Sensorless Control for IPMSM using PWM Excitation ...

Sensorless Control for IPMSM using PWM Excitation: Analytical Developments and Implementation Issues Silverio BOLOGNANI1, Sandro CALLIGARO2, Roberto PETRELLA2, Michele STERPELLONE2 1 Department of Electrical Engineering (DIE), University of Padova (Italy) 2 Department of Electrical, Management and Mechanical Engineering (DIEGM), University of Udine (Italy) [email protected], [email protected], [email protected], [email protected]

Abstract – Persistent, transient and, very recently, pulse width modulated voltage excitation are employed to track saliency of IPM machines, aiming at estimating rotor position. In this paper current transient response introduced by standard (or slightly modified) PWM excitation is considered and analytical relationship between phase current derivatives, inductance and rotor position is derived. A complete mathematical model is developed in the case of IPM machine, by taking also into account the dependence of the rotor position estimation error on the mutual inductance, which is neglected in the past literature adopting the same sensorless approach. Estimation is performed within a single PWM period, differently from previous approaches. Measurement of current derivatives is obtained from dedicated Rogowski coils, oversampling and real-time processing of the measured values. Acquisition issues due to short application times of voltage vector (e.g. during zero and low speed operations) have been overcome by means of a proper edge-shifting technique on the PWM signals. A motor drive system for fractional power high speed IPM motor is considered as a test bench to prove the effectiveness of the proposal. Index Terms – IPM motor, sensorless, signal injection, PWM excitation.

I.

INTRODUCTION

Sensorless control techniques adopted for permanent magnet synchronous machines can be classified in two main categories. The first one relies on the back electro motive force (back-EMF) in order to determine the rotor position angle [1]. It shows good performance in the medium and high speed/frequency range, but back-EMF based schemes are unreliable at low and zero speed, due to the low signal-to-noise ratio or the total lack of useful signal. The other category of sensorless techniques exploits the anisotropy of the rotor, mainly caused by magnetic saturation effect (on surface permanent magnet synchronous motors, SPMSMs) and/or geometric saliency (on interior permanent magnet synchronous motors, IPMSMs). In signal injection based methods (normally used to detect rotor anisotropy) a rotating or pulsating carrier signal is superimposed to fundamental excitation voltage and demodulation of the

resulting high frequency induced currents is used to obtain rotor position information, [2][3]. Those methods are normally recalled as persistent carrier based excitation. Similar methods are based on the measurement of phase current transient response during each inverter voltage space vector. The INFORM method, [4], is well-known and it is based on the injection of and additional voltage signal corresponding to two separate and opposite test vectors during a null space voltage vector. These techniques have some disadvantages because of the increases of the current ripple (i.e. torque ripple) and switching losses. Very recent approaches are based on the measurement of the current transient response introduced by standard (or slightly modified) PWM excitation, i.e. without any additional test voltage, [5]-[9]. References [5] and [6] consider a non-salient PM machine and adopts a persistent excitation within standard space vector PWM exploiting magnetic saturation induced saliencies. No additional signal injection or separate test vector is required, although a small degree of modification is needed when short voltage vectors occur, i.e. near a sector transition or when the modulation index is very low. The method is based on the measurement of current derivatives (i.e. a term related to motor inductance) during the PWM cycle, being related to the sine and cosine of the rotor position as it happens in a sinusoidal encoder. The aim of this paper is to extend that approach. A complete mathematical model is developed in the case of an IPMSM, by taking also into account the mutual inductance dependency with rotor position, which is completely neglected in the aforementioned references. Also estimation is performed within a single PWM period. Measurement of current derivatives is obtained from dedicated Rogowski coils, oversampling and real-time processing of the measured values. Acquisition issues due to short application times of voltage vector (e.g. during zero and low speed operations) have been overcome by means of a proper edge-shifting technique on the PWM signals. A motor drive system for fractional power high speed IPMSM is considered as a test bench to prove the effectiveness of the proposal.

ANALYTICAL MODEL OF IPM MACHINE AND CURRENT DERIVATIVES EXPRESSION

II.

udq  Ridq 

In this section a mathematical model of the IPMSM is recalled in order to illustrate the possibility of estimating rotor position by means of current derivatives measurements under space vector PWM excitation. Differently from the approach presented in [5], the complete model also takes into account for the spatial variation of the total inductance is introduced, thus obtaining a more general result. The analytical model of the IPMSM is considered in the 𝑑𝑞 synchronous reference frame oriented according to the flux linkage due to permanent magnet, including cross-coupling terms.

udq  Ridq 

d dq

 

 

𝑢𝑑𝑞 , 𝑖𝑑𝑞 and 𝜆𝑑𝑞 are voltage, current and flux linkage vectors, and 𝑅 is the phase resistance, i.e. 𝑢𝑑 𝑖𝑑 λ𝑑 𝑢𝑑𝑞 = [𝑢 ], 𝑖𝑑𝑞 = [𝑖 ], 𝜆𝑑𝑞 = [ ] λ𝑞 𝑞 𝑞

(2)

The flux-linkage has been divided into the two terms, the former 𝜆𝑖,𝑑𝑞 (𝑖𝑑𝑞 ) depending on the motor currents, and the latter due to the permanent magnet, 𝜆𝑚𝑔,𝑑𝑞 . 𝐽 is the antidiagonal unity matrix. 𝜆𝑚𝑔,𝑑𝑞 = [

Λ𝑚𝑔 0 ], 𝐽 = [ 1 0

−1 ] 0

 

(4)

In general, self and mutual 𝑑𝑞 inductances depends on the 𝑑𝑞 currents, because of saturation and cross magnetization. Substitution of flux expression inside voltage equation leads to a more complex expression considering the variation of the flux linkage 𝜆𝑖,𝑑𝑞 (𝑖𝑑𝑞 ) with respect to motor currents, i.e.:

u 

i  R  i  resistive voltage drop



  

 me J Ldq idq idq  mg ,dq



(5)

can therefore be defined, i.e.:

 

d i ,dq idq d idq

 i ,d  i d   i ,q   id

i ,d  iq   i ,q   iq 

(6)

 L (i , i ) M qd (id , iq )   d d q   M dq (id , iq ) Lq (id , iq ) 

 

Ldq idq

Voltage equation in the synchronous reference frame therefore becomes: ud  id   Ld (id , iq ) M qd (id , iq )  d id   u   R  i    M (i , i ) L (i , i )   i   q d q  dt  q   q  q   dq d q 0 1   Ld (id , iq ) M qd (id , iq )  id    mg   me        1 0    M dq (id , iq ) Lq (id , iq )   iq   0  

(7)

Finally if the rotating (𝑑𝑞) to stationary (𝛼𝛽) reference frames transformation is applied, the motor model (8) is obtained, where 𝜃𝑚𝑒 is the rotor electrical position. The following inductive terms have been defined:

L L

Matrix 𝐿𝑑𝑞 (𝑖𝑑𝑞 ) is related to motor apparent self and mutual

 Ld (id , iq ) M qd (id , iq )  Ldq idq     M dq (id , iq ) Lq (id , iq ) 

d idq

Differential self and mutual inductance matrix 𝐿̃𝑑𝑞 (𝑖𝑑𝑞 )

(3)

inductances, i.e.:

 me J dq

dt

  dt

(1)

dq  i ,dq idq  mg ,dq  Ldq idq  idq  mg ,dq

idq

 Ridq  Ldq idq

 me J dq

dt

 

 i ,dq idq d idq

Lq  Ld 2 Lq  Ld 2

L L

Lq  Ld 2 Lq  Ld

(9)

2

The contribution of different sources of induced voltage is clearly shown in (8), the most interesting to our purposes is the one proportional to the derivative of stator currents. In fact a proper measurement and processing of current derivatives allows the estimation of the rotor position, as demonstrated in the next sections. III.

ROTOR POSITION ESTIMATION

A symmetric double edge PWM pattern is taken into account, as shown in Fig. 1.

 L  L cos  2 me   M qd sin  2 me   L sin  2 me   M qd cos  2 me   d i     L  L cos  2 me   M qd sin  2 me   dt i    L sin  2 me   M qd cos  2 me  inductive voltage drop

  sin  me     cos  me  

me  mg 

back e.m.f. due to permanent magnet flux

   L  L    L  L  cos  2 me    M qd  M qd  sin  2 me   i   L  L  sin  2me    M qd  M qd  cos  2me      me   L  L    L  L  cos  2 me    M qd  M qd  sin  2 me   i    L  L  sin  2 me    M qd  M qd  cos  2 me    induced voltage due to anisotropy

(8)

derivatives (known values), rotor position and inductances. Rearranging (10) we obtain the following two sets of equations:

Fig. 1. Sampling instants within a symmetric double-edge PWM pattern.

The four (generic) measuring points of motor current derivatives are indicated by the red arrows. Sample 1 is acquired during null voltage vector 𝑉̅0 and sample 2 is taken in the next active vector (i.e. 𝑉̅1 in this example). We have a similar situation with respect to samples 3 and 4, where sampling is done during 𝑉̅7 and the next active vector 𝑉̅2 . This is a different approach with respect to literature, [5], as four current derivative samples are measured within the same PWM period. Hereafter it will be shown that the sampling task is not so easy as described (e.g. due to the presence of parasitic effects and variation of the PWM pattern as a function of the motor operating conditions). A first simplifying hypothesis is to consider rotor electrical speed 𝜔𝑚𝑒 to be relatively low and to assume that rotor position 𝜃𝑚𝑒 is constant within the modulation period. In the next subsection it will be analytically demonstrated that the method is still valid when high speed operation is considered (i.e. when a variation of the rotor position is experienced between the four current samples) by means of a proper simplified model of the rotor position variation, representing one of the original features of this paper. In fact, the set of equations generated by the motor model provides additional speed-dependent terms. It will be proved that they can be eliminated by proper processing of current derivative signals. Hereafter only zero and low speed operation are considered for the moment. A. Low speed operation If the four current derivatives measurements shown in Fig. 1 are considered and resistive voltage drop and backEMFs (both due to permanent magnet and saliency related) are neglected as they are assumed constant within the modulation period, (8) can be rewritten for each sample. The difference between the obtained equations can be considered, leading to (10). Superscripts indicate the difference between the two values within the sampling instants shown in Fig. 1. This represents a set of four equations linking voltage and current

u

(21)

u

(43)

 di (21)   dt  di (43)    dt

  (21) dt   A  u    (43)  (43)   di   B  u   dt 

 di (21)   dt  di (43)    dt

 (21)  dt   C   u    (43)  (43)   di   D  u   dt 

di

di

(21)

(21)

(11)

where A  L  L cos  2 me   M qd sin  2 me  B  C   L sin  2 me   M qd cos  2 me  D  L  L cos  2 me   M qd sin  2 me 

(12)

The solution of the two sets of equations by a simple linear combination provides two signals being respectively proportional to the sine and cosine of twice the (electrical) rotor position, i.e.:  A  D  2  L cos  2 me   M qd sin  2 me    2

L



2

 M qd 2  cos  2 me    

 au (21)  bu (43)  cu (21)  du (43)

 B  C  2  L sin  2 me   M qd cos  2 me    2

L



2

(13)

 M qd 2  sin  2 me    

 au (21)  bu (43)  cu (21)  du (43)

where (21)  di (43) di     dt  dt  (43) (21)  di  di  M  (14)   a b  dt  dt    tan 1   qd  c d   (43) (21) (21) (43)  L  di di di   di   dt dt dt dt

By processing signals (13) (i.e. by inverse trigonometric function 𝑡𝑎𝑛−1 or phase-locked-loop based approach), leads to rotor position estimation. No knowledge of motor inductance is needed except for the mutual terms, due to the presence of the phase delay 𝛼 in the argument of the sine and cosine functions in (13). A compensation is however possible if a map of the motor fluxes is known.

 L  L cos  2 me   M qd sin  2 me   L sin  2 me   M qd cos  2 me   d i       L  L cos  2 me   M qd sin  2 me   dt i    L sin  2 me   M qd cos  2 me 

(21)

 L  L cos  2 me   M qd sin  2 me   L sin  2 me   M qd cos  2 me   d i       L  L cos  2 me   M qd sin  2 me   dt i    L sin  2 me   M qd cos  2 me 

(43)

(10)

B. High speed operation At high speed, a discontinuity of sine and cosine signals is experienced every time the voltage vector crosses sector borders. So it is not related to the variation of the back-EMF in the four ideal sampling instants. However the estimated rotor position as obtained by the ratio of sine and cosine signals is always correct. The analytical proof and results will be presented and discussed hereafter. Also it will be shown that a speed dependent term arises, that can be compensated, leading to the possibility to adopt a more robust approach (e.g. phase locked loop) for rotor position estimation instead of direct 𝑡𝑎𝑛 −1 calculation. In order to continue to deal with linear equations, the following simplifying hypothesis have been introduced (see also Fig. 2):

in Fig. 2 (𝜃𝑚𝑒 , green arrow), to make the analytical calculations easier. In particular this hypothesis allows to consider an inductance matrix (terms 𝐴, 𝐵, 𝐶 and 𝐷 in (11)) that can be expressed in the two equivalent sampling instants (i.e. 𝑠𝑎𝑚𝑝𝑙𝑒 12 and 𝑠𝑎𝑚𝑝𝑙𝑒 34) as a function of the same variations Δ𝐴, Δ𝐵, Δ𝐶 and Δ𝐷, as shown in Fig. 2. Therefore (10) can be rewritten by considering the rotor position variation due to inductance matrix, i.e.:

 the electrical rotor position is assumed to be constant within each couple of ideal sampling instants, i.e. 1 and 2 or 3 and ′ ′′ 4 (𝜃𝑚𝑒 and 𝜃𝑚𝑒 , blue arrows);  the electrical rotor position is assumed to be linearly varying between the first two samples of current derivatives and the last two samples; this means that rotor speed is constant within each period;  a linear variation for each term of 𝐿𝛼𝛽 matrix with Taylor series expansion is considered; this means the two vector equations (8) can be rewritten, each one referring to a mean sampling instant (𝑠𝑎𝑚𝑝𝑙𝑒 12 and 𝑠𝑎𝑚𝑝𝑙𝑒 34 in Fig. 2, blue arrows) corresponding to the mean time of the original sampling instants.

After some manipulations the following linear systems, equivalent to those of (8), can be obtained, allowing the estimation of inductance matrix 𝐿𝛼𝛽 in the position 𝜃𝑚𝑒 :

As rotor position is supposed to vary linearly within the PWM period, the final estimated position will be referred to the mean point between 𝑠𝑎𝑚𝑝𝑙𝑒 12 and 𝑠𝑎𝑚𝑝𝑙𝑒 34, as shown V0

(21)

u (21)   A  A  (21)    u  C  C

B  B  d i    D  D  dt i 

u (43)   A  A  (43)    u  C  C

B  B  d i    D  D  dt i 

(15)

(43)

 di (21)   dt  di (43)    dt

(21) (21)   (21) di  di u   A   B  A    dt dt dt        (43) (43) (43) di   B   (43) di  di  u  A   B dt dt dt   

 di (21)   dt  di (43)    dt

(21) (21)   (21) di  di u   C   D  C    dt dt dt        (43) (43) (43) di   D   (43) di  di  u  C   D dt dt dt   

di

di

(21)

The variation terms Δ𝐴, Δ𝐵, Δ𝐶 and Δ𝐷 can be calculated by Taylor series expansion, as stated before:

V1

V2

V7

V2

V1

V0

t t t

TPWM sample 1 sample 2

 me '  A ' B '   A  A B  B  C ' D '  C  C D  D     

(16)

(21)

sample 3

me

 me ''

T

T

sample 12

sample 4

A B C D   

sample 34

 A '' B ''   A  A B  B  C '' D ''  C  C D  D     

Fig. 2. Sampling instants within a symmetric double-edge PWM pattern (high speed operation).

A  A  L  L cos  2 me ''   M qd sin  2 me '' 

2

 L  L cos  2 me   M qd sin  2 me    2  L sin  2 me   M qd cos  2 me   me T  o  T 

2

where 𝑜(Δ𝑇 2 ) represents all the higher order terms of the series. If the linear terms is considered only, Δ𝐴 is obtained: 2

L



2

 M qd 2  sin  2 me    me T

C  2 D  2

L L L



2

 M qd 2  cos  2 me    me T

2

 M qd 2  cos  2 me    me T

2

 M qd 2  sin  2 me    me T





(19)

Then (22) can be calculated following the same steps used to obtain (13), where i 

di dt

(21)

di dt

(43)



di dt

(43)

di

(21)

dt

(20)

After some algebraic manipulations and considering that Δ𝐴 = −Δ𝐷 and Δ𝐵 = Δ𝐶, the following relations are obtained:

L

A D  2



 M qd 2  cos  2 me     au (21)  bu (43)  cu (21)  du (43) 

2

 di (21) di (43) di (21) di (43)     dt dt dt dt    L 2  M qd 2  cos  2 me    4me T  (43) (21) (43)  di (21) di di  di    dt dt dt  dt 

B  C  2

L



 M qd 2  sin  2 me      au (21)  bu (43)  cu (21)  du (43) 

2

(21)

 di (21) di (43) di (21) di (43)     dt dt dt dt    L 2  M qd 2  sin  2 me    4me T  (43) (21) (21) (43)  di di di  di    dt dt dt  dt 

It can be pointed out that equations (21) are very similar to (13), the last ones obtained in the case no variation of the inductance matrix is considered between the two couples of current derivative samples. The additional term is speed dependent, as previously discussed. A simplified expression can be finally calculated

A D  2 



2

 M qd 2  cos  2 me    

L



2

(23)

 M qd 2  sin  2 me    

 k  au (21)  bu (43)  cu (21)  du (43) 

  di (21) di (43) di (21) di (43)        dt dt dt dt    k  1  2me T (43) (21)  (21) (43)   di di di di         dt dt dt dt   

1

1 i

L



2

(24)

About these last results, one can notice that:  sine and cosine signals experience discontinuities at each sector change; this is due to the different PWM pattern for each sector and, consequently, to the different couples of active vectors and current derivatives;  the ratio between the sine and cosine terms in (23) is still proportional to the 𝑡𝑎𝑛 of the electrical position, as the additional speed-dependent term is an identical gain for both the equations, that simplifies after ratio; that is the reason why, even at high speed and with discontinuities of the sine and cosine signals, the estimated speed is still correct, as it will be shown in the simulation investigation; The additional term speed-dependent terms in (21) can however be compensated in order to obtain pure sinusoidal functions with no discontinuity, thus allowing the application of other position and speed calculation methods, e.g. PLL. Even more complex models can be introduced for the variation of the inductance matrix with rotor position (especially at high speed), e.g. by considering a different value for each current derivative sampling instant (see Fig. 2). It can however be verified by simulation results that the improvements in the estimates are very limited. Therefore this analysis is not reported here. IV.

SIMULATION RESULTS

Reliability of the proposed analysis has initially been verified by comparing the results of a complete electro-mechanical dynamical model simulation of the drive system, including sampled-time control and PWM modulation, and the analytical models discussed in the previous sections.

 M qd 2  cos  2 me     au (21)  bu (43)  cu (21)  du (43) 

(43) (21) (21) (43)  di (21) di (43) di (43) di (21)   di (21) di (43) di (43) di (21)   di di   di (21) di (43) di (43) di (21)  1   di di    B     C     D     A         dt   dt   i   dt dt dt dt dt dt dt dt dt dt dt dt dt  dt      

B  C  2 

L

2

where (18)

The same approach can be adopted for the calculation of Δ𝐵, Δ𝐶 and Δ𝐷, i.e.: B  2



 k  au (21)  bu (43)  cu (21)  du (43) 

(17) 2

A  2  L sin  2 me   M qd cos  2 me   me T

L

 M qd 2  sin  2 me      au (21)  bu (43)  cu (21)  du (43) 

  di (21) di (43) di (43) di (21)   di (21) di (43) di (43) di (21)   di (21) di (43) di (43) di (21)   di (21) di (43) di (43) di (21)        B     C     D     A         dt   dt   dt   dt  dt dt dt dt dt dt dt dt dt dt dt dt        

(22)

In Fig. 3 and Fig. 4 a speed transient under load conditions is considered from 0 to 300 𝑟𝑝𝑚. The aim is to show that rotor position estimation is possible and to highlight the effect of the rotor speed on the sine and cosine signals extracted from current derivatives measurements. One can notice that discontinuities of those signals are present in the result of Fig. 3, due to the effect of speed dependent term that has been neglected in (13). Nevertheless, the estimated rotor position, as obtained by direct 𝑡𝑎𝑛−1 of the ratio of those signals is correct, as discussed in the previous section. The results of Fig. 4 prove that a compensation of that effect is possible as a function of motor speed, leading to the removal of discontinuities. This could provide a beneficial effect in the actual system due to the presence of measurement noise and also allowing the application of more robust rotor position calculation techniques., such as a phase locked loop. Actual machine will however experience saturation and cross-saturation effects that will worsen the obtainable results. Problems related to current derivative measurement would cause estimation performance degradation, too. This topic will be discussed in the next section.

Fig. 3. Rotor position estimation during a speed transient with no compensation for the speed dependent term (simulation).

V.

IMPLEMENTATION ISSUES

A. Current derivative measurement Direct measurement of current derivative by dedicated sensors is for sure the most effective and reliable method. Indirect measurement, through standard current measurement and derivative calculation (either analogically or digitally), is in fact very difficult to obtain mainly due to the resulting low value of the sensitivity and quantization noise. The simplest approach is to use a simple inductor, providing a voltage proportional to the time derivative of the circulating current, but this leads to a number of drawbacks, e.g.:  it is an invasive method, as modification of the measured circuits are introduced (both inductance and resistance);  reduction of the equivalent inductance would lead to sensitivity problems;  there is no galvanic separation between measured and control circuitry. A better solution could be the use of a mutual inductor, whose mutual inductance provides the sensitivity of the measurement. Proper solutions have to be considered in order to reduce any magnetic coupling with external magnetic field and cross-talk among different measuring channels. Also the

Fig. 4. Rotor position estimation during a speed transient with compensation for the speed dependent term (simulation).

coupling of primary and secondary circuits have to be properly chosen and any difference among the channels would lead to a different sensitivity, that needs to be accurately tuned. The solution adopted in this paper is based instead on dedicated Rogowski coils, [10]-[12], mainly due to a simple construction, low cost and high accuracy and bandwidth of the measurements. Accurate modeling of the coils have been taken into account in order optimize the choice of its parameters as a function of the required bandwidth, encumbrance and sensitivity. Fig. 5 shows the arrangement of coils and signal conditioning circuit. Accurate tuning and equalization of the gains of the three current derivative measuring circuits (coils and signal conditioning) have been realized by by feeding all of the three sensors with a triangular current source. The three output voltages 𝑣𝑜𝑢𝑡,𝑥 of the measuring circuitry in response to the test current 𝑖𝑡𝑒𝑠𝑡 are shown in Fig. 6. As clearly visible, the system response during slope inversion of the current is not the one of a second order system, as theoretically considered. However the response time is fully compliant with system requirements and the bandwidth of the coils.

discussed sensorless technique is to guarantee a minimum application time of each space vector in Fig. 1 in all the operating range of the machine. This requirement is related at least to the following considerations:  the application time of one active voltage vector when a low amplitude voltage space vector is applied could be too low (even comparable to the response time of the power switches); therefore the resulting current response is not the one considered in the theoretical analysis and those results are not valid anymore;  the resulting motor phase current oscillations during and immediately after each commutation of the power switches affect the reliability of current derivative measurement; some time has to be waited to make those oscillations decay before measurement can take place. The most critical cases where the application time of one voltage vector becomes too low can be related to two main operating conditions:

B. Edge-shifting of PWM signals One of the main issues for the implementation of the

 low-speed operation: application time of active voltage vectors are quite low;  sector transition of the reference voltage space vector: even if the amplitude of the voltage is high enough, it happens that the application time of an active voltage vector becomes very low (even zero) at each sector transition.

Fig. 5. Experimental arrangement of coils and signal conditioning circuit.

The two cases are graphically recalled in Fig. 7. In those situations, the analyzed sensorless strategy can be applied only if a modification of the PWM pattern is introduced, aiming at guaranteeing enough duration of both zero and active voltage space vector, thus allowing reliable values of current derivative samples. Two main limitation (i.e. constraints) have been considered to any modification strategy of the PWM pattern: the number of commutations within each modulation period and the average value of the reference voltage space vector have to be both maintained.

Fig. 6. Triangular current excitation across the thee-phases and measured current derivative responses.

Fig. 7. Space vector representation and PWM pattern for low amplitude (left) and sector crossing (right) of the reference voltage space vector.

This necessarily requires the introduction of a modification of the PWM pattern by edge-shifting of the PWM signals, [5], which does not preserves symmetry of the pattern. Differently from the recalled reference [5], the four current derivative measurements are performed in the same modulation period, thus providing a more complex edge-shifting algorithm. Quite a huge amount of different cases are possible as a function of the original reference voltage space vector. They have been fully investigated but are not reported in this paper due to space constraints. Edge-shifting of the PWM waveforms introduces a side effect related to the asymmetry of the resulting pattern, that is introduced but not in dept discussed hereafter for the same reason. When adopting a standard double-edge modulation pattern, average values of motor phase currents can be sampled in the symmetry points of the pattern thus removing the most part of the high frequency ripple and avoiding the use of lowpass filters, [13]. Edge-shifting introduces a modification of the pattern that moves the symmetry point and makes the sampling of the average current value more difficult. A compensation strategy implemented in software has been proposed to solve that problem. C. Homopolar currents due to capacitive couplings An IPM synchronous machine represents an asymmetrical three-phase load. The difference between the output voltage of the inverter (as referred to the midpoint of the dc bus) and the phase voltage is due to a homopolar voltage component. As the machine is wye connected, no homopolar current should be present in the ideal case. But, the analysis of actual motor currents during inverters’ switches commutations show that high-frequency oscillations are introduced, with a damped behaviour and time constant in the order of some to tenths of 𝜇𝑠, that can be explained with the presence of a homopolar current component. A simple model is discussed hereafter in order to explain the current behaviour immediately after each inverter commutation, as shown in Fig. 8. Lumped parameters 𝐶𝑖𝑛𝑣 and 𝐶𝑚𝑜𝑡 have been introduced to model the total inverter and motor stray capacitance respectively with respect to earth, that indeed are of distributed nature. From a zero sequence point-of-view, the circuit is a series

Fig. 8. Simplified model of the motor/inverter with parasitic earth capacitances.

RLC circuit and the damping factor is generally low due the small value of the series resistance. Oscillations are therefore expected on both motor currents and neutral to inverter reference voltage, thus requiring a modification of the current derivative sampling instants. VI.

EXPERIMENTAL RESULTS

A motor drive system for fractional power high speed IPMSM based on a last generation digital signal controller is considered as a test bench to prove the effectiveness of the sensorless strategy and the validity of the theoretical analysis. Measurement of current derivatives have been performed by dedicated Rogowski coils as previously discussed, oversampling and processing (averaging over a certain time window) of the measured values. This last feature is needed in order to reduce the effects of parasitic parameters of the system (e.g. capacitive couplings, hard switching transient effects, etc.) on the reliability of the measured values, as discussed in the next section. The tests reported hereafter are mainly intended to highlight some of the problems in the implementation of the estimation method and suggest possible solutions. The results of Fig. 9 refer to the measurement (with oversampling) and processing of the current derivatives within an electrical period of the machine. The four terms 𝐴, 𝐵, 𝐶 and 𝐷 discussed in (12) are shown, together with the sine and cosine functions (13), the estimated rotor position and the estimation error. One can notice that terms 𝐴 and 𝐷 have different offsets and amplitudes, meaning that both differential sum (𝐿̃Σ ) and difference (𝐿̃Δ ) indutances are different. This is the main cause of the offset that is visible in the cosine signal and that is the main source of estimation error. This can be explained by considering that actual motor inductances are non-linear and depend on the flux level, in turn depending on the permanent magnet and instantaneous current contributions. Even if the current ripple is very low, the average value of the motor phase current could be relatively different between the current derivative sampling instants. Moreover position estimation error experiences a relatively high value around 𝜋, probably due to the effect of the permanent magnet. In Fig. 10 the measured current derivatives and currents in the 𝛼𝛽𝑜 reference frame are shown. One can notice the effect of stray capacitance that provides an oscillating behaviour after each commutation. Also a qualitative comparison between the oscillations in the 𝛼 (or 𝛽) and the zero sequence components highlights that a small cross-coupling exists between those axes, as recently modelled in [14] and [15]. In those papers it is in fact demonstrated that, by adopting a proper model for the saturation induced in the stator teeth by the permanent magnet, the inductance matrix in the orthogonal stationary reference is not diagonal and contains some elements linking the

homopolar component to both 𝛼 and 𝛽 axes. However the first transition of the current derivatives in the 𝛼𝛽 components can be explained by considering the actual model of the motor phase inductance, including a parallel capacitance (due to windings) that conducts a high initial current due to the voltage commutation. This means that current derivative measurement and averaging has to be performed only after that transient condition in order to obtain reliable values. Finally in Fig. 11 and Fig. 12 the measured current derivatives are shown when the permanent magnet axis is aligned along the 𝛼 and 𝛽 axis respectively. The two figures allow to highlight a very interesting phenomenon, that is completely neglected in the past literature. The two curves of each figure have been obtained with the permanent magnet north pole aligned along the positive direction of the winding (blue curves) or the negative direction (purple curves) Fig. 11 is considered first, where the rotor position has been chosen in order to have the permanent magnet aligned with the 𝛼 axis. During the application of the zero voltage vectors (𝑉̅0 and ̅ 𝑉7 , 1st and 4th subperiods) the current derivatives are zero, as expected. During the application of 𝑉̅1 (2nd subperiod) and considering

the 𝛼 axis, one can notice that the behavior of current derivative is different if the PM is in-phase or out-of-phase with that axis. When the PM is in-phase with the 𝛼 axis the overall flux along that direction is increasing, due to the superposition of the (positive) PM flux linkage and that produced by the phase current (also positive in that period). The same applies in the 3rd subperiod, during the application of vector 𝑉̅2 . The different behavior seems therefore to be related to the influence of the PM flux linkage, that increases or decreases the total flux linkage. This hypothesis is strengthened by considering the behavior of the 𝛽 component of the current derivative, that is not affected by the in-phase or out-of-phase position of the rotor, as in both the cases it is in quadrature to that axis. A similar situation is present in the results of Fig. 12, where the PM is aligned along 𝛽 axis. In that case the 𝛼 component of the current derivative is not affected by the rotor position.

Fig. 9. Estimation results (average of current derivatives is done in the last 5𝜇𝑠 of each PWM configuration)

Fig. 10. Measured current derivatives and currents (𝜃𝑚𝑒 = 0, 𝑈𝑑𝑐 = 100𝑉, voltage vectors 𝑉̅0 , 𝑉̅1 , 𝑉̅2 , 𝑉̅7 , 𝑉̅4 and 𝑉̅5 ) (from top: 𝛼, 𝛽 and 𝑜 components).

VII.

CONCLUSIONS

In this paper the issue of rotor position estimation in IPMSMs based on persistent PWM excitation and current derivative measurement has been investigated. A complete mathematical model has been developed, by taking also into account the dependence of the rotor position estimation error on the mutual inductance, which is neglected in the past literature adopting the same sensorless approach. Measurement of current derivatives is done by dedicated and optimally designed Rogowski coils, oversampling and processing of the measured values. Simulation results confirm the reliability of the analytical developments. Experimental investigations prove the feasibility of the method but highlights some phenomena related to the presence of parasitic in the actual system and un-modelled effects of the permanent magnet flux linkage that requires further investigations.

APPENDIX Parameters of the IPM motor considered in this paper. Pole pairs, 𝑝𝑝 Stator resistance, 𝑅 Direct inductance, 𝐿𝑑 Quadrature inductance, 𝐿𝑞 PM flux linkage (amplitude), Λ 𝑚𝑔 Rated current, |𝑖|𝑛,𝑟𝑚𝑠 Electromagnetic torque, 𝑇𝑒, @500/18000𝑟𝑝𝑚

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[8] Fig. 11. 𝛼𝛽𝑜 measured current derivatives (PM aligned along phase 𝛼, 𝑈𝑑𝑐 = 300𝑉, voltage vectors 𝑉̅0 , 𝑉̅1 , 𝑉̅2 , 𝑉̅7 , 𝑉̅4 and 𝑉̅5 ). [9]

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[15] Fig. 12. 𝛼𝛽𝑜 measured current derivatives (PM aligned along phase 𝛽, 𝑈𝑑𝑐 = 300𝑉, voltage vectors 𝑉̅0 , 𝑉̅1 , 𝑉̅2 , 𝑉̅7 , 𝑉̅4 and 𝑉̅5 ).

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