This page will feature a wire simulation model useful in LTSpice and an array of other analog simulation tools, such as the Synopsys Saber and Ansys Simplorer tools. This page will be updated as new refinements are added.
We commonly think of a wire as an afterthought in electrical circuit design. It is easy to forget that an electrical wire has parasitic attributes that may adversely affect the operation of a circuit it is servicing.
A partial enumeration of parasitic components is listed here. An image representing a schematic view of a computer simulation model for an electrical wire is shown below.
Wire Type Selected for this Example
For this example we have selected the Coroplast wire type FHLR2GCB2G. This is a shielded high voltage wire metric sized at 10mm2. For this example, we are ignoring the shield and treating it as transparent to the operation of the model. This is a valid consideration since the thermal conductivity of the shield is more than an order of magnitude greater than that of the insulation. It would, however, be correct to subtract the cross-sectional area of the shield from the cross-sectional area of the insulation. But even that falls to the wayside since the thermal characterization of the model is taken directly from the manufacturer’s data sheet.
Description of R1
Of primary interest in the simulation model is the resistance (R1) the model offers to the flow of electrical energy. R is the resistance of R1 in the schematic. Here, 1.82mOhms (0.00182 Ohms) defines the resistance per meter of copper wire having a cross-sectional area of 10mm2. The 10mm2 wire size is metric. Its equivalent in the English system is #12 AWG size wire. The variable “tref” of the equation is the temperature at which the resistance per meter was given in the manufacturer specifications, 21oC. The Tj parameter is the instantaneous temperature of the model within a transient analysis. 3.92mOhms per oC is copper’s coefficient of thermal resistance. For every degree in Celsius temperature increase, the 1.82mOhms coefficient is scaled by that amount.
Description of B2
When electrical energy passes through R1, power is developed, which is spent as thermal energy. The B2 instance captures that energy as the product of R1‘s current and voltage drop.
LTSpice cannot distinguish between thermal and electrical energy. The reader is asked to understand that an analogy exists between the thermal and electrical domains. All domains in the physical world have two coordinated variables–across and through. The across-variable analogy of temperature to electrical is equating potential to temperature. Volts are analogous to degrees Celsius. Likewise, equating the electrical through-variable we have current (Amperes) analogous to Q, or heat transfer.
Consequently, B2 develops an electrical current fed to a reactive network, R2 and C1. While LTSpice will report Volts and Amperes, these should be interpreted as thermal energy with units of temperature. B2 develops a current equal to the power dissipation resulting from R1. This current of B2 develops a temperature potential across R2 referred above ground by the ambient temperature, V1. It is important to distinguish at this point that the ambient temperature is referred to the system ground rather than the input minus pin (inm). The system variable “temp” is system maintained and will always represent the ambient temperature which by default is 27oC.
The self-heating of the Tj net is fed back to the electrical R1 resistor as a voltage, V(Tj). Recall that while LTSpice is reporting this temperature as a voltage, the analogy is perfectly correct. We have to adjust our vision to see it.
Solving for the Thermal Resistance, R2
The wire conducts current through its parasitic resistance. In an ideal world, we would prefer that there would be no resistance but we must accept some. How much parasitic resistance depends on the size of the wire (its cross-sectional area) and its composition (i.e. copper, aluminum, silver, etc.). This means that current is passing through a resistance which defines power or Wattage. If we pass too much current through the wire, there will be too much power and therefore too much heat and the insulation will melt, or worse.
Our simulation model, therefore, must account for these limitations of a wire.
From a high-level perspective, consider that the copper wire core is spending energy in a radial fashion. We are not talking about the wire end-to-end but only its cross-sectional area as if it were merely a circle with no length. Also, we must consider the insulation. Think of the insulation as a pipe with the copper conductor filling its inside. This means that there are two edges for the thermal energy to pass through.
Power will radiate from the center to the first edge, which is the interface between the conductor and insulation. So, the power is developed in the conductor resulting in heat. That heat is passed through the interface to the insulation. It then travels through the insulation, where it meets its second interface–the ambient.
This section will be expanded to say more at a later date.
Solving for the Thermal Capacitance, C1
In the thermal domain there are analogies to the electrical domain. We have already looked at a thermal resistance. There is also a thermal capacitance which defines how long it takes a wire to heat to a steady-state temperature. There are no special rules to distinguish a thermal from an electrical capacitance other than units. For example, in the thermal domain, temperature is analogous to Voltage.
As a matter of interest, there is no analogy in the thermal domain for a thermal inductance.
Text to be added describing theory and empirical method.
Improvements We Would Like to See
The model performs quite well in duplicating the thermal performance published by the manufacturer. However, it has a slight deviation in both DC and transient effects owing to the dynamic nature of the physical attributes of both the insulation and copper’s heat capacity and thermal conductivity. The model assumes static values for these, but in nature, they are dynamic. Having the model accommodate these dynamic attributes dependent on temperature would considerably refine the model’s fidelity to the manufacturer’s published specifications.
Please stay tuned for refinements to this web page.