The inverter block takes as input the reference current IREF provided by the control system, the phase currents IA, IBand IC provided in feedback from the motor block and the electrical position of the motor θe.
The latter is calculated by means of an appropriate function by a special block prior to the inverter representing the resolver, i.e. an inductive displacement transducer which makes it possible to detect the change in magnetic induction flux as a function of the position of the solenoid(s) in the circuit. The resolver makes it possible to transform the mechanical angle output from the motor-transmission dynamics θm into an electrical angle θe.
In fact, the electric motor refers to an abstract reference system in which the segments depicted are representative of specific electrical quantities. The motor is an electromagnetic rotating machine, whereby a sinusoidal voltage is produced. The instantaneous value of this voltage depends on the physical position assumed by the conductors during rotation within the magnetic field and, therefore, on the number of pole pairs present in the machine itself.
In our case, a relationship has been found between the angle of the physical model of the electric motor and the angle between the vectors representing the electrical quantities produced by the machine, described by the relationship 2.6
θe = 2π
Np∗ θm
2π − f loor
NP
θm 2π
(2.6)
where the function f loor(x) rounds each element of x to the nearest integer less than or equal to that element (in our case θm) [22].
The same function is described in Matlab using the block resolver shown in the figure 2.6.
Figure 2.6: Resolver block
Once the electrical angle has been defined, the modules constituting the inverter model are now described.
It consists of three main parts:
– valuation of the three currents IREF,A, IREF,B and IREF,C in the stator phases from the reference current IREF;
– hysteresis PWM modulation of the currents;
– modelling of the H-bridge for static power conversion.
In the first block, Park and Clarke’s anti-transformations are implemented to change the current from rotoric reference to phase current.
The input current IREF can be defined as the real part of a complex variable. The real part coincides with the component of the direct current id, i.e., concordant with the magnetic flux, responsible for the generation of the magnetic field; on the other hand, the imaginary component, set equal to 0, defines the quadrature current iq, i.e., perpendicular to the magnetic field and responsible for the generation of the torque.
Starting from these two components, the Park antitransformate allows to shift the view of the currents from the rotor with velocity ˙θm to the stator, passing from a rotating biaxial system to a static biaxial system, characterized by the currents iα and iβ, synchronous with the stator [23]. The Park antitransformate is defined by the relation 2.7.
iα iβ
=cos θm − sin θm sin θm cos θn
id iq
(2.7)
Clarke’s anti-transform, on the other hand, makes it possible to pass from a stationary two-phase system for the stator to a generic three-phase system, where the triad of instantaneous phase values is determined [24]. Clarke’s anti-transform is defined by the relation 2.8.
IREF,A IREF,B IREF,C
=
1 0
−12 q
3 2
−12 −q
3 2
iα iβ
(2.8)
The Simulink representation of the two anti-transformations are shown in the figures 2.7 and 2.8.
Figure 2.7: Park antitransformate block
Figure 2.8: Clarke antitransformate block
Then the current values evaluated in the electromagnetic model of the motor IA, IB and IC are subtracted from the current values IREF,A, IREF,B and IREF,C in order to obtain a current error Ierr,A, Ierr,B and Ierr,C for the three phases, according to the generic relationship 2.9.
Ierr,i= IREF,i− Ii (2.9)
These three values will be given as input to the hysteresis PWM (Pulse Width Modulation) block.
In this module the current of each phase is evaluated and modulated by means of pulse control defined by multiple switching, in order to obtain continuous regulation of the motor speed when reversing the direction of rotation in the inverter. The commutation takes place around the input signal of each phase within upper and lower limits that define the hysteresis band (in our case 0.5 A). If the input current tends to be higher than the upper limit, switching towards the lower limit is activated, vice versa if the current tends to exceed the bottom limit of the hysteresis band. Due to the dependence of the switching on the value of the input current error and the width of the hysteresis band, it follows that the switching frequency is not constant, but varies according to the shape of the current. This type of commutation is electronic and eliminates the creeping contacts typical of brush motors. In fact, as we shall see in the paragraph 3.2.3, the motor taken into consideration is of the brushless type.
The hysteresis PWM block is shown in the figure 2.9.
Figure 2.9: Hysteresis PWM block
The outputs provided by this block are given as input to the static power converter consisting of an three-phase H-bridge.
In reality, this consists of six transistors supported by six protective diodes.
The transistors work two by two, allowing to vary as needed the voltages of the three phases driving the motor.
Diodes, as mentioned, have a protective function. In fact, when the transistor is switched off, then the circuit is opened, but the current does not cancel instantaneously because energy is stored in the ferromagnetic material of the rotor. The discontinuity formed by the open circuit, however, induces the formation of an electric arc near the junction, where the electric field has a lower intensity, which causes a strong thermal dispersion. As a result, the transistor gets very hot and can burn out, damaging the entire system. Through a protective diode, the potential in the junction increases, so that current circulates in the internal circuit which, due to the Joule effect, will lead to a gradual cancellation of the current.
In Simulink the three-phase H-bridge is modelled using the Universal Bridge block from the Simscape library. This is connected to a DC voltage source which corresponds to the 48 V supply. Each motor phase is connected to two MOSFET-type transistors, one of which is electrical grounded. The mechanism for the activation and deactivation of the transistors is managed by Boolean logic which associates each PWM value with its corresponding Boolean one and its negation. These values represent whether or not current is flowing in a particular phase. This excludes the possibility of short circuits inside the H-bridge [25].
The bridge generates three output voltages which supply the motor in the RL Model block.
The simulink block of the three-phase bridge is shown in the figure 2.10.
Figure 2.10: Three-phase H-bridge block