robust sliding mode speed observer for induction

ERJ Engineering Research Journal Faculty of Engineering Minoufiya University

ROBUST SLIDING MODE SPEED OBSERVER FOR INDUCTION MOTOR DRIVES M. S. Zaky,

M. M. Khater,

H. Yasin,

S. S. Shokralla,

A. El-Sabbe

Electrical Engineering Department, Faculty of Engineering, Minoufiya University, Shebin El-Kom, Egypt ABSTRACT Speed-sensorless control is of utmost importance for high performance operation of induction motor drives. For this purpose, high accuracy of speed estimation is required. Sliding mode observers (SMO) with its salient features are necessary for state estimation of a nonlinear plant such as speed estimation of induction motor drives. In this paper, a speed estimation algorithm based on sliding mode control theory is derived and implemented. Chattering problem is eliminated using continuous sliding mode observer. Lyapunov stability theory is applied to guarantee the convergence of sliding mode observer. Also, the robustness, accuracy and insensitivity to parameter variation of the proposed observer are examined. A floating-point Digital Signal Processor (DSP) TMS320C31 control board with a hardware/software interface is used to implement the proposed sliding mode observer for speed estimation. Experimental results are presented and discussed to compare the measured and estimated speed signals at different operating conditions.

‫ﻟﻘﺪ أﺻﺒﺢ اﻟﺘﺤﻜﻢ ﺑﻄﺮق اﻟﺘﺤﺴﺲ واﻻﺳﺘﺸﻌﺎر اﻟﻐﻴﺮ ﻣﺒﺎﺷﺮ ﻟﻠﺴﺮﻋﺔ ذو أهﻤﻴﺔ ﻋﺎﻟﻴﺔ وذﻟﻚ ﻟﻀﻤﺎن ﺧﺼﺎﺋﺺ أداء ﻋﺎﻟﻴﺔ‬ ‫ وﻣﻦ أﻧﺠﺢ اﻟﻄﺮق اﻟﺘﻲ ﺗﺴﺘﺨﺪم ﻣﻊ اﻷﻧﻈﻤﺔ ﻏﻴﺮ اﻟﺨﻄﻴﺔ – ﻣﺜﻞ ﻣﻨﻈﻮﻣﺔ اﻟﻤﺤﺮك‬،‫ﻟﻨﻈﻢ ﺗﺴﻴﻴﺮ اﻟﻤﺤﺮك اﻟﺘﺄﺛﻴﺮى‬ ‫ ﺗﻠﻚ اﻟﻄﺮﻳﻘﺔ اﻟﺘﻲ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻧﻈﺎم اﻟﺘﺤﻜﻢ ذي اﻟﻨﻤﻂ اﻻﻧﺰﻻﻗﻲ ﺣﻴﺚ أﻧﻬﺎ ﺗﺘﻤﻴﺰ ﺑﺨﺼﺎﺋﺺ ﻣﺘﻌﺪدة ﻣﺜﻞ اﻟﻤﺘﺎﻧﺔ‬-‫اﻟﺘﺄﺛﻴﺮي‬ ‫ إﻻ‬،‫ﺿﺪ ﺗﻐﻴﺮ اﻟﻤﻌﺎﻣﻼت وﺗﺤﻤﻞ اﻻﺿﻄﺮاﺑﺎت آﻤﺎ أﻧﻬﺎ ﻃﺮﻳﻘﺔ ﺑﺴﻴﻄﺔ اﻟﺘﺼﻤﻴﻢ وﺳﻬﻠﺔ اﻟﺘﻄﺒﻴﻖ ﻋﻠﻰ اﻷﻧﻈﻤﺔ اﻟﺮﻗﻤﻴﺔ‬ ‫أن هﺬﻩ اﻟﻄﺮﻳﻘﺔ ﻳﻨﺸﺄ ﻋﻨﻬﺎ ﻗﺪر آﺒﻴﺮ ﻣﻦ اﻟﺘﺬﺑﺬب واﻻهﺘﺰاز ﻓﻲ إﺷﺎرة اﻟﺴﺮﻋﺔ اﻟﻤﺴﺘﺸﻌﺮة آﻤﺎ أﻧﻬﺎ ﺗﺼﻨﻒ ﻣﻦ اﻟﻄﺮق‬ ‫ وﻳﻘﺪم هﺬا اﻟﺒﺤﺚ ﻃﺮﻳﻘﺔ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻧﻈﺎم اﻟﺘﺤﻜﻢ ذي‬،‫اﻟﺘﻲ ﺗﻌﺘﻤﺪ ﺑﺼﻔﺔ أﺳﺎﺳﻴﺔ ﻋﻠﻰ ﻣﻌﺎﻣﻼت اﻵﻟﺔ ﻓﻲ ﺣﺴﺎب اﻟﺴﺮﻋﺔ‬ ‫ وﻳﺘﻌﺮض ﻟﺒﺤﺚ ﻣﺪى دﻗﺔ وﻣﺘﺎﻧﺔ إﺷﺎرة اﻟﺴﺮﻋﺔ‬،‫اﻟﻨﻤﻂ اﻻﻧﺰﻻﻗﻲ اﻟﻤﺘﺼﻞ ﻟﺘﺤﺴﺲ واﺳﺘﺸﻌﺎر ﺳﺮﻋﺔ اﻟﻤﺤﺮك اﻟﺘﺄﺛﻴﺮي‬ ‫ وﻳﺘﻌﺮض آﺬﻟﻚ ﻟﺒﺤﺚ ﻣﺪى ﺗﺄﺛﻴﺮ ﻋﺪم اﻟﺘﻮاﻓﻖ اﻟﺠﻴﺪ ﺑﻴﻦ اﻟﻘﻴﻤﺔ اﻟﻔﻌﻠﻴﺔ‬،‫اﻟﻤﺴﺘﺸﻌﺮة ﻋﻨﺪ ﺣﺎﻻت اﻟﺘﺸﻐﻴﻞ اﻟﻤﺨﺘﻠﻔﺔ‬ ‫ وﻗﺪ ﺗﻢ ﺗﻨﻔﻴﺬ هﺬا اﻟﻨﻈﺎم اﻟﻤﻘﺘﺮح‬،‫ﻟﻤﻘﺎوﻣﺔ ﻣﻠﻔﺎت اﻟﻌﻀﻮ اﻟﺜﺎﺑﺖ وﺗﻠﻚ اﻟﻘﻴﻤﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ ﻓﻲ ﺧﻮارزم ﺣﺴﺎب اﻟﺴﺮﻋﺔ‬ ‫ وﻗﺪ ﺗﺒﻴﻦ ﻣﻦ ﻧﺘﺎﺋﺞ‬،‫ وﻳﻘﺪم اﻟﺒﺤﺚ ﻋﺮﺿﺎ وﻣﻨﺎﻗﺸﺔ ﻟﻠﻨﺘﺎﺋﺞ اﻟﺘﺤﻠﻴﻠﻴﺔ واﻟﻤﻌﻤﻠﻴﺔ‬،‫ﻣﻌﻤﻠﻴًﺎ ﺑﺎﺳﺘﺨﺪام ﻣﻌﺎﻟﺞ اﻹﺷﺎرات اﻟﺮﻗﻤﻴﺔ‬ .‫اﻟﺒﺤﺚ دﻗﺔ وﻣﺘﺎﻧﺔ ﻧﻈﺎم اﻟﻤﺮاﻗﺒﺔ اﻟﻤﻘﺘﺮح ﻋﻨﺪ ﺣﺎﻻت اﻟﺘﺸﻐﻴﻞ اﻟﻤﺨﺘﻠﻔﺔ ﻣﻤﺎ ﻳﺆآﺪ ﻣﺪى ﻓﻌﺎﻟﻴﺔ هﺬا اﻷﺳﻠﻮب‬ Keywords: Sliding Mode Observer, Speed Sensorless, and Lyapunov theory. 1. INTRODUCTION Accurate speed information is always necessary to achieve high performance and high-precision speed control of induction motor drives. Traditionally, a direct speed sensor, such as a resolver or an encoder, is usually mounted to the motor shaft for speed feedback. The use of such direct speed sensor besides being bulky, it adds an extra cost and the drive system becomes expensive. Speed sensor, also, implies additional electronics, extra wiring, extra space, frequent maintenance and careful mounting which detracts from the inherent robustness and the reliability of the drive. For these reasons, the development of alternative indirect methods becomes an important research topic [1]. Therefore, there is a great interest in the research community to develop a high performance induction

motor drive that does not require a direct speed or position sensor for its operation; in other words, to develop a speed-sensorless induction motor drive. Many advantages are expected from speed-sensorless induction motor drives such as reduced hardware complexity, low cost, reduced size, elimination of direct sensor wiring, better noise immunity, increased reliability, and less maintenance requirements. Speed-sensorless motor drives are also preferred in hostile environments and high-speed applications [12]. Recently, serious attempts to eliminate direct speed sensor of induction motor drives are reported. All these attempts employ motor terminal variables and its parameters in some way or another to estimate the speed. The question always arises is to which

Engineering Research Journal, Vol. 30, No. 4, October 2007, PP 425-431 © Faculty of Engineering, Minoufiya University, Egypt

425

M. S. Zaky, M. M. Khater, H. Yasin, S. S. Shokralla, A. El-Sabbe, "Robust Sliding Mode Speed Observer ……"

extent the proposed method is successful without deteriorating the dynamic performance of the drive. Several methods have been recently, proposed for speed estimation of high performance induction motor drives [3-9]. Some of these methods are based on a non-ideal phenomenon such as rotor slot harmonics [3]. Such methods require spectrum analysis, which besides being time consuming procedures; they allow a narrow band of speed control. Another class of algorithms relies on some kind of probing signals injected into stator terminals (voltage and/or current) to detect the rotor flux and consequently, the motor speed [4]. These probing signals, sometimes, introduce a high frequency torque pulses, and hence speed ripple. In some cases a useful data may be distorted due to interference with the high frequency probing signals. Despite the merits of the above methods of speed estimation near zero speeds, they suffer from large computation time, complexity and limited bandwidth control. Alternatively, speed information can be obtained by using the machine model and its terminal quantities like voltage and current [5-9]. These include different methods such as the use of simple open loop speed calculators, Model Reference Adaptive Systems (MRAS) [2], adaptive flux observer [5], Extended Kalman Filters [6], artificial intelligence techniques [7], and sliding mode observer (SMO) [8-9]. Model-based methods are characterized by their simplicity and good performance at high speeds; however their behavior is deteriorated at low speeds mostly, due to parameter variations. Temperature rise under loading conditions has a considerable effect on the motor winding resistance. Any mismatch between the actual winding resistance and its corresponding value in the speed estimation algorithm is reflected on the accuracy of the estimated speed. Therefore, a robust speed estimation algorithm against parameter mismatch becomes mandatory. Sliding mode observer may also be classified as a model-based method. It has many advantages and it is considered one of the robust observers for flux and speed estimation. There is a growing interest of using SMO for speed estimation of induction motor drives [8-13]. Sliding mode observer is based on Variable Structure Control (VSC) theory which offers many good properties, such as good performance against un-modeled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response. These properties are necessary for state estimation of a nonlinear plant such as speed estimation of induction motor drives. In this paper, a speed estimation algorithm based on sliding mode observer for rotor flux and stator current estimations is presented. Lyapunov function is chosen to guarantee the convergence of SMO. 426

Sliding mode observer is implemented in software and linked to a floating-point Digital Signal Processor (DSP) control board for speed estimation procedure. The performance of the proposed SMO is investigated by experiments under different operating conditions. The results show the robustness, accuracy and superiority of the proposed system under stator resistance mismatch. 2. MATHEMATICAL MODEL OF INDUCTION MOTOR The induction motor can be represented by its dynamic model expressed in the stationary reference frame in terms of the stator current and rotor flux as follows;

di ss L m d λ sr ⎞ 1 ⎛ s s = ⎜ u s − R sis − ⎟ dt σL s ⎝ L r dt ⎠ ⎞ d λ sr L m s ⎛ 1 = i s − ⎜ − J ω r ⎟ λ sr dt Tr ⎝ Tr ⎠

(1) (2)

By considering the rotor speed as a system parameter; the dynamic model can be expressed by the following state equation; s s d ⎡ is ⎤ ⎡A11 A12 ⎤ ⎡ is ⎤ ⎡b1 ⎤ s ⎡vs ⎤ = Ax + Bvs ⎢ s⎥ = ⎢ ⎢ ⎥+ dt ⎣λr ⎦ ⎣A21 A22 ⎥⎦ ⎣λsr ⎦ ⎢⎣ 0 ⎥⎦ ⎣ ⎦

(3)

where A11, A12, A21, A22 and b1 are given in the Appendix. The electromechanical equation is given by;

J

dωr = Te − TL − B ω r dt

(4)

where the electromagnetic torque is expressed as;

Te =

2 Lm ⎡⎣ λ sdr i sqs − λ sqr i sds ⎤⎦ 3 Lr

(5)

3. DESIGN OF SLIDING MODE OBSERVER From the induction motor model, the SMO for rotor flux and stator current estimations can be constructed as [8, 9]:

(

pxˆ = Axˆ + Bvs + K1 sgn îss − iss

)

(6)

Where K1 is a gain matrix which can be arranged in the following general form;

K1 = [ K −K] , K = kI T

and k is the switching

gain. The error equation which takes into account the parameter variation can be expressed by subtracting Eqn. (3) from Eqn. (6):

(

pe = Ae + ∆ Axˆ + K 1 sgn îss − i ss where; e = xˆ − x = [ ei

)

T eλ ] , ei = îss − iss , eλ = λˆ sr − λsr

∆A12 ⎤ ⎡ ∆A and ∆A = ⎢ 11 ⎥ ⎣∆A21 ∆A22 ⎦ Defining the switching surface S of the SMO as:

Engineering Research Journal, Minoufiya University, Vol. 30, No. 4, October 2007

(7)


S (t ) = e i = iˆss − i ss = 0

(8)

The sliding mode occurs when the following sliding condition is satisfied; (9) e iT ⋅ pe i < 0 Since the sliding mode condition is satisfied with a small switching gain k, then

eiT = pei = 0

(10)

From which,

0 = A12 e λ + ∆A11îss + ∆A12 λˆ sr − L pe = A e + ∆A î s + ∆A λˆ s + L λ

22 λ

21 s

(

where, L = − K sgn îss − i ss

)

22

r

(11) (12)

From Eqns. (11) and (12), the error equation for the rotor flux in sliding mode condition is obtained as; peλ = ( A22 + A12 ) eλ + ( ∆A21 +∆A11 ) îss + ( ∆A22 +∆A12 ) λˆ sr (13) 3.1. Rotor Speed Estimation Algorithm The matrix ∆A can be obtained, by assuming the rotor speed as a variable parameter and no other parameter variations, as follows: ∆A11 = 0, ∆A12 =

−∆ωr J ˆ r − ωr , ∆A21 = 0, ∆A22 = ∆ωr J, ∆ωr = ω ε

(

V = e Tλ e λ +

1 ∆ωr2 , µε

µ> 0

(15)

where µ is a positive constant. The Lyapunov function must be determined in order to assure the convergence of parameter estimation according to the Lyapunov stability theory. The time derivative of Lyapunov function V can be expressed as; (16) p V = p V1 + p V 2 where, −1 pV1 = LT Λ T A 12 L

pV2 = LTΛTA12−1

(17)

∆ωr ˆ s 2 d ˆr Jλr + ∆ωr ω dt ε µε

(18)

and Λ = I − ε I The condition of (16), being negative definite, will be satisfied if pV1 < 0 and pV2 = 0 . The condition pV1 < 0 is satisfied by choosing

Λ T = − γ A 12 ,

γ>0

(19)

where γ is a positive constant. With this assumption, the condition pV2 = 0 gives

ˆ r = µ γ LT J λˆ sr pω

(20)

This equation can be written in the following form for the speed estimation:

(

3/2

Vs ⎡ B1 ⎤ ⎢0⎥ ⎣ ⎦

+

+ +

Using Eqn. (13), the error equation of the rotor flux observer becomes: (14) pe λ = ( A 22 + A 12 ) e λ + ( I − I ε ) J ∆ω r λˆ sr The Lyapunov function candidate is chosen as;

)

)

ˆ r = µγk ∫ ⎡sgn î dss −isds ⋅λˆ sqr − sgn îqss − isqs ⋅λˆ sdr ⎤ dt (21) ω ⎣ ⎦ The block diagram of speed estimation procedure based on the SMO is shown Fig. 1. The high frequency oscillations called chattering exists in the SMO as an inherent problem associated with VSC theory due to the discontinuous switching function. The negative side effect of chattering is that it involves high gain control action and may excite un-modeled dynamics. Thus, chatter-free becomes essential for a good performance SMO. Chattering elimination philosophy is based on converting the discontinuous switching function to a continuous one [12-13]. This is usually done by using a saturation switching function to replace the discontinuous one in Eqn. (21) as shown in Fig. (2). ia , ib , ic va, vb, vc is _

⎡ îs ⎤ ⎢ ⎥ ⎢⎣λˆ r ⎥⎦

1 s

⎡ a11 a12 ⎤ ⎢a ⎥ ⎣ 21 a 22 ⎦

[I

0]

[ 0 I]

î − i s s

î + s

λˆ r

ˆr ω

[ −I 0]

L

λˆ r

Speed Adaptive Scheme

(

K sgn îs − is

)

Fig. 1 Block diagram of SMO structure

+1

sat

sgn +1

-1 (a) Continuous (saturated)

-1 (b) Discontinuous (sign) ( t t d)

Fig. 2 Switching Functions

4. SYSTEM IMPLEMENTATION A schematic diagram, showing the major components of the experimental system, is shown in Fig. 3. It consists of an induction motor interfaced with a digital control board DS1102 based on a Texas Instruments TMS320C31 Digital Signal Processor for speed estimation. The induction motor is coupled with a dc generator for mechanical loading. The rating and parameters of the induction motor are given in the Appendix. Stator terminal voltages and currents are measured and filtered using analogue circuitry. Hall-Effect sensors are used for this purpose. Measurements are done on two phases only


427


and the corresponding values of the third phase are obtained by calculation. The measured voltage and current signals are acquired by the A/D input ports of the DSP control board. This board is hosted by a personal computer on which mathematical algorithms are programmed and down-loaded to the board for real-time speed estimation. A direct speed measurement is also carried-out for comparison with estimated speed signals. The output switching commands of the DSP control board are obtained via its digital port and interfaced with the inverter through opto-isolated gate drive circuits. 5. RESULTS AND DISCUSSION The main feature intended by the proposed continuous SMO is its ability to obtain a chatter-free system. The system response is tested to show the effectiveness of the proposed SMO speed estimation algorithm at different operating conditions. Figure 4 (a) shows the rotor flux vector contour, while Fig 4 (b) shows the d and q axes of the estimated rotor flux. The amplitude of this flux varies with a narrow band due to the action of the switching function. Figure 5 shows the experimentally measured and estimated speed signals at start-up. The results show a good agreement between measured and estimated speed signals. The estimated speed signal is almost chatter-free. The speed estimation error is 3 rad/sec (2%) after 0.7 sec from starting. This illustrates a high accuracy of the proposed speed estimation procedure. The results show the effectiveness of the proposed continuous switching function for eliminating high frequency oscillations and chattering associated with discontinuous SMO. Also, Fig. 6 shows the measured and estimated speed signals for a trapezoidal speed command. Figure 7 shows the measured and estimated speed signals for step change from 150 to 90 rad/sec. Good speed

estimation is achieved with speed estimation error around 5 rad/sec (5.5%) at speed 90 rad/sec. The system is also tested at speed reversal. Figure 8 shows the measured and estimated speed signals during speed reversal from 150 to -150 rad/sec. A fast convergence of the estimated speed during transients is achieved. The figure shows also a good capability to maintain acceptable estimation around zero speed. Robustness and parameter insensitivity are the most distinguished properties of SMO. These properties are examined for the proposed system. Principally, the introduced speed estimation algorithm depends on the motor parameters. Stator resistance has a major effect on the estimated speed signal. Since motor heating usually causes a considerable variation in the winding resistance, so there is often a mismatch between the actual stator resistance and its corresponding value in the model used for speed estimation. For this purpose, the proposed system is tested under different values of stator resistance to represent this parameter mismatch. Figure 9 shows the experimentally measured and estimated speed signals for a 50% increase in the model value of stator resistance. It is clear that, the speed estimation error becomes around 5 rad/sec (3.33%) just after the instant of stator resistance variation and decays to 3.5 rad/sec (2.5%) after 2 sec. This test proves that the proposed SMO is dependable and gives accurately the same behavior as the measured speed under stator resistance mismatch. The introduced results show that the system exhibits a good robustness and speed estimation accuracy under stator resistance variations; however the proposed SMO with saturation switching function has another advantage of being a chatter-free.

Rectifier

Inverter

Vdc

Induction Motor

Dc Generator

Gate Pulse Generator

Display DSP TMS320C31

Computer

Voltage Sensor

Gate Drive

Current Sensor

Supply

Series or Parallel Communication RAM

Encoder A/D Converter

ROM

D/A Converter

DSP Control Board

Oscilloscope

Fig. 3 Block diagram of experimental system 428


The d axis of flux

(a) Locus of rotor flux vector Estimated fluxes

λˆ qr

λˆ dr

50 Rad/sec/div

Estimated Speed [ Rad/sec ]

Measured Speed

The q axis of flux


Time [ Sec ]

Time [ Sec ]

Fig. 6 Experimental results of Measured and estimated speed signals during trapezoidal speed command

Measured Speed

Measured Speed

(b) Estimation of rotor flux vector Fig. 4 Flux vector diagram

50 Rad/sec/div



50 Rad/sec/div

Time [ Sec ]

Time [ Sec ]

Fig. 5 Experimental results of Measured and estimated speed signals during run-up operation at 150 Rad/sec

Fig. 7 Measured and estimated speed signals during speed change from 150 to 90 Rad/sec


429

50 Rad/sec/div


Measured Speed


Fig. 8 Experimental results of measured and estimated speed signals during speed reversal from 150 to -150 Rad/sec 1.58 RS [ p.u. ]

1.5 RS 1.3

1.052

6. CONCLUSION A sliding mode observer has been introduced and implemented for speed estimation of an induction motor drive. It has been implemented on a Digital Signal Processor (DSP) TMS320C31 platform and experimentally tested in real time. The system has been tested under transient as well as steady-state conditions. The results show the effectiveness of the proposed SMO with a fast convergence of the estimated speed under start-up, step change and speed reversal. The results show also the effectiveness of the proposed continuous switching function for eliminating high frequency oscillations and chattering associated with that of the conventional discontinuous one. The proposed SMO system has been also tested under different values of stator resistance to represent this parameter mismatch. The results show a high dependability and accuracy of SMO under stator resistance mismatch. The introduced results show the effectiveness of the proposed SMO of speed estimation for practical applications of speed sensorless induction motor drives. 7. APPENDIX A. List of symbols Lm Mutual inductance Lr Rotor leakage inductance

RS

0.79

Ls Rs Tr ωr

Time [ Sec ]

i ss = [ i dss i qss ]T

Stator current vector

iˆss = [ iˆdss

Estimated Stator current vector

50 Rad/sec/div

iˆqss ]T

λ rs = [ λ drs λ qrs ]T

Rotor flux vector

λˆ = [ λˆ

Estimated rotor flux vector

s r


Measured Speed

(a) Stator resistance variation

Stator leakage inductance Stator resistance Rotor time constant Rotor angular speed

σ Leakage coefficient Te Electromagnetic torque TL Load torque B Friction coefficient J Moment of inertia

v

s s

s dr

= [u

s ds

λˆ ]

s T qr

u

]

s T qs

Stator voltage vector

ωˆ r

Estimated rotor speed

p = d dt

Differential operator

A11 = aI , A12 = cI + dJ , A21 = eI , A22 = −ε A12 , b1 = bI ⎡1 0 ⎤ ⎡0 −1⎤ I =⎢ ⎥ , J = ⎢1 0 ⎥ ⎣0 1 ⎦ ⎣ ⎦ 2 ⎛ R ⎞ ω Lm L 1 , d = r, e= m a = −⎜ s + ⎟, c = σ σ ε ε L L T L T T s s r r ⎠ r r ⎝ σ Ls Lr 1 L 2m Lr , b= , σ = 1− , Tr = ε= σ Ls Lm Ls L r Rr Time [ Sec ]

B. Induction motor parameters:Fig. 9 Estimated speed signal under 50% increase of stator resistance

430

Rated power (w) Rated voltage (volt) Rated current (Amp.) Rated frequency (Hz) Number of poles

250 380 0.5 50 4

Rs (p.u) Rr (p.u) Ls (p.u) Lr (p.u) Lm (p.u)


0.0658 0.0485 0.6274 0.6274 0.5406


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