The electrical interconnect network is solved for voltage, but Ember™'s thermal network is solved for coefficients of the temperature formula *T*(*x*) on every element.

Closed-Form Temperature Trajectories Capture Key Thermal Behaviour

Ember™'s closed-form solution captures the complex shape of the temperature trajectory below with only 9 nodes! Temperatures calculated based on rms current density (the "*T∞* formula" values on the graph) cannot model real heat conduction in complex interconnect. Notice that simply varying the material properties (e.g. from Si02 dielectric to low-k SiLK™) can shift the location of the hottest interconnect segment.

The reference temperature *Tref* for the resistor is the weighted average of the temperatures of all the resistors to which it is capacitively coupled, plus **the local substrate temperature**, which forms the boundary condition. A self-consistent solution for a design with multiple, coupled networks is found by iteration between nets.

### Ember™ Transforms Electric Circuit Elements to Solve the Heat Equation

Electromigration analysis decomposes the interconnect into a network of π equivalent circuits representing (a) resistors and (b) vias. Ember™'s replaces electrical component values with transformed component values that express the closed-form solution to the heat equation.

### Closed-Form Solution to the Heat Equation

The time-independent heat equation on a resistor can be solved in closed form, under reasonable assumptions:

Ember™ uses circuit analysis and closed-form models to forecast interconnect temperature rises due to Joule self-heating.

### Electrical-Thermal Analogy

Ember™ exploits the well-known mathematical analogy between electrical circuit quantities and thermal quantities, re-using the electric circuit topology for thermal purposes.