Recent Developments of Electrical Drives - Part 16 potx

For small stator currents, this flux is mainly generated by the high-grade permanent magnets, buried within the rotor.. In a lot of drives, using field-oriented control, the rotor flux v

II-1 A GENERAL DESCRIPTION

OF HIGH-FREQUENCY POSITION

ESTIMATORS FOR INTERIOR

PERMANENT-MAGNET SYNCHRONOUS MOTORS

Frederik M.L.L De Belie, Jan A.A Melkebeek, Kristof R Geldhof,

Lieven Vandevelde and Ren´e K Boel

Electrical Energy Laboratory (EELAB), Department of Electrical Energy, Systems and Automation (EESA), Ghent University (UGent), Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium

frederik.debelie@ugent.be, Jan.Melkebeek@ugent.be, Kristof.Geldhof@ugent.be

Lieven.Vandevelde@ugent.be, Rene.boel@ugent.be

Abstract This paper discusses fundamental equations used in high-frequency signal based interior

permanent-magnet synchronous motor (IPMSM) position estimators For this purpose, an IPMSM model is presented that takes into account the nonlinear magnetic condition, the magnetic interaction between the two orthogonal magnetic axes and the multiple saliencies Using the novel equations, some recently proposed motion-state estimators are described Simulation results reveal the position estimation error caused by estimators that neglect the presence of multiple saliencies or that consider

the magnetizing current in the d-axis only.

Introduction

Vector control of a high-dynamical, high-performance interior permanent-magnet syn-chronous motor (IPMSM) requires the stator flux linkage vector For small stator currents, this flux is mainly generated by the high-grade permanent magnets, buried within the rotor

In a lot of drives, using field-oriented control, the rotor flux vector is considered instead of the stator flux linkage vector Moreover, the rotor flux direction can be approximated by the rotor position, measured with a mechanical sensor

During the last 15 years, motion-state estimation methods have been developed with the intention to remove the expensive mechanical transducer, which, due to temperature variations and mechanical vibrations, produces measurements of low reliability Modern sensorless drives try to estimate the motion states from measurements of electrical variables Filtering techniques and observing strategies are used to estimate the back-EMF vector and from that the rotor speed and angle However, for a slow rotor motion, small signals have to

be measured or calculated that are disturbed strongly by noise produced by normal operation

of the PWM and motor As a result, the precision of such estimators in the low speed region

is insufficient to control the motor in a stable and efficient way

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 141–153.

2006 Springer.

position estimators For this purpose, the small signal dynamic flux model, presented in [9], is used which takes into account the nonlinear magnetic condition and the magnetic interaction between the direct and the quadrature magnetic axis An addition to the model is given to tackle the presence of multiple saliencies By using the novel equations some recently proposed motion-state estimators are described It is shown that the higher the inductance difference between the two orthogonal magnetic axes, the higher the position estimation resolution Furthermore, simulation results reveal the position estimation error caused by estimators that disregard the existence of multiple saliencies or that consider the magnetizing

current vector in the d-axis only.

General description of a PMSM

Small signal dynamic flux model

To obtain accurate position estimations at low speed, in recently proposed estimation meth-ods a high-frequency voltage is supplied, which generates a high-frequency variation of small amplitude in the stator, flux linkage [1–8] This implies that, to describe position esti-mators, a small signal dynamic flux model can be used In an IPMSM, without the damper effects e.g due to short-circuited windings or eddy currents, the flux variation generated

by the high-frequency voltage mainly occurs in the main flux φ m instead of the leakage flux path As a result, the small signal dynamic model of a saturated synchronous machine, presented in [9], can be used This model is given by the flux equation

m (t )=

⎛

⎜

⎝L qmtcos 2μ + L qmosin

2(L qmt − L qmo) sin (2μ)

1

2(L dmt − L dmo) sin (2μ) L dmtsin 2μ + L dmocos2μ

⎞

⎟

⎠i qd

written in a reference frame (qd) fixed to the physical quadrature and direct axis and

with i m the magnetizing current, with, see Fig 1, L qmo , L dmothe chord-slope magnetizing

inductances and L qmt , L dmt the tangent-slope magnetizing inductances in quadrature and direct magnetic axis respectively,μ the angle between the q-axis and the vector i mand

denoting the small variation of a vector In a current controlled drive, the vector i m and the angle μ are regulated to a constant during steady state By using the flux equation

(1) the saturation level in both magnetic axes is assumed to be determined by i mand as a result the proposed small signal dynamic flux model includes cross saturation or magnetic

Figure 1 Magnetizing characteristic with i moandφ mo the average modulus of i mandφ mrespectively.

interaction between d- and q-axis However, this model neglects possible stator leakage flux

saturation

In some high-frequency signal based sensorless drives, a small high-frequency stator current is supplied instead of a voltage Therefore, fundamental equations used in position estimators are given for current as well as voltage sources Nevertheless, it will follow from the discussion that both methods can be described in a similar way

High-frequency current source

An estimation algorithm using a high-frequency current source measures the high-frequency flux response For these estimators, the flux equation (1) is written in a complex notation,

with the real axis parallel to the q-axis, as

m (t) = l · i qd

with the complex inductance l given by

where

− j2μ+L q + L d

2 L d = L dmo + L dmt

2 L d = L dmo − L dmt

and measure the current response For these estimation methods, the flux equation (1) is written as

m (t) = r · φ qd

with the complex reluctance r given by

where

L q L d + (L q L d − L d L q) cos(2μ) − L q L d

(12)

2 +L q + L d

− j2β(t)

written as

(L q L d + (L q L d − L d L q) cos(2μ) − L q L d)

×



2 −L q − L d

j 2 

m (t)

−



2 −L q + L d

j 2 μ

m (t )

(16)

Discussion

As most estimators are based on a current response to a high-frequency voltage variation, the following discussion will be restricted to such strategies However, as the equations for

an estimator, using a high-frequency current source, are similar to those in (10)–(16), the following discussion applies to both cases

From the reluctance r in (11) it can be seen that, in addition to a current change in

phase withφ due to L in (13), two important components in the current variation can be

distinguished As follows from (14), a part of the current change is proportional to the

inductance difference between q- and d-axis and is phase shifted from φ m over−2β.

Another current variation, according to (15), is linked with the differences between the

chord-slope inductance and the tangent-slope inductance in both q- and d-axis.

If the saturation level is low, the chord-slope inductance equals almost the tangent-slope inductance Consequently, the inductance in (15) becomes small and a phase shift between

i mandφ mis the result of the inductance difference in (14) only Clearly, the component

in (14) reflects the reluctance variation along the air gap with extrema in both orthogonal magnetic axes

The reluctance r in pu, in the case of an unsaturated salient-pole synchronous machine,

is shown in Fig 2(a) It is given for various values of the inductance difference between the two magnetic axes The trajectories of i m for a circular trajectory ofφ m, shown

in Fig 2(b), are elliptical with axes of symmetry in q- and d-directions, corresponding to the point of minimum and maximum modulus of r respectively Furthermore, for a given

value ofβ, the higher the difference between the q- and d-inductance, the higher the angle

betweeni mandφ m

Figure 2 Reluctance r in pu and current response to small flux variations in the case of magnetic

saliency withβ as parameter.

Figure 3 Reluctance r in pu and current response to small flux variations in the case of saturation

withβ .as parameter

If the PMSM has a uniform air-gap permeance, most controllers disregard the reluctance

variation along the air gap Consequently, the direction of the q-axis fixed to the rotor can

be chosen deliberately Furthermore, the reciprocity property, mentioned in [10], implies that

As a result, the difference in (14) becomes zero and the inductance (15) reduces to

This means that a noticeable phase shift betweeni mandφ mis caused by (18) only

In the case of a saturated smooth air-gap synchronous machine the reluctance r in pu is presented in Fig 3(a) for a given modulus of i m This figure shows that the direction of i m

influences the phase shift betweeni mandφ mfor the sameβ The trajectory of i mfor

a circular trajectory ofφ mis shown in Fig 3(b) The figure shows elliptical trajectories with a maximumi m in the direction of i m, corresponding the point in Fig 3(a) with the

maximum modulus of r.

Multiple saliencies Due to the construction of the rotor, the reluctance variation along the air gap can display

global extrema in q- and d-axis as well as several local extrema Such a reluctance variation

Figure 4 Reluctance r in pu of an IPMSM with and without multiple saliencies with β as parameter.

is called a multiple saliency Assuming sinusoidal reluctance variations, these multiple saliencies can be modeled in a similar way as the reluctance variation with extrema in

q- and d-axis only As a result, by using previous discussion, the equation in (14) can be

replaced for modeling multiple saliencies by

i

withϕ ia possible space phase shift

To illustrate the model with multiple saliencies, the trajectory of r for an unsaturated

salient-pole machine withβ as parameter is calculated by using (19) instead of (14) Two

cases, with and without an extra sinusoidal reluctance variation having four extrema per

pole pitch (i equals to 2 and 4 in (19)), are shown in Fig 4 This reluctance trajectory is also

observable in [7] and mentioned in [8] In [7], by applying finite element simulations, almost the same trajectory as in Fig 4 can be observed In [8] the effect of multiple saliencies is measured as a variation in the stator current instead of an inductance However, these results are not modeled such as in (19)

Recently proposed estimators

Approximated small signal dynamic flux model

In an IPMSM the magnetizing current is mainly generated by the permanent magnets For

this reason, in some recently proposed sensorless drives, such as [1–8], the angle of i mis approximated byπ/2 For μ equal to π/2, the equation (10) results in

m (t )= 1

2 − L qmo − L dmt

− j2β(t)

By defining

the relationship (20) can also be written as

L2− L2e − j2β(t)

j θ r i αβ

mo + i αβ

L2− LL2 j θ r φ αβ

mo + φ αβ

m

mo + φ αβ

m



∗ e j 2 θr (24)

mo + φ αβ

m = Lj θ r i αβ

mo + i αβ m



+ Le j 2 θr

mo + i αβ m

 (25) results in the matrix notation

m (t )= L + L cos (2θ r) L sin (2θ r)

· i αβ

This equation is well-known as it shows the sinusoidal variation of the magnetic reluctance along the air gap with the pole pitch as period

Stator voltage equation

By supplying a high-frequency voltage to the motor terminals, a high-frequency stator flux linkage variation of small amplitude is generated In an IPMSM, without damper effects e.g due to short-circuited windings or eddy currents, this flux variation mainly occurs in the main flux instead of the leakage flux path For this reason and by disregarding the voltage drop across the stator resistance and leakage inductance, the motor voltage equation, in a two-dimensional stationary reference system (αβ), can be approximated by

s (t)=φ

αβ

m (t)

withυ s the complex stator voltage Transformation of (27) to the synchronous reference

frame (qd) results in

s (t )= φ

qd

m (t)

qd

High-frequency voltage pulse train

In modern IPMSM drives, a pulse-width-modulated (PWM) inverter is used This means that, at normal operation, a voltage pulse train at high frequency is supplied to the motor

terminals From equation (28) it follows that the current variation is piecewise linear As

a result, according to the model in (10), a magnetizing current variation occurs, which

depends on the direction of the main flux variation in the reference frame (qd) and on the magnetizing current For a current controlled drive, the current i m can be approximated

by the desired i mcalculated within the controller Moreover, in the synchronous reference

frame, the variation of i mequalsi s , with i sthe complex stator current, as the equivalent magnetizing current due to the magnets is constant

The high-frequency flux variation, generated by using a PWM, can be used to estimate the rotor angle Calculating the main flux variation with (28) and transforming the measured stator current to an estimated synchronous reference frame ( qd) , it follows from (10) that

an estimation of the reluctance r can be obtained Substituting i m with its desired value calculated within the controller, the angleμ and the inductances in (7)–(8) can be

approxi-mated, which result, together with the estimated r, in an estimation of the angle β As the

angle of the main flux variation can also be calculated in the stationary reference frame by

using (27), a new estimation of the q-axis is obtained.

If the motion-induced voltage is known,φ mcan be calculated by using (28) However, for a slow rotor motion, a back-EMF of small amplitude has to be measured or calculated which is strongly disturbed by noise produced by normal operation of the PWM and motor

As for most drives the mechanical time constant is higher than the electrical one, the rotor speed and the motion-induced voltage can be assumed to be constant during a sufficiently small time period Therefore, by subtracting the stator voltage generated by two successive PWM pulses, back-EMF measurements are avoided Together with (10), this results in the following system

m ,2 − φ dq

m ,1

qd

s ,2 − υ qd

s ,2 − i dq

s ,1

t = r(β2)φ qd

m ,2

t − r(β1)φ qd

m ,1

In the sensorless drive presented in [1], the reluctance r is estimated by using equation (30).

This method is called indirect flux detection by online reactance measurement (INFORM)

as introduced by Schr¨odl However, in such an estimatorμ is approximated by π/2, which

means that the reluctance r is estimated by using equation (22) instead of (10) Furthermore,

aβ2value is used that is equal toβ1+ π As a result, the system of (29) and (30) together

with the relationship in (22), results in

s ,1 = −υ qd

s ,2 − i dq

s ,1

L

L2− L2e − j2β2

2υ qd

as the reluctance r in (22) varies periodically with 2 β Furthermore, in the INFORM method

the estimation of r is repeated in the two other stator phases The average reluctance of the

three phases approximately coincides with

3)+ r(β −2π

3)

voltage In most of these strategies, calculations are performed in a stationary reference frame by using the equations in (24) and (27) The stator voltage is given by

s (t) = V αβ

with V sa complex voltage and withω ithe pulsation of the injected high-frequency voltage

υ s ,i With the voltage in (34), the voltage equation in (27), at a high frequency, is written

as

υ αβ s ,i(ω i t)=d φ

αβ

m ,i(ω i t)

Furthermore, at a sufficiently high frequency, the rotor angle can be assumed to be constant Consequently, equation (24), for a sufficiently high frequency, results in

s ,i(θ r , ω i t)

L2− L2e j 2 θ r υ αβ s,i(ω i t )



In some methods a voltage rotating at a high pulsationω iis added to the stator voltage

The high-frequency current response, obtained by using (36), will include a positive and negative rotating component

i αβ

s ,i(θ r , ω i t) = I0e j ω i t + I1e j (2 θ r −ω i t) (38) with

ω i

(39) Transformation of the current response to a reference frame that rotates atω iresults in

i ω i

s ,i(θ r , ω i t) = I0+ I1e j 2( θ r − ω i t) (40)

In other estimation methods a high-frequency voltage in an estimated quadrature axis is added to the fundamental voltage

υ αβ s ,i( ˆθ r , ω i t) = V icos (ω i t) e j ˆ θ r (41)