobe. - -We can easily cascade shift registers without any ill side-effects, and even hundreds of channels should be no problem -for this setup. The only reason we chose to stick to a 32-channel board is the mechanics of it. We thought it would be -easier to have several small boards instead of having one huge board with loads of connectors and cables coming off it. - -The BOM cost per channel for our system is 3ct for a reasonable MOSFET, about 1ct for one eighth of a shift register -plus less than a cent for one resistor between shift register and MOSFET. In the end, the connectors are more expensive -than the driving circuitry. - -.. _MOSFET: https://en.wikipedia.org/wiki/MOSFET -.. _STM32F030F4P6: http://www.st.com/resource/en/datasheet/stm32f030f4.pdf - -Hardware design -=============== - -From this starting point, we made a very prototype-y hardware design for a 32-channel 12V LED tape driver. The design is -based on the STM32F030F4P6_ driving the shift registers as explained above. The system is controlled through an RS485_ -bus that is connected up to the microcontroller's UART using an MAX485_-compatible RS485 transceiver. The LED tape is -connected using 9-pin SUB-D_ connectors since they are cheap and good enough for the small current of our short segments -of LED tape. The MOSFETs we use are small SOT-23_ logic-level MOSFETs. In various prototypes we used both International -Rectifier's IRLML6244_ as well as Alpha & Omega Semiconductor's AO3400_. Both are good up to about 30V/5A. Since we're -only driving about 2m of LED tape per channel we're not going above about 0.5A and the MOSFETs don't even get warm. - -.. _RS485: https://en.wikipedia.org/wiki/RS-485 -.. _MAX485: https://datasheets.maximintegrated.com/en/ds/MAX1487-MAX491.pdf -.. _IRLML6244: https://www.infineon.com/dgdl/?fileId=5546d462533600a4015356686fed261f -.. _AO3400: http://aosmd.com/pdfs/datasheet/AO3400.pdf -.. _SUB-D: https://en.wikipedia.org/wiki/D-subminiature -.. _SOT-23: http://www.nxp.com/documents/outline_drawing/SOT23.pdf - -Switching nonlinearities ------------------------- -During testing of our initial prototype, we noticed that the brightness seemed to jump around when fading to very low -values. It turned out that our extremely simple LED driving circuit consisting of only the shift register directly -driving a MOSFET, which in turn directly drives the LED tape was maybe a little bit too simple. After some measurements -it turned out that we were looking at about 6Vpp of ringing on the driver's output voltage. The picture below is the -voltrage we saw on our oscilloscope on the LED tape. - -.. raw:: html - - <figure> - <img src="images/driver_ringing_strong.jpg" alt="Strong ringing on the LED voltage waveform edge at about - 100% overshoot during about 70% of the cycle time."> - <figcaption>Bad ringing on the LED output voltage caused by wiring inductance. Note that the effect on the - actual LED current is less bad than this looks since the LED's V/I curve is nonlinear.</figcaption> - </figure> - - -Dynamic switching behavior: Cause and Effect -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -A bit of LTSpice_ action later we found that the inductance of the few metres of cable leading to the LED tape is the -likely culprit. The figure below is the schematic used for the simulations. - -.. raw:: html - - <figure> - <img src="images/driver_output_ltspice_schematic.jpg" alt="The LTSpice schematic of one output of the driver, - taking into account the shift register's output ESR and the wiring ESL."> - <figcaption>The schematic of the simulation in LTSpice</figcaption> - </figure> - -As tested, the driver does not include any per-output smoothing so the ~.5A transient on each BCM cycle hits the cable -in full. Combined with the cable inductance, this works out to a considerable lag of the rising edge of the LED -current, and bad ringing on its falling edge. Below is the voltage on the LED output from an LTSpice simulation of our -driver. - -.. raw:: html - - <figure> - <img src="images/overshoot_sim_r0.svg" alt="The result of the LTSpice simulation of our driver output. The LED - current shows similar ringing to what we measured using the oscilloscope. Interestingly, the gate voltage shows - strong ringing, too."> - <figcaption>The result of our LTSpice simulation. This simulation assumes 1µH of wiring inductance and 50Ω of - output impedance on the part of the shift register. The ringing at the gate visible in the gate voltage graph is - due to feed-through of the ringing at the output through the MOSFET's parasitic Cgd.</figcaption> - </figure> - -We were able to reduce the rining and limit the effect somewhat by putting a 220Ω series resistor in between the shift -register output and the MOSFET gate. This resistor forms an RC circuit with the MOSFET's nanofarad or two of gate -capacitance. The result of this is that the LED current passing the wire's ESL rises slightly more slowly and thus the -series inductance gets excited slightly less, and the overshoot decreases. Below is a picture of the waveform with the -damping resistor in place and a picture of our measurement for comparison. The resistor values don't agree perfectly -since the estimated ESL and stray capacitance of the wiring is probably way off. - -.. raw:: html - - <figure> - <img src="images/driver_ringing_weak.jpg" alt="Weak ringing on the LED voltage waveform edge at about 30% - overshoot during about 20% of the cycle time."> - <figcaption>Adding a resistor in front of the MOSFET gate to slow the transition damped the ringing somewhat, - but ultimately it cannot be eliminated entirely. Note how you can actually see the miller plateau on the - trailing edge of this signal. - </figcaption> - </figure> - -.. raw:: html - - <figure> - <img src="images/overshoot_sim_r100.svg" alt="The result of the LTSpice simulation of our driver output with an - extra 100 Ohms between shift register output and MOSFET gate. Similar to the oscilloscope measurement the - ringing is much reduced in its amplitude."> - <figcaption>The LTSpice simulation result with the same parameters as above but with an extra 100Ω between the - shfit register's output and the MOSFET's gate.</figcaption> - </figure> - -A side effect of this fix is that now the effective on-time of the LED tape is much longer than the duty cycle at the -shift register's output at very small duty cycles (1µs or less). This is caused by the MOSFET's `miller -plateau`_. For illustration, below is a graph of both the excitation waveform (the boxy line) and the resulting LED -current (the other ones) both without damping (top) and with 220Ω damping (bottom). As you can see the effective duty -cycle of the LED current is not at all equal to the 50% duty cycle of the excitation square wave. - -.. raw:: html - - <figure> - <img src="images/asymmetric_iled.svg" alt="The result of an LTSpice simulation of the LED duty cycle without and - with damping. Dampening widens the LED current waveform from 50% duty cycle with sharp edges to about 80% duty - cycle with soft edges."> - <figcaption>Simulated LED duty cycle with and without damping. The damping resistance used in this simulation - was 220Ω.</figcaption> - </figure> - -.. raw:: html - - <figure> - <img src="images/asymmetric_vgate.svg" alt="The gate voltages in the spice simulation above. The undamped - response shows sharp edges with the miller plateau being a barely noticeable step, but with strong ringing on - the trailing edge. The damped response shows RC-like slow-edges, but has wide miller plateaus on both edges - adding up to about 50% of the pulse width."> - <figcaption>The MOSFET gate voltage from the simulation in the figure above. You can clearly see how the miller - plateau (the horizontal part of the trace at about 1V) is getting much wider with added damped, and how the - resulting gate charge/discharge curve is not at all that of a capacitor anymore.</figcaption> - </figure> - - - -In conclusion, we have three major causes for our calculated LED brightness not matching reality: - -* Ringing of the equivalent series inductance of the wiring leading up to the LED tape -* Miller plateau lag -* The damping resistor and the MOSFET gate forming an RC filter that helps with wire ESL ringing but worsens the miller - plateau issue and deforms the LED current edges. - -Added up, these three effects yield a picture that agrees well with our simulations and measurements. The overall effect -is neglegible at long period durations (>10µs), but gets really bad at short period durations (<1µs). The effect is -non-linear, so correcting for it is not as simple as adding an offset. - -.. _LTSpice: http://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html -.. _`miller plateau`: https://www.vishay.com/docs/68214/turnonprocess.pdf - -Measuring LED tape brightness -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -In order to correct for the nonlinearities mentioned above, we decided to implement a lookup table mapping BCM period to -actual timer setting. That is, each row of the table contains the actual period length we need to set the -microcontroller's timer to in order to get our intended brightness steps. - -To calibrate our driver, we needed a setup for reproducible measurement of the relative brightness of our LED tape at -different settings. Absolute brightness is not of interest to us as the eye can't perceive it. To perform the -calibration, the LED driver is set to enable each single BCM period in turn, i.e. brightness values 1, 2, 4, 8, 16 etc. - -The setup we used to measure the LED tape's brightness consists of a bunch of LED tape stuck into a tin can for -shielding against both stray light and electromagnetic interference and a photodiode looking at the LED tape. We used -the venerable BPW34_ photodiode in our setup as I had a bunch leftover from another project and because they are quite -sensitive owing to their physically large die area. - -.. raw:: html - - <figure> - <img src="images/linearization_setup.jpg" alt="The led measurement setup consists of several PCBs and a - breadboard linked with a bunch of wires and a big tin can to shield the LEDs and the photodiode. A large sub-D - connector is put into the top of the tin can as a feed-through for the LED tape's control signals and the - photodiode signal. In the background the control laptop is visible."> - <figcaption>The LED brighness measurement setup. The big tin can contains a bunch of LED tape and the - photodiode. The breadboard on the right is used for the photodiode preamplifier and for jumpering around the LED - tape's channels. The red board next to it is the buspirate used as ADC. The board on the bottom left is a - TTL-to-RS485 converter and the board in the middle is the unit under test.</figcaption> - </figure> - -The photodiode's photocurrent is converted into a voltage using a very simple transimpedance amplifier based around a -MCP6002_ opamp that was damped into oblivion with a couple nanofarads of capacitance in its feedback loop. The MCP6002_ -is a fine choice here since I had a bunch and because it is a CMOS opamp, meaning it has low bias current that would -mess up our measurements. For many applications, opamp bias current is not a big issue but when using the opamp to -directly measure very small currents at its input it quickly swamps out the signal for most BJT-input types. - -The transimpedance amplifier's output is read from the computer using the ADC input of a buspirate USB thinggamajob. In -general I would not recommend the buspirate as a tool for this job since it's ADC is not particularly good and it's -programming interface is positively atrocious, but it was what I had and it beat first wiring up one of the dedicated -ADC chips I had in my parts bin. - -The computer runs a small python script cycling the LED tape through all its BCM period settings and taking a brightness -measurement at each step. Later on, these measurements can be plotted to visualize the resulting slope's linearity, and -we can even do a simulation of the resulting brightness for all possible control values by just adding the measured -photocurrents for a certain BCM setpoint just as our retinas would do. - -.. raw:: html - - <figure> - <img src="images/driver_linearity_raw.svg" alt=""> - <figcaption> - A plot of the measured brightness of our LED tape for each BCM period. The brightness values are normalized - to the value measured at the LSB setpoint (brightness=1/65535). Ideally, this plot would show a straight - line with slope 1. Obviously, it doesn't. The bend in the curve is caused by the above-mentioned duty cycle - offset adding an offset to all brightness values. Shown is both the raw data (light), which has essentially zero - measurement error and a linear fit (dark). - - The plot is in log-log to approximate how the human eye would perceive brightness, i.e. highly sensitive at - low values but not very sensitive at all at large values. - </figcaption> - </figure> - -While it would be possible to fully automate the optimization of BCM driver lookup tables, we needed only one and in the -end I just sat down and manually tweaked the ideal values we initially calculated until I liked the result. You can see -the resulting brightness curve below. - -.. raw:: html - - <figure> - <figure class="side-by-side"> - <img src="images/uncorrected_brightness_sim.svg" alt=""> - <figcaption> - Calculated brightness curve for the uncorrected BCM setup. As you can see, at low setpoints the result - is about as smooth as sandpaper, which is well in line with our observations. At high setpoints the - offset gets swamped out and the nonlinearity in the low bits is not visible anymore. - </figcaption> - </figure><figure class="side-by-side"> - <img src="images/corrected_brightness_sim.svg" alt=""> - <figcaption> - Brightness curve for the corrected BCM setup extrapolated using actual measurements. Looks as buttery - smooth in real life as it does in this plot. - </figcaption> - </figcaption> - </figure> - </figure> - -.. _BPW34: http://www.vishay.com/docs/81521/bpw34.pdf -.. _MCP6002: http://ww1.microchip.com/downloads/en/DeviceDoc/21733j.pdf - -Controlling the driver ----------------------- - -Now that our driver was behaving linear enough that you couldn't see it actually wasn't we needed a nice way to control -it from a computer of our choice. In the ultimate application (our staircase) we'll use a raspberry pi for this. Since -we already settled on an RS485_ bus for its robustness and simplicity, we had to device a protocol to control the driver -over this bus. Here, we settled on a simple, COBS_-based protocol for the reasons I wrote about in `How to talk to your -microcontroller over serial <serial-protocols>`_. - -To address our driver nodes, we modified the Makefile to build a random 32-bit MAC into each firmware image. The -protocol has only five message types: - -1. A 0-byte *ping* packet, to which each node would reply with its own address in the - first 100ms after boot. This can be used to initially discover the addresses of all nodes connected to the bus. You'd - spam the bus with *ping* packets, and then hit reset on each node in turn. The control computer would then receive - each device's MAC address as you hit reset. -2. A 4-byte *address* packet that says which device that the following packet is for. This way of us using the packet - length instead of a packet type field is not particularly elegant, but our system is simple enough and it was easy to - implement. -3. A 64-byte *frame buffer* packet that contains 16 bits of left-aligned brightness data for every channel -4. A one-byte *get status* packet that tells the device to respond with... -5. ...a 27-byte status packet containing a brief description of the firmware (version number, channel count, bit depth - etc.) as well as the device's current life stats (VCC, temperature, uptime, UART frame errors etc.). - -Wrapped up in a nice python interface we can now easily enumerate any drivers we connect to a bus, query their status -and control their outputs. - -.. _COBS: https://en.wikipedia.org/wiki/Consistent_Overhead_Byte_Stuffing - -Conclusion ----------- - -.. raw:: html - - <figure> - <figure class="side-by-side"> - <a href="images/olsndot_schematic.png"> - <img src="images/olsndot_schematic.png" alt="A picture of the LED driver schematic"> - </a> - <figcaption>The LED driver <a href="images/olsndot_schematic.png">schematic</a></figcaption> - </figure><figure class="side-by-side"> - <a href="images/olsndot_pcb.png"> - <img src="images/olsndot_pcb.png" alt="A picture of the LED driver PCB layout"> - </a> - <figcaption>The LED driver <a href="images/olsndot_pcb.png">PCB layout</a></figcaption> - </figure> - </figure> - -Putting some thought into the control circuitry and software, you can easily control large numbers of channels of LEDs -using extremely inexpensive driving hardware without any compromises on dynamic range. The design we settled on can -drive 32 channels of LED tape with a dynamic range of 14bit at a BOM cost of below 10€. All it really takes is a couple -of shift registers and a mildly bored STM32 microcontroller. - -Get a PDF file of the schematic and PCB layout `here <olsndot_v02_schematics_and_pcb.pdf>`__ or download the CAD files -and the firmware sources `from github <https://github.com/jaseg/led_drv>`_. You can view the Jupyter notebook used to -analyze the brightness measurement data `here <http://nbviewer.jupyter.org/github/jaseg/led_drv/blob/master/doc/Run_analysis.ipynb>`__. - |