Sub-Domain Model for Induction Motor With More Accurate Realization of Tooth-Saturation

Incorporation of magnetic saturation and its aftereffects have been challenging for sub-domain based analytical motor models. The proposed model utilizes the direct phenomenal impact of magnetic saturation on spatial variation of the core's permeability to formulate the effective tooth width. Implementation of the proposed method does not need an additional domain or magnetic vector-potential dependent term to include the saturation and hence, provides an efficient approach. The developments derived from the proposed method are seen in better estimation of motor's power factor, breakdown torque, and additional iron loss due to tooth pulsation and surface eddy currents. Prediction capability of the model is demonstrated with the experimental and finite element results of an 11 kW, four-pole squirrel cage induction motor for a wide range of operations.


Sub-Domain Model for Induction Motor With More
Accurate Realization of Tooth-Saturation Rajendra Kumar and Narayan C. Kar , Senior Member, IEEE Abstract-Incorporation of magnetic saturation and its aftereffects have been challenging for sub-domain based analytical motor models.The proposed model utilizes the direct phenomenal impact of magnetic saturation on spatial variation of the core's permeability to formulate the effective tooth width.Implementation of the proposed method does not need an additional domain or magnetic vector-potential dependent term to include the saturation and hence, provides an efficient approach.The developments derived from the proposed method are seen in better estimation of motor's power factor, breakdown torque, and additional iron loss due to tooth pulsation and surface eddy currents.Prediction capability of the model is demonstrated with the experimental and finite element results of an 11 kW, four-pole squirrel cage induction motor for a wide range of operations.
Index Terms-Induction machine, iron loss, magnetic saturation, magnetic vector potential, motor performance, sub-domain.Frequency of spatial periodicity for domain-v.T em Electromagnetic torque.

T Shaf t
Torque available at rotor-shaft.

P Loss
Sum of all the motor's losses.

J x
Bessel function of first kind and order-x.

Y x
Bessel function of second kind and order-x.

I. INTRODUCTION
P RECISE and rapid evaluation of motor performance is a crucial aspect of a modeling approach used for motor design and optimization.One of the key electromagnetic phenomenon responsible for the deviation in actual and simulated 0885-8969 © 2024 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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TABLE I IMPACT OF VARIOUS ELECTROMAGNETIC PHENOMENON FOR SCIM
motor modal characteristics is non-uniform magnetic saturation of motor's various segments [1].Others are fringing-aided non-uniform airgap permeance, rotor bar temperature, stator end-leakage permeance, skin and proximity effects [2], [3], [4], [5], [6].Table I shows the relative impact of all these phenomena on the characteristic indices of a squirrel cage induction motor(SCIM).
Apart from restricting the effective utilization of active iron to produce the maximum steady state torque, the saturation also results time harmonics due to the magnetic non-linearity of iron, additional magnetic flux-pulsation due to enlarged tooth-width and extra surface core losses [7], [8].Cumulative impact of all these aspects attributes disparities in power factor, efficiency, and torque profile.There exist four different motor-modeling approaches used prominently in the iterative process of motor design viz, analytical [9], semi-analytical [10], lumped magnetic equivalent circuit (MEC)-based [11], and direct expression based [12].All these techniques consider saturation in different degrees and the complexity of its realization.
Finite element analysis (FEA), though, accounts the saturation comprehensively with all its consequences with the least modeling complexity, needs significant time and virtual space for a SCIM due to inevitable numerical and electrical transients introduced by rotor time-constant [13].The analysis becomes worse when motor performance needs to be assessed for a wide range of operation.On the other hand, primitive direct expressions-based modeling is straightforward.However, inviolable assumptions made to derive these formulations., infinite iron permeability, uniform rotor bar current distribution and empirical relations used to express flux paths limit their efficacy for motors designed with high airgap flux density i.e., operating point close to the saturation in B − H plane [14].For example, Carter's coefficients used in traditional methods rely on empirical coefficient curves [8], [15], resulting 20-30% lower inductances [16].Second to FEA, sub-domain methods with slots defined as separate sub-domain are considered the most accurate approach to consider slotting-effect [9].Similarly, with the simple leakage flux distribution patterns inside the slot, the algebraic formulas of leakage inductances, result in error upto 35% [16], [17] in semi-closed slots.
Analytical and semi-analytical approaches, i.e., MEC, sub-domain model (SDM) and, a hybrid of both, offer a good compromise between modeling complexity and accuracy and, therefore, provide viable basis for the motor's initial design, optimization, and dynamic performance evaluation.Among these, SDMs are the most suitable because, a systematic representation of saturation and non-homogeneous airgap, demands finely meshed complex reluctance network with MECs [18], [19], [20].
SDMs are extensively explored with permanent magnet synchronous machines with analytical solutions of the Laplace and Poison equations [21], [22].The first SDM model for SCIM was presented in [23] with solving the Helmholtz equation for rotor bars at a single frequency.Infinitely permeable slot less stator, enclosing a single current sheet, is used in this work.The model is further improved for slotted stator and magnetomotive force (MMF) harmonics by adding stator-slot sub-domain and connection matrix-based current density calculation [24].In recent developments, [25] presented an improved version of [24] by defining slip frequency for each space harmonic of stator MMF.However, the permeability of iron is considered infinite; hence, the saturation is neglected.Further use of this harmonic model for optimizing motor geometry is reported in [26].3D SDMs are developed to effectively include the impact of skewing [27], [28].The models overlook all the non-linearities.With a convenient approach, [18] proposes equivalent air magnetic permeability to incorporate the impact of rotor slot dip angle which is otherwise considered zero or needs additional sub-domains.The developed model neglects the saturation completely and therefore change in tooth's magnetic permeance due to saturation is not considered while calculating the slot's magnetic resistance using the proposed equivalent air magnetic permeability.In [29], stator and rotor slot pitches are replaced with a homogeneous region of equivalent relative permeabilities (ERP) to account the excess MMF required to magnetize the teeth.However, these ERPs are determined using single values of iron permeabilities and hence, do not account the teeth-saturation.Major shortcomings of all the existing SDMs, are seen in relinquishment of toothsaturation and, time harmonics generated therefrom.
Addressing the above-mentioned shortcomings, the work presented in this paper caters following objectives: r Realization of saturation in SDM for SQIM employing the direct phenomenal impact of magnetic saturation on spatial variation of the permeability of the iron-core and motor geometry.
r Incorporation of current harmonics superimposed by the magnetic non-linearity of teeth.
r Determination of additional iron losses produced by the saturation and the consequent stator current harmonics.The organization of the paper is as follows: Section II details the theoretical framework of the SDM and the proposed method.The performance of the proposed method is described in Section III.Section IV shows estimation accuracy of the method with measurement data.Section V concludes the work with notable observations.

II. PROPOSED ANALYTICAL FORMULATIONS FOR IM
This section presents the analytical formulations of the two foremost consequences of magnetic saturation, i.e., modulation of teeth-width and, injected current harmonics in stator current.The formulation utilizes space distribution of airgap magnetic flux density.

A. Incorporation of Tooth-Width Deviation
The performance of an IM is always susceptible to slot dimensions.Functional values of these dimensions get deviated, from their actual physical measurements, due to the saturation.The dimensions in the vicinity of high flux densities, such as slot openings (B s0 , B r0 ), experience maximum impact.These slot-openings significantly affect motor performance in terms of starting current, breakdown torque, current and slip at rated output power.Since the flux density distribution, at a time, is not uniform, in space for all the slots under a pole; the change in slot-openings depends upon the slot's location.Angular variation of the radial airgap flux density, set up by a rotating MMF, for a three-phase IM is expressed as [7] where,∀z Since, for a particular load, the space distribution of the airgap magnetic flux density remains same for all the times, except for its phase (which changes as ωt).For, time-instant, t = 0, (3) With the magnetization non-linearity, the permeability of the iron also varies with the tooth-location (as shown in Fig. 1(c)), which can be translated as angular position dependent permeability as: for all other positions (4) where, θ i is mean position of i-th stator slot.A saturated tooth exhibits permeability close to the air which, analytically is interpreted as increased width of the slop-opening as simplifying (5), Since, the teeth are discretely placed, and the impact of the saturation applies only to the tangential displacement Δθ = τ s B S0 .So, the realization of ( 6) with respect to (4) needs a continuous for all other positions (7) Using (7), the expanded slot opening corresponding to the permeability distribution of ( 4) is expressed as Fig. 1(d) shows the variation in slot-width as per (8).If permeability is considered infinite, g(θ) is always zero and, in such case, B S0 (θ) is equal to the physical value of slot-opening.
With saturation present, the effective equivalent slot-opening is determined with the average of B S0 (θ) over the half pole pitch as similarly for rotor, These updated values of slot-openings with ( 9)-( 10) are used in the development of SDM described in Section III.The model is initially run with physical value of the slot openings with infinite permeability.The process repeats with flux-density based relations ( 7)-( 9) until the MMF required for the teeth gets compensated with the elongated slot-openings.With heavy tooth-saturation, the MMF drop across the teeth becomes prominent apart from the tooth-tip MMF.At this situation, according to Ampere's law, an encirclement enclosing a tooth of height H St , slot-opening B S0 and total slot ampere turn F S at location θ is governed by (11) as where, μ r (θ), μ r (θ), B t (θ) and B t (θ) are the magnetic permeabilities and flux densities at the tooth tips and at 1/3 rd of the tooth height, respectively.At the locations of heavy saturation, B t (θ) ≈ B t (θ) and, μ r (θ) ≈ μ r (θ) and hence, right hand side term of (11) can be replaced for an equivalent permeability μ r (θ) as ( 12) Thus, when evaluating (5) for motors that show heavy saturation, the permeability of ( 4) is scaled as (13).

B. Incorporation of Saturation-Caused Stator Current Harmonics
Cumulative effect of the time dependent B/H ratio for all the motor-segments leads to time harmonics in current.Being the function of instantaneous flux densities, these time harmonics cannot be evaluated with an average saturation coefficient [7], [30] or with the rms current based analytical models [31], [32].Unlike space harmonics, all these saturation harmonics travel with the synchronous speed, they induce significant rotor bar and eddy currents, which lower the motor efficiency.A hybrid of MEC and subdomain ensures a precise solution, however, with prevalence of the complexity.The proposed method puts forward an alternate approach to extract the current harmonics using the B-distribution obtained with the sub-domain iteratively.Followings are assumptions made in the proposed method:   r Since, in an IM, the resultant airgap flux primarily corre- sponds to the no-load current's magnetizing component, the impact of leakage flux is neglected.
r End winding leakage flux and therefore, the MMF magne- tizing the end-winding region is assumed zero.The spatial distribution of the MMF, at a particular time instant, is given as For the airgap flux density distribution (1), the distribution MMF in θ-direction is written with Ampere's law as (for zero rotor current): In (15), the elemental MMF terms (i.e.,l X H X (θ)) are the Bdependent, volume averaged values which are determined with the B-distribution of (1) as for stator teeth: for rotor teeth: Fig. 3(a) shows the comparison of l St H St (θ) values with fundamental component of B t for different teeth under a pole with and without saturation.For stator and rotor yoke regions, unlike the teeth, the length for flux-lines (in tangential direction) is not same for all the encirclements (as shown in Fig. 2).So, the terms l Sc H Sc (θ)) and l Rc H Rc (θ)) are approximated with angular position dependent flux line lengths in yoke regions as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where, Evaluations of ( 15)-( 18) use B-distribution in various segments of the motor, which Section III describes in detail.After all the right-hand side terms of ( 15) are estimated for a particular angular position, F (θ) is mapped for a half cycle comprising a pole and extrapolated for next half cycle perceiving the half wave symmetry as shown in Fig. 3(b).F (θ) comprises the harmonics generated by the stepwise distribution of the MMF (with discrete placements of the conductors in slot) as well as the harmonics generated due to non-uniformity of H St and H Rt .So, a twostep Fourier transformation (FFT) based approach is proposed to obtain the motor's phase current analogous to F (θ) in following manner: r At first, FFT is applied on the F (θ) determined with (15)- (18).
r Using the fundamental component of the FFT (F 1 ∠φ 1 ), magnitude and phase of the phase-A MMF are determined with following relation: r Current of phase-A (rms) is determined with (19) as [8] r For these balanced set of sinusoidal currents, F (θ) is determined using ( 15)-( 21) with the permeability of all the iron parts equal to that of an unsaturated one.FFT is applied on the F (θ).
r Since, harmonics generated due to discrete placements of the conductors are same for both the cases i.e., F (θ)and, F (θ), the difference in each harmonic component of the two FFTs i.e., FFT of F (θ) and, F (θ) yields the additional space harmonics generated by the saturation.
r For known values of magnitudes and phase angles of these additional space harmonics, currents harmonics corresponding to these harmonics are found with the same transformation i.e., (21) without changing the wave number-p as ( 23)-( 24): The current for phase-A is shown in Fig. 3(c).These updated three-phases' currents with the harmonics are applied in sub-domain analysis with iterative convergence to assess the additional iron losses and the motor performance.Both of the two formulations of this section make use of B-distribution in motor's various segments and, the magnetization profile of the electrical steel.With known MVPs and B − H curve, they can be used to improve the SDM as described in next section.

III. IMPLEMENTATION OF THE PROPOSED FORMULATIONS WITH SDM
A 2-D SDM is developed in polar coordinate system with current carrying regions excited with z-directional magnetic vector potentials.The SDM is first solved with the conventional method and assumptions of [24], [25], [29].The solutions obtained with the SDM are modified further, iteratively, to incorporate the impact of saturation following the formulations of Section II.

A. Sub-Domains and General Solutions of MVPs
The motor geometry, under a pole, is divided into total 7 SDs as shown in Fig. 2. Spatial periodicities of the periodic and nonperiodic regions are, respectively, equal to 2π and, 2πK, where K is the ratio of pole-pitch to the tangential spread of SD.For rotor and stator semi-slot regions (III, V), Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Depending upon the source of excitation, governing equations of MVPs in the SDs are given as ) General solution of (26), with separation of variables, is written as [33]: where, A n,k (r, θ) for different regions are given by ( 28)-( 31): for periodic SDs- for rotor bars- for rotor and stator semi-slots i.e., v = III, V- for stator slots- The source term for Poisson's (26b), J ext is formulated as [24] J v (θ, t) = J m cos(vpθ − nω s t) (32) Unknown coefficients of ( 28)-( 31) are determined with boundary conditions of continuity of normal component of B and, discontinuity of tangential component of H across an interface with deviation equal to surface current density.[24], [25] discuss all the interface conditions in detail so, they are not repeated.The radial and tangential components of the flux densities in every region is obtained as For voltage excitation, motor currents are determined with ECPs.The ECPs are determined with known MVPs (for inductances) [9] along with classical analytical expressions (for resistances and end-leakage inductances) [7].

B. Implementation of the Proposed Approaches With SDM
The implementation employs two cascaded stages, one for each approach.The first stage incorporates the impact of saturation on slot openings.For this, the SDM is run first with assumption of infinitely permeable iron.Flux densities achieved with this are used to determine the degree of saturation for each tooth (Fig. 1(b)-(d)) and, therefore its inclusion as an adjustment in slot openings as per ( 9)-( 10).An average equivalent value of the slot openings, determined with (9), replaces its preceding value starting from the physical slot opening.At second stage, the excess magnetization current and, impact of current harmonics are introduced using the flux densities and, B − H characteristics of material as described in subsection-B of Section II.Fig. 4 shows the steps involved in the process.The MMF harmonics calculated with (23) include the components corresponding to the teeth (t), yoke (y), and airgap (t) as shown Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. in (34), so they were segregated to determine the airgap flux density distribution in presence of saturation before running the SDM.
The MMF harmonics for the teeth, yoke, and airgap, (35), are determined with the ( 14)-( 20).In the next step, three components of current corresponding to (35) are calculated for each harmonic.Since, the MMF represented by ( 35) is the resultant of all the three phases, its reflection in the currents of the three phases, in terms of current harmonics, is determined with expressions ( 23)- (24).Unlike other space harmonics (v) generated by the stepwise distribution of MMF which rotate at sub-synchronous speeds with corresponding pole number equal to v.p, the harmonics caused by saturation rotate with synchronous speed with their pole number in spatial distribution equal to v.p.So, after the current magnitudes of these harmonics are calculated, the field distribution in presence of these harmonics is determined by running the SDM for these harmonics individually.Here, the excitation current density variation for each harmonic, caused by saturation, is considered a sinusoidal function such that, the ratio of spatial and time domain frequencies is constant for all the harmonics as J X v = J X mv sin(vpθ − vωt).Finally, the vector potentials calculated corresponding to the current density harmonics J X v are added together with the fundamental to obtain the net airgap field, in effect of saturation.With star (Y) connected stator windings, the resultant MMFs of 5th, 7th and other non-triplen saturation harmonics remain largely unchanged because the corresponding rotor MMF is cancelled by the extra stator current, a phenomenon similar to the fundamental MMF on load.However, in absence of stator current path for triplen harmonics, 3 rd order saturation MMF generates rotor currents whose MMF cannot be cancelled out.This ultimately alters the resultant airgap flux density distribution.In the absence of stator winding third harmonic currents, this impact of rotor's 3 rd harmonic MMF is thus load dependent.Therefore, a recursive procedure is adopted for the 3 rd harmonic with following steps: r At first, similar to the other non-triplen saturation har- monics, the SDM is run with the stator winding region (VI) excited with 3 rd MMF current harmonic density distribution of (9).Concerned governing equation with r Rotor current magnitude (I mvr ) is evaluated for this ex- citation as per (37), (38) with ω rv = sω rv as saturation harmonics travel with synchronous speed.r Stator current density magnitude (J mvr ) equivalent to this rotor current magnitude is calculated by MMF equivalence [8] as corresponding space distribution is given by ( 39) r Here, the magnitude J mv remains constant for given funda- mental A V I whereas, J mr changes with loading, i.e., with slip, along with its angle γ 3 .So, to make the iterative process univariable, γ 3 is written in terms of J mv , J mvr , slip and known equivalent circuit parameters-rotor resistance (R r ) and leakage inductance (L lr ).As shown in Fig. 5, the phase angle of rotor MMF (γ 3 ) determined with the current magnitudes of J mv , J mvr i.e., I mv , I mvr using law of sines for triangle is given by ( 40)-( 43) r In absence of any stator windings 3 rd harmonic current, 3 rd harmonic current induced in rotor bars are consequent to the resultant of stator and rotor 3 rd harmonic MMFs only.
For field distribution, the condition can be interpreted as running the model with net current excitation as J mv + J mvr .Therefore, in the next step, the model is run again with updating the excitation term of region IV i.e., right hand term in (36) as J mv + J mvr .
r Steps (ii) and (iii) are followed again to determine the revised current density J mvr which is then fed back for solving the next iteration.The iteration is repeated till the difference between the two consecutive rotor current density magnitudes are less than 2%.It is important to mention here that all these calculation-steps are used only to determine the field distribution in presence of saturation third harmonic.The actual third harmonic and other triplen harmonics in stator current are taken zero.

C. Calculation of Surface Core Loss, Motor Efficiency and Electromagnetic Torque
After the fundamental and harmonic core-losses, surface core loss, P SCL , is a major constituent of electromagnetic loss of an IM [2].P SCL is additional eddy current on the surfaces of stator and rotor teeth.Saturation at slot-openings affects P SCL by changing the duty cycle of permeance variation.With linear relation of P SCL and slot opening, P SCL is determined with modified slot opening as where, as per [2], Core loss, electromagnetic torque and net copper-loss are determined, respectively, with (45), ( 46), ( 47) and ( 48) where, σ II is the conductivity of rotor-bar material.Using (32), where, R s (T, f) is per phase stator resistance with temperature and frequency corrections.R b and R er are the resistances of single rotor bar and end-rings, respectively.With total loss of the motor, P Loss (I, ω), taken as sum of (43), ( 45) and (48), efficiency of the motor at the operating point, specified with speed and stator current, is determined with (47) as Only the eddy-currents and hysteresis loss terms are considered in core-loss estimation.Excess loss is neglected in this work [28].Similarly, the variation in rotor-bar current density due to the reaction field generated by eddy currents is not counted in (35).

IV. IM PERFORMANCE EVALUATION AND VALIDATION
Validation of the proposed approach is carried out with the experimental test results of [29] for an 11 kW laboratory IM.Table II shows the nameplate details of the motor.FEA simulations are performed with ANSYS Maxwell for the motordimensions given in Annexure A. Fig. 7 shows the B − H curve of the motor's iron laminations.

A. Comparison of Proposed Method With FEA Simulations
To verify accuracy of the method, at first, motor performance is evaluated for rated voltage operation.The motor model is run with sinusoidal excitation corresponding to 285 V line-voltage and, 100 Hz frequency.For relative comparison, results with 2D FEA and, SD with infinite permeability are used.Figs.9(a)-(d) show the characteristics obtained with the three methods along with the average deviations of the two methods with respect to the FEA.Only the region of stable operation of motor between 2,730 and 3,000 rpm is considered.As Fig. 9(a) shows, the assumption of infinite permeability underestimates the current complying zero MMF-drop across iron segments.For the motor in this work, the currents fall short with an average of -16.22% for the selected speed range.This leads to overestimation of all other parameters as in Figs.9(b)-(d) for a given supply voltage, frequency, and rotor speed.Among all parameters, power factor exhibits the highest susceptibility with average deviation of +17.35%, which gets reduced to +2.07% with the presented method.As shown in Fig. 8(a), the impact of proposed method is seen in compensation of reactive component of current with increased flux-path permeance with B-dependent B eq S0 and, addition of MMF corresponding to iron-magnetization.The proportion of these two components of compensation varies with the motor-speed.As shown in Fig. 8(a), at the speed of the breakdown torque, 2,730 rpm, B eq S0 is maximum and equal to its physical value (1 p.u.).Therefore, at this speed, the adjustment in reactive current solely inherent to the surplus MMF corresponding to iron-magnetization as per ( 14)- (19).As the rotor-speed increases, the drop in current results in higher magnetization voltage and flux densities.This eventually reduces the B eq S0 and, permeance of the magnetic network of the motor.Therefore, at lower speed, the adjustment in current pertains to structural modulation of permeance network.Both of these components, ultimately, govern the resultant of current enclosed by the flux-path i.e., F (θ).   III.In [12], iron-loss coefficients for M19-29GA are reported for a wide range of induction and frequency, and these coefficients are used to determine P CL .Observed average errors, with the method, for power factor, current and magnetic loss are 2.71, −2.67 and 8.78%, respectively.The magnetic loss is taken as a sum of P CL and, P SCL .P SCL is determined by modifying the conventional expression as (31) whereas; the calculation of P CL pursues the classical two-term model (45).

B. Verification With Experimental Results and Discussion
To compare the accuracy of proposed method with experiments results, measurement data of variable frequency operation are utilized [12], [29].Schematic and test-bench used to record these data are shown in Figs.6(a)-(b).The test data cover rated torque operation up to the base speed of 2,925 rpm and, field weakening thereafter up to 4,600 rpm.In these data, impact of temperature is accounted with constant values of temperature coefficient of resistances of copper and iron for the temperature rise of 40 • C. Also, the stator of the motor is shrink-fitted inside the aluminum housing with circumferential stress less than 10 MPa, hence, considering the linear trend of the 0.1 p.u./MPa gradient for the range of 0-10 MP, the effect of mechanical stress on the iron permeability as well as core-loss is assumed negligible [27].Separate measurements of P CL and, P SCL are not possible so their combined values, denoted as P ML , are extracted from the no-load input power and, friction-windage loss.However, the coefficient of friction-windage loss is taken constant for all the speeds, which is calculated by dividing the friction-windage loss by ω r 3 .Here, the friction-windage loss, P F W L , is determined as intercept of the voltage 2 -versus no-load-power plot on the power axis.A total of six no-load operating points are taken, which are uniformly distributed between 20 and 125% of the rated voltage of 285 V at 100 Hz.Furthermore, for the comparison of torque values obtained from (46) with the same   With these data, compatibility of the model is checked for three aspects of performance estimation: efficiency, input impedance and magnetic losses.Figs.[13][14][15] show the results obtained from the measurements and proposed method.These results draw following conclusions: r For a given current, measured voltage across motor ter- minals closely matches with the same calculated using proposed method, with an average error of 2.01%.This ensures collective accuracy of the method for current and power factor estimations.The errors for the same at rated operation with methods [26], [27] are 6.75% and 5.65%, respectively.The error can be further reduced with more discrete realization of ( 15)-( 17) by fragmenting the slots and yokes regions.However, this will increase the implementation time.
r The error in efficiency evaluation lies between 1.71 and 0.9% for various loading conditions.Higher values of error are observed for the points of light loading.The difference in efficiency can be attributed to the several other factors that are not accounted in this work such as variation of material properties due to circumference stress, skin effect in rotor-bars and additional load dependent losses.The errors in the efficiency calculation at rated operation with methods [26], [27] are 2.7% and 2.13%, respectively.r The proposed method enumerates magnetic losses with accuracy more than 92%.Perceiving the accuracy of the method in determining the magnetic field density as shown in Fig. 9, the eight percent estimation deficit of magnetic loss mainly corresponds to the limitation of classical twoterm core-loss model.Conventional SDMs of [26], [27] display errors greater than 50% for these computations.The two-term model does consider the impact of phase angles of various B-harmonics while determining the hysteresis loss [8].The accuracy for the magnetic loss, therefore, can be further improved by appending the proposed model with more realistic hysteresis loop models along with accurate formulations of pulsation loss, stray load loss, rotor crossbar current losses and, other high frequency iron-losses.

V. CONCLUSION
This research manuscript presents simple and easy-toimplement analytical formulations of magnetic saturation for sub domain modeling of an induction motor.Two major impacts of saturation are addressed which include increment in specific permeances of different magnetic segments and, generation of current harmonics.The formulations included these aspects in iterative manner using the using the solutions of MVPs and magnetizing property of the lamination-sheet.The accuracy of the method is verified with FEA simulations and measured results for an 11 kW laboratory IM.It is observed that the results obtained with the proposed method are in close agreement with the simulation and measured values with average errors in efficiency, input impedance and magnetic loss estimations equal to 1.71, 2.01 and, 8.23%, respectively.The straightforward execution of the method makes it a viable alternative to complex hybrid SDM-MEC models with an evaluation efficacy comparable to FEM.

Fig. 1 (
Fig. 1(a)-(b) depicts this variation.As shown in the Fig. 1(b), the half-wave symmetric magnetization, gets confronted by teeth non-uniformly, with each tooth denoting a unique position in B − H plane, at a time, as shown in Fig. 1(c).Since, the tangential component of the flux density at tooth-tip is very small, which gets reduced further by the saturation due to increased specific permeance at tooth-tip, only the variation in radial flux density is considered.The radial flux density at the tip of tooth, towards the airgap, is:

Fig. 1 .
Fig. 1.Genesis of assorted slot-openings.(a) Fux density distribution for angular span of one pole.(b) Magnetization stage of teeth at t = 0 s for half pole pitch.(c) Variation of magnetic permeability with magnetization.(d) Change in slot opening corresponding to the teeth saturation.

r
Stator and rotor teeth are the most affected areas and, show the non-linearity in B − H relation.

Fig. 2 .
Fig. 2. Approximation of flux line paths for angular and radial passages.

r
Stator and rotor yokes lie in the linear range of magnetiza- tion.rMean-length of the flux path is considered for the stator and yoke regions.

Fig. 3 .
Fig. 3. Current harmonics with varying saturation-extent of teeth.(a) Stator tooth MMF pattern (l St H St (θ)) for one pole-pitch.(b) Angular variation of the resultant ampere-turns F (θ), with and without saturation.(c) Motor current correspond to the resultant ampere-turns, with and without saturation.

Fig. 4 .
Fig. 4. Steps of the SDM implementation with proposed formulations.

Fig. 13 .
Fig. 13.Comparison of the measured and calculated terminal voltages for measured currents.

Fig. 14 .
Fig. 14.Comparison of efficiency values.(a) Experimental and analytically calculated with proposed method.(b) Corresponding percentage error distribution.

Fig. 15 .
Fig. 15.Comparison with measurement results.(a) Stator leakage inductance, (b) rotor leakage inductance, (c) motor efficiency at rated torque and rated power, and (d) torque variation with motor current.