Engineering Electromagnetics I

Objective

Part 1

• A wire of radius 1 cm carries a current of 100 A, as shown in Figure 1. Compute, analytically, the magnetic flux density distribution in the region surrounding the wire. Plot magnitude of the flux density as a function of the distance from the wire.

Figure 1: Wire carrying current for Part 1.

• Model the system using Matlab's PDE toolbox. You can download the PDE Toolbox tutorial here (this has been provided to you as part of the pre-lab handout).Compare the magnetic flux density obtained from numerical simulation, with that obtained analytically. Comment on your results.

Part 2

Figure 2 below shows a pair of high voltage power lines adjacent to a house. Use the Matlab PDE toolbox to model this environment - choose reasonable dimensions and values for all geometric and field quantities. You can assume that the frequency of current flowing through the power lines is very low - and a static model provides a good approximation. Compute the magnetic fields in and around the power lines/house combination. Vary the distance between the pair of power lines and the house and the permeability of the house siding material and oberve the resulting field variations.  Provide a detailed explanation (including all the mathematics) of how you designed your model. Use your simulation results to provide conclusions that you have drawn from this study.



Figure 2: High voltage power lines adjacent to a house.

Part 3

• Using the Matlab PDE toolbox simulate the magnetic field of a 4 cm x 10 cm rectangular permanent bar magnet with a magnetic flux density of 0.5 T (shown in Figure 3 below). Provide a detailed explanation (including all the mathematics) of how you designed your model.

Figure 3: A permanent bar magnet showing the magnetic field lines.

Part 4: Numerical Simulation of a Magnetic Flux Leakage Nondestructive Evaluation Process

Background

• Magnetic Flux Leakage (MFL) is widely used for the Nondestructive Evaluation (NDE) ferromagnetic objects such as gas transmission pipelines, railroad rails and wheels, etc. The objective of an NDE process is to non-invasively inspect an object for defects, without impairing its usefulness. NDE forms a vital part of the maintenance procedure for almost all types of infrastructure.
• The MFL technique is a two-step process. In the first step, the test-object is magnetized, either by placing it within a magnetic field or by passing current through the object. In the next step, the surface of the specimen is scanned using a magnetic flux sensitive device such as a Hall probe. In the presence of surface-breaking defects, some of the magnetic flux density "leaks" out of the specimen in the region of the flaw, thus giving rise to the name "magnetic flux leakage." The Hall probe utilizes the "Hall effect" - a phenomenon wherein a current carrying conductor placed inside a magnetic field develops a transverse voltage. The Hall probe is connected to a Gaussmeter which converts the Hall voltage into an equivalent magnetic flux density indicated in Gauss or Tesla.
• The magnitude of the leakage flux density is related to the depth of the flaw and its magnitude distribution in 2-D space is related to the shape of the flaw. A 2-D surface plot of leakage flux density that is obtained over the flaw area is known as an MFL "image" of the flaw. Figure 4 shows the MFL image obtained from a slot-shaped flaw inside a piece of ASTM 836 steel bar-stock.

Figure 4: ASTM 836 Steel Bar-stock machined with a slot-shaped defect, and the corresponding MFL image.

EEMAG Lab Project Tasks

• The ability to numerically simulate NDE processes is very useful - much time and money can be saved by predicting measurements before investing in hardware.
• Using the Matlab PDE Toolbox, model the MFL NDE of a steel bar embedded with a slot shaped defect, as shown in Figure 5. Obtain the magnetic vector potential contours and the leakage magnetic flux density at a probe lift-off of 1 mm above the bar.

Figure 5: Model of the MFL NDE process for Part 4.

• Study the effect of varying the probe lift-off, defect depth and magnetizing current on the leakage flux density. Comment on your results.