Reduced Order Model for HTSC - PM Interaction for Flywheel

Bearing Applications Clay Hearn1, Sid Pratap1, Robert Hebner1, Dongmei Chen2, and Raul Longoria2

Introduction

Bond graphs are highly useful for modeling systems across multiple energy domains,

such as the mechanical and magnetic coupling of the PM-HTSC system [2]. The far

left 1-junction of the bond graph below represents the mechanical dynamics of the

levitated PM mass. The G gyrator elements represent the electromechanical

coupling forces which levitate the PM mass and induce currents within the nested

HTSC rings. Each discrete ring of the bulk HTSC is represented by a 1-junction and

connected to the other rings via mutual inductance matrix, represented by the large I

element. A nonlinear resistance term is also tied to each HTSC ring element to

describe the rapid rise in resistivity once the critical current density, Jc, is exceeded

[3].

1 The Center for Electromechanics at The University of Texas at Austin 2 The Department of Mechanical Engineering at The University of Texas at Austin

References 1. Smythe, William R., Static and Dynamic Electricity, Hemisphere Publishing, New York, 1989, pp.

234, 290-291

2. Karnopp, Dean C., Margolis, Donald L., and Rosenberg, Ronald C. System Dynamics: Modeling

and Simulation of Mechatronic Systems. 3rd Ed. pp. 228-237. John Wiley and Sons, New York,

2000.

3. J. Rhyner. Magnetic properties and AC losses of superconductors with power law current-voltage

characteristics. Physica C, vol. 212, pp. 292-300, 1993

4. D.R. Alonso, T.A. Coombs, and A.M. Campbell. “Numerical Analysis of High-Temperature

Superconductors with the Critical State Model”. IEEE Transactions on Applied Superconductivity,

vol. 14, no. 4, pp. 2053-2063, Dec. 2004.

Initial Pass

HTSC Top Surface

HTSC BottomSurface

PM and mass

HTSC

IDEC Position Probe

Guide Rod

𝑑𝑧𝑝𝑚

𝑑𝑡= 𝑣𝑝𝑚

𝑑𝑣𝑝𝑚

𝑑𝑡= −𝑔 −

𝐹𝑟(𝑣𝑝𝑚 )

𝑀+

1

𝑀

𝑑𝜙𝑠𝑖

𝑑𝑧𝑠𝑖𝐼𝑖

𝑛

𝑖=1

𝑑𝐼1𝑑𝑡𝑑𝐼2𝑑𝑡⋮𝑑𝐼𝑛𝑑𝑡

= −𝑳−1

𝑅1 𝐼1

𝑅2 𝐼2 ⋮

𝑅𝑛 𝐼𝑛

−

𝑑𝜙𝑠1

𝑑𝑧𝑑𝜙𝑠2

𝑑𝑧⋮

𝑑𝜙𝑠𝑛

𝑑𝑧

𝑣𝑝𝑚

(1)

Model Development

The HTSC puck is modeled as discrete, nested, superconducting rings shorted

on themselves. The permanent magnet is modeled by discrete current loops to

represent the equivalent surface currents and the resultant magnetic fields in

free space. This model assumes that the permanent magnet puck is concentric

to the bulk HTSC, which allows the use of an axisymmetric model.

Based on conventional formulations [1], the magnetic potential due to a current

loop can be calculated at any axial and radial location with respect to loop in

free space. From this calculation of magnetic potential, the magnetic flux, and

change in flux with respect to axial location of the permanent magnet can be

calculated for each discrete superconducting ring of the bulk puck.

Z axis

𝑟𝑖

𝑧𝑠 𝑎𝑠

𝐼𝑠

𝑧𝑖

𝐴𝑠𝑖 =𝜇0𝐼𝑠𝜋𝑘𝑠𝑖

𝑎𝑠𝑟𝑖

12 1 −

1

2𝑘𝑠𝑖

2 𝐾 𝑘𝑠𝑖 − 𝐸 𝑘𝑠𝑖

𝑘𝑠𝑖 = 4𝑎𝑠𝑟𝑖

𝑎𝑠 + 𝑟𝑖 2 + 𝑧𝑖 − 𝑧𝑠

2

𝜙𝑠𝑖 = 𝐴𝑠𝑖 𝑑𝑙𝑖 = 2𝜋𝑟𝑖𝐴𝑠𝑖

Calculation of magnetic potential in free space

Modulus of elliptical integrals of 1st and 2nd kind

Magnetic flux which links ring i of bulk HTSC

Reduced Order Dynamic Model and Bond

Graph Representation

Our flywheel sizing studies for grid storage highlight the well-recognized benefit

that accrues when the losses are reduced in flywheels. To help provide the

engineering knowledge to design robust systems, this project presents a

reduced order model for a permanent magnet (PM) and high temperature

superconductor (HTSC) in an axisymmetric frame to determine static and

dynamic force response. This model is formulated as a bond-graph to be used

for system models, where the nonlinear force-displacement interactions are

important for stability analysis and control design. The development of the

reduced order model is based on the mechanical and electromagnetic

interaction between a permanent magnet and bulk HTSC. Performance of the

proposed reduced order model is compare to FEM analysis and experimental

tests to confirm the static and transient performance.

Publications 1. C.S. Hearn, S.B. Pratap, D. Chen, and R. Longoria. Reduced Order Dynamic Model of Permanent

Magnet and HTSC Interaction in an Axisymmetric Frame. Submitted to IEEE Transactions on

Mechatronics, Aug. 27, 2012

Comparison to FEA

To verify performance of the reduced order model, comparisons were made to the finite

element method (FEM). The FEM algorithm [4] which utilizes the critical state model, was

used for this analysis. The system in this study consisted of a cylindrical PM moving through

a bulk HTSC ring at a constant velocity. The hysteretic nature of the force-displacement

profile is captured by both analysis techniques.

Experimental Validation

An experiment was performed to

validate the model performance by a

drop test. This procedure creates a

dynamic response by means of a step

input, where the weight of the PM is

quickly transferred to the magnetic

interaction between the PM and HTSC.

Tests were performed at different load

weights and initial heights above the

bulk HTSC. Time domain and

frequency response predictions from the

model matched experimental results