**3.2. Experimental characteristics pulse-width modulation - control current**

170 Performance Evaluation of Bearings

calculations were as follows:

 inductance of the electromagnet winding *L = 80mH,*  resistance of the power amplifier circuit *R = 3.1*

supply voltage of the power amplifier *Uz = 80V*,

0 10 20 30 40 50 60

frequency *fPWM=1667Hz TPWM= 600*

*UZ = 80 V, L = 80 mH*

developed.

voltage

0

1

2

3

4

*I [A]*

calculations.

load capacity.

Numerical calculation procedures that allow for the generation of instantaneous and mean time histories of control currents in the assumed *PWM* cycles, for the following actuator parameters: amplifier supply voltage *Uz*, inductance of the bearing bush winding *L*, resistance of the power amplifier circuit *R*, frequency of control pulses *fPWM* as a function of values of the pulse-width modulation coefficient *W* of the control pulse, have been

In Figure 4 a and b exemplary *pulse-width modulation - control current* characteristics of the

Nominal, experimentally identified values of the parameters of the actuators assumed in the

*.*

 *I [A]*

 *L = 80 mH f PWM = 1667Hz*

bearing actuator system calculated for selected values of its parameters are shown.

*s.*

 *a. b.* 

 *R = 2.0* 

 *R = 3.1* 

 *R = 4.0* 

**Figure 5.** *Pulse-width modulation – control current* characteristics of the bearing actuator

of the designed bearing system actuator. It is of great practical importance.

actuator has to operate under pulse-width modulations *W*

a. - as a function of the amplifier circuit resistance, b.- as a function of the power amplifier supply

*W [%]*

The characteristics were calculated for experimentally identified parameters of the actuator systems in order to enable their verification for a real object at each stage of the

0

1

2

3

4

To show a tendency of changes, the characteristics generated for bigger and smaller parameters than the nominal ones have been presented in the figures as well. This allows one to forecast and evaluate a possibility of introducing changes in the values of parameters

From the characteristics shown in Figure 5a and b, it follows that for the structure of the power amplifier under investigation and for the real parameters of bush windings, the

*50%* in order to ensure a suitable

0 10 20 30 40 50 60

*W [%]*

 *Uz = 30 V*

 *Uz = 80 V*

*Uz=200 V*

In order to identify the properties of each actuator path, suitable algorithms, software for a microprocessor of the measurement-control card and a host computer of the bearing system have had to be prepared. Independent auxiliary procedures that allow for experimental testing the actuator paths of the bearing have been developed. These procedures form a program for operation of the measurement-control card, which is the main element of the bearing digital control system [11].

An active magnetic bearing system comprises actuators, which are realised in the form of windings of individual pairs of bush electromagnets that interact with digitally controlled power amplifiers. A scatter of real values of electric parameters of the winding for individual pairs of electromagnets, to which a technological scatter of the amplifier structure elements is added, will introduce the asymmetry of properties for the control axes.

From the point of view of simplification of the structure designing and the controller parameter tuning in order to obtain the system stable operation, individual bearing actuators should be characterised by the symmetry of properties. A fulfilment of this condition will allow for making the mechanical characteristics of the bearing system independent of the actuator properties for each axis.

Control systems of the bearing response are a very important part of a machine with active magnetic bearings because they decide about the dynamic characteristics of the rotor. In order to design and build a control system fulfilling the requirements of the object under investigation, the knowledge of its real model is needed.

An active journal magnetic bearing with a digital control system has been built according to the presented idea. A new concept of control systems has been elaborated. It allows one to program the control of the bearing actuator characteristics. To achieve this goal, a programming procedure for the control processor of the bearing has been developed. The programmed testing procedure of an actuator allows one to reach experimental characteristics for its various parameters (frequency and width *PWM*, power supply, air gap between the rotor and the stator) for each control axis [5].

The experimentally determined *pulse-width modulation W - control current I* characteristics show differences in the bearing properties for two control axes *x* and *y* (Figure 6a). As a result, they render the asymmetry of mechanical properties of the system for these axes when the feedback loop is closed and the bearing control system is turned on.

Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 173

From the *pulse-width modulation - control current* characteristics that were experimentally determined (Figure 6a), it follows that the dynamics of the current changes increases already at the pulse-width modulations W50% in an actual actuator. These phenomena are connected with the structure of control systems, power amplifier transistor gates and they need a necessary correction. The theoretical characteristics are used to evaluate a scatter of properties of real actuator paths of the bearing. This evaluation allows one to introduce necessary programmed corrections of characteristics of real actuator systems to achieve the convergence between the characteristics determined experimentally and those calculated

This correction results in obtaining the symmetry of properties of individual actuators of the designed bearing which interact with the bush electromagnet windings *EM1, EM2, EM3,* 

After this correction, *pulse-width modulation - control current* characteristics of actuator paths of the bearing have been determined experimentally. The results of experimental investigations and numerical simulations have exhibited a good convergence. Figure 7 shows these characteristics for the bearing actuator path interacting with the axis *x - EM3* for

The proposed way of the programmed correction of the characteristics has been verified

In the analysis of model investigation results, the same quantities have been chosen as those recorded in a real magnetic bearing system, namely: pulse-width modulation, displacement, static equilibrium position, journal trajectory, phase portrait, Bode's plot, etc. It has enabled their direct verification with the experiment. The verification has been carried out on the test

different values of the power amplifier supply voltage.

**3.3. Numerical calculations and verifications** 

**Figure 8.** Structure of the built radial bearing

experimentally and carried out for a real bearing structure [4,5,6].

stand with the built radial bearing, whose part is presented in Figure 8.

numerically [6].

*EM4* (Figure 6.b).

**Figure 6.** a. - Experimental *pulse-width modulation W - control current I* characteristics of the bearing actuator systems before programmed correction, b. - Theoretical and experimental characteristics after programmed correction.

For the given parameters of a real bearing system, there is always a scattering of properties of the actuating system for each pair of electromagnets. The developed procedure enables one to tune these characteristics in order to obtain the symmetry of operation of actuating systems for each pair of electromagnets and for each bearing control axis *y* and *x*.

An alternative to compensation for the asymmetry of bearing actuator properties by designing a proper structure and algorithm of the controller is to develop a method of its programmed correction. On the assumption that the source of the identified asymmetry lies in a scatter of actuator parameters, a correction method that leads to minimisation of the effect of the scatter of their characteristics for each bearing control axis has been proposed.

**Figure 7.** Theoretical (solid line) and experimental (dots) *pulse-width modulation - control current* characteristics of the bearing actuator systems after the programmed correction for various values of the power amplifier supply voltage

From the *pulse-width modulation - control current* characteristics that were experimentally determined (Figure 6a), it follows that the dynamics of the current changes increases already at the pulse-width modulations W50% in an actual actuator. These phenomena are connected with the structure of control systems, power amplifier transistor gates and they need a necessary correction. The theoretical characteristics are used to evaluate a scatter of properties of real actuator paths of the bearing. This evaluation allows one to introduce necessary programmed corrections of characteristics of real actuator systems to achieve the convergence between the characteristics determined experimentally and those calculated numerically [6].

This correction results in obtaining the symmetry of properties of individual actuators of the designed bearing which interact with the bush electromagnet windings *EM1, EM2, EM3, EM4* (Figure 6.b).

After this correction, *pulse-width modulation - control current* characteristics of actuator paths of the bearing have been determined experimentally. The results of experimental investigations and numerical simulations have exhibited a good convergence. Figure 7 shows these characteristics for the bearing actuator path interacting with the axis *x - EM3* for different values of the power amplifier supply voltage.

The proposed way of the programmed correction of the characteristics has been verified experimentally and carried out for a real bearing structure [4,5,6].
