VM185

VM185
AC Analysis of a Slot Embedded Conductor

Overview

Reference:

A. Konrad, "Integrodifferential Finite Element Formulation of Two-Dimensional Steady-State Skin Effect Problems", IEEE Trans. Magnetics, Vol. MAG-18 No. 1, January 1982, pg. 284-292.

Analysis Type(s):

Coupled-field Analysis (ANTYPE = 3)

Element Type(s):

2D Coupled-Field Solid Elements (PLANE13)

2D 8-Node Electromagnetic Solid Elements (PLANE233)

Input Listing:

vm185.dat

Test Case

A solid copper conductor embedded in the slot of an electric machine carries a current I at a frequency ω. Determine the distribution of the current within the conductor, the source current density, the complex impedance of the conductor, and the AC/DC power loss ratio.

Figure 280: Slot Embedded Conductor Problem Sketch

Material Properties

Geometric Properties

µ_o = 4 π x 10^-7 H/m

µ_r = 1.0

ρ = 1.724 x 10^-8 ohm-m

a = 6.45 x 10^-3 m

b = 8.55 x 10^-3 m

c = 8.45 x 10^-3 m

d = 18.85 x 10^-3 m

e = 8.95 x 10^-3 m

I = 1.0 A

ω = 45 Hz

Analysis Assumptions and Modeling Notes

The slot is assumed to be infinitely long, so end effects are ignored, allowing for a two-dimensional planar analysis. An assumption is made that the steel slot is infinitely permeable and thus is replaced with a flux-normal boundary condition. It is also assumed that the flux is contained within the slot, so a flux-parallel boundary condition is placed along the top of the slot.

The problem requires a coupled electromagnetic field analysis using the VOLT and AZ degrees of freedom. All VOLT DOFs within the copper conductor are coupled together to enforce the correct solution of the source current density component of the total current density. The eddy current component of the total current density is determined from the AZ DOF solution. The current may be applied to a single arbitrary node in the conductor, since they are all coupled together in VOLT.

The complex impedance of the slot is calculated in POST1 from the equation

where V = voltage drop, and are real and imaginary components of the source current density (obtained from the solution results in the database file). The real component of the impedance represents the AC resistance R_ac per unit length. The DC resistance per unit length R_dc is calculated as ρ/A. The AC/DC power loss ratio is calculated as R_ac/R_dc. The problem is solved first using PLANE13 elements then using PLANE233 elements.

Results Comparison

	Target[1]	Mechanical APDL	Ratio
PLANE13
	10183	10123.54	0.994
	27328	27337.36	1.000
Impedance (Ohm/m) x 10^-6	175 + j471	175 + j471	0.997, 1.001
Loss Ratio	2.33[1]	2.387	1.025
PLANE233
	10183	10187.33	1.000
	27328	27326.18	1.000
Impedance (Ohm/m) x 10^-6	175 + j471	176 + j471	1.004, 1.00
Loss Ratio	2.33[1]	2.40	1.031

Target solution based on graphical estimate.

Figure 281: Flux Lines using PLANE13 Elements

Figure 282: Total Current Density using PLANE13 Elements

Figure 283: Eddy Current Density using PLANE13 Elements

Figure 284: Total Current Density using PLANE233 Elements