text: the main symbols which will be used in the detailed disclosure are listed below . v r [ v ] module of maximum phase voltage that the inverter can apply to the stator ; m se [ h ] maximum value of mutual induction between stator winding and rotor winding ; θ [ rad ] electrical position of the reference system dq that is synchronous with the rotor ; i a , i b , i c [ a ] currents of the stator phases ; i d , i q [ a ] components of the stator current in the rotating two - phase reference system dq ; i d s , i q s [ a ] components of the stator current in the fixed two - phase reference system dq s ; i dref , i qref [ a ] references of the components of the stator current in the reference system dq ; φ d , φ q [ wb / m 2 ] components of the gap flux in the reference system dq ; l d , l q [ h ] “ equivalent inductance ” parameters in the reference system dq ; v d s , v q s [ v ] components of the reference stator voltage in the fixed reference system dq s . a first part of the disclosure is dedicated to the theoretical analysis of a linear wound rotor asynchronous motor . in this part of the description the analysis of a linear wr - sm is carried out using the synchronous rotating d - q reference system . considering the d axis aligned with the rotor pole , and ignoring the stator resistance , the stator voltage equations of the wr - sm machine are { v d = ⅆ φ d ⅆ t - ωφ q v q = ⅆ φ q ⅆ t + ωφ d ( 1 ) φ d = l d i d + m se i e ( 2 ) φq = l q i q ( 3 ) t = 3 / 2 p [ m se i e i q +( l d − l q ) i q i d ] ( 4 ) in which the first term represents the torque due to the excitation flux , whereas the second term is the torque contribution due to the motor anisotropy . as mentioned above , the study of the machine is developed on the i d - i q plane . on this plane the limits set by the current supply and the voltage supply can be represented as follows . the limit on the capacity to provide current from an inverter associated with the motor is represented by where i r is the module of the rated stator phase current supplied by the inverter . the limit set by the maximum available voltage can be represented by i q 2 + ( l d l q ) 2 ⁢ ( i d + m se ⁢ i e l d ) 2 = ( v r ω ⁢ ⁢ l q ) 2 ( 6 ) where v r is the magnitude of the maximum voltage that the inverter can apply to the motor . the current limit ( 5 ) defines a circle centred on the i d - i q plane origin . the voltage limit ( 6 ) defines an ellipse , centred on the following coordinates i d ⁢ ⁢ 0 = - m se ⁢ i e l d ; i q ⁢ ⁢ 0 = 0 ( 7 ) with semi - axes in the i d and i q directions equal to , respectively , as in this machine l d & gt ; l q , the ellipse is oriented as shown in fig2 . it can be noted in ( 7 ) that the position of the ellipse centre depends on the excitation current . the centre moves in the direction of the negative i d or towards the origin as the excitation current is increased or decreased , respectively . for a given value of the excitation current , by increasing the rotating speed the voltage ellipse becomes smaller and smaller until it collapses in its origin . on the i d - i q plane , the torque expression ( 4 ) is represented by a hyperbola , having the following asymptotes as l d & gt ; l q , the vertical asymptote is located on the semi - plane i d & lt ; 0 , and its position depends on the excitation current . a given torque value can be obtained for any combination of stator currents ( i d , i q ) that lies inside the limits defined by the current circle and the voltage ellipse . the position of the centre of the voltage ellipse defines the maximum rotating speed . if the centre lies inside the current circle , the rotating speed is theoretically unlimited ( ω max =∞). if the centre is positioned outside the current circle , the maximum theoretical speed corresponds to the intersection of the current circle with the smaller voltage ellipse and is given by ω max = v r m se ⁢ i e - l d ⁢ i r ( 10 ) the control of the excitation current makes it possible to move the centre of the ellipses and the torque asymptote in order to obtain the best performance of the machine over a wide speed range . best performance of the machines can be defined in several ways , such as : a control technique is illustrated below that gives maximum torque per ampere of the stator current for any rotation speed . at low speeds , maximum torque per ampere is always obtained with an excitation current equal to its maximum value ( i e = i em ). fig3 a , 3 b and 3 c show torque hyperbolas obtained for different values of the excitation current . the higher the excitation current , the higher the torque that can be generated with the same stator current . in these terms , fig3 c represents the best operating conditions for the machine at low speed . at high speed , the operation of the machine at maximum torque per ampere can be obtained by regulating the excitation current so as to achieve unity power factor operating conditions . the combination of i d , i q and i e that generates the highest torque for any given speed value can be found by using a standardised p . u . representation of the machine . the current components in the p . u . representation are ĩ d = i d / i r and ĩ q = i q / i r . below , the analysis will be carried out assuming the magnitude of the stator current to be equal to the rated current of the inverter [ 4 ]. consequently , the p . u . current components become where θ is the phase angle of the vector representing the stator current . a = m se ⁢ i e l d ⁢ i r , ĩ d coordinate of the centre of the ellipse ( 14 ) with a = 1 the centre of the ellipses corresponds to the intersection of the circle representing the current limit with the axis ĩ d , whereas with a & lt ; 1 and a & gt ; 1 the centre of the ellipses is respectively inside and outside the circle representing the current limit circle . t 0 = 3 / 2 pl d i r 2 ( 18 ) t = a ⁢ ⁢ sin ⁢ ⁢ ϑ + ( 1 - 1 r ) ⁢ sin ⁢ ⁢ ϑ ⁢ ⁢ cos ⁢ ⁢ ϑ , ( 19 ) by introducing ( 11 )-( 14 ) and ( 19 ) in ( 6 ), the equation of the ellipses representing the voltage limit in the p . u . representation becomes below , three different operation regions of the wound rotor synchronous motor are examined : a low - speed operation region , a high - speed operation region and a transition - speed region interposed between the low - speed operation region and the high - speed operation region . for the maximum value of the excitation current ( i e = i em ), the ellipses representing the voltage limit are centred on ĩ d =− a m , where with this excitation current , the current vector angle θ b that gives the maximum torque is obtained by setting : cos ⁢ ⁢ ϑ b = - a + a 2 + r ′ 2 4 ⁢ r ′ , ( 23 ) solving ( 23 ) for θ b and substituting in ( 19 ) gives the maximum torque t b . b b =√{ square root over (( 1 − cos 2 θ b )+ r 2 ( cos θ b + a ) 2 )}{ square root over (( 1 − cos 2 θ b )+ r 2 ( cos θ b + a ) 2 )}. ( 25 ) equating ( 25 ) to ( 21 ) gives the highest speed ω b at which maximum torque t b can be obtained . point b , shown in the p . u . graph of fig5 , represents the base operating point of the machine . when angular speed exceeds ω b no further maximum torque t b can be obtained because of the limit set by the available voltage . however , for any angular speed ω & gt ; ω b it is possible to find a combination of armature current components and excitation current , defined by suitable θ and a values , which enables the maximum torque to be generated . it has been ascertained that , taking into account the voltage limit , this operating condition is achieved at unity power factor . with reference to the vector graph in steady - state conditions , shown in fig4 , the unity power factor operating condition can be represented by the following equations by inserting the current limit i d = i r cos θ and i q = i r sin θ in ( 26 ) leads to by solving ( 27 ) for θ it is possible to determine the phase angle θ h of the armature current vector that ensures a unity power factor at any angular speed ω & gt ; ω b . the solution is torque can be calculated by introducing ( 28 ) and ( 30 ) in ( 19 ). it is possible to check that the resulting torque corresponds to constant power operation . furthermore , the parameter a h assumes the asymptotic value a h = 1 for ω →∞, which means that the centre of the voltage ellipses is placed on the current circle ( i . e . a h = 1 → m se i eh = l d i r ). machine operation with unity power factor is prevented at low speed by the maximum value of the excitation current . as can be seen from ( 29 ), the lower the speed , the greater the excitation current . in general , the maximum excitation current is reached at a speed ω c greater than ω b . therefore the speed range between ω b and ω c , in which unity power factor operation cannot be achieved , must be further examined for completing the disclosure of the control technique . in this speed range , considered as a transition region between constant torque and constant power operation , the best performance is obtained at the points of the current limit circle defined by the intersection with the voltage ellipses . the current vector angle can be calculated from ( 20 ) to obtain by way of example , fig5 represents the operating condition in the p . u . coordinates for a machine with r = 3 , a m = 1 . 8 . point b is obtained with b b = 6 . 42 , whereas unity power factor operation is obtained for values of b less than b c = 4 . 95 . at very high speeds the excitation current is reduced to a value corresponding to a h = 1 . in order to show the behavior of the most representative machine entities , a numeric simulation from zero to maximum machine speed has been carried out . fig6 shows , from top to bottom , the torque , the power , stator current angle and the excitation current as a function of speed . fig7 shows the torque , the power , the power factor and the supply voltage in the transition speed range between a constant - torque operation region and a constant - power operation region . in fig6 it can be seen that the excitation current is kept at its maximum value until point c corresponding to the end of the transition region , and subsequently decreases at higher speeds . in fig7 it can be seen that in the transition range the power factor increases from 0 . 85 to 1 . torque is constant for speeds lower than the speeds corresponding to point b whilst power is constant for speeds greater than the speed corresponding to point c . the operation of wr - sm at constant speed theoretically extends to an unlimited speed . fig8 shows a flow - chart illustrating the phases of the method according to the invention . this flow - chart comprises a first section ( section 1 hereinafter ) regarding the acquisition of the electromagnetic entities , the transformations of reference system and the measuring of instantaneous speed , a second section ( hereinafter section 2 ) regarding the generation of reference values of the electromagnetic entities and a third section ( hereinafter section 3 ) regarding the estimate of the magnetic flux and the current regulation . with reference to the diagram shown in fig9 , the acquired stator currents are transformed by the matrices d and t shown below in the fixed two - phase reference system dq s and the two - phase rotating system dq that is synchronous with the rotor . the position of the rotating reference system dq is identified by the rotor position by means of ankle θ . the position of the rotor in electric degrees is measured by a suitable measuring transducer ( encoder ). speed is calculated as a temporal derivative of the position , according to the formula set out below . as the system is discrete , the calculation is carried out at the finite differences . ω k average speed in calculation interval δt δt calculation interval , equal to a whole multiple of the calculation period t c , so δt = nt c δθ k rotation that occurred in the interval δt with reference to fig1 , in function of rotation speed ω and maximum voltage v r that the inverter is able to apply to the motor , there is calculated the combination of machine currents i dqref , i eef for which the machine absorbs a module stator current equal to the required reference i ref and produces the maximum generatable torque . the references of the stator currents i dqref and the rotor currents i eref are generated that will be used by the current regulators for the generation of the machine currents and for the estimate of the machine parameters in section 3 . to calculate the references , the estimated machine parameters l d , l q and the estimated fluxes φ dq calculated in section 3 are used . the control strategy is based on the knowledge of the linear model of the synchronous machine as defined by the equations ( 1 )-( 31 ) reported above . the real machine operates in conditions of saturation of the magnetic circuit and therefore in a non - linear condition . the model of the machine operating in conditions of saturation is obtained by using the linear model in which the values of the parameters are updated instantaneously using the virtual values calculated by the estimator block that is in section 3 . the estimated values of the parameters depend on the machine &# 39 ; s actual operating conditions . the machine operating in saturation conditions is thus represented by a linear model with estimated variable parameters . instantaneously , by using the model of the machine in saturation conditions , depending on the available voltage v dc and the rotation speed ω , the machine &# 39 ; s operation field is divided into the three operation zones examined previously , i . e . : in the low - speed operation zone the value of the excitation current corresponds to the maximum permissable value i e = i em . there is a sole operation point at maximum torque , which is point b , shown in fig5 . operation in this zone is possible up to speed ω b . in the intermediate speed operation zone the value of the excitation current is equal to the maximum permitted value i e = i em . for each speed there is a locus i dq — in which torque is at maximum — described by the circle arc comprised between points b and c , shown in fig5 . operation in this zone is possible up to speed ω b & lt ; ω & lt ; ω c . in the high - speed operation zone the values of the excitation current i e and of the stator current components i dq are regulated in such a way as to obtain maximum torque per stator current . operation in this zone is possible for speeds greater than ω c . the operation in this zone has no upper speed limit . management of the control strategy of the three zones occurs according to the block diagram in fig1 . two selection blocks are present , in which , depending on the direct supply voltage of the inverter v dc , the speed ω and the reference of the module of the stator current i ref , the operation zone is identified in which the machine works . depending on the operation zone and on the constraints present ( voltage limit , stator current limit , rotor current limit ), at any instant of motor operation , by using the analytical expressions shown above , it is possible to identify the configuration of the supply i dq , i e that complies with the constraints and produces maximum torque . in the high - speed operation zone , in which unity power factor operation is set , fine adjustment of the excitation current is carried out so as to improve performance of the system , overcoming any imprecisions in the parameters estimated by the estimator illustrated in section 3 . this fine adjustment consists of adding a corrective value to the value of the excitation current calculated according to the diagram in fig1 . fine adjustment is based on the principle of reaching the condition in which the vectors representing stator current and stator voltage are in phase or , in an equivalent mode , the vectors representing stator current and stator flux are in quadrature . the latter condition in the rotating two - phase reference system dq is expressed by the following condition the equation ( 32 ) represents magnetic stator energy e ms . setting operation at e ms = 0 corresponds to not supplying reactive power to the motor by means of the stator currents . e ms = 0 operation is obtained by using a proportional - integral regulator ( pi ) applied to the calculated value e ms . the corrective value of the excitation current i e is added to the result obtained using the calculation system disclosed above and its contribution is limited by a saturator , as shown in fig1 . in the estimator block of this section the components of the flux - linkage with the stator φ d , φ q are estimated . once the flux is known , the values of the parameters “ fictitious inductances ” l d , l q are calculated . these parameters are used in section 2 , in the non - linear model of the synchronous machine . in the estimator block the back electromotive force induced in the stator is calculated . this value is used by the regulating block to enable optimal adjustment of the stator current . in the regulator block the reference value of the voltage that the inverter has to apply to the stator and which enables control of the current at the reference value calculated in section 1 is furthermore calculated . the estimate of flux - linkage with the stator in the two - phase fixed reference system dq s is based on the following equation : { φ d s = ∫ ( v d s - r s ⁢ i d s ) ⁢ ⅆ t φ q s = ∫ ( v q s - r s ⁢ i q s ) ⁢ ⅆ t ( 33 ) the implementation of the estimator in a digital control system requires ( 34 ) to be expressed in discrete form . at the k - th cycle a calculation of the flux value at the next cycle ( k + 1 ) is made . the flux value resulting from this calculation is then used in the subsequent cycle in terms of fictitious inductances in section 2 . the value of the flux at the present instant φ dq ( k ) s and of the voltage applied in the present instant v dq ( k ) s are calculated in the previous instant ( k − 1 ) in section 3 . the current i dq ( k ) s derives from measurements taken at the present instant . in order to stabilise the estimate of the fluxes , a term is introduced that represents the value of the “ normal operation flux ”. this correction is based on the fact that during normal operation the result of the estimator must converge in the value of the “ normal operation flux ”. the value of the “ normal operation flux ” φ dqω s is calculated in the fixed two - phase reference system as the equation of the estimator ( 34 ) must therefore be modified as the parameter g influences the estimator &# 39 ; s response . high g values cause a more stable and less precise response . low g values determine a less stable and more precise response . the discrete equations used for making the flux estimator are the estimated fluxes are shown in the rotating two - phase reference system dq by the transformation matrix [ t ] the values of the “ fictitious inductances ” corresponding to the estimated fluxes are given by { l d ⁡ ( k + 1 ) = φ d ⁢ ⁢ ( k + 1 ) n d ⁡ ( k ) l q ⁡ ( k + 1 ) = φ q ⁢ ⁢ ( k + 1 ) n q ⁡ ( k ) ( 41 ) where n d ( k ) , n q ( k ) are the equivalent currents of axis d and axis q . { n d ⁡ ( k ) = i d ⁡ ( k ) + i eref ⁡ ( k ) ′ n q ⁡ ( k ) = i q ⁡ ( k ) ( 42 ) i ′ eref ( k ) is the reference of the rotor current in relation to the stator . the flux estimator block diagram is shown in fig1 .