COMSOL 6.4 - Active Magnetic Bearing

An active magnetic bearing is used to control vibrations in a rotor by using a feedback control mechanism. The rotor motion is monitored by a displacement sensor located very close to the location of the bearing. A schematic representation of the active magnetic bearing system is shown in Figure 3-12:

There is a wide variety of control mechanisms that can be used, but here we focus on a PID controller. The coil current for such a controller is given by

where us is the displacement measured from the sensor and Kp, Ki, and Kd are the proportional, integral, and derivative gain of the controller, respectively.

The air gap forces between the stator and rotor pole pair is given by

where B is the magnetic flux density, Aproj is the projected pole area, and μ0 is the permeability of free space. Assuming that the magnetic resistance of air is much larger than that of the poles, we can use Ampère’s law to write the magnetic flux density in the air gap as

where N is the number of turns per pole, h is the air gap, and I is the current of the coil. Using this, the air gap force can be written as

This shows that magnetic force in the bearing is proportional to the square of the current, which is undesirable for the control purpose. To overcome this difficulty, a pair of opposing electromagnets are used with an equal but opposite bias force. In this case the net force on the rotor will be

where I1 and h1 are the coil current and air gap in one electromagnetic, and I2 and h2 are the coil current and air gap in the opposing electromagnet. Now if I1 and I2 are composed of a steady bias current Ib and a control current Ic, we get

Then the air gap force can be written as

for h1 = h2 = h0. This expression is now linear with respect Ic. In general, the air gap on both sides will not be equal. As the rotor moves by the distance us, the gaps change so that h1 = h0 − us and h2 = h0 + us. As a result, the expression for the air gap force, in general, is

In the above expression, a different bias current is assumed in the two opposite electromagnets. The force constant in the expression is also generalized to Fc. In general, the air gap force is a nonlinear function of current and displacement, which is suitable for a time-dependent analysis. For frequency-domain and eigenfrequency analyses, a linearized expression for the air gap force is more suitable. If we assume that us<< h0 and Ic << Ib, then the above expression can be approximated as

where F0 is the static force produced due to the difference in bias current on positive and negative axes. It is given by

Thus, the difference in the bias current can be used to levitate the rotor against the static load such as the weight of the rotor. The bearing stiffness ki due to the current is given by

and the bearing stiffness ku due to displacement is given by

Note that the displacement stiffness is negative, thus the bearing is inherently unstable without a control current. For a stationary analysis, a PID controller reduces to a proportional controller. As a result, the control current is given as Ic = −Kpus. In this case, the expression for the air gap force reduces to

Thus, to make the bearing stable (that is, to have a positive effective stiffness), the proportional gain should satisfy the following condition

For a frequency-domain analysis, a control current can be expressed in terms of displacement and frequency as

Using this expression for the control current in the linearized expression for the air gap force and dropping the static force term, we get

The coefficient in front of us can be considered as the effective impedance of the electromagnetic bearing.