An Improved Dynamic Model for the Simulation of ...

An Improved Dynamic Model for the Simulation of Three-Phase Induction Motors with Deep Rotor Bars D. Lin and P. Zhou Ansoft Corporation, Pittsburgh, PA 15219, USA

Abstract-The skin effect of rotor current is one of key factors which affect the simulation accuracy of three-phase induction motors with deep rotor bars. This paper proposes an improved dynamic model considering the skin effect of three-phase induction motors at arbitrary operation, from the starting to rated-load operation. The simulation results using the proposed model are compared with those from the 2-dimensional (2D) transient finite element analysis (FEA).

I.

INTRODUCTION

The skin effect of a three-phase induction motor is traditionally considered by introducing the rotor resistance and inductance correction factors, noted as Kr and Kx, respectively. For rectangular rotor bar conductors, they are computed from sinh(2ξ ) + sin( 2ξ ) ⎧ ⎪ K r = ξ ⋅ cosh(2ξ ) − cos(2ξ ) ⎪ (1) ⎨ ⎪ K = 3 ⋅ sinh(2ξ ) − sin(2ξ ) ⎪⎩ x 2ξ cosh(2ξ ) − cos(2ξ )

with

ξ = hc ⋅ π ⋅ μ 0 ⋅ f r ⋅ σ r ⋅ (bc / bs ) = 0.001987 ⋅ hc ⋅

(2)

f r ⋅ σ r ⋅ (bc / bs )

where bc and hc are the rotor bar width and height respectively, bs is the slot width, σr is the bar conductivity, and fr is the rotor slip frequency. For other rotor slot shapes, the rotor bar resistance and the rotor slot leakage inductance are computed in two steps. The first step is to decide the equivalent resistance and inductance conductor heights, hr and hx respectively, by ⎧hr = hc / K r (3) ⎨ ⎩hx = hc ⋅ K x

The second step is to compute the resistance and the inductance based on the relevant equivalent conductor height. Equations (1)-(3) are widely used in the steady-state parameter computation in the frequency domain even though the parameter evaluation for non-rectangular slot shapes is approximated. Many dynamic models have been developed to consider the rotor bar skin effect in the time domain [1]-[4]. Following transmission line theory, the rotor bar is divided into a finite number of sections and each section is modeled by a 2-port network with T, π, or L configuration. Each section current is equivalent by a line current, or a sheet current which is located

in the section center for the T configuration, or at the section bottom for the L configuration. For the π configuration, the section current is equivalent by two equal line currents or sheet currents which are located on the top and at the bottom of the section. If enough sections are used, theses models with various configurations can simulate the skin effect with sufficient accuracy at various high frequencies. However, at a low frequency, such as the slip frequency at the no-load or full-load operation, the models may not be as accurate as the model using DC resistance and DC inductance because the current density distribution in each section is not assumed to be uniform. In this paper, the current density distribution in each section is assumed to be uniform, based on which a more accurate dynamic model is proposed. The accuracy of the model with uniform or non-uniform sections is discussed and compared with the accuracy of the T configuration presented in [1]-[3]. II.

SKIN EFFECT MODELING

To consider the skin effect in the time domain, the rotor bar is divided into several sections, and each section is modeled by a 2-port network. The current density distribution in a section is assumed to be uniform. The self inductance of each section and the mutual inductance between sections are computed. In order to be able to compare the accuracy between the proposed model and the analytical solution (1)-(2), a rectangular rotor slot is considered. Assume a rectangular rotor bar is split into two sections, as shown in Fig. 1. The ratios of the top and the bottom section heights to the total bar height are defined as ⎧k1 = Δh / hc . (4) ⎨ ⎩k 2 = (hc − Δh) / hc = 1 − k1

Fig. 1. The rectangular rotor bar split into 2 sections

The self inductance of the top section is Δh (5) L1 = μ 0l ⋅ = k1 ⋅ Ls 20 . 3bs The mutual inductance between the top and bottom sections is Δh 3 (6) M = μ0l ⋅ = k1 ⋅ Ls 20 . 2bs 2 The self inductance of the bottom section can be decomposed into two components: L0 which corresponds to the leakage flux through the top section, and L2 which corresponds to the leakage flux through the bottom section, and they are given below Δh ⎧ ⎪ L0 = μ 0l ⋅ b = 3k1 ⋅ Ls 20 = 2M ⎪ s (7) . ⎨ h h ( ) − Δ c ⎪L = μ l ⋅ = k 2 ⋅ Ls 20 0 ⎪⎩ 2 3bs The resistances of the top and the bottom sections are 1 l ⎧ ⎪ R1 = σ Δh ⋅ b = Rs 20 / k1 ⎪ r c (8) . ⎨ 1 l ⎪R = = Rs 20 / k 2 ⎪⎩ 2 σ r (hc − Δh) ⋅ bc In (5)-(8), Rs20 and Ls20, are the DC resistance and DC leakage inductance of the rotor bar, respectively, and are given below hc ⎧ ⎪ Ls 20 = μ 0l ⋅ 3b ⎪ s (9) ⎨ l 1 ⎪R = ⎪⎩ s 20 σ r hc ⋅ bc with l being the rotor axial length. The coupling circuit for the top and the bottom sections is shown in Fig. 2. The bottom section can be split to two sections consecutively. The equivalent circuit for the rotor bar with m sections is shown in Fig. 3, where ki is the ratio of the i-th section height to the total bar height, and satisfies

A

A

(10)

i =1

It can be seen from (5)-(7) that if each section current is equivalent by a line current, then L1=M when the line current is located in the section center, and Fig. 2 becomes the T configuration [3]. If the line current is put at the bottom of each section, then L1=M=L0, and Fig. 2 becomes the L configuration. In Fig. 3, all inductances in series with the section resistances are negative, which can be avoided by using mutual inductances in the circuit, as shown in Fig. 4. It is approved below that the circuit model of Fig. 3 is accurate at low frequency. When the rotor current I distributes uniformly in the whole rotor bar, the i-th section current will be (11) I i = ki ⋅ I . The input active power is m

m

i =1

i =1

P = ∑ (ki I ) 2 ( Rs 20 / ki ) = I 2 Rs 20 ∑ ki = I 2 Rs 20 .

(12)

M

L1 R1

L0+L2 R2

L1-M

L2

R1

R2

B

B

Fig. 2. The circuit of rotor bar with 2 sections

3k1Ls20/2

3k1Ls20/2

3kiLs20/2

3kiLs20/2

3kmLs20/2

A -k1Ls20/2

-kiLs20/2

-kmLs20/2

Rs20/k1

Rs20/ki

Rs20/km

B Fig. 3. The circuit of rotor bar with m sections

3k1Ls20

3kiLs20

A k1Ls20

3k1Ls20/2

Rs20/k1

kiLs20

3kiLs20/2

Rs20/ki

kmLs20 Rs20/km

B Fig. 4. The circuit of rotor bar with mutual inductances

The reactive power in the last two sections is Q = I 2 ⋅ 2πf r ⋅ Ls 20 ⋅ [k m 2 (1.5k m − 0.5k m + 1.5k m −1 )

+ (k m −1 + k m ) 2 (1.5k m −1 ) + k m −12 (−0.5k m −1 )]

= I 2 ⋅ 2πf r ⋅ Ls 20 ⋅ [(k m' −1 ) 2 (1.5k m' −1 − 0.5k m' −1 )] (13)

m

∑ ki = 1 .

L0-M

M

where k m' −1 = k m −1 + k m .

(14) Equation (13) indicates that the last two sections can be combined into one section if the current density distributes uniformly in the sections. Continue to combine the last two sections, one can finally obtain Q = I 2 ⋅ 2πf r ⋅ Ls 20 . (15) Equations (12) and (15) show that when the rotor current distributes uniformly in the whole rotor bar, the equivalent port resistance and inductance of Fig. 3 become the DC resistance and inductance, respectively, which means that the proposed model is also applicable at low slip rotor frequencies. The equivalent bar resistance and inductance at different frequencies can be obtained from the input impedance of Fig.3. The ratio of the equivalent bar resistance to the DC bar resistance is Kr, and that for the inductance is Kx. For a given accuracy of Kr and Kx, the required number of sections m may

change with the section distribution, as discussed in the following sections in detail.

0

1

2

3

4

5 ξ

0.0% -10.0%

When the rotor bar is split into uniform sections, the ratio of the i-th section height to the total bar height is (i = 1, 2, …, m). (16) ki = 1/m The port input impedance ZAB at different rotor frequencies, or at different ξ from (2), can be derived from Fig. 3. Then the resistance and inductance correction factors are ⎧ K r = Re( Z AB ) / Rs 20 . ⎨ (17) ⎩ K x = Im(Z AB ) /(2πf r ⋅ Ls 20 ) The results computed from (17) with various numbers of uniformly distributed sections are compared with the analytical solutions in Fig. 5. The errors of the resistance correction factor between the results of the proposed model and the analytical solutions with uniformly distributed sections are shown in Fig. 6 and Table I, compared with those from the T-configuration [1]-[3]. The errors of the inductance correction factor with uniformly distributed sections are compared in Fig. 7 and Table I. kx 1

Proposed model (m=2)

-30.0%

T confiuration (m=2) Proposed model (m=4)

-40.0%

T confiuration (m=4) Proposed model (m=8)

-50.0%

T confiuration (m=8)

-60.0%

Fig. 6. The errors of Kr with uniformly distributed sections from the proposed model compared with those from the T configuration [1]-[3] 60.0% Proposed model (m=2)

50.0%

T confiuration (m=2) Proposed model (m=4)

40.0%

T confiuration (m=4) Proposed model (m=8)

30.0%

T confiuration (m=8) 20.0% 10.0% 0.0% 0

1

5 4

Kx - Analytical solution

3

ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0 ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0 ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0

m=2

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

1

0

0 0

1

2

3

4

ξ

5

m=4

(a) With m=2 uniformly distributed sections kx 1

kr 5

0.8

4 Kx - Proposed model Kx - Analytical solution

0.6

3

m=8

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

1

0

0 0

1

2

3

4

ξ

5

(b) With m=4 uniformly distributed sections kx 1

IV.

kr 5

0.8 0.6

3

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

1

0

0 0

1

2

3

4

5

4

5

ξ

Proposed model ΔKx (%) ΔKr (%) 0.0 0.0 -2.3 0.9 -10.8 9.9 -15.0 21.4 -25.7 35.3 -37.4 56.0 0.0 0.0 -0.6 0.3 -2.6 2.4 -4.3 4.9 -8.0 8.5 -12.1 13.8 0.0 0.0 -0.1 0.1 -0.7 0.6 -1.1 1.2 -2.1 2.1 -3.2 3.3

T configuration ΔKr (%) ΔKx (%) 0.0 12.5 -2.5 13.0 -21.0 24.6 -39.0 61.4 -51.5 106 -60.5 153 0.0 3.1 -0.6 3.2 -6.0 5.7 -13.2 14.7 -21.4 27.3 -30.1 43.2 0.0 0.8 -0.2 0.8 -1.5 1.4 -3.5 3.5 -6.0 6.5 -9.2 10.2

MODEL ACCURACY WITH GEOMETRICALLY DISTRIBUTED SECTIONS

When the rotor bar is split into geometrically distributed sections, such that each section is doubled in height as the previous section, that is (i = 2, 3, …, m). (18) ki = 2 ki-1 Combining (18) with (10), one obtains (i = 1, 2, …, m). ki = 2 i −1 /(2 m − 1) (19)

4 Kx - Proposed model Kx - Analytical solution

3

Fig. 7. The errors of Kx with uniformly distributed sections from the proposed model compared with those from the T configuration [1]-[3]

Kx - Proposed model 0.6

2

TABLE I. THE ERRORS OF THE RESISTANCE AND INDUCTANCE CORRECTION FACTORS WITH UNIFORMLY DISTRIBUTED SECTIONS FROM THE PROPOSED MODEL AND FROM THE T CONFIURATION [1]-[3]

kr

0.8

-20.0% ΔKr

MODEL ACCURACY WITH UNIFORMLY DISTRIBUTED SECTIONS

ΔKx

III.

ξ

(c) With m=8 uniformly distributed sections Fig. 5. The results of Kx and Kr with uniformly distributed sections from the proposed model compared with the analytical solutions

The results of Kr and Kx with various numbers of geometrically distributed sections are compared with the analytical solutions in Fig. 8.

kx 1

40.0% Proposed model (m=2) T confiuration (m=2) Proposed model (m=3)

30.0%

T confiuration (m=3) Proposed model (m=4) ΔKx

The errors of the resistance correction factor between the results of the proposed model and the analytical solutions with geometrically distributed sections are shown in Fig. 9 and Table II, compared with those from the T-configuration [1]-[3]. The errors of the inductance correction factor with geometrically distributed sections are compared in Fig. 10 and Table II.

T confiuration (m=4)

20.0%

10.0%

kr 5

0.0%

0.8

0

4

1

2

3

4

5 ξ

Kx - Proposed model Kx - Analytical solution

0.6

Fig. 10. The errors of Kx with geometrically distributed sections from the proposed model compared with those from the T configuration [1]-[3]

3

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

TABLE II. THE ERRORS OF THE RESISTANCE AND INDUCTANCE CORRECTION FACTORS WITH GEOMETRICALLY DISTRIBUTED SECTIONS FROM THE PROPOSED MODEL AND FROM THE T CONFIURATION [1]-[3]

1

0

0 0

1

2

3

4

ξ

5

ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0 ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0 ξ = 0.0 ξ = 1.0 ξ = 2.0 ξ = 3.0 ξ = 4.0 ξ = 5.0

m=2

(a) With m=2 geometrically distributed (1/3 + 2/3) sections kx 1

kr 5

0.8

4 Kx - Proposed model Kx - Analytical solution

0.6

m=3

3

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

1

0

0 0

1

2

3

4

ξ

5

m=4

(b) With m=3 geometrically distributed (1/7 + 2/7 + 4/7) sections kx 1

kr 5

0.8

Proposed model ΔKr (%) ΔKx (%) 0.0 0.0 -1.7 1.0 -4.5 8.9 -0.3 10.4 -8.7 9.6 -21.7 17.6 0.0 0.0 -0.8 0.6 -1.1 4.7 2.0 4.0 -0.7 3.5 -2.6 6.6 0.0 0.0 -0.6 0.4 -0.4 3.4 2.2 2.3 0.5 1.7 0.0 3.2

T configuration ΔKr (%) ΔKx (%) 0.0 16.7 0.8 16.4 -0.8 16.0 -13.6 25.9 -28.7 46.5 -41.2 73.6 0.0 10.6 1.3 10.1 3.7 7.2 -0.2 9.6 -2.0 14.8 -3.4 18.5 0.0 8.7 1.4 8.1 4.3 4.7 2.0 5.3 1.8 7.6 2.0 8.5

4 Kx - Proposed model Kx - Analytical solution

0.6

V.

3

Kr - Proposed model Kr - Analytical solution

0.4

2

0.2

1

0

0 0

1

2

3

4

5

ξ

(c) With m=4 geometrically distributed (1/15 + 2/15 + 4/15 + 8/15) sections Fig. 8. The results of Kr and Kx with geometrically distributed sections from the proposed model compared with the analytical solutions 5.0% 0.0% 0

1

2

3

4

5 ξ

ΔKr

-5.0% -10.0% -15.0%

Proposed model (m=2) T confiuration (m=2) Proposed model (m=3) T confiuration (m=3)

-20.0%

Proposed model (m=4) T confiuration (m=4)

-25.0%

Fig. 9. The errors of Kr with geometrically distributed sections from the proposed model compared with those from the T configuration [1]-[3]

MODEL ACCURACY WITH OPTIMIZED TWO NON-UNIFORM SECTIONS

From Table I one can observe that when the rotor bar is split to two uniform (1/2 + 1/2) sections, the errors of Kr and Kx are within about 10% for ξ ≤ 2. However, with two non-uniform (1/3 + 2/3) sections, the maximum value of ξ can extend to 4 for about the same errors as shown in Table II, which is equivalent to that with four uniform sections (see Table I for m = 4). For the computation accuracy of the stator winding currents and the rotor torque in the motor starting process, the error of Kr is more sensitive than the error of Kx, this is due to the following factors: i) The rotor resistance consists of the bar resistance and the end ring resistance, and the bar resistance usually takes the major part, while the rotor bar leakage inductance usually takes a minor part of the rotor total leakage inductance which may include slot leakage inductance, phase-belt leakage inductance, end-ring leakage inductance, and skew leakage inductance. ii) The skin effect causes the rotor bar resistance further to increase significantly (increase about 3 times when ξ = 4), and causes the rotor bar leakage inductance further to decrease significantly (decrease about 60% when ξ = 4).

kx 1

400 300 200

Current (A)

iii) The rotor resistance affects the torque computation not only via the rotor current, but also via the rotor resistance itself. Therefore, it is reasonable to optimize the ratio k1 by minimizing the maximum error of Kr for 0 ≤ ξ ≤ ξs provided the maximum error of Kx does not change too much, where ξs is the value of ξ at the starting operation. When ξs = 4, the optimized top section height ratio k1 = 0.31. The results of Kr and Kx with such two non-uniform sections are shown in Fig. 11, and their errors are shown in Fig. 12.

100 0 0

200

400

600

800

1000

-100 -200 -300

kr 5

Current waveform from transient FEA Magnitude current from proposed model

-400 Speed (rpm)

0.8

Fig. 13. The computed stator current from the proposed model with optimized two non-uniform sections compared with the transient FEA result

4 Kx - Proposed model Kx - Analytical solution

0.6

3

Kr - Proposed model

700

Kr - Analytical solution

0.4

2 600

1

0

0 0

1

2

3

4

500

ξ

5

Fig. 11. The results of Kr and Kx with two non-uniform (0.31 + 0.69) sections from the proposed model compared with the analytical solutions

Torque (Nm)

0.2

400 300 200 100

15.0%

For Kr

0

For Kx

10.0%

Torque from transient FEA Torque from proposed model

0

200

400

600

800

1000

Speed (rpm)

Fig. 14. The computed rotor torque from the proposed model with optimized two non-uniform sections compared with the transient FEA result

Error

5.0%

0.0% 0

1

2

3

4

ξ

-5.0%

-10.0%

Fig. 12. The errors of Kr and Kx with two non-uniform (0.31 + 0.69) sections from the proposed model

The maximum error of Kr with the optimized section ratio decreases from about 9% to about 5% compared with the case of m = 2 in Table II for ξ ≤ 4, meanwhile, the maximum error of Kx does not change too much, from about 10% to about 12%. For induction motors with cast-aluminum squirrel-cage rotors operating at 60Hz voltage sources, ξ ≤ 4 means the maximum rotor bar height hc can be as large as 54mm based on (2), where σr = 2.3×107 Siemens/m for cast aluminum at 75 Centigrade degrees. Such a bar height value is generally large enough for most medium and small size induction motors. VI.

APPLICATION

The proposed model is applied to calculate the performance of a 6-pole, 11kW three-phase induction motor with a cast-aluminum squirrel-cage rotor operating at 50Hz, 380V voltage sources. The computed results are compared with the two-dimensional (2D) transient FEA solutions. The results of the stator winding currents and the rotor torque varying with the rotor speed are compared in Fig. 13 and Fig. 14, respectively. In Figs. 13-14, the FEA results are obtained by the following two steps: 1) the motor is analyzed at the synchronous speed first; 2) after the steady state is reached in step 1, the rotor speed is slowly decreased to the zero speed.

VII. CONCLUSION An improved dynamic model for the transient simulation of three-phase induction motors with deep rotor bars is proposed. Comparing to the model presented in [1]-[3], this improved model reduces errors by about 2/3 in average for various uniformly or geometrically distributed sections. By minimizing the error of the rotor resistance, the optimized model with two non-uniform sections is also presented, which makes the model general for double cage rotors. This model is theoretically accurate when the rotor frequency approaches to zero. Therefore, the proposed model is also applicable for the performance computation at low rotor slip frequencies, such as at the no-load and full-load operations. The proposed model is also suitable for non-rectangular rotor slots, as long as the parameter computation in (5)-(8) is based on the real section sizes. REFERENCES [1] D. S. Babb, and J. E. Williams, “Network analysis of ac machine conductors,” AIEE Transactions (Power Apparatus and Systems), Vol. 70, pp. 2001-2005, 1951. [2] E. A. Klingshirn, and H. E. Jordan, “Simulation of polyphase induction machines with deep rotor bars,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89, No. 6, pp. 1038-1043, 1970. [3] W. Levy, C. F. Landy, and M. D. McCulloch, “Improved models for the simulation of deep bar induction motors,” IEEE Transactions on Energy Conversion, Vol. 5, No. 2, pp. 393-400, June 1990. [4] J. Li, and L. Xu, “Investigation of cross-saturation and deep bar effects of induction motors by augmented d-q modeling method,” IEEE Industry Application Conference, Thirty-Sixth IAS Annual Meeting, Vol. 2, pp. 745-750, 2001.