It’s time to dig a little deeper into the incandescent lamp behavior I introduced in Part 3 of this blog series. My goal is to select a set of equations that best describe the elements of the lamp’s behavior that I want to quantify during simulation. Recall my comment in Part 3 that a lamp has several characteristics worth analyzing including electrical properties, thermal properties, aging properties, and photometric properties. For this post I will focus on the lamp’s relationship between electrical and thermal power. If you have attended any of our VHDL-AMS Introduction workshops or training classes, you will no doubt recognize this discussion.

Electrically, an incandescent lamp behaves like a temperature dependent resistor. The filament’s resistance increases with temperature, which in turn affects the lamp’s electrical power characteristics. From a thermal perspective, as the filament heats up, it dissipates power through thermal conductance, thermal capacitance, and radiation. So the lamp consumes electrical power, and dissipate thermal power. Since the lamp must obey conservation of energy laws, we have:

Electrical power = Thermal power

This is a simple, high-level, mathematical relationship. Let’s add some detail by deriving equations for each power type. Deriving an equation for the electrical power starts with the standard, textbook definition:

Electrical power = Voltage x Current

But we need to account for the change in filament resistance. We can do so by considering Ohm’s Law:

Voltage = Current x Resistance

We know that filament resistance varies with temperature. For our discussion, let’s assume resistance is related to temperature as follows:

Filament resistance = Rcold x (1.0 + Alpha x (Tfil – Tcold))

Where Rcold is the filament’s cold resistance, Tcold is the filament’s cold temperature, Tfil is the filament’s heated temperature which increases over time as the lamp approaches full brightness, and Alpha is the filament’s resistive temperature coefficient. Given this equation for the filament resistance, Ohm’s Law for the lamp becomes:

Voltage = Current x Filament resistance

Now let’s discuss the thermal power which is represented as a heat flow in the lamp. As I mentioned above, we will assume there are three elements to the lamp’s thermal power: thermal conductance, thermal capacitance, and radiation. The total thermal power, or total heat flow, is the sum of heat flows from each element. Let’s consider these heat flows in order. Heat flow due to thermal conductance can be characterized as:

Hflow(conductive) = (Tfil – Tamb) / Rth

Where Tfil is the filament’s temperature, Tamb is the ambient temperature, and Rth is the filament’s thermal resistance. This equation tells us that, for a given thermal resistant, conductive heat flow increases with higher filament temperatures. Heat flow due to thermal capacitance can be characterized as:

Hflow(capacitive) = Cth x d(Tfil) / dt

Where Cth is the thermal capacitance and Tfil is the filament’s temperature. Using this relationship, we set capacitive heat flow to increase proportionally with the filament temperature’s rate of change. Finally, radiated heat flow can be characterized as:

Hflow(radiated) = Ke x (Tfil**4 – Tamb**4)

Where Ke is the radiated energy coefficient, Tfil is the filament’s temperature, and Tamb is the ambient temperature. Recalling that thermal power is simply the sum of all heat flows in the lamp, and using these three heat flow definitions, the lamp’s total heat flow is:

Hflow(total) = Hflow(conductive) + Hflow(capacitive) + Hflow(radiated)

Now that we have derived equations representing the incandescent lamp’s electrical and thermal power relationships, we can move on to the third step in our modeling process (outlined in Analog Modeling – Part 2), which is:

- Implement the equations and relationships using syntactically correct statements based on your modeling language of choice

Which I will do using my modeling language of choice, VHDL-AMS, in my next blog post: Analog Modeling – Part 5.

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