The heat transfer process, known as thermal conduction, can be quantified in terms of appropriate rate equations. The rate equation in this heat transfer mode is based on Fourier’s law of thermal conduction. This law states that the time rate of heat transfer through a material is proportional to the negative gradient of temperature and the area, at right angles to that gradient, through which the heat flows. Equation 1 can be derived, and is valid as long as all the parameters are constant throughout the sample.

where:

• q is the vector of local heat flux density [W/m²]
• k is the conductivity of the material [W/mK]
• ΔT is the temperature gradient [K/m²]

From Equation 1, heat flow can be modeled using a thermal circuit as shown in Figure 3-1, where heat sources are represented by current or voltage sources, heat flow is represented by current, temperature is represented by voltage, and absolute thermal resistances are represented by resistors.

where:

• T_j is the junction temperature of the device
• T_c is the temperature at its case
• T_h is the differential temperature along its path
• R₁ is the thermal resistance along the path to the die
• R₂ is the thermal resistance between the two reference measurements
• R₃ is the unknown resistance along the path to the ambient temperature

Again, it is difficult to account for the thermal resistances and external influences to determine T_j. Although the exact thermal resistances may not be known, the thermal equivalent model of a voltage divider circuit and the consistent ratio of temperature drops between measurements can be used to accurately calculate the die temperature. To understand this ratio, consider the voltage divider circuit in Figure 3-2, with known voltage measurements and unknown resistances.

The voltage across R₁ is 2 V and the voltage across R₂ is 4 V, and the ratio of these voltage drops is 1/2. This voltage drop ratio remains constant even if the source voltage changes, because the current through R1 equals the current through R2. Equation 2 and Equation 3 can be applied to this circuit according to Ohm's law.

Solving for I in Equation 2 and Equation 3, setting them equal to each other, and manipulating the resulting equation gives Equation 4:

From Equation 2 through Equation 4, the exact resistance values in the circuit are no longer needed because a ratio of the resistance values has been established using the voltage drops across the two resistors instead. As long as the resistances remain constant, the ratio of the voltage drops remains constant. This ratio can be used to calculate a missing voltage drop, and therefore a missing voltage, when just one voltage drop is known. Consider the circuit in Figure 3-3 with the same unknown resistances but a different, unknown voltage source and new voltage measurements.

The voltage drop ratio calculated from the initial circuit can now be used to find the new, unknown voltage source. The voltage drop across R₂ is 6 V. The voltage drop across R₁ is found by multiplying 6 V by the previously calculated ratio of ?, which results in 3 V. Adding 3 V to the 12 V measured at V₂ reveals the new voltage source, V₁, is 15 V. This example of using differential voltage measurements to find the value of the unknown voltage source is the same concept used to determine the unknown temperature of a heat source using differential temperature measurements.