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NORPIE 2004, Trondheim, Norway Saliency Modeling in Radial Flux Permanent Magnet Synchronous Machines Sigurd Ovrebo ([email protected]) Smartmotor AS, Stiklestadveien 1 N-7041 Trondheim, Norway Prof. Roy Nilsen, Prof. Robert Nilssen Norwegian University of Science and Technology, 7491 Trondheim, Norway d − axis Abstract—Sensorless control of Permanent Magnet Synchronous Machines is popular for several reasons: cost saving and system reliability. The basis for most low and zero speed sensorless control is that there exist a difference in the direct and quaderature inductance of the machines. This difference is also reefed to as saliency. In this paper the different contributors to the resulting saliency is modeled. This type of modeling is useful for several reasons: increased understanding of the fundamentals of sensorless control, incorporating the combined effects of leakage and main inductance variations, incorporating the effects of loading of the machine and finally it gives valuable insight when the saliency ratio is attempted increased by machine design. Ld q − axis Lq Figure 1 Salient pole synchronous machine Index Terms—Sensorless control, saliency modeling, PMSM I. S INTRODUCTION ensorless control of electric machines has been a research topic for more than two decades. The objective for this work is to eliminate the need for a position sensor by using the machine as the position sensor. The position sensor, cabling and connectors have been a source of failure for motor control applications. For small drives the sensor contributes considerable to the overall cost. The basis for most low and zero speed sensorless control [1-5] is the presents of a difference in the d and q inductances. This difference is referred to as saliency. There are several sources of saliencies in PM machines: rotor inherent saliency, saturation based saliency (yoke, teeth) [3], rotor stator teeth harmonics [1], lamination direction based saliency [7], eddy current based saliency [8], rotor eccentricity based saliency. In this paper a single saliency model is developed from the rotor inherent saliency and the saturation induced saliency. The term saliency can be understood when considering a salient pole synchronous machine structure (Figure 1). In this structure the rotor has an inherent saliency due to the shape of the rotor. In the d-axis the air-gap is small while there is a large air-gap in the q-axis. The d-axis inductance will in this case be larger then the inductance in the q-axis. The injection based sensorless control schemes [1-5] extract rotor position information from the difference in the inductance. II. SALIENCY MODELING In Sensorless Control of PMSM the rotor inherent saliency or the saturation based saliency is usually used as the basis for the saliency. The presented saliency model is simplified as only these sources of saliency are included. In order to illustrate the saturation based saliency a simplified model of a 30 kW IPMSM machine was simulated in FEMLab. The relative permeability function (from supplier) was included in the FEM simulation. In Figure 2 the machine is unloaded and only the flux from the permanent magnets contributes to the field. Three regions of the machine can be seen to have low relative permeability values: the permanent magnet (in rotor), the stator teeth (in the main flux path) and the stator yoke (in the main flux path). The stator teeth are more saturated compared to the portions of the stator yoke that is saturated. d − axis q − axis Yoke saturation Tooth saturation Figure 2 Relative permeability, IPMSM (8-pole) FEM simulation NORPIE 2004, Trondheim, Norway The modeling of saliency is done by modeling the inductance in a dq-rotor oriented reference frame. The inductance is separated in two parts: the leakage inductance and the main inductance. In this way the model can be used to incorporate the saliency sensed by carrier signals that is concentrated in the leakage flux paths or main flux path (or both). The sensorless schemes [1-5] will in general sense the resulting inductance but the high frequency carrier flux distribution determines the components of main- or leakage flux. In the first stage of the modeling the machine will be modeled at no load (only flux from the permanent magnets). The load dependency is included in the last section of the modeling. Figure 3 illustrates why the main and leakage inductance should be modeled separately: the main flux and the leakage flux are 90 electrical degrees spatially shifted. In order to simplify the modeling some simplifications are made: • Only fundamental components are considered • The flux from the magnets and the windings are assumed sinusoidal distributed over the air gap. The distributed windings are simplified by representing them as a two axis model. • Saturation effects are assumed to be sinusoidal distributed (superposition applied). • Relative permeability assumed to wary linear with flux density These simplifications may seem crude. Some lamination materials may have extremely non linear relation between relative permeability and flux density. The machine in the test setup has laminations from Congent (M235-35). The relative permeability is derived from the BH curves from the supplier. Figure 4 shows the relative permeability for the laminations (measurement errors in the 100 and 200 Hz curves). From the 50 Hz curve an approximate linear relation can be obtained in the region 0.4 to 1.3 T. The non linearity in the lower region is related to the first magnetization curve of the material. When the machine model is considered this effect is neglected as the laminations have been exposed to a β s Is Leakage flux regions ξ Figure 4 Relative permeability for 50,100,200 and 400 Hz excitation frequencies (Congent M235-35 lamination-supplier data). magnetic field and the time constant for the material to return to its original sate is very large. A. No load model In the no load model only the flux from the permanent magnets are considered. The inductance will be modeled in dq-rotor reference frame and the main and leakage inductance will be modeled separately. 1) Main inductance In the first stage the machine is considered as a uniform structure with no magnets or saturation. Ld −main = L0 Lq−main = L0 (1) (2) The sources of saliency for a no load condition is shown in Figure 5. These effects are included in the model as: Ld −main = L0 − L0 rd − L0ts − L0 ys (3) L0 - main inductance (symmetrical machine) L0rd - rotor design (low relative permeability in magnets and induced currents in magnets) L0ts - stator tooth saturation ψs L0 ys - yoke saturation αs The yoke saturation is assumed to have minimal impact on the q-axis inductance. If high frequency carrier is used there may be induced currents in the magnets due to the carrier flux. The effect of induced currents in the magnets is reduced d-axis inductance. This effect is included in the model by the term L0rd (combined effect of low permeability and induced currents). The induced current is highly dependent on the carrier frequency [8]. Figure 3 Current distribution, current and flux space vectors NORPIE 2004, Trondheim, Norway βs stator teeth saturation Stator Leakage Flux q d permanent magnets θ αs Rotor stator yoke saturation Figure 6 Leakage flux path 3) Resulting no load model In the resulting no load model the leakage and main inductance are summed up in each axis: Ld = Ld −main + Ld −leakage Figure 5 Sources of saliency – no load 2) Leakage inductance If we consider a flux vector (space vector) the leakage flux is located 90 electrical degrees spatially shifted compared to the main flux distribution. When the dq- inductances are modeled the reference system is referred to the main flux. The effects on the leakage inductance are there fore shifted 90 electrical degrees. The basis for the leakage inductance model is also a uniform structure with no saturation: Ld −leakage = Lσ (4) Lq−leakage = Lσ (5) The leakage flux paths are illustrated in Figure 6. Three regions may lead to modeling of the inductance: stator teeth, stator yoke and rotor surface. Variations (or saturation) in the rotor surface is neglected in this model as the surface of the test machine is uniform and the saturation level is modest. If we consider a stator flux vector oriented in the magnet d-axis the leakage flux path for this main vector can be influenced by yoke saturation (saturation due to flux from the permanent magnets). This effect is included in the model as a reduction of the d-axis leakage inductance by a factor Lσ ys . If a stator current vector is oriented in the q-axis the leakage inductance may be influenced by tooth saturation (saturation due to flux from the permanent magnet). This effect is included in the model as a reduction of the q-axis leakage inductance by a factor Lσ ts : Ld −leakage = Lσ − Lσ ys (6) Lq−leakage = Lσ − Lσ ts (7) Ld = L0 − L0 rd − L0ts − L0 ys + Lσ − Lσ ys Lq = Lq−main + Lq −leakage (9) Lq = L0 + Lσ − Lσ ts The difference between (8) Ld and Lq is the basis for saliency in the machine. The position dependent parts of the inductance can be illustrated by expressing the inductance on vector form: L = ΣL + ∆1Le j 2θ (10) where: θ - rotor position ΣL = ΣL = Ld + Lq 2 2 L0 − L0 rd − L0 ts − Lys + 2 Lσ − Lσ ts − Lσ ys ∆1L = ∆1L = (11) 2 Lq − Ld 2 − Lσ ts + Lσ ys + L0 rd + L0 ts + L0 ys (12) 2 If the main inductance variation is used as the reference the variation in the leakage inductance influence the magnitude of position dependent inductance. If the leakage yoke saturation term ( Lσ ys ) in (12) is larger than the leakage tooth saturation Lσ - leakage inductance (symmetrical machine) Lσ ys - yoke saturation effect on leakage inductance term ( Lσ ts ) the leakage terms adds to the main inductance Lσ ts oppose the position dependent terms from the main inductance. - tooth saturation effect on leakage inductance saliency. On the other hand if Lσ ts > Lσ ys the leakage terms NORPIE 2004, Trondheim, Norway B. Load dependency Several factors may change from an unloaded to a loaded machine: flux and saturation levels, flux and saturation orientation and saturation regions. In this model the orientation of the resulting stator flux is considered in the first stage. In the next stage saturation from the load current is included. In order to simplify the model the control optimal control of the PMSM is not considered. The control strategy of the machine is assumed to be Stator Load dependent Leakage Flux Saturated Regions id = 0 and iq = is* . Ψs I q Lq ∆θ Rotor Ψm Figure 8 Saturation due to leakage flux from load current Figure 7 Load dependency on stator flux The stator flux increases when the machine is loaded (Figure 7). The resulting flux vector is shifted by a term ∆θ due to the loading of the machine. ⎡ ⎤ iq Lq ⎢ ⎥ ∆θ ≈ a sin ⎢ ψ M 2 + (iq Lq ) 2 ⎥ ⎣ ⎦ L = ΣL' + ∆ 4 Le jξ (13) The shift in the stator flux results in a shift of the stator yoke and teeth saturation regions. In addition to the main flux effects the leakage effects must also be included. Figure 8 illustrates typical saturation regions due to the leakage flux from the load current. This type of saturation will result in a load dependent term in the q-axis leakage inductance ( Llds ): Lq−leakage = Lσ − Lσ ts − Llds (14) Lq = L0 + Lσ − Lσ ts − Llds (15) The spatial shift in the stator flux is incorporated in the model by shifting the saturation based saliency by the angle ∆θ : L = ΣL' + ∆ 2 Le − j 2θ + ∆ 3 Le − j 2(θ +∆θ ) (16) Where the upper index ‘ indicates that there may be a change in the saturation level due to the loading (as the stator flux increase) and the terms in (16) can be expressed as: ' ΣL = L'd + L'q 2 ' ' 1 ⎛ 2 L0 − L0 rd − L0 ts − Lys + ⎞ ' ΣL = ⎜ ⎟ 2 ⎜⎝ 2 Lσ − L'σ ts − L'σ ys − Llds ⎟⎠ L ∆ 2 L = 0 rd 2 − L'σ ts + L'σ ys + L'0 ts + L'0 ys − Llds ∆3L = 2 In this model the saturation is assumed sinusoidal distributed and linear. The different terms may be combined to form a resulting single saliency: (17) (18) (19) ξ = a tan( ∆4L = (20) sin( ∆θ ) ∆ 3 L ) ∆ 2 L + cos( ∆θ ) ∆ 4 L ( ∆ 2 L + cos( ∆θ )∆ 3L ) 2 (21) + (sin( ∆θ ) ∆ 3 L) 2 (22) The resulting model in (20), (21) and (22) describes the relation between the different sources of saliency in the IPMSM and load current in the q-axis. III. EXPERIMENTAL WORK A 30 kW radial flux IPMSM (motor data in Table 1) was used in the experimental work. The machine was designed for high fundamental flux (500-600 Hz) operation during field weakening. Table 1 IPMSM parameters PN [kW] UN [V] IN [A] nN [rpm] Rs [Ω] Ld[mH] Lq[mH] p 30 110 181 2500 0.011 0.123 0.381 4 The self and mutual inductance was estimated by applying a transient excitation [4] to the machine. The excitation sequence (t1-t5) is illustrated in Figure 9. The time interval between t1/t2 and t4/t5 was 12.5 us and t2/t3 and t3/t4 was 25 us. NORPIE 2004, Trondheim, Norway 011 100 011 a A inverter Control signals to inverter leg A,B,C IPMSM Ia 100 A V Ua n B V Uc V Ub b c C Figure 10 Measurements of self and mutual inductance t Figure 9 Transient excitation in the zero vector Figure 10 shows the connection between the inverter and the IPMSM. The pulses were applied across the a-phase and neutral point while the b and c phase was open. The current in the a-phase, the applied voltage and the induced voltage in the b and c-phase was measured. Current and voltage measurements where sampled at time instances t2, t3 and t4. From these measurements the self and mutual inductance can be derived as: t4 − t2 I a (t2 ) − 2 I a (t3 ) + I a (t4 ) t 4 − t2 Lab = U b I a ( t 2 ) − 2 I a ( t3 ) + I a ( t 4 ) t4 − t 2 Lac = U c I a (t2 ) − 2 I a (t3 ) + I a (t4 ) Laa = U a (23) (24) (25) The rotor was turned by hand and the inductance as a function of the position was estimated by current and voltage measurements. In Figure 11 there is a large difference in the position dependent part of the self and mutual inductance. In classical machine modeling [9] the self and mutual inductance are modeled as: La (θ ) = Laσ + La 0 − Lg cos(2θ ) L Lab (θ ) = − a 0 − Lg cos(2θ − 120) 2 Where: Laσ - leakage inductance phase a La 0 -average inductance (minus leakage) phase a Lg -position dependent inductance phase a (26) Laσ (θ ) = Laσ 0 − ( Laσ ts − Laσ ys ) cos(2θ ) (28) La (θ ) = Laσ 0 + La 0 + ( Laσ ts − Laσ ys − Lg ) cos(2θ ) (29) Where: Laσ 0 -average leakage inductance phase a Laσ ts -tooth saturation dependent term phase a Laσ ys -yoke saturation dependent term phase a The b and c-phase did not conduct any current and the leakage inductance in these phases where not measured. The expression in (27) is therefore assumed to be correct for the mutual inductance term. The classical modeling uses the average inductance as the basis for the model. In this paper a maximum inductance was used. In order to relate the two models (29) was used in the stationary inductance matrix and transformed to the dqrepresentation. By comparing the dq-inductance the relation between the models where found (Table2). la [mH] t4 t5 0.4 0.3 0.2 0.1 0 (27) 200 300 400 500 600 700 800 900 200 300 400 500 600 700 800 900 200 300 400 500 600 700 800 900 0.1 lab [mH] t3 0 -0.1 -0.2 -0.3 -0.4 0.1 lac [mH] t1 t2 In the saliency modeling in the previous section variations in the leakage inductance where assumed. If these variations are included in the model the self inductance can be expressed as: 0 -0.1 -0.2 -0.3 -0.4 rotor position[el.deg] Figure 11 Self and mutual inductance measured with transient excitation Table 2 Relation between Theoretical model and measurements NORPIE 2004, Trondheim, Norway Theoretical model Modified model(measurements) L0 + Lσ 3La 0 + Laσ 2 L0 rd + L0ts + L0 ys Lg Lσ ts Laσ ts 2 Lσ ys Laσ ys [1] [2] [3] [4] [5] 2 From the measurements done in this work it is difficult to estimate all the components in the saliency model. Table 3 shows the terms that can be derived from the measurement. Table 3 Measured inductance terms La 0 Laσ Laσ st − Laσ yt [mH] [mH] [mH] 0.125 0.025 0.0375 V. REFERENCES standard Lg Ld Lq [mH] [mH] [mH] 0.0875 0.1 0.325 The magnitude of the position dependent terms in the mutual and self inductance indicates that there is a position dependency in the leakage inductance. These variations can be incorporated in the machine model by using saliency modeling as presented in this paper. Table 3 shows the estimated inductance terms for the test machine used in this work obtained by transient excitation (Figure 9). The term ( Laσ st − Laσ yt ) describes the position dependent term in the leakage inductance. In classical machine modeling this term will equal zero. IV. CONCLUSION A new saliency model is presented in this paper. The model incorporates the combined saturation effect in the stator teeth/yoke, induced currents in the magnets and load dependency. The model suggests that the leakage inductance should be modeled with a position dependent part. Measurements on an IPMSM show that there can be a difference in the position dependent term in the self and mutual inductance when transient excitation is used to estimate the inductance. This difference can be incorporated in the machine model by adding position dependent part to the leakage inductance. 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