Analysis of a PDE Thermal Element Model for Electrothermal Circuit Simulation

Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 14)


In this work we address the well-posedness of the steady-state and transient problems stemming from the coupling of a network of lumped electric elements and a PDE model of heat diffusion in the chip substrate. In particular we consider the thermal element model presented in [1] and we prove that it can be controlled by any combination of voltage sources (imposing the average current in a region of the chip) and current sources (imposing the Joule power per unit area produced in a region) connected to its temperature nodes. This result justifies the implementation of the element as a linear n-port conductance as carried out in [2].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Culpo, M., de Falco, C.: A pde thermal model for chip-level simulation including substrate heating effects. Preprint MS-08-10, School of Mathematical Sciences (2008)Google Scholar
  2. 2.
    Culpo, M., de Falco, C., Denk, G., Voigtmann, S.: Automatic thermal network extraction and multiscale electro-thermal simulation. In: Proceedings of the SCEE 2008 Conference (Submitted) (2008)Google Scholar
  3. 3.
    roadmap commitee, I.: International tecnology roadmap for semiconductors 2007. Tech. rep., ITRS (2007)Google Scholar
  4. 4.
    Culpo, M., de Falco, C.: Dynamical iteration schemes for coupled simulation. In: Proceedings of the GAMM2008 meeting (Submitted) (2008)Google Scholar
  5. 5.
    Grasser, T., Selberherr, S.: Fully coupled electrothermal mixed-mode device simulation of sige hbt circuits. Electron Devices, IEEE Transactions on 48(7), 1421–1427 (Jul 2001). DOI 10.1109/16.930661CrossRefGoogle Scholar
  6. 6.
    Igic, P., Mawby, P., Towers, M., Batcup, S.: Dynamic electro-thermal physically based compact models of the power devices for device and circuit simulations. Semiconductor Thermal Measurement and Management, 2001. Seventeenth Annual IEEE Symposium pp. 35–42 (2001). DOI 10.1109/STHERM.2001.915142Google Scholar
  7. 7.
    Bartel, A.: Partial Differential-Algebraic Models in Chip Design Thermal and. Semiconductor Problems. VDI-Verlag (2004)Google Scholar
  8. 8.
    Quarteroni, A., Valli, A.: Numerical approximation of Partial Differential Equations. Computational Mathematics. Springer (1997)Google Scholar
  9. 9.
    Evans, L.: Partial Differential Equations. American Mathematichal Society (1998)Google Scholar
  10. 10.
    Deheuvels, R.: Formes quadratiques et groupes classiques. Presses Universitaires de France, Paris (1981)zbMATHGoogle Scholar
  11. 11.
    Silvester, J.: Determinants of block matrices. The Mathematical Gazette 84(501), 460–467 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Università della CalabriaArcavacata di RendeItaly
  2. 2.Bergische Universität WuppertalWuppertalGermany
  3. 3.Dublin City UniversityDublin 9Ireland

Personalised recommendations