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Table 4 Calculating the required thermal properties of a heatsink

From: An economical and highly adaptable optogenetics system for individual and population-level manipulation of Caenorhabditis elegans

Using the following thermal model (eq. 1, [85], Fig. 2F) and calculating or estimating all the individual parameters (eq. 2–4), the required thermal properties of a heatsink can be calculated.
\( {R\theta}_{J-\mathrm{Ref}}=\frac{\Delta {T}_{J-\mathrm{Ref}}}{P_D} \) (1)
Where:
\( {\displaystyle \begin{array}{c}{R\theta}_{J-\mathrm{Ref}}=\mathrm{Thermal}\ \mathrm{resistance}\ \left({}^{\circ}\mathrm{C}/\mathrm{W}\right)\ \mathrm{from}\ \mathrm{the}\ \mathrm{LED}\ \mathrm{junction}\ \mathrm{to}\ \mathrm{a}\ \mathrm{reference}\ \mathrm{point}\\ {}={R\theta}_{\mathrm{Junction}-\mathrm{Thermalpad}}+{R\theta}_{\mathrm{Thermalpad}-\mathrm{Solderpad}}+{R\theta}_{\mathrm{Solderpad}-\mathrm{Adhesives}}+{R\theta}_{\mathrm{Adhesive}-\mathrm{heatsink}}\end{array}} \) (2)
TJ − Ref = (TJ, junction temperature) – (TRef, reference point temperature, °C) (3)
\( {\displaystyle \begin{array}{c}\mathrm{PD}=\mathrm{Power}\ \mathrm{dissipation}\ \left(\mathrm{W}\right)\\ {}=\mathrm{LED}\ \mathrm{forward}\ \mathrm{current}\ \left(\mathrm{If}\right)\ast \mathrm{LED}\ \mathrm{forward}\ \mathrm{voltage}\ \left(\mathrm{Vf}\right)\end{array}} \) (4)
Typically, the maximal junction temperature can be found in the datasheet for the LED. For a Royal-Blue Luxeon Rebel LED on a SinkPAD-II that is used in this paper, the listed temperature is 150 °C. Ideally, the operating temperature should be well below that limit, as even reaching this temperature for a fraction of time could influence the properties of the LED [92,93,94,95]. Therefore, we decided to set the maximum junction temperature to 100 °C. Next, the power dissipation of the same LED can be calculated by multiplying the forward voltage rating of the LED with the drive current in Amperes (eq. 4). Since we use a fixed 700-mA LED driver and know the maximal voltage drop of the LED (3.5 V), Pd is easily calculated (eq. 4). Finally, the maximal thermal resistance should be calculated by taking in the maximum working ambient temperature into account, which was determined to be 25 °C (20 °C room temperature + 5 °C margin, eq. 3). Importantly, note that our set-up is located in a climate-controlled room. If this is not the case, a wider margin for temperature fluctuations and higher maximum temperature based on local environmental conditions should be considered. Rewriting eq. 1 and calculating gives (eq. 5):
\( {R\theta}_{J-\mathrm{Ref}}=\frac{\Delta {T}_{J-\mathrm{Ref}}}{P_D}=\frac{\Delta {T}_{J-\mathrm{Ref}}}{If\ x\ Vf}=\frac{100-25}{0.7\times 3.5}=30.6{}^{\circ}\mathrm{C}/\mathrm{W} \) (5)
This means that the total thermal resistance of the system can be maximally 30.6 °C/W (eq. 2) when one aims for junction temperatures that are maximally 100 °C. In order to see what that means for the required heatsink, all the known thermal resistances (Fig. 2F) can be collected from the datasheets of the LED (junction to thermal pad: 6.0 °C/W and thermal pad to solder pad: 0.7 °C/W) and used adhesives (4.5 °C/W). Rewriting eq. (2) gives (eq. 6):
\( {\displaystyle \begin{array}{c}{R\theta}_{\mathrm{Adhesive}-\mathrm{Heatsink}}={R\theta}_{J-\mathrm{Ref}}-{R\theta}_{\mathrm{Junction}-\mathrm{Thermalpad}}-{R\theta}_{\mathrm{Thermalpad}-\mathrm{Solderpad}}-{R\theta}_{\mathrm{Solderpad}-\mathrm{Adhesives}}\\ {}=30.6-6.0-0.7-4.5\\ {}=19.4{}^{\circ}\mathrm{C}/\mathrm{W}\end{array}} \) (6)
Theoretically, a heatsink in this set-up can have a maximal thermal resistance of 19.4 °C/W and any heatsink with lower resistance than that can be used. We used a finned heatsink with thermal resistance of 5.2 °C/W to keep the set-up compact and still achieve sufficient cooling. Next, we continued with an empirical approach to verify whether appropriate heat dissipation could be obtained with this set-up. Therefore, we measured the actual junction temperature and the voltage drop of the LED mounted to the OptoArm (Fig. 2D). With a Fluke TM80 module and associated test probe, we measured the temperature at the specified test location (Fig. 2F) of the LED for about 1 min until the temperature stabilized. At the same temperature, we also determined the actual voltage drop of the LED. We measured a temperature of 55.68 ± 0.76 °C (n = 10) and a forward drop of 2.972 ± 0.004 V (n = 5). By using eq. (7), the actual junction temperature can be derived:
\( {\displaystyle \begin{array}{c}{T}_{\mathrm{Junction}}={T}_{\mathrm{testpoint}}+\left({R\theta}_{\mathrm{Junction}-\mathrm{Thermalpad}}+{R\theta}_{\mathrm{Thermalpad}-\mathrm{testpoint}}\right)\times \left({V}_f\times {I}_f\right)\\ {}=55.68{}^{\circ}\mathrm{C}+\left(6.0{}^{\circ}\mathrm{C}/\mathrm{W}+0.5{}^{\circ}\mathrm{C}/\mathrm{W}\right)\times \left(2.972\ V\times 0.7\ A\right)=69.2{}^{\circ}\mathrm{C}\end{array}} \) (7)
With an estimated junction temperature of 69.2 °C, the system fulfills our criteria for a system with appropriate thermal cooling. This is further evidence by the temperature measured after 10 min of constant illumination at the maximal capacity: 55.81 ± 1.09 °C (=junction temperature of 69.3 °C, n = 10).