14 34 46

Proc. of Int. Conf. on Advances in Power Electronics and Instrumentation Engineering, PEIE

Computational Analysis of 3-Dimensional Transient Heat Conduction in the Stator of an Induction Motor during Reactor Starting using Finite Element Method 1

A. K. Naskar1 and D. Sarkar2 Seacom Engineering College, Howrah, India Email: naskar73@gmail.com

2

Bengal Engineering & Science University, Howrah, India Email: debasissrkr@yahoo.co.in

Abstract— In developing electric motors in general and induction motors in particular temperature limit is a key factor affecting the efficiency of the overall design. Since conventional loading of induction motors is often expensive, the estimation of temperature rise by tools of mathematical modeling becomes increasingly important. Excepting for providing a more accurate representation of the problem, the proposed model can also reduce computing costs. The paper develops a three-dimensional transient thermal model in polar co-ordinates using finite element formulation and arch shaped elements. A temperature-time method is employed to evaluate the distribution of loss in various parts of the machine. Using these loss distributions as an input for finite element analysis, more accurate temperature distributions can be obtained. The model is applied to predict the temperature rise in the stator of a squirrel cage 7.5 kW totally enclosed fan-cooled induction motor. The temperature distribution has been determined considering convection from the back of core surface, outer air gap surface and annular end surface of a totally enclosed structure.

Index Terms— FEM, Induction Motor, Thermal Analysis, Design Performance, Transients

I. INTRODUCTION Considering the extended use of squirrel cage induction machine in industrial or domestic applications both as motor and generator, the improvement of the energy efficiency of this electromechanical energy converter represents a continuous challenge for the design engineers, any achievements in this area meaning important energy savings for the world economy. Thus to design a reliable and economical motor, accurate prediction of temperature distribution within the motor and effective use of the coolant for carrying away the heat generated in the iron and copper are important to designers[18].

Traditionally, thermal studies of electrical machines have been carried out by analytical techniques, or by thermal network method [1], [8]. These techniques are useful when approximations to thermal circuit parameters and geometry are accepted. Numerical techniques based on finite element methods [5],[7] and [10]-[20] are more suitable for analysis of complex system. Rajagopal, M.S, Kulkarni, D.B, Seetharamu,K.N, and Ashwathnarayana P. A [4], [6] have carried out two-dimensional steady state and transient thermal DOI: 02.PEIE.2014.5.14 Š Association of Computer Electronics and Electrical Engineers, 2014

analysis of TEFC machines using FEM. Compared to the finite difference method, the finite element method can easily handle complicated boundary configurations and discontinuities in material properties.

The finite element method was first introduced for the steady state thermal analysis of the stator cores of large turbine-generators by Armor and Chari [2]. However, their works are restricted to core packages far from the ends and they do not consider the influence of the stator coil heat. Sarkar and Bhattacharya [9] also described a method based on arch-shaped finite elements with explicitly derived solution matrices for determining the thermal field of induction motors.

In this paper, the finite element method is used for predicting the temperature distribution in the stator of an induction motor using arch-shaped finite elements with explicitly derived solution matrices. A 100-element three-dimensional slice of armature iron, together with copper winding bounded by planes at mid-slot, midtooth and mid-package, are used for solution to a transient stator heating problem, and this defines the scope of this technique. The model is applied to one squirrel cage TEFC machine of 7.5 KW and the temperatures obtained are found to be within the permissible limit in terms of overall temperature rise computed from the resulting loss density distribution.

II. POLYPHASE INDUCTION MOTOR MODEL AND BOUNDARY CONDITIONS The details of the induction motor are shown in Fig. 1. In this analysis, the 3-dimensional slice of core iron and winding has been chosen for modeling the problem and the geometry is bounded by planes passing through the mid-tooth, the mid-slot and the package centre. This is shown in Fig.2, taken from the shaded region A of Fig.1. The temperature distribution is assumed symmetrical across these three planes, with the heat flux normal to the three surfaces being zero. From the other three boundary surfaces, heat is transferred by convection to the surrounding gas. It is convected to the air-gap gas from the teeth, to the back of core gas from the yoke iron, and to the core end gas from the annular end surface of core. The boundary conditions may be written in terms of δT / δn , the temperature gradient normal to the surface.

(1) Axial centre of package δ T = 0 δ np Mid-slot surface δ T = 0 (2) δ nS Mid-tooth surface

δT =0 δ nt

(3)

Air-gap surface h ( T − T AG

) = − Vr

δT δ n AG

δT δ nD T δ Back-of-core surface, h ( T − TBC ) = − Vr δ nBC Annular end surface of core h ( T − TD ) = − Vz

(4) (5) (6)

A. Finite Element Formulation The governing differential equation for transient heat conduction is expressed in the general form as

δ T ⎞ Vθ δ 2 T δ 2T ~ δT 1 δ ⎛ ⎜⎜ Vr r ⎟⎟ + 2 V + + Q − Pm Cm =0 z 2 2 r δr ⎝ δ r ⎠ r δθ δZ δt

(7)

To find the solution of (7) together with the boundary and initial conditions by Galerkin’s weighted residual approach, we first express the approximate behaviour of the nodal temperature within each element according to equation (8). Since substitution of this approximation into the original differential equation and boundary conditions results in some error called a residual, the method of weighted residual requires that the integral of the projection of the residual on a set of weighting functions is zero over the solution region.

The approximate behaviour of the potential function within each element is prescribed in terms of their nodal values and some weighting functions N1, N2… such that

35

Fig.1. Half sectional end & sectional elevation of a 7.5 kW squirrel cage induction motor

Fig. 2. Slice of core iron & winding bounded by planes at mid-slot, mid-tooth and mid-package

T =

∑

N i Ti

(8)

i = 1 , 2.... m

The required equation governing the behaviour of an element is given by the expression:

∫∫

vol

⎡ δ ⎛ δ T ( e ) ⎞ δ ⎛ Vθ δ T ( e ) ⎞ δ ⎛ δT ( e) ⎞ ~ δT (e) ⎤ ⎜⎜ Vr ⎟⎟ + ⎜⎜ 2 ⎟⎟ + ⎜⎜Vz ⎟⎟ + Q − PmCm Ni ⎢ ⎥ dvol = 0 (9) r r r z z t δ δ δ θ δ θ δ δ δ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦

Integrating all the terms through integration by parts, the equation takes the form

∫∫

Ni

D( e)

−

δ T ( e) δ ⎛ δ T (e) ⎞ ⎜ Vr ⎟ r dθ drdz = ∫ Vr Ni r d θdz δ r ⎜⎝ δ r ⎟⎠ δr S (e) 2

∫∫ V

r

D( e)

=

∫

S2 (e)

Vr

δ T δ Ni r d θ d rdz δr δr (e)

δ T( e) δ T( e) δ Ni nr Ni d ∑ − ∫∫ Vr . r dθ d rdz δr δ r δ r (e) D 36

(10)

B. Arch-Element Shape Functions Consider the arch-shaped prism element of Fig. 3, formed by circle arcs radii a, b, radii inclined at an angle 2∝, and prism faces at positions z = -c and z = c.

Fig.3: Three-dimensional arch-shaped prism element suitable for discretisation of induction motor rotors

The shape functions can now be defined in terms of a set of non-dimensional co-ordinates by nondimensionalising the cylindrical polar co-ordinates r, θ and z using

r ρ = ; ν = a

π 2 ;τ = z c α

θ −

(11)

The arch element with non-dimensional co-ordinates is shown in Fig. 4.

Fig.4: The non-dimensional arch element

The temperature at any point within the element be given in terms of its nodal temperatures, by

T = T A N A + TB N B + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + T H N H Where the N’s are shape functions chosen as follows: 37

(12)

( ρ − b/ a) (ν −1)(τ −1) 4 ( 1 − b/ a) ( ρ − b/ a ) (ν + 1)(τ −1) NF = − 4( 1 − b/ a ) ( ρ − 1) (ν + 1 )(τ + 1) N = ( ρ − 1) (ν + 1)(τ −1) NC = G − 4 ( 1 − b / a) 4 ( 1 − b / a) ( ρ − 1 ) (ν − 1 )(τ + 1) N = ( ρ − 1) (ν − 1)(τ −1) ND = H 4 (1 − b / a ) − 4( 1 − b / a )

( ρ − b/ a) (ν − 1)(τ +1) − 4 ( 1 − b / a) ( ρ − b / a ) (ν + 1 )(τ +1) NB = 4( 1 − b / a )

NA =

NE =

(13)

It is seen that the shape functions satisfy the following conditions: (a) That at any given vertex ‘A’ the corresponding shape function NA has a value of unity, and the other shape functions N B , NC, ……, have a zero value at this vertex. Thus at node j, Nj = 1 but Ni= 0 , i ≠ j. (b) The value of the potential varies linearly between any two adjacent nodes on the element edges. (c)

The value of the potential function in each element is determined by the order of the finite element.

The order of the element is the order of polynomial of the spatial co-ordinates that describes the potential within the element. The potential varies as a cubic function of the spatial co-ordinates on the faces and within the element.

C. Approximate Numeric Form According to Galerkin’s weighted residual approach, the weighting functions are chosen to be the same as the shape functions. Substituting (12) and (13) into (10), gives

⎡ δ T ( e ) δ ⎛ δ T ( e ) ⎞ Vθ δ T ( e ) δ ⎛ δ T ( e ) ⎞ ⎤ ⎜ ⎟+ ⎜ ⎟ + ⎥ dV 0 = ∫ ⎢ Vr δ r δ Ti ⎜⎝ δ r ⎟⎠ r 2 δ θ δ Ti ⎜⎝ δ θ ⎟⎠ ⎦ V ⎣ P C δT ( e ) δ ⎛ δT ( e ) ⎞ ⎜⎜ ⎟⎟ − Q N + m m ∫ 2[ N ]{T }( e ) N i − 2T0 N i dV + Vz δ z δ Ti ⎝ δ z ⎠ 2Δt V

[

∫ (h ⎣ N ⎦ {T }

(e)

S2

(e)

)

N i − h T∞ N i d ∑

]

(e)

for i = A, B,…., H These equations, when evaluated lead to the matrix equation

[ [ SR ] + [ Sθ ] + [SZ ] +[ST ] +[ SH ] ] [ T ] =[ST ][T0 ] +[ R ] +[ SC ]

(14)

(15)

III. DISCRETIZED MODEL FOR FEM APPLICATION The stator of an induction motor, under transient conditions is designed to maintain all temperatures below class A insulation limits of 105oC hot spot. The hottest spot is generally in the copper coils. Thermal conductivity of copper and insulation in the slot are taken together for simplification of calculation [2].

In the case of transient stator heating caused by reactor starting, the transient analysis procedure is able to provide an estimate of the temperatures throughout the volume of the stator at an interval of time required to bring the motor from rest to rated speed by providing reduced voltage and current during the starting period and as the motor has reached a sufficiently high speed near to the operating speed, rated voltage and current are provided by short circuiting the reactors during the starting action of the induction motor.

Assuming that the machine is at rest with its stator winding at normal ambient temperature, respective voltage and current are injected to the stator winding of the machines. The temperatures within the volume of the stator are calculated at all nodal points for a period of time required for the reactor starting action.

38

In this analysis, because of symmetry, the 3-dimensional slice of core iron and winding, chosen for modeling the problem are divided into arch shaped finite elements as shown in fig 5.

Fig.5. Slice of core iron & winding bounded by planes of mid-slot & mid-tooth divided into arch shaped Finite Elements

IV. CALCULATION OF HEAT LOSSES Heat losses in the tooth and yoke of the core are based on calculated magnetic flux densities (0.97 wb / m2 and 1.293 wb/m2 respectively) in these regions. Tooth flux lines are predominantly radial and yoke flux lines are predominantly circumferential. The grain orientation of the core punching differs in these two directions and therefore influences the heating for a given flux density. Copper losses in the winding are determined from the length as well as the area required for the conductors in the slot.

Iron loss of stator core per unit volume = 0.0000388 W/mm3.

Iron loss of stator teeth per unit volume= 0.0000392 W/mm3.

A. Stator copper loss In reactor starting of the induction motor, the equivalent circuit of which is shown in Fig.6, we are interested to calculate the temperature distribution in the stator during the starting period. For the purpose of starting we will take the starting voltage at 50% of full voltage to start with and calculations will be done on that voltage till the reactor acts as impedance in the motor circuit. Finally, the temperature distribution within the stator due to reduced voltage reactor starting are calculated by splitting the entire slip range (i.e. from s=1 to full load slip s=0.04) into small intervals.

Fig.6. Equivalent Circuit of Induction Motor

39

x1=8.15Ω ; Ic=0.176 Amp; r1=2.04Ω; Im=2.41Amp r2/ = 2.39 Ω ; V1 = 415 V ; s = 0.0425

To calculate winding impedance of the reactor Voltage across the reactor is=415/√3 V The stator winding current per phase during starting =

0.5 × 415 = 22.37A. (4.43 + j8.15)

Line current = 22.37 × 3 =38.75A, which is the output current of the reactor. From VA balancing the input current of the reactor is=38.75/2 =19.38A and 50% of total impedance of the reactor will be =415/(√3×19.38×2)= 6.18 Ω To calculate stator current at starting when reactor is connected in the circuit from s=1.0 to s=0.2, At start s=1.0 Resistance of the circuit= (‫ݎ‬ଵ +‫ݎ‬ଶᇱ /s) = (2.04+2.39/1) =4.43Ω Reactance in the circuit (x1) =8.15Ω Impedance of the circuit (z1) = √(4.43)2+ (8.15)2 =9.3Ω As the stator is delta connected and 50% of full voltage is applied across the stator winding, the stator current at s=1 will be I1=415/ (9.3+6.18)=415/15.48 =26.85A To calculate stator current at different slips when motor is directly connected to the supply from slip s=0.2 to full load slip s=0.0425, At slip s=0.2 Resistance of the circuit= r1+r’2/s = (2.04+2.39/0.2)= 13.99Ω When full voltage is provided across the stator winding by short-circuiting the reactors, the stator current at s = 0.2 will be I1=415/ (13.99+j8.15) =25.63A The stator currents, stator copper losses and the time required for starting action at different slips are calculated and tabulated as shown in Table I. TABLE – I. THE DIFFERENT VALUES OF STATOR C URRENT, STATOR COPPER LOSS /SLOT /UNIT VOLUME AND TIME REQUIRED FOR STARTING ACTION AT DIFFERENT SLIPS IN REACTOR STARTING

B. Convective heat transfer co-efficient [2,8] Three separate values of convection heat transfer co-efficient have been taken for the cylindrical curved surface over the stator frame and the cylindrical air gap surface and the annular end surfaces. The natural convection heat transfer co-efficient on cylindrical curved surface over the stator frame is taken as h=5.25 w/m2 oC. 40

The heat transfer co-efficient on forced convection for turbulent flow in cylindrical air gap surface is taken as h=60.16 w/m2 oC. The heat transfer co-efficient on forced convection for turbulent flow in annular end surface is taken as, h = 34.67W / m2 oC. C. Thermal constants [3,9] For a transient problem in three-dimensions, the following properties are taken for each different element material as shown in Table II. TABLE II. T YPICAL SET OF M ATERIAL PROPERTIES FOR INDUCTION MOTOR STATOR Copper &Insulation

Magnetic Steel Wedge Vr

33.070

2.007

Vθ VZ

0.8260 2.874

1.062 358.267

Pm

7.86120

8.9684

Cm

523.589

385.361

V. RESULTS AND DISCUSSIONS Since the hottest spots are found to be in the stator copper as envisaged from the calculated temperatures for the three-dimensional structure during the reactor starting period as shown in Table III, the temperature variation with time in each node of copper is taken as an index to understand the temperature profile during the transient. It is to be noted that the temperature is found to be maximum at the nodes pertaining to copper in the axis of symmetry. The temperature rise is steady at different stator currents under the reactor starting region at different slips from s = 1 to s = 0.2. It is also to be noted that under DOL run the motor has reached a steady speed under full load condition and as such there is a slight decrease of hot spot temperatures persisting across the axis of symmetry after the reactors are shorted.

As a consequence, the temperature variation with time at hottest spots has been depicted in graphs as shown in Fig. 7 to investigate the magnitude of the temperature variation with time at different nodal points along the stator copper winding.

TABLE III. SOLUTION FOR T HREE DIMENSIONAL STRUCTURE

67

400C

103

400C

137

400C

138

400C

139

400C

175

400C

47.695 C 47.707 0 C 47.781 0 C 47.099 0 C 47.112 0 C 48.092 0 C 47.668 0 C 0

49.906 C 49.933 0 C 50.073 0 C 49.466 0 C 49.550 0 C 50.532 0 C 50.105 0 C 0

51.488 C 51.531 0 C 51.737 0 C 51.320 0 C 51.458 0 C 52.322 0 C 51.925 0 C 0

52.610 C 52.670 0 C 52.935 0 C 52.753 0 C 52.924 0 C 53.622 0 C 53.265 0 C 0

41

53.368 C 53.443 0 C 53.758 0 C 53.819 0 C 54.007 0 C 54.524 0 C 54.210 0 C 0

53.799 C 53.887 0 C 54.244 0 C 54.544 0 C 54.739 0 C 55.067 0 C 54.802 0 C 0

53.925 C 54.023 0 C 54.415 0 C 54.950 0 C 55.148 0 C 55.276 0 C 55.068 0 C 0

Q=0.00296779 I=25.63Amp Starting time= 1.1975 secs Q=0.00105249 I=15.26Amp Starting time= 1.222 secs

Q=0.00213725 I=21.75Amp Starting time= 1.7 secs

Q=0.0025078 I=23.56Amp Starting time= 1.99 secs

Q=0.0027563 I=24.7Amp Starting time= 2.33 secs

Q=0.0029148 I=25.4Amp Starting time= 2.702 secs

44.563 0 C 44.567 0 C 44.587 0 C 44.061 0 C 44.014 0 C 44.741 0 C 44.408 0 C

Q=0.00303065 I=25.9Amp Starting time= 3.09 secs

Q=0.003257 I=26.85Amp Starting time= 4.29 secs

400C

Q=0.003125 I=26.3Amp Starting time= 3.48 secs

Initial Temperature

31

Q=0.0031967 I=26.6Amp Starting time= 3.88 secs

Node Nos.

Temperatures exceeding 44 0C in 1st time step with convection in three dimensional structure of totally enclosed machine for different stator current during Reactor starting.

54.427 C 54.533 0 C 54.946 0 C 55.584 0 C 55.761 0 C 55.842 0 C 55.621 0 C 0

53.982 C 54.095 0 C 54.531 0 C 55.403 0 C 55.597 0 C 55.437 0 C 55.312 0 C 0

Fig.7. Corresponding Temperatures vs Time

VI. CONCLUSIONS The numerical models using finite element method developed in this paper referring to a 7.5 kW induction motor proved to be accurate offering useful results for engineers involved in the design of electric machines.

The results based on the proposed method show close agreement for totally enclosed machines with calculated heat transfer coefficients at the back of core surface, air-gap surface and annular end surface of the stator winding. This analysis gives the designers a better idea of where the hot spot is and how the heat is carried away at the outer surfaces with the help of both conduction and convection modes of heat transfer, the magnitude of which is determined in terms of coefficients usually derived from machine parameters.

VII. APPENDIX A. Formulation Of The Heat Convection Matrix On Cylindrical Curved Surface The heat convection term in (13) for i = A is

∫ h [ N ] {T }

(e)

S 2( e )

=h

∫ (T

A

N i ds

)

N A2 + TB N A N B + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + TH N A N H ds

S 2( e )

Now performing the integration in non-dimensional notation

h

∫

2 A

N ds

1

=h∫

−1

1

∫

−1

(τ + 1)2 (v −1)2 (aρcdταdv) 16

42

(16)

1 haρcα 1 (τ + 1)2 dτ ∫−1 (v − 1)2 dv ∫ 1 − 16 4 = haρcα 9 1 1 (τ + 1) (v − 1) × (τ + 1) (v + 1) (aρcdταdv) N A N B ds = h ∫−1 ∫−1 −4 4

=

h

∫

=

haρcα − 16

=

2 ha ρ c α 9

∫ (v 1

−1

2

)

(17)

− 1 dv ∫ (τ + 1) dτ 1

2

−1

(18)

Evaluating the other terms, we obtain

⎡ ⎢ ⎢ ⎢ ⎢ [ S H ] = h a ρc α ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

4 9

2 9 4 9

0 0 ⎤ ⎥ 0 0 0 0 ⎥ 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ 4 2 0 0 ⎥ 9 9 ⎥ 4 0 0 ⎥ 9 0 0 ⎥ ⎥ 0 ⎥⎦ 2 9 1 9

0 0

1 9 2 9

(19)

B. On annular end surface By performing the integration over ( ρ ,v) space,

h

∫

N A2 ds = h

S 2( e )

=

h

=

=

2 ha 2α 2 3(1 − b / a )

∫

1

1

−1

−1

∫ ∫

⎛1 b4 2b b 2 ⎜⎜ − − + 4 3a 2 a 2 ⎝ 4 12 a

N A N B ds = h

ha 2α

− 4(1 − b / a )

2

ha 2α 2 3(1 − b / a )

(ρ − b / a )2 (v − 1)2 2 4 (1 − b / a )

1

1

−1

b/a

∫ ∫

( a ρ cd τα dv )

⎞ ⎟⎟ ⎠

(20)

(ρ − b / a )2 (v 2 − 1) 2 − 4(1 − b / a )

⎛1 b4 b2 2b ⎜⎜ − − + 4 3a 2 a 2 ⎝ 4 12 a

⎛1 b4 2b b 2 ⎜⎜ − − + 4 3a 2 a 2 ⎝ 4 12 a

( aρ cd τα dv )

⎞⎛ 4 ⎞ ⎟⎟⎜ − ⎟ ⎠⎝ 3 ⎠

⎞ ⎟⎟ ⎠

(21)

43

[SH ] =

⎡ 2 ha 2 α ⎢ [ 2 − b / a) 3 1 ( ⎢ 4 ⎢ b 1 ⎢ ×( − 4 12 a 4 ⎢ ⎢ b2 2b + ) ⎢ − a 3 a2 2 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

[

ha 2 α

3 (1 − b / a )

×

2

b4 1 − 4 12 a 4 b2 2b − + ) 3a 2a 2

[

]

2 ha 2 α × 2 3 (1 − b / a )

b4 1 − 4 12 a 4 b2 2b − + ) 3a 2a 2

ha 2 α

3 (1 − b / a )

2

b4 1 − 12 12 a 4 b3 b + − ) 6a 6a 3

×

(

[

]

[

×

[

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0 0 0

0 0 0 0

]

2 ha 2 α × 2 3 (1 − b / a )

(

]

2 ha 2 α

3 (1 − b / a )

2

b4 1 − 12 12 a 4 b3 b + − ) 6a 6a 3

(

]

2 ha 2 α

3 (1 − b / a )

(

2 ha 2 α × 2 3 (1 − b / a )

b4 1 − 12 12 a 4 b3 b + − ) 6a 6a 3

[

b4 1 − 12 12 a 4 b3 b + − ) 6a 6a 3

(

(

]

[

2

×

b4 1 − 12 4a 4 b2 2b 3 + − ) 3 3a 2a 2 (

[

ha 2 α

3 (1 − b / a )

2

b4 1 − 12 4a 4 b2 2b 3 + − ) 3 3a 2a 2

] ×

(

]

[

]

2 ha 2 α × 2 3 (1 − b / a )

b4 1 − 12 4a 4 2b 3 b2 + − ) 3 3a 2a 2 (

SYM

]

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(22)

C. Heat Convection Vector On cylindrical curved surface From (13), the first term of the heat convection vector

∫

h T ∞ N A ds 2 = h T ∞

S 2( e )

= =

∫

(τ

s 2( e )

h T

aρcα − 4

∞

h T∞ aρcα − 4

∫ (τ 1

−1

∫ (τ 1

−1

+ 1)(v − 1) a ρα cd τ dv −4

+ 1 )(v − 1 ) d τ dv

+ 1 )d τ

∫

1 −1

(v

− 1 ) dv

= h T∞ a ρ c α

(23)

Evaluating the other terms, we obtain

[ s C ] = h T∞

⎡ 1⎤ ⎢ 1⎥ ⎢ ⎥ ⎢ 0⎥ ⎢ ⎥ 0 a cρ α ⎢ ⎥ ⎢ 1⎥ ⎢ ⎥ ⎢ 1⎥ ⎢ 0⎥ ⎢ ⎥ ⎣⎢ 0 ⎦⎥

(24)

44

D. On annular end surface The first term of the heat convection on ( ρ ,v) space

∫

h T∞ N

ds 2

A

S 2( e )

= h T∞

1

1

b /a

−1

∫ ∫

(ρ

− b / a )(v − 1 ) a ρα cd ρ dv − 2 (1 − b / a )

1 ⎛ 1 h T∞ a 2α b ⎞ 2 ρ ρ d ρ = − ⎟ ⎜ ∫−1 (v − 1)dv a ⎠ − 2 (1 − b / a ) ∫b / a ⎝

=

h T ∞ a 2α ⎛ 1 b b3 ⎜⎜ − + (1 − b / a ) ⎝ 3 2 a 6 a 3

⎞ ⎟⎟ ⎠

(25)

Evaluating the other terms, we obtain

[ sC

h T ∞ a 2α ] = (1 − b / a )

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

b 1 − + 3 2a b 1 − + 3 2a b3 1 + − 6 3a 3 b3 1 + − 6 3a 3 0 0 0 0

b3 6a 3 b3 6a 3 b2 2a 2 b2 2a 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(26)

VIII. LIST OF SYMBOLS Vr, Vθ and Vz

Thermal conductivities in the radial, Circumferential and axial directions Material density, Pm Material specific heat Cm T Potential function (Temperature) V Medium permeability (Thermal conductivity) watt /m oC q Flux (heat flux) watt / mm2.

Q Forcing function (Heat source) T Surface temperature TAG Air-gap gas temperature TBC Back of core gas temperature n Outward normal vector to the bounding curve Σ Core-end gas temperature.

TD dΣ Differential arc length along the boundary [SR], [Sθ ] and [SZ] Symmetric co-efficient matrices (thermal stiffness matrices) [SH] Heat convection matrix.

[T] Column vector of unknown temperatures.

[R] Forcing function (heat source) vector.

[ST] Column vector of heat convection.

[SC] Column vector heat convection.

[T0] Column vector of unknown (previous point in time) temperatures.

45

REFERENCES [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

G. M. Rosenberry, Jr., “The transient stalled temperature rise of cast aluminum squirrel-cage rotors for induction motors,” AIEE Trans., vol. PAS-74, Oct. 1955.

Armor, A.F., and Chari, M. V. K., “Heat flow in the stator core of large turbine generators by the method of threedimensional finite elements, Part-I: Analysis by Scalar potential formulation: Part - II: Temperature distribution in the stator iron,” IEEE Trans,Vol. PAS-95, No. 5, pp.1648-1668, September 1976.

Armor, A. F., “Transient Three Dimensional Finite Element Analysis of Heat Flow in Turbine-Generator Rotors”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS 99, No.3 May/June. 1980.

Rajagopal,M.S.,Kulkarni,D.B.,Seetharamu,K.N.,and shwathnarayana P.A.,“Axi-symmetric steady state thermal analysis of totally enclosed fan cooled induction motors using FEM”, 2nd Nat Conf.on CAD/CAM,19-20 Aug,1994.

A. Bousbaine, W. FLow, and M. McCormick, “Novel approach to the measurement of iron and stray losses in induction motors,” IEE Proc. Electr. Power Appl., vol. 143, no. 1, pp. 78–86, 1996.

Rajagopal,M.S., Seetharamu,K.N .and Ashwathnarayana.P.A., “ Transient thermal analysis of induction Motors , IEEE Trans ,Energyconversion, Vol.13, No.1,March1998.

Hwang, C.C., Wu, S. S. and. Jiang,Y. H., “Novel Approach to the Solution of Temperature Distribution in the Stator of an Induction Motor”, IEEE Trans. on energy conversion, Vol. 15, No. 4, December 2000.

O.I. Okoro, “Steady and Transient states Thermal Analysis of a 7.5 KW Squirrel cage Induction Machine at Rated Load Operation”, IEEE Trans. Energy Convers., Vol. 20, no. 4, pp. 730-736, Dec. 2005.

Sarkar, D., Bhattacharya, N.K. “Approximate analysis of transient heat conduction in an induction motor during star-delta starting”, in Proc. IEEE Int. Conf. on industrial technology (ICIT 2006), Dec., PP.1601-1606.

Nerg, J., “Thermal Modeling of a High-Speed Solid-Rotor Induction Motor” Proceedings of the 5th WSEAS International Conference on Applications of Electrical Engineering, Prague, Czech Republic, March 12-14, 2006 (pp90-95).

Ruoho S., Dlala E., Arkkio A., “Comparison of demagnetization models for finite-element analysis of permanent magnet synchronous machines”, IEEE Transactions on Magnetics 43 (2007) 11, pp. 3964–3968 Islam J., Pippuri J., Perho J., Arkkio A., “Time-harmonic finite-element analysis of eddy currents in the form-wound stator winding of a cage induction motor”, IET Electric Power Applications 1 (2007) 5, pp. 839–846.

Lin R., Arkkio A., “3-D finite element analysis of magnetic forces on stator end-windings of an induction machine”, IEEE Transactions on Magnetics 44 (2008) 11, Part 2, pp. 4045–4048.

E. Dlala, "Comparison of models for estimating magnetic core losses in electrical machines using the finite-element method", IEEE Transactions on Magnetics, 45(2):716-725, Feb. 2009.

Ruoho S., Santa-Nokki T., Kolehmainen J., Arkkio A., “Modeling magnet length within 2d finite-element analysis of electric machines”, IEEE Transactions on Magnetics 45 (2009) 8, pp. 3114-3120.

Pippuri J.E., Belahcen A., Dlala E., Arkkio A., “Inclusion of eddy currents in laminations in two-dimensional finite element analysis”, IEEE Transactions on Magnetics 46 (2010) 8, pp. 2915–2918.

Nannan Zhao, Zhu, Z.Q., Weiguo Liu., “Thermal analysis and comparison of permanent magnet motor and generator”, International Conference on Electrical Machines and Systems (ICEMS), pp. 1 – 5. 2011.

Hai Xia Xia; Lin Li; Jing Jing Du; Lei Liu.,”Analysis and calculation of the 3D rotor temperature field of a generator-motor”, International Conference on Electrical Machines and Systems (ICEMS), 2011, pp.1 – 4.

Ganesh B. Kumbhar, Satish M. Mahajan., “Analysis of short circuit and inrush transients in a current transformer using a field-circuit coupled FE formulation”, IJEPES- Volume 33, Issue 8, Pages 1361–1367, October 2011.

B.L. Rajalakshmi Samaga, K.P. Vittal.,“Comprehensive study of mixed eccentricity fault diagnosis in induction motors using signature analysis”, IJEPES- Volume 35, Pages 180-185, 2012.

46