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) |
∆T_{J − 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), P_{d} 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 Rθ 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). |