QUESTION:
The attached article (01432505.pdf in 2D_AC_resistance_time_int_pot_term_conditions.zip) presents an analytical solution of AC resistance of a straight solid circular conductor . It concludes that the AC resistance increases with current frequency (as a function of square root of the frequency). I also found some other papers showing the same profiles. I wonder whether you could take a look at the article as attached and have some comments on this issue. I also wonder whether you provide any related benchmark problems to verify the accuracy of ANSYS EM calculation.
ANSWER:
I use two methods to obtain AC resistance:
1) Use the time integrated electric potential available in the magnetic scalar potential elements (PLANE13, PLANE53, SOLID97). An example is included in the attached file named:
2D_AC_resistance_time_int_pot_term_conditions.zip
The parameterized input file (wire11.inp) creates a 2D quarter symmetry model of the cross section of a solid circular conductor (currently using r = 1mm). Using terminal conditions - the time integrated electric potential (which is actually the integral of physical voltage) and the applied current - I can evaluate the complex impedance of the conductor at each investigated frequency. The attached macro (indctnc0.mac) contains the APDL that evaluates AC resistance by converting the time integrated electric potential to physical voltage for use in the impedance calculation (z=v/i).
The real part of the calculated complex impedance (the AC resistance) is plotted versus frequency in the file001.png. Compare the results shown in this plot to those for a 1mm wire shown in Figure 4 in 01432505.pdf. It looks to me like the agreement between the ANSYS result and those obtained in the paper by Gatous and Filho is quite good.
I included a simple rho*L/A calculation of the DC resistance in my input file (parameter r_DC) which evaluates to 5.41e-3 ohm/m. At the lowest investigated frequency (2096 H
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QUESTION: The attached article (01432505.pdf in 2D_AC_resistance_time_int_pot_term_conditions.zip) presents an analytical solution of AC resistance of a straight solid circular conductor . It concludes that the AC resistance increases with current frequency (as a function of square root of the frequency). I also found some other papers showing the same profiles. I wonder whether you could take a look at the article as attached and have some comments on this issue. I also wonder whether you provide any related benchmark problems to verify the accuracy of ANSYS EM calculation. ANSWER: I use two methods to obtain AC resistance: 1) Use the time integrated electric potential available in the magnetic scalar potential elements (PLANE13, PLANE53, SOLID97). An example is included in the attached file named: 2D_AC_resistance_time_int_pot_term_conditions.zip The parameterized input file (wire11.inp) creates a 2D quarter symmetry model of the cross section of a solid circular conductor (currently using r = 1mm). Using terminal conditions - the time integrated electric potential (which is actually the integral of physical voltage) and the applied current - I can evaluate the complex impedance of the conductor at each investigated frequency. The attached macro (indctnc0.mac) contains the APDL that evaluates AC resistance by converting the time integrated electric potential to physical voltage for use in the impedance calculation (z=v/i). The real part of the calculated complex impedance (the AC resistance) is plotted versus frequency in the file001.png. Compare the results shown in this plot to those for a 1mm wire shown in Figure 4 in 01432505.pdf. It looks to me like the agreement between the ANSYS result and those obtained in the paper by Gatous and Filho is quite good. I included a simple rho*L/A calculation of the DC resistance in my input file (parameter r_DC) which evaluates to 5.41e-3 ohm/m. At the lowest investigated frequency (2096 Hz), the AC resistance using terminal conditions (the first values in table array R_AC) was 5.44e-3 ohm/m, which agrees nicely with the nominal DC value. The plot in file001.png shows a bend in the r_AC curve beginning at around 8 kHz extending to about 25 kHz. Thereafter, r_AC increases linearly on the log-log plot created in ANSYS. This is consistent with Figure 4 in Gatous and Filho. It`s a little hard to see, but in Gatous and Filho, r_AC @ 1e6 Hz is perhaps a little more than 0.04 ohm/m, which is pretty much what I see in the ANSYS plot. So all in all, I think the terminal conditions based evaluation of AC resistance agrees well with published results. I should tell you, however, that development warned me that there are theoretical limitations to this approach (which is why it is not officially supported), so please use it with caution and always try to find some way of validating the results you obtain before trusting them. Development also asserts that a model of a single conductor with no return current constitutes a violation of the solenoidal flow condition; with the consequence that no reliable inductance calculation can be obtained with such a model (the inductance of such a system is undefined). If one were to model ever more surrounding free space around the single conductor, one would find that the resulting value of inductance calculated from terminal conditions changes without bound. I have done this myself and know it to be true. The workaround is to model 2 conductors (`go` and `return`). This 2 conductor model represents a system that CAN exist in nature and therefore has a defined value of inductance. The second conductor can be implied by symmetry (flux parallel condition half way between the 2 conductors), allowing you to use only one conductor in your model. In 3D models, the solenoidal condition is satisfied by extending `go` and `return` conductorsconnected to the coil to a flux parallel boundary of the model. The conductors should intersect this boundary perpendicularly. The issue of solenoidal flow is the reason I decided NOT to attempt to compare inductance calculations from this single conductor ANSYS model with value(s) reported in Gatous and Filho. I didn`t take the time to determine what, if any, aspect of this issue they considered in their inductance calculations. 2) The other modeling strategy I use to calculate AC resistance is to use the circuit coupled options of PLANE53 and SOLID97. The attached file: 2D_AC_resistance_crkt_cpld.zip is a circuit coupled version of the quarter symmetry model of the solid conductor discussed above. Circuit coupling allows a different, more straightforward strategy for calculating AC resistance versus frequency (no need to convert time integrated electric potential to physical voltage). The results are the same as those using terminal conditions and the time integrated electric potential. 3) A related topic is the variation of volumetric joule heating with respect to depth below the surface. In attached file: Skindepth_benchmark.zip results at the surface are compared tothose one skin depth below it, and compare favorably with expectations (they should differ by a factor of 1/e^2). Please see the correspondence text file for a discussion I had with another user and try reading the input file into ANSYS. You may consider performing a frequency sweep with this model to investigate the frequency dependency noted by the authors of the paper you sent me. |
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