http://people.csail.mit.edu/jaffer/Work/thermal
	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 f_c = 1 / t_c.

Ei       Ri,k      Ek
 o---+---/\/\---+---o
     | >-Ii,k-> |
     |          |
    _|_        _|_
Ci  ---        --- Ck
     |          |
    _|_        _|_
    ///        ///

  _
 / \_/ >--+------+
  fc      |   ___|___
       ___O___ |   |_
Ei      |   |  |     |    Ek
 o---+--+   +--+-||--+-+---o
     | >-Ii,k->  Ci,k  |
     |                 |
    _|_               _|_
Ci  ---               --- Ck
     |                 |
    _|_               _|_
    ///               ///

Simulating

This technique requires storage for (scalars) charge Q_i and voltage E_i for each (capacitor) node.

• Calculate each node's voltage from its charge and capacitance: E_i = Q_i / C_i
• For each pair of connected nodes indexed by i and k, where i < k:
  • calculate the transfer charge Q_t = (E_k - E_i) · C_i,k = (E_k - E_i) · t_c · G_i,k
  • add Q_t to Q_i (the charge accumulating in C_i)
  • subtract Q_t from Q_k (the charge accumulating in C_k)
• Repeat this procedure T · f_c times, where T is the desired length of the simulation in seconds.