Math: Basics

Background math for the polarimeter.

Ideal polarizer

The intensity of light after two polarizers with polarization planes at an angle of `phi` relative to `I_0` is described as ` I_phi = cos ^ 2 phi * I_0` Check the graph below for the dependency of intensity on the angle. You will recognize the maximums at 0°=360° and 180° and minimums at 90° and 270°. It's 180° (or `pi`)-periodic, due to the square of cosine.

Ideal photoresistor

CdS (Cadmium Sulphide) photoresitors show an inverse dependency of resistance on incident light. The higher the luminance, the lower the resistance, ideally described as a a power function in I `R_I = c * I_phi^gamma` with `c=1700 and gamma=-0.65`. How to get the actual parameters `c` and `gamma` is hinted at in the pink graph, it will be shown with real data in the next math chapter (to come...).

Voltage divider

An example for a voltage divider circuit to measure the resistance of an electronic part is shown on the MCP3208 page[1]. The fixed resistance `R_0` is connected as first in series with the variable resistance `R_I`. The formula for the resulting voltage in relation to the incoming voltage is `V_I = V_0 * (R_I /( R_0 + R_I)) = V_0 * (1 - (R_0/(R_0+R_I)))`
In case of `R_I -> 0 => V_I -> 0` and `R_I -> oo => V_I -> V_0`.

ADC readings

The ADC, here a MCP3208, compares an incoming voltage with a base voltage and returns the clipped ratio of these, which can also be expressed in terms of `R_0` and `R_I`, see B-3:
`counts = floor(2^p*(V_I/V_0)) = floor( 2^p * (1 -(R_0/(R_0+R_I))))`
`2^p` denotes the number of steps, with `p` being the bit depth. In case of MCP3008 `p = 10 => 1024`, for MCP3208 `p = 12 => 4096`.

This graph is interesting as it shows that until ~2700 counts the change of `R_I` needed to get the next higher count value is less than 10 `Omega`, a high resolution with respect to R. After 4000 counts, the difference in R needed for the next higher counts value increases drastically. Good on the one hand, because we can determine the range of R for a certain counts value with almost no error. Bad on the other hand, because we may miss out resolution for small changes of I with low intensities.

Putting it together

Plugging in all formulas into one we get the dependency of counts on angle
`counts_phi= floor( 2^p*(1-(R_0/(R_0+( c*(cos^2 phi* I_0)^gamma))))`
With `I_0=1, c=1700, gamma=-0.65, p=12, R_0 in {2000, 5000, 10000}` we get following curves:

Conclusions

With 5kOhms resistor the counts range spreads over ~ 3000 values, while there are just 900 0.1°steps between a minimum and a maximum, e.g from 0° to 90°, suggesting a good resolution of count vs. step position.
A higher resistance `R_0` is not recommended, as it may introduce errors due to "bleeding" currents before sampling has completed. Especially it is not advisable to put the photoresistor first in row.
Peaks for high values (=little light) are much better defined than the valleys for low values (=much light). This is caused by big values of R for low intensities, partly compensated by the ratio in the voltage divider formula (B-3).

smartyPies.com