THE SOLUTION OF TWO-DIMENSIONAL HEAT CONDUCTION PROBLEMS FOR PREDICTING OPERATING TEMPERATURE AND POWER HANDLING CAPABILITIES OF HYBRID CIRCUITS

The problems of heat conduction in hybrid thin and thick film circuits have been investigated using both analytical and numerical approaches. The calculations were restricted to two dimensions by assuming zero temperature gradient across the thickness of the substrate (slightly more than half millimeter). All normally recognized parameters were taken into consideration, such as film temperature, power dissipated, substrate area, etc. Allowances have also been made for lead conduction and connection technology, for horizontal or vertical circuit assembly, and for flush or stand-off mounting on the mechanical support. The measured results for practical cases show close agreement with the theoretical calculations. This approach provides a simple tool for the calculation, at an early design stage of power handling capacities of hybrid circuits using composite resistive configurations. Very little difference was found between temperatures predicted from the theoretical approach and those measured in practical cases of circuits under load. The availability of such accurate design information means that the stability, life, and reliability of hybrid circuits can be predicted with considerable accuracy at an early design stage.


INTRODUCTION
To investigate analytically heat conduction problems in hybrid circuits, models were restricted to two dimensions assuming that the thickness of the substrates was negligible for the usual applications in relation to the dimensions of the plane of the substrate; also because the material mployed (alumina ceramic) had a relatively high thermal conductivity. We also assumed that coefficients of convection were identical for front and back surfaces of the substrates and that radiation effects linearized for the temperature intervals used were included in the convection coefficient itself. Finally, we assumed that the temperature of the powered resistive films was uniform over the whole area at the hottest f'tim temperature.
Key wordst a ro K thickness of substrates in mm;. radius of substrates in mm; radius of resistive film in mm thermal conductivity of substrate in mW mm/mm2 C; For a ring of thickness t included between radii r and r + Ar (Figure 1) The power dissipated from substrate zones not covered by film is: The power dissipated from the resistive film is: -iJ1 (iwa))-H (iwro )) -iJ, (iwro ) ) -O( (iwa) (Jo (iwro ))-H (iwa)) + (-iJ, (iwa))(iH(o (iwro )) (Jo (iwa)X-tl )(iwa)} + (-iJ1 (iwa)}{iH1)(iwa) } (Jo (iwro))(-H' )(iwa)) + -iJl(iwa))(iHl)(iwro))  Table we observe that the values of Pw/T . show a tendency to diminish with increasing power but these effects, probably due to radiation emission, can reasonably be disregarded in the temperature range considered, because they introduce an error of less than -+5% in predictions of temperature and this is the limit of the validity of our theory. The results of Table I, column (a), were plotted against D on a log log scale ( Figure 3). Assuming the value from the data sheets for AL 772 as the thermal conductivity of the substrate, (K 36 mW mm/mm 2 C) experimental values for the convection coefficient of the substrate (h) were calculated by extrapolation since from Eq. (12):  Assuming exactly the same conditions as in the circular case ( Figure 2) but applied to rectangular geometries: T(x,y) T(x,y) Since the geometry is symmetrical, Eq. (16) was solved numerically for the first quadrant. Using a finite increment method" T(x,y) AT(x,y) x Ax r(x + I-I/2,y) T(x I/e,y) T(x + Hx,y) + T(x Hx,Y) 2T (2c2b)/rr a:(2Xo 2yo)/r r) x/2iKtD F(a,ro,W) rm=Tf hX/s (D-l) It must still be shown that Eq. (21) and Eq. (12) are comparable at the limits of theoretical validity (+5%).

Experiments with Rectangular Substrate
Alumina substrates of thickness 0.635 mm. with central rectangular resistive Tantalum nitride films were produced at 1" x 1", l"x 2", 2" x 2". 3 The film temperature was measured at the hottest resistive film point by an infrared pyrometer in identical conditions to those imposed for the circular substrates.
The Pw/Ty" data plotted against D, on a log log scale, is shown in Figure 4. Also for rectangular substrates the values of Pw/T 1, are independent of the power across the resistive films at the theoretical limits (-+5%). Calculation at D >> 10 and extrapolation at D gives the experimental values for the convection coefficient. These were substituted in Eq. (12) (continuous curves 1,2 and 3 in Figure 4) and in Eq. (21) (asterisks). The correlation of the points and curves allowed Eq. (12) and statements of Eq. (22) to be used for all further calculations.

Experiments with Substrates Mounted
Horizontally Temperature measurements were repeated with substrates mounted horizontally and values of the parameter h were calculated. Plotting h for vertically and horizontally mounted substrates on a log log scale against 1/A s ( Figure 5)

Experiments with Leads
To evaluate lead effects, 6 standard GTE leads were soldered onto the periphery of the circular substrates of area 660.52 mm The substrates were soldered to a copper heat sink as in Figure 6 to simulate a real situation. The copper heat sink was used to ensure that the ends of the leads were effectively at the ambient temperature so that Eq. (12) could be used. Experimental measurements for vertical position are recorded in Table I, Table I, column c. The values obtained are close to those of experiment without leads and within the measurement tolerance. This is due to the poor thermal conductivity of the PCBs which cannot insure ambient temperature at the ends of the leads. To obtain an equivalent thermal resistance for leads on PCBs, experiments were repeated with 22 leads mounted on the substrate periphery and, from calculations, RL 5.40 C/mW.

MEASUREMENTS WITH HYBRID CIRCUITS
The validity of the theory and of Eq. (12) was verified for numerous hybrid circuits from GTE production in both thin and thick film. Two circuits were chosen in particular for detailed examination since they show interesting properties. The first is the thick film circuit in Figure 7. This circuit, used 2" x A" alumina substrate, R 10 kohm -+ 0.05%.
The area of each resistor is calculated for dissipating 50 mW without exceeding 150 C at 70 C ambient. Experiments were performed by powering resistors at 100 mW each at 26 C ambient for easy measurements on an infrared pyrometer and the results are shown in Table III. The predicted resistor temperatures for each position were calculated from Eq. (23) and Eq. (12) assuming linear temperature gradient for the substrate from the position of the hottest resistor to that of the coolest (at each end). 5. CONCLUSION ACKNOWLEDGEMENTS A method has been described to calculate the highest temperature for resistors on hybrid circuits and also to estimate the distribution of temperature for circuits with multiple resistors, only when component location is roughly symmetrical. The method allows calculation of the minimum area of substrate for a given power on the circuit. The effects of trimming are not considered but employing the approach of D. W. Walter 6 it is also possible to compute the resulting hot spots since the temperature of the untrimmed resistor is known. The precision of the method is +-5%. The availability of such information on the temperature of resistors means that the stability, life and reliability of hybrid circuits can be predicted from the appropriate data extrapolated from life tests on both thin and thick film technologies.