From 4a98183253e29a66be4def63a381e084b36a5c7e Mon Sep 17 00:00:00 2001 From: jaseg Date: Sat, 12 May 2018 12:43:39 +0200 Subject: Initial commit --- content/posts/led-characterization.rst | 304 +++++++++++++++++++++++++++++++++ 1 file changed, 304 insertions(+) create mode 100644 content/posts/led-characterization.rst (limited to 'content/posts/led-characterization.rst') diff --git a/content/posts/led-characterization.rst b/content/posts/led-characterization.rst new file mode 100644 index 0000000..e856a09 --- /dev/null +++ b/content/posts/led-characterization.rst @@ -0,0 +1,304 @@ +--- +title: "Led Characterization" +date: 2018-05-02T11:18:38+02:00 +draft: true +--- + +Preface +------- + +Recently, I have been working on a `small driver`_ for ambient lighting using 12V LED strips like you can get +inexpensively from China. I wanted to be able to just throw one of these somewhere, stick down some LED tape, hook it up +to a small transformer and be able to control it through Wifi. When I was writing the firmware, I noticed that when +fading between different colors, the colors look *all wrong*! This observation led me down a rabbit hole of color +perception and LED peculiarities. + +The idea of the LED driver was that it can be used either with up to eight single-color LED tapes or, much more +interesting, with up to two RGB or RGBW (red-green-blue-white) LED tapes. For ambient lighting high color resolution was +really important so you could dim it down a lot without flickering. I ended up using the same driver stage I used in the +`multichannel LED driver`_ project for its great color resolution and low hardware requirements. + +.. figure:: https://upload.wikimedia.org/wikipedia/commons/d/d6/RGB_color_cube.svg + :width: 100% + :alt: An illustration of the RGB color cube + + An illustration of the RGB color cube. `Picture by Maklaan + `_ (`CC BY-SA 3.0`_), from Wikimedia Commons. + +To make setting colors over Wifi more intuitive I implemented support for HSV colors. RGB is fine for communication +between computers, but I think HSV is easier to work with when manually inputting colors from the command line. RGB is +close to how most monitors, cameras and the human visual apparatus work on a very low level but doesn't match +higher-level human color perception very well. When we describe a color we tend to think in terms of "hue" or +"brightness", and computing a measure of those from RGB values is not easy. + +Colors and Color Spaces +----------------------- + +`Color spaces`_ are a mathematical abstraction of the concept of color. When we say "RGB", most of the time we actually +mean `sRGB`_, a standardized notion of how to map three numbers labelled "red", "green" and "blue" onto a perceived +color. `HSV`_ is an early attempt to more closely align these numbers with our perception. After HSV, a number of other +*perceptual* color spaces such as `XYZ (CIE 1931)`_ and `CIE Lab/LCh`_ were born, further improving this alignment. In +this mathematical model, mapping a color from one color space into another color space is just a coordinate +transformation. + +.. figure:: https://upload.wikimedia.org/wikipedia/commons/4/4e/HSV_color_solid_cylinder.png + :width: 50% + :align: center + :alt: An illustration of the HSV color space as a cylinder + + An illustration of the HSV color space as a cylinder. `Picture by SharkD + `_ (`CC BY-SA 3.0`_), from Wikimedia Commons. + +The wrong colors I got when fading between colors were caused by this coordinate transformation being askew. Thinking +over the problem, there are several sources for imperfections: + +* The LED driver may not be entirely linear. For most modulations such as PWM the brightness will be linear starting + from a certain value, but there is probably an offset caused by imperfect edges of the LED current. This offset can be + compensated with software calibration. I built a calibration setup for driver linearity in the `multichannel LED + driver`_ project. + .. FIXME picture with ringing on edges + +* The red, green and blue channels of the LEDs used on the LED tape are not matched. This skews the RGB color space. + In practice, the blue channel of my RGB tape to me *looks* much brighter than the red channel. + +* The precise colors of the red, green and blue channels of the LEDs are unknown. Though the red channel *looks* red, it + may be of a slightly different hue compared to the reference red used in `sRGB`_ which would also skew the RGB color + space. + +These last two errors are tricky to compensate. What I needed for that was basically a model of the *perceived* colors +of the LED tape's color channels. A way of doing his is to record the spectra of all color channels and then evaluate +their respective XYZ coordinates. If all three channels are measured in one go with the same setup the relative +magnitudes of the channels in XYZ will be accurate. + +To map any color to the LEDs, the color's XYZ coordinates simply have to be mapped onto the linear coordinate system +produced by these three points within XYZ. LEDs are extremely linear in their luminous flux vs. current characteristic +so this model will be adequate. The spectral integrals mapping the channels' measured responses to XYZ need only be +calculated once and their results can be used as scaling factors thereafter. +.. FIXME: Add led/current graph here + +Measuring the spectrum +---------------------- + +In order to compensate for the cheap LED tape's non-ideal performance I had to measure the LED's red, green and blue +channels' spectra. The obvious thing would be to go out and buy a `spectrograph`_, or ask someone to borrow theirs. The +former is kind of expensive, and I did not want to wait two weeks for the thing to arrive. The latter I could probably +not do every time I got new LED tape. Thus the only choice was to build my own. + +Luckily, building your own spectrometer is really easy. The first thing you need is something that splits incident light +into its constituent wavelengths. In professional devices this is called the *`monochromator`_*, since it allows extraction +of small color bands from the spectrum. The second thing is some sort of optics that project the incident light onto a +screen behind the monochromator. In professional devices lenses or curved mirrors are used. In a simple homebrew job a +pinhole as you would use in a `camera obscura`_ does a remarkably nice job. + +For the monochromator component several things could be used. A prism would work, but I did not have any. The +alternative is a `diffraction grating`_. Professional gratings are quite specialized pieces of equipment and thus +rather expensive. Luckily, there is a common household item that works almost as well: A regular CD or DVD. The +microscopic grooves that are used to record data in a CD or DVD work the same as the grooves in a professional +diffraction grating. + +Household spectra +----------------- + +From this starting point, a few seconds on my favorite search engine yielded an `article by two researchers from the +National Science Museum in Tokyo`_ providing a nice blueprint for a simple cardboard-and-DVD construction for use in +classrooms. I replicated their device using a DVD and it worked beautifully. Daylight and several types of small LEDs I +had around did show the expected spectra. Small red, yellow, green, and blue LEDs showed narrow spectra, daylight one +continuous broad one, and white LEDs a continuous broad one with a distinct bright spot in the blue part. The +single-color LED spectra are quite narrow since they are determined by the LED's semiconductor's band gap, which is +specific to the semiconductor used and is quite precise. White LEDs are in fact a blue LED chip covered with a so-called +*phosphor*. This phosphor is not elementary phosphorus but an anorganic compound that absorbs the LED chip's blue light +and re-emits a broader spectrum of more yellow-ish wavelengths instead. The final LED spectrum is a superposition of +both spectra, with some of the original blue light leaking through the phosphor mixing with the broadband yellow +spectrum of the phosphor. +.. FIXME: Cardboard spectrograph pictures + +Now that I had a spectrograph, I needed a somewhat predictable way of measuring the spectrum it gave me. + +Measuring a spectrum +-------------------- + +Pointing a camera at the spectrograph would be the obvious thing to do. This produces pretty images but has one critical +flaw: I wanted to acquire quantitative measurements of brightness across the spectrum. Since I don't have a precise +technical datasheet specifying the spectral response of any of my cameras I can't compare the absolute brightness of +different colors on their pictures. Some other sensor was needed. +.. FIXME: Spectrum picture + +Measuring light intensity +~~~~~~~~~~~~~~~~~~~~~~~~~ + +Looking around my lab, I found a bag of `SFH2701`_ visible-light photodiodes. Their +datasheet includes their spectral response so I can compensate for that, allowing precise-ish absolute intensity +measurements. Just like LEDs, photodiodes are extremely linear across several orders of magnitude. The datasheet of the +classic `BPW34`_ photodiode shows that this photodiode's light current is exactly proportional to illuminance over at +least three orders of magnitude. The `SFH2701`_ datasheet does not include a similar graph but its performance will be +similar. The `SFH2701`_ photodiodes I had at hand were perfect for the job compared to the vintage `BPW34`_ since their +active sensing area is really small (0.6mm by 0.6mm) compared to the BPW34 (a whopping 3mm by 3mm). If I were to use a +`BPW34`_ I would have to insert some small apterture in front of it so it does not catch too broad a part of the +spectrum at once. The `SFH2701`_ is small enough that if I just point it at the projected spectrum directly I will +already get only a small part of the spectrum inside its 0.6mm active area. + +To convert the photodiode's tiny photocurrent into a measurable voltage I built another copy of the `transimpedance +amplifier`_ circuit I already used in the `multichannel LED driver`_. A `transimpedance amplifier`_ is an +amplifiert that produces a large voltage from a small current. The weird name comes from the fact that it works kind of +like an amplified resistor (which can be generalized as an *impedance* electrically). Apply a current to a resistor and +you get a voltage. A transimpedance amplifiert does the same with the difference that its input always stays at 0V, +making it look like an ideal current sink to the connected current source. + +Transimpedance amplifiers are common in optoelectronics to convert small photocurrents to voltages. In this instance I +built a very simple circuit with a dampened transimpedance amplifier stage followed by a simple RC filter for noise +rejection and a regular non-inverting amplifier using another op-amp from the same chip to further boost the filtered +transimpedance amplifier output. I put all the passives setting amplifier response (the gain-setting resistors and the +filter resistor and capacitors) on a small removable adapter so I could easily change them if necessary. I put a small +trimpot on the virtual ground both amplifers use as a reference so I could trim that if necessary. +.. FIXME: Add transimp amp schematic and build pics + +Given a way to measure intensity what remains missing is a way to scan a single photodiode across the spectrum. + +Scanning the projection +~~~~~~~~~~~~~~~~~~~~~~~ + +A cheap linear stage can be found in any old CD or DVD drive. These drives use a small linear stage based on a +stepper-driven screw to move the laser unit radially. Removing the laser unit and connecting a leftover stepper driver +module I was left with a small linear stage with about 45 steps per cm without microstepping enabled. The driver I used +was an `A4988`_ module that required at least 8V motor drive voltage. I used a small micro USB-input boost converter +module to generate a stable 10V supply for the motor driver, with the USB's 5V rail used as a logic supply for the motor +driver. + +.. FIXME: Add picture of photodiode stage here + +The `SFH2701`_ can easily be mounted to the linear stage using a small SMD breakout board glued in place with thin wires +connecting it to the transimpedance amplifier. The DVD drive linear stage is not very strong so it is important that +this wire does not put too much strain on it. + +Above the photodiode, I mounted a small piece of paper on the linear stage to be used as a projection screen to align +the linear stage in front of the spectrometer viewing window. A line on the screen paper points to the photodiode die in +parallel to the linear stage allowing precise alignment. + +The whole unit with photodiode preamplifier, linear stage, photodiode and stepper motor driver finally looks like this: + +.. FIXME: pics of linear stage unit electronics + +The projection of the spectrum can be adjusted by moving the light source relative to the entry slot and by moving +around the grating DVD. + +The capture process +~~~~~~~~~~~~~~~~~~~ + +To capture a spectrum, first the light source has to be mounted near the spectrograph's entry slot. The LED tape I +tested I just taped face-down directly into it. Next, the grating DVD has to be adjusted to make sure the spectrum +covers a sensible part of the photodiode's path. Mostly, this boils down to adjusting the photodiode distance and height +to match the vertical extent and wiggling the grating DVD to adjust the projection's horizontal position. + +After the optics are set-up, the photodiode preamplifier has to be adjusted. In my experiments, most LED tape at 5GΩ +required a high-ish amplification. The goal in this step is to maximize the peak response of the preamp to be just +shy of its VCC rail to make best use of its dynamic range. To adjust the pre-amp, I took several very coarsely-spaced +measurements to give me an estimate of the peak while I did not yet know its precise location. + +Since stray daylight totally swamped out the weak projection of the LED's spectrum I shielded the entire setup with a +small box made of black cardboard and two black t-shirts on top. This shielding proved adequate for all my measurements +but I had to be careful not to accidentially move the DVD that was stuck into the spectrograph with the shielding +t-shirts. + +For capturing a single spectrum I wrote a small python script that will automatically move the stepper in adjustable +intervals and take two measurements at each point, one with the LED tape off that can be used for offset calibration and +one with the LED tape on. All measurements are stored in a sqlite database that can then be accesssed from other +scripts. + +I built a small script that shows the progress of the current run and an jupyter notebook for data analysis. The jupyter +notebook is capable of live-updating a graph with the in-progress spectrum's data. This was quite useful as a sanity +check for when I made some mistake easy to spot in the resulting data. + +After one color channel is captured, the LED tape has to be manually set to the next color and the next measurement can +begin. + +Data analysis +~~~~~~~~~~~~~ + +Data analysis consists of three major steps: Offset- and stray light removal, wavelength and amplitude calibration and +color space mapping. + +Offset removal +************** +The first task is to remove the offset caused by dark current as well as stray light of the LED's bright primary +reflection on the DVD. The LED is very bright and only a small part of its light gets reflected by the grating towards +the photodiode screen. The remaining part of the light is reflected onto the table in front of the DVD spectrograph. +Though I covered all of this with black cardboard, some of that light ultimately gets reflected onto the photodiode. +This causes a large offset, in particular in the blue part of the spectrum since in this part the photodiode is closest +to the spectrograph's opening. + +The composite offset can be approximated with a second-order polynomial that is fitted to all the data outside of the +main peak's area. Since at this point the wavelength of each data point is still unknown this is done with a rough first +estimate of the three colors' peaks' locations and widths. + +Wavelength- and amplitude calibration +************************************* +The photodiode's response is strongly wavelength-dependent. In particular in the blue band, the photodiode's sensitivity +gets very poor down to about 20% at the edge to ultraviolet. This effect is strong enough to move the apparent location +of the blue peak towards red. + +The problem is that in order to remove this non-linearity, we would already have to know the wavelength of the measured +light. Since I don't, I settled for a two-step process. First, a coarse wavelength calibration is done relative to the +red peak and the short-wavelength edge of the blue peak. The photodiode measurements are then sensitivity-corrected +using this coarse measurement. Then all three channel peaks are measured in the resulting data and a fine wavelength +estimate is produced by a least-squares fit of a linear function. This fine estimate is then used for a second +sensitivity correction of all original measurements and the scale is changed from stepper motor step count to +wavelength in nanometers. + +.. FIXME: calibration for brightness imbalance due to wedge-shaped projection of spectrum + +Color space mapping +******************* +Finally, to achieve the objective of measuring the LED tape's channels' precise color coordinates the measured spetra +have to be matched against the color spaces' *color matching functions*. The color matching functions describe how +strong the color space's idealized *standard observer* would react to light at a particular wavelength. Going from a +measured spectrum to color coordinates XYZ works by integrating over the product of the measurement and each color +coordinate's color matching function. + +The result are three color coordinates X, Y and Z for each channel R, G and B yielding nine coordinates in total. When +written as a matrix conversion between XYZ color space and LED-RGB color space is as simple as multiplying that matrix +(or its inverse) and a vector from one of the color spaces. + +If you view the three channels' color coordinates as vectors in XYZ space, the set of colors that can be produced with +this LED tape is described by the `parallelepiped`_ spanned by the three channel vectors. + +The last task is to decide on a scaling factor to map XYZ space to RGB space. Both are limited to values between 0.0 and +1.0. The LEDs cannot go below off or above fully on. For any LED tape there will be a set of colors that are outside +the range that this tape can produce. + +A scaling factor can be used to increase the number of XYZ coordinates that can be mapped to RGB colors the tape *can* +produce by stretching the RGB parallelepiped along its major axis. Up to a point the number of possible colors (the +gamut) increases at expense of maximum brightness. When the parallelepiped is stretched far enought for all three +channel vectors to be outside the 1,1,1 XYZ-cube, maximum brightness continues to decrease but the gamut stays constant. + +Firmware implementation +----------------------- +In the end, the above measurements yield two matrices: One for mapping XYZ to RGB, and one for mapping RGB to XYZ. I +chose the CIE 1931 XYZ color space as a basis for the firmware because it is most popular. Mapping a color coordinate in +one color space to the other is as simple as performing nine floating-point multiplications and six additions. Mapping +Lab or Lch to RGB is done by first mapping Lab/Lch to XYZ, then XYZ to RGB. Lab to XYZ is somewhat complex since it +requires a floating-point power for gamma correction, but any self-respecting libc will have one of those so this is +still no problem. Lch also requires floating-point sine and cosine functions, but these should still be no problem on +most hardware. + +My implementation of these conversions in the ESP8266 firmware of my `Wifi LED driver`_ can be found `on Github`_. + +.. _`on Github`: https://github.com/jaseg/esp_led_drv/blob/master/user/led_controller.c +.. _`Wifi LED driver`: {{}} +.. _`small driver`: {{}} +.. _`multichannel LED driver`: {{}} +.. _`sRGB`: https://en.wikipedia.org/wiki/SRGB +.. _`CC BY-SA 3.0`: https://creativecommons.org/licenses/by-sa/3.0 +.. _`Color spaces`: https://en.wikipedia.org/wiki/Color_space +.. _`HSV`: https://en.wikipedia.org/wiki/HSL_and_HSV +.. _`CIE Lab/LCh`: https://en.wikipedia.org/wiki/Lab_color_space +.. _`XYZ (CIE 1931)`: https://en.wikipedia.org/wiki/CIE_1931_color_space +.. _`camera obscura`: https://en.wikipedia.org/wiki/Pinhole_camera +.. _`article by two researchers from the National Science Museum in Tokyo`: http://www.candac.ca/candacweb/sites/default/files/BuildaSpectroscope.pdf +.. _`spectrograph`: https://en.wikipedia.org/wiki/Ultraviolet%E2%80%93visible_spectroscopy +.. _`monochromator`: https://en.wikipedia.org/wiki/Monochromator +.. _`diffraction grating`: https://en.wikipedia.org/wiki/Diffraction_grating +.. _`SFH2701`: https://dammedia.osram.info/media/resource/hires/osram-dam-2495903/SFH%202701.pdf +.. _`BPW34`: http://www.vishay.com/docs/81521/bpw34.pdf +.. _`transimpedance amplifier`: https://en.wikipedia.org/wiki/Transimpedance_amplifier +.. _`A4988`: https://www.pololu.com/file/0J450/A4988.pdf +.. _`parallelepiped`: https://en.wikipedia.org/wiki/Parallelepiped -- cgit