Category: Electrical Engineering

Differential equations are the heart of a vast array of engineering theory, and as such, one can create models of entirely different branches of physics by using circuit theory.

The usefulness of this concept here is that electronics engineers can gain insight into mechanical systems by using their circuit knowledge. They can also use standard electronics tools like SPICE to incorporate mechanical outputs, such as motors or temperature rise.
In the recent past, this was useful for designing analog computers to solve mechanical problems.

Electrical Fundamentals

The fundamental electrical components are resistors, capacitors and inductors.
Their voltage-current relationships are:
$$v(t) = i(t)R$$
$$i(t) = C \frac{dv}{dt}$$
$$v(t) = L \frac{di}{dt}$$

Mechanical Dynamics Fundamentals

The key equations of motion are Newton’s laws, the main one being:
$$F(t) = ma(t) = m\frac{dv}{dt}$$
In a rotational system, we have:
$$M(t) = I\frac{dw}{dt}$$

Thermal Fundamentals

The steady state thermal equation is:
$$T = P\theta + T_{env}$$

Dynamically there is Newton’s law of cooling:
$$\frac{dT}{dT} = \frac{T_{env} – T}{t_0}$$

Steady State Circuit

The steady state temperature rise equation is identical to the solution of a DC resistor circuit.

The equivalents are:
Voltage (V) = temperature (°C)
Current (A) = power (W)
Resistance ($$\Omega$$) = thermal resistance (°C/W)

The solution is then very intuitive:
Temperature rise is caused by heat flow through a non-zero thermal resistance, exactly how voltage difference is caused by current flow through non-zero electrical resistance.

The power is a current source, the resistance a resistor, and the ambient temperature is the reference voltage. The resulting voltage at the current source is the new temperature of the device.

Typically a component datasheet will provide a $$\theta_{j-a}$$ which is the thermal resistance from thr component junction (j) to the ambient (a) environment. This can easily be added to a SPICE simulation.

Dynamic Circuit

Having established thr equivalence above, we can choose an electrical element to represent the thermal time constant.

Since temperature is similar to voltage and the thermal time constant slows it down, a capacitor is the appropriate electrical element (inductors slow down current).

A capacitor added in parallel to the current source results in an equation identical to Newton’s law of cooling:
$$i_c(t) = C \frac{dv}{dt}$$
$$i_c(t)R = RC \frac{dv}{dt}$$
$$v(t) = RC \frac{dv}{dt}$$
$$\frac{dv}{dt} = \frac{v_c(t) – v_{ref}}{RC}$$
$$\frac{dT}{dt} = \frac{T(t) – T_{env}}{t_0}$$
Therefore, the thermal time constant is equivalent to RC, which is the familiar electrical time constant.