In this page, we will deal with some examples that are made up of two different domain, electrical domain and mechnical domain. Each of the domain is described by different governing equation. As a result, you will have a system equation that are made up of multiple equations, each of which came from different governing equation.

The two domains do not run independently. A coupling law ties them together, and in a motor that law has two directions. The current in the coil produces a torque on the shaft, and the rotating shaft produces a voltage back into the circuit. So the electrical equation contains a mechanical variable, the mechanical equation contains an electrical variable, and the two must be solved together.

• Example : DC Motor

Example : DC Motor

Let's look into one of the simplest electro mechanical system in which a the electrical input (Voltage) is converted to mechnical output (Torque). The drawing below shows the two parts of the model. The coil is a resistor R in series with an inductor L, and the rotating shaft carries an inertial load J with a viscous damping coefficient k_d.

The governing equation for the mechanical part for this system can be described as shown below, based on Newton's second law.

From this governing equation, we get a differential equation as shown below. (Some of you may not be familiar with the parameter J. This parameter is called 'Moment Of Inertia'. It is a kind of inertia of an object in rotational movement. The property is very similar to Mass (m) in linear motion. For more formal explanation of this parameter, refer to Moment Of Inertia page)

Now let's think of the governing equation about the electrical part. It can be described as follows based on Kirchhoff's law.

From the governing equation shown above and with a little bit of rearrangement, we get the equation as shown below.

By combining the two equations based on two different governing equation, we can get a system equation as shown below.

Notice the symbol for the torque constant. The mechanical diagram writes the motor torque as k_ti, while the equations write k_ii. Both names mean the same constant, the torque produced per ampere. The back-emf constant k_e is the voltage produced per unit of angular velocity. In SI units, with N m/A for k_t and V s/rad for k_e, the two constants have the same value. The reason is the energy balance. The electrical power k_eωi taken by the back-emf equals the mechanical power k_tiω delivered to the shaft.

Let's write the system in a form a numerical solver can use. Take the shaft angle θ, the angular velocity ω = dθ/dt and the current i as the three state variables. Then dθ/dt = ω, dω/dt = (k_ti - k_dω)/J and di/dt = (V - Ri - k_eω)/L. The model is third order, and it needs the initial values θ(0), ω(0) and i(0).

For a constant input voltage V, the motor settles at a constant speed. In the steady state both derivatives vanish, so k_ti = k_dω and Ri + k_eω = V. Solving the two gives ω_ss = k_tV / (Rk_d + k_tk_e) and i_ss = k_dV / (Rk_d + k_tk_e). With the Laplace transform, the transfer function from voltage to speed is Ω(s)/V(s) = k_t / ((Js + k_d)(Ls + R) + k_tk_e).

• Two governing laws, one system : Newton's second law for rotation describes the shaft, and Kirchhoff's voltage law describes the coil.
• The coupling runs in both directions : The current creates the torque k_ti, and the speed creates the back-emf k_edθ/dt that opposes the supply voltage.
• Back-emf limits the speed : As the motor speeds up, the back-emf grows and the current falls, so the speed settles at ω_ss instead of growing without limit.
• The speed follows a second-order model : Viewed from V to ω, the transfer function has a second-order denominator. Its two poles come from the coil and the load, mixed by the coupling term k_tk_e. The shaft angle adds one more integration.