Motor Equation:

The general motor torque equation can be expressed as:

In words, the torque of the motor equals the torque of the load plus the product of inertia and the rate of change of angular velocity with time. The units in this equation must of course be consistent. In the MKS system of measurement the torques are expressed in newton meters, the inertia in Kg-m2 /sec, and angular velocity in sec-1 (actually dimensionless-radians/sec).

Now of course beneath the seeming simplicity of this equation is the little detail that the motor and load torques are complex, and often very nonlinear. However, SPICE is good at arriving at numerical solutions to nonlinear differential equations. Thus the torques, and even the inertia can be nonlinear functions of time, w, the integral of w or angle, as well as other parameters.

However in the general case:

Given that all of these items might be functions of time, a circuit that will solve for w is shown in Figure 1 following:

Figure 1
Motor torque equation block circuit #1

In Figure 1 we see the simple realization of this function, together with test circuitry consisting of voltage sources v2, v3 and v4 together with resistors R2, R3 and R4. Device A1 is an x-spice integrator block. In this circuit, the torques are constants as is the inertia. The graph of the circuit is shown in the following Figure 2.

Figure 2
Motor torque equation block circuit test #1

This rather unimpressive output is just a ramp, as one would expect the integral of a constant to be.

The circuit can be more interesting with a variable loading. This is shown in the circuit of Figure 3.

Figure 3
Motor torque equation block circuit test #2

In this test circuit we have added a variable inertia, by means of nonlinear arbitrary source B2. B2 provides a load torque which is a constant plus a value proportional to the square of the angular velocity, such as might be provided by a fan type load. The output graph is shown in Figure 4.

Figure 4
Motor torque equation block circuit test #1

Here we see that the load which increases with velocity eventually causes a steady state velocity to be reached.

Now one could use many different loading conditions and inertial load variables, but the point here is to create a building block to be used in mechanical simulations of rotary devices, and motors in particular. Consequently, the circuit in Figure 5 following should be converted into a subcircuit and added to the library.

Figure 5
Motor Torque Equation Block

The circuit shows the model, together with a subcircuit part that represents this divide in a large block beneath it. A netlist for the circuit is:

A simple circuit has been created which can be used together with some other devices to help in the modeling of mechanical motor type problems to solve the motor equation.