Sensorless Capability of Permanent Magnet Synchronous Machines Due To Saturation- And Reluctance-Based Coupling Effects Andreas Eilenberger, Erich Schmidt, Manfred Schr¨odl Vienna University of Technology Institute of Electrical Drives and Machines Gusshausstrasse 25-29, A-1040 Vienna Email:
[email protected]
Abstract—The paper shows the theoretical derivation of sensorless capability of a permanent magnet synchronous machine due to saturation- and/or reluctance-based coupling effects. Only due to an introduction of coupling terms in the rotor-oriented inductance tensor a dependency of double the electrical rotor position can be calculated.
I. I NTRODUCTION A lot of publications handle with sensorless capability of salient permanent magnet synchronous machines (PMSM) [1] and therefore assume unequal inductances in direct and quadrature direction of a rotor-oriented two-axis inductance tensor. In most cases the coupling terms between the fluxlinkage space phasor in direct direction to the current space phasor in quadrature direction and vice versa are neglected. This work gives a theoretical derivation of sensorless capability of PMSM’s also due to these coupling terms which consider saturation- and/or reluctance-based effects. These fictive components of the inductance tensor in the rotor-oriented reference frame are represented as rotor position dependent phase inductances which also includes higher harmonics up to second order to sufficiently describe the saliency of a PMSM [2], [3]. Furthermore some simulation results are given for better comprehension. Andreas Eilenberger June 10, 2010 II. F UNDAMENTALS A. Phase Inductances According to the ideas in [2] the self- and mutual phase inductances l, m ψ = l s,uvw s,uvw luu mvu ψu ψv = muv lvv muw mvw ψw
mwu mwv lww
i s,uvw iu iv iw
(1) (2)
are represented by their Fourier series expansion in dependence of the angular rotor position γ up to second order
elements with luu lvv
= l0 + l2 cos (2γ) , = l0 + l2 cos (2γ + 2π/3) ,
(3) (4)
lww mwv = mvw
= l0 + l2 cos (2γ + 4π/3) , = m0 + m2 cos (2γ) ,
(5) (6)
muw = mwu mvu = muv
= m0 + m2 cos (2γ + 2π/3) , = m0 + m2 cos (2γ + 4π/3) .
(7) (8)
B. Stator-Oriented Inductances The linear transformation matrix Tuvw for transforming αβ three-phase to equivalent two-phase quantities and Tαβ uvw for opposite direction Tuvw αβ Tαβ uvw
2 = 3
1 1 − √2 3 0 2
1 = − 21 − 12
0
√
3 2√ − 23
−√12 − 23
,
,
(9)
(10)
yields to the self- and mutual stator-oriented inductances lαα , lββ and mαβ in the stator-oriented fixed reference frame expressed with the self- and mutual phase inductances luu , lvv , lww , mwv , muw and mvu in a three-phase winding configuration (zero-sequence self and mutual inductances are neclegted) αβ l s,αβ (l0,2 , m0,2 ) = Tuvw αβ · l s,uvw (l0,2 , m0,2 ) · Tuvw lvv +lww 2 1 (l − m − m ) + m + uu uv uw vw 3 3 2 = √1 muv − muw − lvv −lww 2 3 ! √1 muv − muw − lvv −lww 2 3 . lvv +lww − mvw 2
(11)
(12)
Furthermore with above representations eq. 3-8, the self- and mutual stator-oriented inductances follow to l0 − m0 + 12 l2 + m2 cos 2γ lαα mαβ = 1 mαβ lββ sin 2γ l + m 2 2 2 ! 1 + m2 sin 2γ 2 l2 . l0 − m0 − 12 l2 + m2 cos 2γ With the components of the inductance tensor quantity 1 lαα = l0 − m0 + l2 + m2 cos 2γ, 2 1 l2 + m2 cos 2γ, lββ = l0 − m0 − 2 1 mαβ = l2 + m2 sin 2γ. 2
ψ s,dq , the complex stator current space phasor i s,dq and the complex stator flux linkage space phasor due to the permanent magnets ψ pm,dq in the rotor-oriented fixed reference frame dq and the tensor quantity l s,dq as follows ψd = ψd + ψq , ψ s,dq = ψ q ψpm ψ pm,dq = = ψpm , 0 id = id + iq , i s,dq = iq ψ = l s,dq i s,dq + ψ pm,dq , s,dq ψpm id ψd ld 0 , + = 0 lq 0 iq ψq
l s,αβ
(13) (14) (15)
(26) (27) (28) (29) (30)
the subsequent flux linkage equations are achieved
C. Rotor-Oriented Inductances 1) With decoupled stator flux components: With the linear dq transformation matrix Tαβ dq (γ) and Tαβ (γ) for linear coordinate transformations between the stator- and rotor-oriented fixed reference frame and vice versa cos γ sin γ αβ , (16) Tdq (γ) = − sin γ cos γ cos γ − sin γ Tdq , (17) αβ (γ) = sin γ cos γ and the trigonometrical identities 1 + cos 2γ cos2 γ = , (18) 2 1 − cos 2γ sin2 γ = , (19) 2 sin 2γ sin γ cos γ = , (20) 2 the self- ldd , lqq and mutual ldq , lqd inductances in the rotororiented fixed reference frame, the coefficients la , lb , ma and mb of the Fourier series expansion of the phase inductances (eq. 3-8) and the inductance tensor quantity l s,dq ldd lqd ld m l s,dq = = , (21) ldq lqq m lq follows to dq l s,dq (l0,2 , m0,2 ) = Tαβ dq (γ) · l s,αβ (l0,2 , m0,2 ) · Tαβ (γ) (22) 0 l0 − m0 + 21 l2 + m2 . = 0 l0 − m0 − 21 l2 + m2
With
ld lq m
1 l2 + m2 , 2 1 l2 + m2 , = l0 − m0 − 2 = 0. = l0 − m0 +
(23)
ψd ψq
= ld id + ψpm , = lq iq .
(31) (32)
The self- lαα , lββ and mutual mαβ inductances in the statororiented fixed reference frame (eq. 11) follows with the rotororiented self inductances ld and lq to αβ l s,αβ (ld,q ) = Tdq αβ (γ) · l s,dq · Tdq (γ)
=
l −l ld +lq + d 2 q cos 2γ 2 ld −lq sin 2γ 2
(33)
ld −lq sin 2γ 2 ld +lq ld −lq − cos 2γ 2 2
!
.
Furthermore subsequent abbreviations ld + lq , lm = 2 ld − lq l∆ = , 2
lm = l∆ =
lm 0 l∆ 0
0 lm
, 0 , l∆
(34) (35)
simplify to l s,αβ =
lm + l∆ cos 2γ l∆ sin 2γ
l∆ sin 2γ lm − l∆ cos 2γ
.
(36)
2) With coupled stator flux components: With the tensor quantity l s,dq the components of the stator flux linkage space phasor ψd , ψq in the rotor-oriented reference frame and introducing mutual inductances m = ldq = lqd according to the rotor-oriented fixed reference frame leads to ψd
= ld id + miq + ψpm
(37)
ψq
= lq iq + mid .
(38)
(24) (25)
According to the two-axis theory [4], [5] and defining the components of the complex stator flux linkage space phasor
Again with eq. 33 and eq. 18-20 the self- and mutual inductances lαα , lββ and mαβ in the stator-oriented fixed reference frame (eq. 11) follow with the self inductances ld , lq and the mutual inductance m in the rotor-oriented fixed reference
mwv
frame to l s,αβ (ld,q , m) =
l −l ld +lq + d 2 q cos 2γ − m sin 2γ 2 ld −lq sin 2γ + m cos 2γ 2 ld −lq sin 2γ + m cos 2γ 2 ld −lq ld +lq − cos 2γ + m sin 2γ 2 2
=
lm + l∆ cos 2γ − m sin 2γ l∆ sin 2γ + m cos 2γ ! l∆ sin 2γ + m cos 2γ lm − l∆ cos 2γ + m sin 2γ
!
(39)
(40)
and yields to the components of the tensor quantity l s,αβ as stator-oriented self- and mutual inductances lαα
= lm + l∆ cos 2γ − m sin 2γ,
(41)
lββ mαβ
= lm − l∆ cos 2γ + m sin 2γ, = l∆ sin 2γ + m cos 2γ,
(42) (43)
= mvw = m0 + m2 cos (2γ + ε) , (56) q 2 2 1 l + m2 cos (2γ + ε) , (57) = − lm + 3 3 ∆ muw = mwu = m0 + m2 cos (2γ + 2π/3 + ε) , (58) q 1 2 2 l + m2 cos (2γ + 2π/3 + ε)(59) = − lm + 3 3 ∆ mvu = muv = m0 + m2 cos (2γ + 4π/3 + ε) , (60) q 2 2 1 l + m2 cos (2γ + 4π/3 + ε)(61) = − lm + 3 3 ∆ D. Stator-Oriented Flux Linkage Space Phasor The stator flux linkage space phasor ψ s,αβ in the statororiented fixed reference frame αβ follows to i h ψ s,αβ = Tdq αβ (γ) · l s,dq · i s,dq + ψ pm,dq dq αβ = Tdq αβ (γ) · l s,dq · Tdq (γ) · i s,αβ + Tαβ (γ) · ψ pm,dq
= l s,αβ (ld,q , m) · i s,αβ + Tdq αβ (γ) · ψ pm,dq .
ψα ψβ
With comparison of eq. 13 and eq. 41 we get the relation lαα lββ mαβ
q 2 + m2 cos(2γ + ε), l∆ q 2 + m2 cos(2γ + ε), = lm − l∆ q 2 + m2 sin(2γ + ε), l∆ = = lm +
(44) (45) (46)
with tan ε =
m . l∆
(47)
The self- and mutual phase inductances follow with uvw l s,uvw (lαα,ββ , mαβ ) = Tαβ (48) uvw · l s,αβ (lm,∆ , m) · Tαβ
=
2 − 13 lαα + √13 mαβ 3 lαα − 31 lαα + √13 mαβ 16 lαα + 21 lββ − √13 mαβ 1 1 − 31 lαα − √13 mαβ 6 lαα − 2 lββ ! − 13 lαα − √13 mαβ 1 1 , 6 lαα − 2 lββ 1 1 √1 mαβ l + l + αα ββ 6 2 3
luu
lvv
lww
= l0 + l2 cos (2γ + ε) , (50) q 2 2 2 = lm + l + m2 cos (2γ + ε) , (51) 3 3 ∆ = l0 + l2 cos (2γ + 2π/3 + ε) , (52) q 2 2 2 l + m2 cos (2γ + 2π/3 + ε) , (53) lm + = 3 3 ∆ = l0 + l2 cos (2γ + 4π/3 + ε) , (54) q 2 2 2 l + m2 cos (2γ + 4π/3 + ε) , (55) = lm + 3 3 ∆
iα lm + l∆ cos 2γ l∆ sin 2γ = l∆ sin 2γ lm − l∆ cos 2γ iβ iα −m sin 2γ m cos 2γ + m cos 2γ m sin 2γ i β ψ pm cos γ − sin γ + sin γ cos γ 0 i l∆ 0 lm 0 α + = 0 l∆ 0 lm iβ iα cos 2γ − sin 2γ sin 2γ cos 2γ −iβ iα 0 m cos 2γ sin 2γ + m 0 − sin 2γ cos 2γ iβ ψ pm cos γ − sin γ . (62) + sin γ cos γ 0
The expression of the stator-oriented stator flux linkage ψ s,αβ with the rotor position independent stator-oriented equivalent inductance tensors l m and l ∆ follow to ψ s,αβ
αβ ∗ = l m i s,αβ + l ∆ Tdq αβ (2γ) i αβ + m Tdq (2γ) i s,αβ
with
(63) +Tdq αβ (γ) ψ pm,dq h i = l m + (l∗∆ + m) · Tαβ dq (2γ) · i s,αβ + ψ pm,αβ .
(49)
to a modified approach for the Fourier series expansion of the phase inductances (see eq. 2)
m=
0 m
m 0
.
(64)
The expression of the stator-oriented stator flux linkage space phasor ψ s,αβ with the rotor position dependent stator-oriented equivalent inductance tensor l s,αβ (2γ) follows to ψ s,αβ = l s,αβ (2γ) · i s,αβ + ψ pm,αβ ,
(65)
and yields to the coherence of the rotor position dependent inductance tensor l s,αβ and the rotor position independent inductance tensors l m and l ∆ with l s,αβ = l m + (l∗∆ + m) · Tαβ dq (2γ).
(66)
III. S IMULATION AND M EASUREMENT R ESULTS Fig. 1 depicts the complex solution of the stator-oriented flux linkage space phasor divided by the stator-oriented current space phasor. Fig. 2 depicts the phase shift ε, difference
ld=2; m=0
ld=1.5, m=0
ld=1; m=0.25
ld=1; m=0.5
0.6 0.4 0.2 0 −0.2
Fig. 3.
−0.4
Picture of the used 400 Nm PMSM for measuring ε
−0.6 0
Fig. 1.
0.5
Simulation of ℑ
1
n
ψ
s,αβ
/i s,αβ
1.5
o
ov. ℜ
2
n
ψ
s,αβ
/i s,αβ
o
with
ψpm = 0, lq = 1 and sinusoidal current supply at various direct ld , quadrature lq and mutual m inductances
between measured and sensorless calculated angular rotor position, over the magnitude of the quadrature current space phasor of the PMSM according to fig. 3. The sensorless calculation of the angular rotor position was done with the so called INFORM method which is based on the second order element of the Fourier series expansion.
step was the well known transformation to the fictive selfand mutual stator-oriented inductances lαα , lββ and mαβ with neglected self- and mutual zero sequence inductances. Third step was introducing the self inductances in direct and quadrature direction at rotor-oriented reference frame, with the solution of non existing mutual inductances in the rotororiented reference frame due to the given approach of the Fourier series expansion up to second order elements. The second order coefficients also show the dependency of the angular rotor position. Next important step is introducing mutual inductances in the rotor-oriented reference frame due to saturation effects in the stator and/or rotor. With this approach we obtain to the same coefficients of the Fourier series expansion but with an additional angular phase shift ε. That means that the sinusoidal spatial distribution of a given phase inductance is shifted by a phase angle ε due to saturation effects. We also get an angular rotor position dependency due to the these effects. V. C ONCLUSION The paper discusses an analysis of the rotor-oriented reference frame mutual inductances, which results in coupled stator flux components, with an additional phase shift according to the Fourier series expansion of given self- and mutual phase inductances. R EFERENCES
Fig. 2. Angular phase shift ε over the magnitude of the quadrature current load with iq at CH1: 1 p.u./div. and ε at CH2: 22.5 deg./div.
IV. D ISCUSSION Starting with the measureable self- and mutual phase inductances luu , lvv , lww , muv , mvw and mwu we firstly introduce the ideas of [2] with a representation by their Fourier series expansion in dependence of the angular rotor position. Next
[1] M. Schr¨odl, Sensorless Control of A.C. Machines, VDI-Verlag, VII, Fortschritt-Berichte VDI : Reihe 21 Elektrotechnik, Nr. 117, ISBN 318-141721-1, January 1992 [2] S. I. Nabeta, Etude des Regimes Transitoires des Machines Synchrones par la Methode des Elements Finis, Doctoral thesis (in French), INP Grenoble, 1994 [3] E. Schmidt, Finite Element Analysis of Electrical Machines, Transformers and Electromagnetic Actuators, postdoctoral lecture qualification, Vienna University of Technology, 2007. [4] R.H. Park, Two-Reaction Theory of Synchronous Machines Generalized Method of Analysis - Part I, Transactions of the American Institute of Electrical Engineers (AIEE), New York, N.Y., July 1929, Vol. 48, pp. 716-727 ˇ epina, Raumzeiger als Grundlage der Theorie der elektrischen [5] J. Stˇ Maschinen, ETZ-A, 1967, Bd.88, pp. 584-588
