Qsci User Manual - an electrostatic conductive shells solver -
Contents
1. 3 3 1 loop center sups 6 nth 50 number of elements sups 6 vers 0 1 1 orientation versor 3 7 Creating your own geometry To create your own geometry you have to write down a function returning 1 position rq for sources dimension Npoints 3 2 position rc for control points dimension Npoints 3 3 matrices xxc yyc zzc of control points positions in meshgrid format they will be used to plot surfaces 4 matrix of element area Amat or length Lmat for line elements in the same format as xxc You can call this function something like my_own_geometry_creator m and modify the geometry_creator m file in order to take it into account 14 Also if you want a surface charge plot you have to create another function that reshapes the vector of charges generated by the solver matching the same meshgrid format as xxc yyc zzc Amat The dirty way a hint There s actually another way if you are not interested in plotting the surface you could simply add a surface from the input file and give it a type attribute sups type 666 This will result in an unknown surface and thus will probably not generate errors Then you must assign values for the sources and control points position to rq and rc attributes of your surface Maybe you could write a little script to fill sups rq and rc and place it in the MAIN m file right after the INPUT calling The method should result in a conductor that is invisible2. 2 nL 50 number of elements spanwise sups 2 V sups 2 Q free H 300 Q 3 means that the real phisical charge is q Q epsilonO 2 655 nC Don t skip places that is don t jump from sups 1 to sups 3 forgetting to fill sups 2 or Qsci could get angry You can find more about the geometry creation in section 3 and reading the example INPUT files included in the sources To every element you can assign a value for the electric potential that is a Voltage or the total electric charge Not setting a voltage results in a freely floating conductor whose potential is determined by the field If you have set a voltage any setting on the total conductor charge will be ignored If you have not set a voltage and you didn t set a total charge either the conductor is supposed globally electrically neutral For example to set a voltage of 21V to the 3rd object we write sups 3 V 21 while setting the object number 3 with a floating voltage and with a total charge of 3 4 y reads sups 3 V free sups 3 Q 3 4 Again please note that every charge shown and inputted to Qsci is in the truth a value that have been divided by en This is because I didn t feel like working with numbers in the order of 1E 12 Just mind this and you ll get straight 2 2 Plotting Properties Surface Charge Surface charge is automatically computed and plotted in a single plot con taining all the created geometr
3. b is the right hand known term NOTE THAT since I didn t like to have a matrix storing values ranging from 1 12 that is 1 e9 to unitary values I lumped the ey coefficient into the charges value In the input file you must enter a value that is the real physical charge you want to impose divided by eol 16 4 2 Capacity computation First of all note that Capacity computation may be quite expensive for Qsci must solve the linear system N times where N is the number of objects Computing the capacity of our object is achieved by computing the so called elastance matrix followed by it s inversion and one last little step Let s say we have three objects with total charges q1 q2 q3 We can write Vi 81141 1242 1393 V2 2141 S22G2 2393 V3 831941 322 53393 that is Vi S11 12 13 q Vo S2 S22 S23 q2 V3 S31 532 33 43 To compute the elastance matrix s Qsci puts a unitary charge on a surface null charge on the others and computes induced potentials After finding s Qsci inverts it s s This is called capacitance matrix Then Capacities are readily found Ca ey for iF j Ca Ci Cij J The coefficients Ci represent the capacity of the it object with respect to a reference huge sphere at the borders of the universe while the coefficients Ci are the capacity of the capacitor made by the it and j objects A Capacity matrix made with those capacities is displ
4. in surface plots but that affects the field contour planes should be alright Haven t tried this out anyway 15 4 Numerical Method 4 1 Lumped Charges Method A brief overview of the numerical method The underlying idea of the method is a conductor is an equipotential body right So let s place a number of charges on every conducting surface and then impose that the electric potential generated by all the charges on the control points of a the same surface be the same Let s place ourselves on the i control point The electric potential in duced by all the charges follows the superposition principle q q2 2 1 1 2j Vi Areo Lrii T2i T3i TNi 43 IN where rj represents the distance between the j source and the i control point Now if the control point belongs to an object whose potential was imposed through the input file then V is a known value otherwise the voltage V will will be kept as unknown Conductors whose voltage is not set require an additional condition that is the value for total charge This constraint for the k t body reads qi q3 ke Be oes Mal Q qn E l A C q Vimpos d We end up with a system in the form E 4 P L where A stores the influence coefficients ee C multiplies floating and thus un ji known potentials D is made of ones for charges relative to floating potentials and thus imposed total charge picking only the charges belonging to the kt body and
5. of Isolated Sphere and Capacitor 5 5 Electric Field plot e BA AS 5 6 POVray fields rendering 2 e ea Pe a ii 10 12 13 13 14 16 16 17 1 Introduction Well the first time I ve been studying electromagnetism I found a little weird that surface charge could eventually be non uniformly distributed on a conducting surface Of course after some time and a little more knowledge it turned out to be a why on Earth should it be otherwise but still I find fascinating to watch at charge distribution induced by other bodies and seeing how merging a floating voltage conductor into an electrostatic field can deflect the field lines to make its iso potential dream true Qsci was essentially born to quench this need for plotting Since it didn t want to be just a fashion tool Qsci outputs the total charge and Voltage on every conductor Qsci allows you to create some conductive shells or bodies via geometry creating functions You can assign to the surfaces a value for the electric potential or leave them floating and assign a total charge value and Qsci calculates e charge density on the surfaces by default e intensity of electric potential on request e electric potential on request e capacities between objects The electric field and potential are computed and plotted on three orthog onal planes Also Qsci may export the fields and geometry to POVray pov format Geometry creation plotting and exp
6. AN SERA DELL LLL LIA LALA LALALA LE O AS ALLA MALE EN ANAN AN DN AN RN NN PAN AAN NN NN NN am ESPANA ANA AN UN ARAS RAR AAA RAR AR RAR AR ARA Tha Pal ANA hs eka Wel ARAN ANA ANN AAN ANN ANA ANA ARAN AN AAA AR AAN AAA ANA IA AAA AR ARAN ANA AN AN ANNE GANA AAA AAN ENANA LLL LALA LALALA LLL GAAL AAA AN AAN A AAA IAN AR ANA ANA AN YAA PIAR AE AN LLL LL ALS TL LS PAL LS SAA NAAA PINES AID AAA A ANNA AAA AA el PLL PRS KOST T T lt L T T eg ye A Fal A LLL Pal hela Mae tsa Pala eae el Wel Pala AAA Wala Pal AAA AAA Thai Pak LT ORO OOT OOO RP AAN AAN AE RIAN ERE AAN AAA RRNA ER AAN RL OE AR AN AN AAN AN AN ANN ARAN ARAN AAN RN IN en hin TE SO CRO T OTT OLT OTT A T T OT eg alata ala LS aha Th SO IL ENANA A OO REVERE ye RAVEN C ANA 0 OL OTT OTO T 7 OLLA A A A CR AE R R ARAS Sources and control points Figure 14 20 5 2 Sharp Edge Effect Here s a nice pic about the intensification of charge density around sharp corners Figure 15 Charge density around a sharp edge 21 5 3 POVray geometry Here s a geometry similar to the the previous example s exported in a pov file and then rendered Figure 16 A geometry rendered with POVray 5 4 Capacity of Isolated Sphere and Capacitor As a quick benchmark I calculated the capacity of an isolated sphere whose value is known to be 4r Reo by theory I m proud to announce that on this simple case Qsci s result is exact to the 0 6 with around 200 distributed charge
7. POVray and the image viewer feh the following line in the INPUT m file will automatically launch a POVray rendering and visualize the image with feh Press Esc to quit exportPOVray showMeResults 1 PS I m not sure whether feh is available under Windows or not In the Examples section you can see the potential and electric field generated by a sphere and rendered with the exported POVray file The exported file stores grid points and values for the solution on such points Note that exported points are rescaled so that the biggest axis have a unitary length and the other axis is scaled accordingly The solution is both rescaled and traslated as a result solution in POVray plot always goes from 0 to 1 This is simply to obtain a nice plot in many different situations To export the geometry add in the INPUT m file the following line exportPOVray exportGeo 1 To modify camera light shining and color settings you must modify a few parameters in the POVray_geo_export m file 3 Elementary Geometries In this section Pll show you the implemented elementary geometries and provide an example of their inserting into the INPUT file Let s start with a picture of the geometries all together Green circles represent charges sources while black stars represent control points K AEE SE SE SE SE SESE SEESE ESEE E ESE SESE SE SESE SEESE LEE SESE ESE E GESE E SE gt E Figure 1 Elementary geometries 3 1 Sphere Th
8. Qsci User Manual an electrostatic conductive shells solver Stefano Boccelli April 2015 http boccelliengineering altervista org Hi everybody Qsci is a MATLAB script that compute the electrostatic field generated by conductive bodies You can place spheres cones wires planes cylinders in the space impose a voltage on them or let them float and impose an electric charge value Qsci computes and plots the electric potential and electric field on planes the charge density over the shells and capacities between conductors Qsci may export Electric field Potential and the geometry in POVray format Qsci is under LGPLv3 license Hope you enjoy Contents Introduction Input File 2 1 Geometry Construction sa a a a ae a a ee 2 2 Plotting Properties lt x Lge ee et ee Poe ee a ox 2 3 Capacitance computation A BN Se do 2 4 Exporting in pov format aa AS Elementary Geometries GI Sphera A sae oh A ed AV NAET o te ds LAE ae Bs E Ee ets Ay to e Got CUONG Bat cee R AS RE eh AE Se ata be Se eee os Hen eet tee o Dti eee ee ee eo DINES oe Ys iv Se ewes Sea tee BS Wt RES cee Shee T es 30 Wire Doop s e A e tag Rated i RaSh oe Goa 3 7 Creating your Own geometry sooo a Numerical Method 4 1 Lumped Charges Method os adie eb di A 4 2 Capacity computation ooo a Examples 5 1 Melting Pok oo a eed as ad a ad hs a 5 2 Sharp Edge Bec tas ert a e ee K Be Be ee a 5 3 POVray geometry ooa 5 4 Capacity
9. ayed in the command window Remember what I said about q and 9 The capacities must be multiplied by ey in order to obtain the real physical value 17 5 Examples In this section some colored plots 5 1 Melting Pot Here s a simulation of a melting pot of the objects that Qsci implements Object Imposed Voltage V Resulting Voltage V sphere cylinder plane cone line 5 free free 10 0 1 395 0 034 Table 1 Imposed and simulated electric potentials for objects Charge density for object 3 Epsilon q Area Figure 9 Charge density distri bution on the plane Figure 10 Charge density dis tribution around the edge of the cone 18 Figure 11 Charge density dis tribution on the cylinder l Figure 12 Charge density dis tribution upon the sphere ZZ Me 7 i Figure 13 Eletctostatigpotential contour lines Charges and Control Points GIGAS AAA A AN 2 23 2 A A r OT 7 LA YY INGA NERF LL LALALA LA LAS LALA NEP ANNA ELA LL KOS Wein Tiel Wel Weal Wein LALALA LOA TT Weigh EL AAA A A AA TRA AA ANA RAS ILLA SAA PRP LLL DIAZ LL ALL LL LL AO A ELO LL LLL LALALA LAIA af urine A A LALALA IA LA LL LS LOLA LAA EA LA GARA A GAR ARAN ANAYA ERE RARER ANNAN EAN PAN EAN ANNAN EA AE eN AN a BC SESE ENN EEN Fh ety aaa AA LA EAA RAIN AR LA L GOE E LALALA KO RR ER LADA RS LALALA AER NA LTL er AZ G UL R ANI RA ANNAN ANA AA AN FIAR AAN AI AIN EAT RANA A
10. ical entities Plotting together multiple sur faces results in plots that are not very clear in terms of surface charge you may desire a plot of some object on their own This can be done by setting somewhere in the INPUT file lines like sups 4 plot_me_alone 1 this will result in the production of a surface density plot for the object number 4 Note that surface density is implemented only for surface like objects setting plot_me_alone 1 for a line object results in a plot of a completely black and meaningless line Electrostatic field and Electrostatic Potential In order to plot the electrostatic field or the potential you have to ask Qsci to do so Qsci will show the fields on three planes merged in the volume and oriented as xy yz and xz First of all tell Qsci that you want him to compute Potential and or Electrostatic field by writing the following in the INPUT file plotField plotPotential 1 plotField plotElectricField 1 plotField plotPotential enables a contour plot of the Electric Po tential over planes plotField plotElectricField enables both a contour plot for the intensity of the the Electric field and also three cool surface plots Not setting one of those variable is the same as setting them to zero Then you must specify how long and where you want plot planes to be In the INPUT file set Put the intersection of the 3 planes in 4 2 1 plotField xorig 4 plotField yorig 2 plotField zo
11. ion 3 3 Cone Cones are pretty simple shapes for they have coarse grid at the base and very fine grid near the tip The implemented cones are always truncated If you want a sharp edge you choose a very thin top radius 10 Figure 5 Sources and Control Points for a sharp edge cone A cone is created by here I choosed the number 4 but it s arbitrary a ad Surface 4 gt sups 4 type cone sups 4 Rbase 0 7 bottom radius sups 4 Rtop 0 01 radius for top sups 4 Len 4 length height sups 4 xyz 2 1 5 0 origin of the mid height point sups 4 nth 20 of angular divisions sups 4 nh 80 of height divisions sups 4 vers 0 0 1 Axis direction 11 3 4 Plane Creating a plane is a little more tricky A plane can be oriented only as the Cartesian planes The orientation is specified via the parameter sups jj orient that can be set to xy yz or xz The other values x1min xlmax and x2min x2max specify the plane extension along its surface that is if you set sups jj orient yz x1 will represent the y coordinate and x2 will represent z Then the vale x3val represent the not varying coordinate that is x in our example If unsure just try it oF GF GF de dt GEG ot gt oF oH OF HR oH Oe oe OR oe OF GE ot GF oF KK He HH He oF OF OF 054 gk gk oe ged ER dt ok ot of de ct dt oF dE ic
12. is is a generated sphere Charges are spread all over the surface For plotting reasons surface density plots avoid plotting the poles this is for purely programming reasons for the computed area was inaccurate near the poles and the surface density thus had a numerically wrong peak Exceed the number of elements if you want to investigate the polar regions In order to create a sphere you need to specify the following parameters ata lita Sphere sups 1 type sphere 8 ER ets R Z K LL A 4 Figure 2 Control Points and Sources Figure 3 Plotted surface sups 1 Rad 0 6 sups 1 xyz l K sups 1 nth 20 sups 1 nphi 40 The number of elements nth and nphi represent respectively the number of divisions from North to South and the number from West to East that is latitude and longitude 3 2 Cylinder The cylinder object creates an object whose axis is in the direction of the specified vector The length is specified by the parameter L Figure 4 Sources and Control Points for the cylinder You create a cylinder by here I gave it the number 2 but it s arbitrary le Cylinder sups 2 type cylinder sups 2 Rad 0 3 Radius sups 2 Len 6 Length sups 2 xyz 0 2 1 Origin mid point sups 2 nth 10 of angular divisions sups 2 nh 40 of length divisions sups 2 vers 0 7 2 0 axis direct
13. on of Q the universal symbol for charge and gusci Italian word for shells Qsci and gusci sound almost the same Licensing Qsci is a weekend project I wouldn t rely too much on Qsci results if I were you Qsci is Open Source and is under LGPL license v3 2 Input File The input file specifies geometry plotting properties and whether to compute Capacities or not This file is called INPUT m 2 1 Geometry Construction Qsci initializes a class called sups in Italian superfici surfaces and the input file almost consists in filling it Every type of elementary geometry is defined by certain attributes e spheres are defined by radius center position and number of elements in which dividing it e planes are defined by the direction the length and the number of ele ments e and so on Let s say we want to add 2 objects a sphere and a line write in your input file something like T MY CONDUCTORS a a Surface 1 sphere sups 1 type sphere sups 1 Rad 0 6 sups 1 xyz 1 2 1 sups 1 nth 20 number of elements in theta direction sups 1 nphi 40 number of elements in phi direction sups 1 V 5 Volts sups 1 Q 0 this is going to be ignored because we have set the voltage ee ea Object 2 wire sups 2 type line sups 2 xyzA 2 2 4 starting point sups 2 xyzB 2 2 3 ending point sups
14. orting requests are managed through an input file called INPUT m Which geometries can I create At the moment I ve implemented spheres planes cones cylinders straight wires and loops The firsts are surfaces and Qsci will output a surface density plot for them while the latters are 1D elements and you will not see a linear density plot You can implement your own geometry see the appropriate chapter of this user manual Ud really love to implement a cow or a cactus like surface so if someone would kindly implement that for me Pd include him her in the thanks to section What about the numerical magic I really love lumped parameters methods they are so intuitive and quickly implemented Qsci disposes sources electric charges on the surfaces or lines and calculates the potential field in some control points A linear system is solved and solution is served Measuring Units PAY ATTENTION HERE Qsci works in IS units but please be aware that to avoid working with values of 1E 12 every value for the charge is actually q e That is if you set a charge Q 20 units you are actually setting a physical charge value of q Qeo 20 0 C Computed Capacities are typically huge it s because of the choice of not dividing by 9 To obtain real capacities multiply the output by o What does Qsci mean Since it was conceived as a script to plot surface charge distribution in elec trostatics Qsci is the contracti
15. ot ok oF od T A oF dt dtd of oF oF HH HH HHI HH ok de ce GF oF EG Gt OF oF oF GF ce Ge oF oF GF oF oF oF oF GF ot dt dt cE Gt ot ot oF AH gk ge ge Geo oF oF dt ok of ot ok ct oF dt cc oF GF oF Tak GE ge gt oto ot dt gt dt e a as o AN 1 0 54 Figure 6 Sources and Control Points for a plane Control points are placed around 45 degrees from each axis closer to their respective source than to others That s for numerical stability choosing control points in the middle of the square defined by four sources resulted in an unstable problem and a badly oscillating surface charge on the surface e AAA Plane sups 3 type plane sups 3 ximin 5 sups 3 x1imax 5 sups 3 x2min 3 sups 3 x2max 4 sups 3 x3val 2 sups 3 n1 60 sups 3 n2 40 sups 3 orient yz 12 sups 3 d_cc 0 02 This create a plane where y goes from 5 to 5 z from 3 to 4 and x has the value 2 3 5 Line Line objects are straightforward A is the starting point and B the end Straight line Figure 7 This is how a straight line looks like N Line sups 5 type line sups 5 xyzA 2 2 4 starting point sups 5 xyzB 2 2 3 ending point sups 5 nL 50 3 6 Wire Loop Here are input parameters for a wire loop 13 sups 6 type loop sups 6 Rad 1 h loop radius sups 6 xyz
16. rig 1 The x coordinate must go from the origin x 4 ahead for Lx 10 meters The y coordinate must go from the origin y 2 backwards for Ly 10 meters The z coordinate starting from 1 forward for 5 m plotField Lx 10 plotField Ly 10 plotField Lz D I want the plotting resolution to be 100 points along x 105 points along y 130 points along x plotField nx 100 plotField ny 105 plotField nz 130 Note that the electric field computation and plot is less accurate than electric potential near surfaces for the electric field blows up much faster than the potential does when approaching charges 2 3 Capacitance computation Activating the computation of capacities requires only adding the following line into the INPUT m file capacityCalc computeCapacities 1 Avoiding this is equivalent to setting the parameter value to zero 2 4 Exporting in pov format Qsci allows you to export results and geometry for renderings with POVray the well known Open Source ray tracing software You can ask Qsci to export the plotted Electric Field intensity or the Electric Potential by adding to the INPUT m file one or both the following lines exportPOVray exportPotential 1 exportPOVray exportElectricField 1 Note that those lines will work only if the corresponding plotField vari ables have been set to one you must first enable plots in order to export them If you have installed
17. s The plane plates capacitor is a little harder to simulate and results vary a lot 20 may be the case 5 5 Electric Field plot Here s a cool shaded plot of the Electric Field intensity near a wire loop and a cylinder Note that the field tries to blow up near the wire and the cylinder surface 22 Figure 17 Electric Field intensity 5 6 POVray fields rendering This how a POVray rendering of the exported file looks like Theese are the electric field and electric potential of a spherical conductive shell Note that the potential is constant along the shell and the electric field is zero Also note how faster the electric field blows up while approaching the surface 23 Figure 18 Electric Potential of a sphere Figure 19 Electric Field of a sphere The sphere has 30 lumped charges distributed in equatorial direction 24
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