Electromagnetics 314 – Practical 2 – Electromagnetics Solenoid modeling / Elektromagnetika Elektromagneti ka 314 – Prakties 2 – Solenoı̈ede modellering Prof MM Botha, Stellenbosch University / Stellenbosch Universiteit
Overview The aim of this practical is to use physically rigorous, numerical simulation software (FEKO) to – •
Verify our theoretical calculations of the fields and inductances of two different solenoids
•
Evaluate the validity of our theoretical assumption regarding a constant field inside the linear solenoid and no external fields, for inductance calculation purposes
Write a report on your work and findings. Report can be hand-written and concise. Report should include
the following elements: •
Introduction, stating the purpose of the work
•
Details of the theoretical work
•
Simulation results (inductance values, graph of field strength, screenshots if needed)
•
Comparison of theoretical and simulated results. Can you explain why the theory always yields a smaller inductance result than the rigorous simulation for the linear solenoid? (Hint: think of the circuit definition of stored energy as well as the definition of stored energy in a magnetic field). Can you explain the level of agreement in the Biot-Savart vs. FEKO results? Can you explain why accuracy differs for the two toroid inductance results?
•
Conclusion on the suitability of the theoretical approaches that were used. Benefits and drawbacks. How does the accuracy of the theoretical approaches vary with physical parameters?
Here follows step-by-step instructions to achieve these objectives. Work individually or in groups of two, in case there are not enough computers in the lab. Everyone must hand in their own, individual report.
Oorsig Die doel van die praktika is om fisies volledige, numeriese simulasie sagteware (FEKO) te gebruik om – •
Ons teoretiese berekeninge van die velde en induktansies van twee induktore te verifieer
•
Evalueer die geldigheid van ons teoretiese teoretiese aannames aangaande konstante veld binne binne die liniêre solenoïed met zero eksterne velde, vir induktansie berekening doeleindes
Skryf ’n verslag oor jou werk en bevindings. Kan handgeskrewe wees, bondig asb. Verslag moet vo lgende
elemente insluit: •
Inleiding wat doel van die werk beskryf
•
Details van teoreties werk
•
Simulasie resultate (induktansie waardes, grafiek van verldsterkte, “screenshots” soos gepas)
•
Vergelyk teorie en simulasie. Kan jy verduidelik waarom die teorie altyd ’n kleiner induktansie resultaat lewer as die simulasie, vir die lineêre solenoïed? (Wenk: dink aan gestoorde energie in die stroombaankomponent sowel as in die mag neetveld). Kan jy die vlak van o oreenkoms tussen Biot-Savart en FEKO verduidelik? Kan jy verduidelik waarom akkuraatheid verskil vir die twee torus induktansies?
•
Gevolgtrekkings oor toepaslikheid van die teoretiese benaderings wat gebruik is. Voordele en nadele. Hoe varieer die akkuraatheid van die teorie met betrekking to fisiese parameters?
Hier volg stap-vir-stap instruksies om hierdie doelwitte te bereik. Werk individueel of in groepe van twee, ingeval daar nie genoeg rekenaars is. Elkeen moet hul eie, individuele verslag ingee.
Step 1: Theoretical calculations Linear solenoid Consider a linear solenoid of radius a and length h (
∈ − ℎ2 ℎ2
), with N turns.
;
(a) Rigorously calculate the magnetic field strength along the z-axis, using the Biot-Savart law. You may regard the current as a uniform,
-directed surface current.
(b) Assuming a constant field inside the solenoid, use Ampère’s law in integral form to evaluate the flux density and based on that, calculate the inductance (see G+H §7.5).
Toroidal solenoid with core Consider a toroidal solenoid with radius a and circular core cross-section with radius b and N turns. The
core permeability is . (a) Calculate the magnetic field strength inside the toroid, using Ampère’s law in integral form. (b) Now assume for simplicity, that the magnetic field inside the toroid is everywhere the same as it is at
= . Calculate the inductance of the toroid (see G+H §7.5).
Stap 1: Teoretiese berekeninge Liniêre solenoïed Beskou die liniêre solenoïed van radius a en lengte h (
∈ − ℎ2 ℎ2 ;
), met N windings.
(c) Bereken die magneetveld sterkte langs die z-as, d.m.v. die Biot-Savart wet. Beskou die stroom as uniform,
-gerigte oppervlakte stroom digtheid.
(d) Aanvaar konstante veld binne die solenoïed en gebruik Ampère se wet in integraal vorm om vloed digtheid te bereken; bereken dan daaruit die induktansie (sien G+H §7.5).
Torus solenoïed met kern Beskou ’n torus solenoïed met radius a en sirkelvormige kern deursnit met radius b en N windings. Die kern
permeabiliteit is .
(c) Bereken magnetiese veldsterkte binne die torus, d.m.v. Ampère se wet in integraalvorm. (d) Vir gemak, aanvaar dat die magnetiese veld binne die torus oral selfde is as by induktansie van die torus (sien G+H §7.5).
= . Bereken
Step 2: Verify the theoretical calculations for the linear solenoid through simulation with FEKO / Verifieer die teoretiese berekenings vir die liniêre solenoïed deur simulasie met FEKO Background FEKO is a frequency domain simulation code, solving electromagnetic field quantities in phasor format. In other words, FEKO solves the electromagnetic f ields, given a source with steady-state sinusoidal excitation at a specified frequency. In the case of an inductor, you know from previous work that the inductance can be calculated from the phasor quantities, as
⇒ � ̃ ⇒ ( )=
( )
( )=
( )
=
( )
With FEKO the source excitation voltage can be specified and then the source current is calculated. The magnitude of the impedance can be calculated at a given frequency (
/(2 )). This data can then be
=
processed to yield the inductance, according to the equation above. Note: in FEKO, the model can be: •
Rotated by clicking and dragging,
•
Zoomed by the mouse wheel and
•
Panned by holding down the wheel or CTRL and dragging the mouse.
Ag ter gr ond FEKO is ’n frekwensie gebied simulasie kode, wat elektromagnetiese velde oplos in fasor formaat. M.a.w., FEKO los velde op, gegee bronne met bestendige toestand sinusvormige aandrywing by die gespesifiseerde frekwensie. In die geval van ’n induktor weet jy van vroeër dat die induktansie bereken kan word van die fasor waardes, as
⇒ � ̃ ⇒ ( )=
( )
( )=
( )
=
( )
In FEKO kan die aandryfspanning gespesifiseer word en dan kan die bronstroom bereken word. Die grootte van die impedansie kan dan bereken word by ’n gegewe frekwensie (
=
/(2 )). Die data kan dan
geprosesseer word om die induktansie te lewer,volgens vergelyking hierbo. Let wel: in FEKO, kan die model: •
Geroteer word deur te kliek en te sleep,
•
Gezoem word deur middel van die muiswiel en
•
Gepan word deur die muiswiel af te hou of “CTRL” af te hou en die muis te sleep.
As Afrikaanssprekendes probleme het om die Engelse sagtewareinstruksies wat volg te verstaan, laat asb. weet. Die sagteware is net in Engels beskikbaar.
Simulation procedure 1) Use FEKO Suite 6.1 2) Start by copying all the provided files into a single, working directory.
3) Open file EM314_practical02_model1.cfx in CADFEKO. This file already contains a number of parametric variables, as well as the frequency specification and near field calculation request, for the simulation. 4) Create a helix with radius a, length h and N turns, using the “Construct --> Helix” primitive. Helix must be centred around the origin, with axis along the z-axis. Use the “Origin” option to centre the helix by specifying the origin as (x,y,z) = (0,0,-h/2).
o
5) Create the feed wire as a 180 arc to connect the two end of the solenoid. It must have a radius of h/2 and must lie in the positive-x part of the xz-plane. Use the “Construct --> Elliptic arc” primitive. NB: use the workplane tab to create this correctly, by specifying the workplane origin at (x ,y,z) = (a,0,0) and by rotating appropriately around the local axes.
6) Union everything together, using the “Construct --> Union” command, to ensure electrical connectivity (first select the arc and helix in the model tree, before applying Union).
7) Now add a wire port to the centre of the feed wire arc by right-clicking on “Ports” in the model tree and following the instructions.
8) Now add a 1 V excitation at Port1, under “Excitations” in the model tree.
9) Next, mesh the structure, under “Mesh -> Create mesh”. Mesh parameters are already set up.
10) Launch the FEKO solver and wait for it to finish.
11) Launch POSTFEKO from within CADFEKO, to view the results.
12) View the magnetic field distribution (should automatically be displayed in a preset, 3D view window). Note qualitatively, the amount of fringing fields around the solenoid as well as the nearly uniform field inside the solenoid. On the right hand side, you can play around with quantitative view settings. Under the “Result” tab you can play around with the arrow settings and under “Display --> Individual range” you can change the range for more dramatic, illustrative colour variations. Hold down CTRL-Shift and click to obtain field value annotations. 13) Create a source graph to view the magnitude of the impedance on a 2D graph, as a f unction of frequency. It is just a single point. This is done by initialising a Cartesian graph “Home --> Cartesian” and then plotting the source data with “Home --> Source data” (make sure you are plotting the
magnitude of the impedance!).
14) Under “Reporting --> Export data” export the graph data to a text file. Process the values from the text file to obtain the simulated inductance value. Compare with theory. 15) Plot the magnitude of the H-field z-component along the z-axis using the Cartesian graph option. I.e. initialise a Cartesian graph “Home --> Cartesian” and then plot the field data with “Home --> Near field” (make sure you are plotting the correct component at the correct locations!).
16) Now compare with your theoretical calculation. Export the graph data to a text fi le. Use the provided MATLAB script to compare your simulated data with the theory (just paste the exported data block and impedance into the appropriate places in the *.m file and run). Note: your FEKO data will have to be multiplied by the magnitude of the impedance, to correspond to the theory result for a 1 A current source excitation (thus also do the theoretical result for I = 1 A). See the *.m-file. 17) Do not close POSTFEKO! Go back to CADFEKO and increase both h and N by a factor 2. Re-mesh and re-solve the problem. 18) Go back to POSTFEKO that is still open. The data is automatically updated. Again calculate the inductance. Looking at both inductance results: why is there a discrepancy between theory and simulation? Can you explain quantitatively why the theory yields a smaller inductance result than the simulation? Think about the circuit-theory expression for stored energy in an inductor.
Step 3: Verify the theoretical calculations for the toroidal solenoid through simulation with FEKO 1) Open file EM314_practical02_model2.cfx in CADFEKO. This file already contains a number of parametric variables and material definitions, as well as the frequency specification and near field calculation request, for the simulation. 2) Create the toriodal solenoid wire with radii a and b and N turns, using the “Analytical curve” primitive. a. Specify the parametric variable range as
∈
b. U(t) : cos(2*pi*t) * ( a + b*sin(2*pi*N*t) ) c.
V(t) : sin(2*pi*t) * ( a + b*sin(2*pi*N*t) )
{0; 1}.
d. N(t) : b*cos(2*pi*N*t)
3) Create a cross-sectional, circular disk that will be spun to obtain the toriodal core volume. Use the “Construct --> Ellipse” primitive. Under the workplane tab, set the ellipse origin to (x,y,z) = (a,0,0). The disk should have radius c and be rotated to lie in the y=0 plane. Affect appropriate rotation under the “Workplane” tab.
4) Now select the ellipse in the model tree. Then click on the “Construct --> Spin” operation. Spin the o
ellipse through 360 around the z-axis, to obtain the core volume of the toroid.
5) Select the volume. Apply the “Transform --> Simplify” operator to it, to remove the internal face.
6) Select the volume in the details tree, under “Regions”: a.
Set its material to “magnetic1”
b. Set the solution method to “VEP”.
7) Now add a wire port to the middle of the solenoid wire by right-clicking o n “Ports” in the model tree and following the instructions.
8) Now add a 1 V excitation at Port1, under “Excitations” in the model tree. 9) Next, mesh the structure, under “Mesh -> Create mesh”. Mesh parameters are already set up.
10) Launch the FEKO solver and wait for it to finish. 11) Launch POSTFEKO from within CADFEKO, to view the results. 12) View the magnetic field distribution (should automatically be displayed). Note qualitatively its characteristics and how it corresponds to the theory. Note the extent of the field external to the toroid. On the right hand side, you can play around with quantitative view settings. Under the “Result” tab you can play around with the arrow settings and rendering. Under “Display --> Individual range” you can change the range for more dramatic, illustrative colour variations. Hold down CTRL-Shift and click to obtain field value annotations. 13) Create a source graph to view the magnitude of the impedance on a 2D graph, as a f unction of frequency. It is just a single point. 14) Under “Reporting --> Export data” export the graph data to a text file. Process the values from the text file to obtain the simulated inductance value. Compare with theory. 15) Do not close POSTFEKO! Go back to CADFEKO and change the core material to “magnetic2”, under “Regions” in the details tree. 16) Re-mesh the structure and re-run the solver. 17) When the solution is done, return to POSTFEKO. Again view the near field. Again go through the steps to obtain the simulated inductance value. Compare to theory. Has the comparison improved? Can you explain why?
