**3.1. Numerical characteristics pulse-width modulation - control current**

Each pair of the bush electromagnets of the built journal bearing interacts with a digitally controlled power amplifier with a variable pulse width *PWM.* The control pulse-width modulation *W* is counted by the controller on the basis of measurements of the position of the journal with respect to the bush. The control current that supplies the windings of individual electromagnets is an explicit function of this modulation [4,9].

Power transistors and discharging diodes have been used to build the amplifiers (Figure 4). They are characterized by specified values of operating parameters. The properties of transistors and diodes, as well as the winding inductance and resistance of a given pair of poles, determine the parameters of the whole bearing actuator path, in which control current is generated.

The dynamics of changes of the current in windings is strictly dependent on the force generated by electromagnets in the gap between the journal and the bush and it affects directly the dynamics of the mass suspended in the bearing. The knowledge of actual characteristics of the actuator paths is an important factor in designing the structure and the algorithm of the controller, whose task is to ensure stable operation of the bearing.

According to these assumptions, a development of the program that allows for numerical simulation of the theoretical equations describing the phenomena occurring in the actuator electric circuit versus time (i.e., in subsequent periods of control pulses) for nominal parameters of the power amplifier elements, electromagnet windings, supply voltage and control frequency, was required.

**Figure 4.** Power amplifier circuit and an idea of current changes in the bearing

168 Performance Evaluation of Bearings

elaborated.

been applied.

is generated.




The model gives also a possibility of analysis of permissible levels of disturbances *FZ, Ru, No* that provide a proper margin of the bearing system stability for a given type of the

The numerical procedures representing actual characteristics of the actuator were developed and verified, and then applied in the simulation model of the bearing. This allows for modelling the magnetic bearing system quickly and accurately. The numerical simulation of the active magnetic bearings system by means of the Hewlett-Packard HPVEE software was

In the designed digital control system of a journal active magnetic bearing, the suitable software that allows for investigations of the *pulse-width modulation - control current* characteristics that determine the bearing system properties while its actuators are built, has

Each pair of the bush electromagnets of the built journal bearing interacts with a digitally controlled power amplifier with a variable pulse width *PWM.* The control pulse-width modulation *W* is counted by the controller on the basis of measurements of the position of the journal with respect to the bush. The control current that supplies the windings of

Power transistors and discharging diodes have been used to build the amplifiers (Figure 4). They are characterized by specified values of operating parameters. The properties of transistors and diodes, as well as the winding inductance and resistance of a given pair of poles, determine the parameters of the whole bearing actuator path, in which control current

The dynamics of changes of the current in windings is strictly dependent on the force generated by electromagnets in the gap between the journal and the bush and it affects directly the dynamics of the mass suspended in the bearing. The knowledge of actual characteristics of the actuator paths is an important factor in designing the structure and the

According to these assumptions, a development of the program that allows for numerical simulation of the theoretical equations describing the phenomena occurring in the actuator

algorithm of the controller, whose task is to ensure stable operation of the bearing.

**3.1. Numerical characteristics pulse-width modulation - control current** 

individual electromagnets is an explicit function of this modulation [4,9].

the magnetic bearing, initial position of the journal with respect to the bush;

operating point on the characteristics, feedback coefficient K*fb* 

controller. Thus, in a sense, it constitutes the synthesis of a robust controller.

system *R*, value of the bearing constant *K*.

An operation of the actuator was analyzed. Each cycle of the control pulse *PWM* of a given frequency *fPWM* and a pulse-width modulation *W* forces two modes of the amplifier operation, namely:


For a given pulse-width modulation, a value of the control current *Iav i* that generates the bearing magnetic force results from its averaged value in the *PWM* cycle and has been expressed by relation (1):

$$I\_{avi} = W\_i \frac{I\_{0i} + I\_{\text{max}i}}{2} + (1 - W\_i)X\_i \frac{I\_{\text{max}i} + I\_{0i+1}}{2} \tag{1}$$

*I0 i –* initial current for the *i – th* cycle of the amplifier charging.

*I0 i+1 –* initial current for the *i+1 – th* cycle of the amplifier charging.

The coefficient *Xi* occurring in formula (1) determines a ratio of the discharging time for a given cycle to the time during which the power amplifier electronic elements cause that the control current value diminishes to zero. In the cycles in which the current discharges to *Imin i = I 0 i+1* , the coefficient *Xi =1.*

Individual operation sequences of the digitally controlled pulse power amplifier and the windings powered by it have to be analyzed. Numerically simulated sequences of the ideal actuator operation enable one to generate model characteristics, which are the basis for evaluation of an influence of changes in values of individual parameters of the actual system on their shape.

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 developed.

Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 171

*50%*, the dynamics of changes in the control current

For a pulse-width modulation *W*

supply voltage *UZ* (Figure 5b).

bearing digital control system [11].

independent of the actuator properties for each axis.

investigation, the knowledge of its real model is needed.

between the rotor and the stator) for each control axis [5].

depends on resistance in the power amplifier circuit. The total value of resistance includes electromagnet winding resistance and dynamic resistance of the amplifier electronic elements in subsequent cycles of its pulsating operation, i.e. during charging and discharging (Figure 5a). An increase in the dynamics of the current changes in the bearing system under consideration, i.e. for given, actual parameters of the bush windings and the power amplifier, can be achieved through an increase in the value of the power amplifier

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

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

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

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

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

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

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

elements is added, will introduce the asymmetry of properties for the control axes.

In Figure 4 a and b exemplary *pulse-width modulation - control current* characteristics of the bearing actuator system calculated for selected values of its parameters are shown.

Nominal, experimentally identified values of the parameters of the actuators assumed in the calculations were as follows:


**Figure 5.** *Pulse-width modulation – control current* characteristics of the bearing actuator a. - as a function of the amplifier circuit resistance, b.- as a function of the power amplifier supply voltage

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 calculations.

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

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 actuator has to operate under pulse-width modulations *W50%* in order to ensure a suitable load capacity.

For a pulse-width modulation *W50%*, the dynamics of changes in the control current depends on resistance in the power amplifier circuit. The total value of resistance includes electromagnet winding resistance and dynamic resistance of the amplifier electronic elements in subsequent cycles of its pulsating operation, i.e. during charging and discharging (Figure 5a). An increase in the dynamics of the current changes in the bearing system under consideration, i.e. for given, actual parameters of the bush windings and the power amplifier, can be achieved through an increase in the value of the power amplifier supply voltage *UZ* (Figure 5b).
