Scheme for Thermal Simulation

For systems with a few time constants, spreadsheet programs can provide quick and easy simulation of heat and temperature over short durations. As systems grow more complex or if simulation of a typical meteorological year is required, spreadsheets become unwieldy and brittle.

Scheme for Linear System Analysis shows how to solve linear systems in the frequency domain; this page details how to solve them in the time domain.

Lumped element thermal simulations can be modeled as electrical networks composed of grounded capacitors (thermal mass) and resistors (thermal resistance). With all their poles lying in the left half-plane, such systems are unconditionally stable; and quantization errors are attenuated by the integrations associated with the capacitors.

Switched Capacitors

Each resistor can be modeled as a switched-capacitor operating at the sampling frequency fc = 1 / tc.

Resistor (in Circuit)Equivalent Switched-Capacitor
Ei       Ri,k      Ek
     | >-Ii,k-> |
     |          |
    _|_        _|_
Ci  ---        --- Ck
     |          |
    _|_        _|_
    ///        ///
 / \_/ >--+------+
  fc      |   ___|___
       ___O___ |   |_
Ei      |   |  |     |    Ek
 o---+--+   +--+-||--+-+---o
     | >-Ii,k->  Ci,k  |
     |                 |
    _|_               _|_
Ci  ---               --- Ck
     |                 |
    _|_               _|_
    ///               ///
Ii,k = (Ek - Ei) / Ri,k Ii,k = (Ek - Ei) · Ci,k · fc

Equating the Ii,k currents at sampling frequency fc gives:    Ci,k = 1
Ri,k · fc
= tc · Gi,k


This technique requires storage for (scalars) charge Qi and voltage Ei for each (capacitor) node.

Copyright 2003 Aubrey Jaffer

I am a guest and not a member of the MIT Computer Science and Artificial Intelligence Laboratory.  My actions and comments do not reflect in any way on MIT.
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