Numerical Solution to Laplace s Equation

Transcription

1 Numercal Soluton to Laplace s Equaton Carleton Unversty Department of Electroncs ELEC 3105 Laboratory Exercse 1 January 019

2 PRE-LABORATORY EXERCISE You need to complete the pre-lab and have the TA sgn off your pre-lab wor before startng the computer laboratory exercse. Pre-1: Numercal soluton to Poson s and Laplace s equaton Please refer to the course lecture sldes related to Poson s and Laplace s equatons for addtonal detals on the technque. A summary s provded here. The startng equaton s: (P-1) whch s nown as Poson s equaton. It s a pont functon whch mples that the second dervatve (Gradent squared here) of the potental functon at a partcular pont n space must equal the negatve rato of the charge densty at the pont dvded by the delectrc constant at that same pont. Should the charge densty be zero then the equaton smplfes to Laplace s form: 0 (P-) A consderable amount of effort goes nto solvng ths equaton. For nstance once you solve for the potental you can determne the magntude and drecton of the electrc feld through: E (P-3) Once you now the electrc potental and electrc feld you can pretty well calculate anythng else related to electrostatc. The pre-lab wll examne solvng Laplace s equaton usng two dfferent technques. The frst s a drect approach solvng the second order dfferental equaton. The second nvolves a numercal soluton usng a fnte dfference approach. Both technques are dscussed n detal n class. Pre-1: Solvng the dfferental equaton Laplace s equaton s a second order dfferental equaton. In Cartesan coordnates t s: x y z 0 (P-4) The same functon s subected to dervatves wth respect to x y z and when the second dervatves are formed and then summed the resultant must be zero. Only then can the orgnal functon be a vald soluton to the equaton. Under normal crcumstances fndng the functon that satsfes (P-4) can be dffcult and when ths occurs other approaches are used to solve the equaton (such as numercal ndcated below). For ths pre-lab we wll consder a smple soluton to (P-4). Consder the parallel plate capactor shown n fgure Pre-1. The lower plate s at 10 volts and resdes n the (x y) plane. The upper plate s at 80 volts also resdes n the (x y) plane and ntersects the z axs a

3 dstance d from the orgn. We wll treat d (capactor plate separaton) as small such that we may approxmate the capactor plates as nfnte n extent n the (x y) planes. As a result the potental functon s ndependent of the x and y coordnates. Ths statement has to do wth the translatonal symmetry that s present wth regards to the x and y coordnates. As you move about n the (x y) plane KEEPING z CONSTANT the envronment always loos the same. Thus n equaton (P-4) the dervatves wth respect to x and y are zero as (for ths geometry) the potental s ndependent of x and y. The potental does vary n movng along the z drecton. The potental s 10 volts at z = 0 and s 80 volts at z = d. Queston Pre-1.1: Solve the dfferental equaton (P-4) for the parallel plate capactor of fgure Pre-1. It s a second order dfferental equaton so the general soluton wll have two constants. Determne these constants by mang use of the now voltage values at z = 0 and z = d. Tae d = mm. Plot several equpotental lnes and from these draw n the electrc feld lnes. What s the numercal value (magntude and drecton) of the electrc feld? 1 mar Fgure Pre-1: Parallel plate capactor geometry Queston Pre-1.: Two concentrc metal shells are shown n fgure Pre-: The nner shell has a radus of 1 cm and s at 50 volts the outer shell has a radus of cm and s at 150 volts. The regon between the metal surfaces s charge free and ar. Express Laplace s equaton n sphercal coordnates. Indcate whch dervatves of the potental functon wll be zero and why they are zero. Solve the remanng dfferental equaton and plot several equpotental lnes for the regon between the metal shells. Draw the electrc feld lnes. 1 mar Queston Pre-1.3: What approach would you use to solve the second order dfferental equaton f the geometry of the capactor plates do not conform to the unt vector drectons of a coordnate system? 1 mar

4 Fgure Pre-: Concentrc metal shells geometry Pre-: Fnte dfference soluton to Laplace s equaton n 1-D At ths tme t s a good dea to revew the course lecture sldes related to the numercal soluton to Poson s and Laplace s equaton. A revew of the numercal technque s presented here for a geometry whch results n a 1-D varaton n the potental functon. The parallel plate capactor geometry shown n fgure Pre-1 s such a geometry. The potental vares only the z drecton and s constant n the (x y) plane. Now consder the parallel plate capactor geometry redrawn n fgure Pre-3. The z axs between the capactor plates has been segmented and each pont the z axs s assgned an ndex (). The spacng between grd ponts s unform and equal to h. The capactor plate separaton s d. Fgure Pre-3: Parallel plate capactor geometry for numercal technque Consder now any two adacent grd ponts say ponts 4 and 5. The dfference n voltage between these two ponts s The separaton along the z axs between these ponts s z h. By defnton the frst dervatve of the potental wth respect to the z axs s: z lm h 0 z h z h (P-5) If at the moment we gnore the lm as h0 we see that ( z h) ( z) s the dfference n voltage between adacent grd ponts separated by z h. Thus an approxmaton to the frst dervatve can be obtaned by. So now we have a way to calculate the frst dervatve by examnng voltage values of z z adacent pont. But actually Laplace s equaton s made up of second dervatves. A second dervatve s nothng more than the dervatve of the dervatve. So let s frst obtan the dervatve between each grd pont par as shown n fgure Pre-4. Note that the dervatve ponts are offset from the potental ponts by h/. We can now obtan the dervatve of the dervatve usng the green grd ponts.

5 ( z h) ( z) z z z. The dervatve of the dervatve s also offset by h/ n grd pont z z z locaton. Ths brngs the second dervatve grd pont locaton bac on top of the orgnal grd pont locaton. We are almost there but we wll start all over agan. Let s get the dervatve between ponts 4 and 5 and also between ponts 5 and 6: and 6 h (P-6) z h z Let s get the dervatve of the dervatve between ponts 4 5 and 6: z 65 z z z h 5 4 h h h (P-7) For the parallel plate capactor problem there are no varatons n the potental wth respect to x and y and the regon between the plates s charge free. Thus z 0 whch when usng (P-7) gves: h after rearrangng 5 (P-8) Ths expresson ndcates that the voltage at grd pont 5 s the average value of the voltage one grd pont up and grd pont down. Ths expresson can be turned nto a numercal technque through the followng algorthm: Dvde the space nto an equal number of grd ponts. Mae certan that grd ponts are assgned to surfaces that are at fxed voltages (le the plates of the capactors see fgure Pre-3) Assgn an arbtrary voltage to each grd pont that s not fxed. Try to select voltage values n the range of the fxed values. Update the voltage on each grd pont by formng the average of ts nearest neghbours. Usng the updated values for the voltages update them agan by formng the average of nearest neghbours. Repeat the updatng process untl the voltage values at each grd pont no longer change. Usually you wll specfy the number of decmal ponts for the accuracy and once the requred number of decmal ponts are resolved the updatng process s stopped. The fnal voltage values are the voltage values at the grd ponts.

6 Pre-4: Potental frst dervatve and second dervatve Queston Pre-.1: For the parallel plate capactor gven n fgure Pre-3 use the numercal technque to obtan the voltages at the grd ponts accurate to 1 decmal place. Mae a good startng guess to the voltages. Tae d = 4 mm. 1 mar Queston Pre-.: Develop an XL spread sheet to solve the parallel plate capactor numercally to 3 decmal places. (If you wsh you may wrte a MATLAB program nstead). 1 mar Queston Pre-.3: Instead of usng 1 grd ponts use 10 grd ponts. Modfy your program to solve numercally Laplace s equaton for the parallel plate capactor to 5 decmal places. 1 mar Queston Pre--4: Any numercal technque utlzed requres an estmate of ts accuracy. Examne the course lecture sldes text boos on numercal technques and obtan an estmate for the error nvolved n usng ths approach to solvng Laplace s equaton. 1 mar Pre-3: Fnte dfference soluton to Laplace s equaton n -D and 3-D The numercal approach presented above can be easly extended nto -D and 3-D. We need to develop the fnte dfference approxmatons to each of the second order dervatves n equaton (P-4). We have already wored out the dervatve part for the z drecton. We mposed a grd along the z axs and formed the frst and second dervatve. Now n 3-D we need to establsh grd ponts along the other two axes. We thus end up wth a volume of grd ponts wth each grd pont dentfed by the ndces ( ). We then

7 form the second dervatves for each addtonal drecton. Fgure Pre-5 shows one of the grd ponts extracted (pont ) and ts sx nearest neghbours. Pre-6: 3-D grd ponts about center ( ) pont The resultant combnaton of the three second order dervatves of equaton (P-4) results n the followng expresson: h h h z y x (P-9) When dealng wth Laplace s equaton the above equaton s equal to zero and thus can be smplfed and rearranged to yeld an expresson for the voltage at pont ( ) as the average of ts nearest neghbours (3-D Grd): (P-10) In the stuaton where the geometry can be analysed n -D say x and y the averagng would nvolve only 4 nearest neghbours wth the grd usng ndces and (P-11) The same numercal algorthm presented above can be appled to the -D and 3-D grd. The dffculty n usng ths approach n -D and 3-D comes from the booeepng requred to eep all the grd pont averagng correctly lned. Queston Pre-3.1: For the structure shown n fgure Pre-7 use a -D numercal grd approach to obtan a mappng of the potental nsde the electrode regon. Top plate s at 150 and bottom plate at 0. To eep the problem manageable use a grd wth a 10 mm spacng. Obtan the voltages on the grd ponts accurate to 1 decmal place and use ether XL or MATLAB to solve. 1 mar

8 Pre-7: Potental well electrode structure Queston Pre-3.: From the potental values determned above draw n the electrc feld vectors. 1 mar

9 Lab 1: Numercal Soluton of Laplace s Equaton ELEC 3105 Updated ANSYS Lab 1. Before You Start Ths lab and all relevant fles can be found at the course webste. You wll need to obtan an account on the networ f you do not already have one from another course. Wrte your name n the sgn n sheet when you arrve for the lab. You can wor alone or wth a partner. One lab wrte-up per person. Show unts n all calculatons all graphs requre a legend.. Obectves The obectve of ths lab s to llustrate the use of a powerful numercal technque nown as the fnte element method to solve Laplace s equaton for selected problems. The lab wll run n the Department of Electroncs undergraduate laboratory room ME475. The software pacage we wll use s ANSYS Electroncs Destop Maxwell D/3D Solver from Ansys Corporaton. Ths software wll enable you to vsualze the electrc feld vectors and voltage equpotental lnes n cross sectons of structures consstng of conductors and nsulators. 3. Bacground The fnte element method (FEM) s a numercal technque for fndng approxmate solutons to partal dfferental equatons [1]. Consder the example of a -D soluton and ts correspondng mesh shown n Fgure 1. The lnes represent the drecton and magntude of flux densty smulated usng FEM n the soluton mage and the trangles (or sub regons) represent a sngle calculated soluton n the mesh mage. As an analogy compare a peg fle wth large pxels mang the mage blurry and a peg fle wth smaller pxels allowng the mage to become sharper. Therefore the smaller the sub regon the more accurate the entre soluton. A numercal soluton s always an approxmaton of an analytcal soluton whch s based on mathematcal theory.

10 Fgure 1: The -D soluton (left) and mesh (rght) [1] Consder Laplace s equaton descrbng the potental n a -D regon: x y 0 (1) A soluton can be found usng FEM by approxmatng the sze of d. Smaller trangles are used where the potental (x y) s rapdly varyng and larger trangles are used where the potental s varyng slowly. The potental s approxmated wthn each trangle as a polynomal expanson n x and y. A numercal algorthm s used to solve for the coeffcents of the polynomal n each trangle such that the nodes of adacent trangles have the same potental. Conductng surfaces are constant potental surfaces - the user ntally sets the value of the potental at the conductor. Electrc energy s stored n the electrc feld. The energy stored s gven by the expresson (unts Joules). W E 1 D Ed where D E s the electrc flux densty (C/m ) E s the electrc feld ntensty (/m) and the dot product s used n the ntegrand. The energy stored n a capactor C s gven by (unts J Joules): () W E 1 C( ) (3) where s the potental dfference between the conductors of the capactor. The capactance of a structure can be evaluated as (unts F Farads): W E C (4) ANSYS Maxwell D/3D can calculate the energy W E over the -D cross-secton and then calculate the approxmate value of the capactance C per unt length (F/m) of the structure usng a capactance matrx.

11 You wll be analyzng fve dfferent structures: Problem 1 - Feld n a hollow Problem - Feld at a sharp or rased pont Problem 3 - Parallel wre transmsson lne Problem 4 - Parallel wre transmsson lne wth plastc coatng Problem 5- Rectangular potental well You wll be ased to plot the voltage and electrc feld lnes for these structures. The relaton between electrc feld and voltage s found by usng the relaton below (unts J/C or ). [] (pg.60) B B W AB E dl E dl cos Q (5) unt A whch descrbes the potental of pont A wth respect to pont B defned as the wor done W n movng a unt charge Qunt from A to B. The electrc feld and the potental are perpendcular. In the case of the structures n ths lab equaton 5 can be smplfed by choosng a path ntegral such that cos(θ) = 1. If the electrc feld s constant n the regon of ntegraton then all that s left to calculate s the ntegral wth respect to the dsplacement. Based on these specal crcumstances the resultng equaton s l A E (6) l where s the dfference n potental between two ponts and s the dstance between the ponts. The structures n ths lab have pre-defned voltages. Keep trac of ther values as you go through the lab. l

12 4. Runnng ANSYS Maxwell D Note: It s always a good dea to regularly save your proects to prevent losng progress. If the nstructons below are not clear for you research what you are tryng to accomplsh on the nternet to try and fnd a soluton. Assume a plate thcness of 1mm unless otherwse stated. 1. Start the ANSYS Electroncs Destop program and select Proect then Insert Maxwell D Desgn. Now select the Maxwell D menu opton and clc on Soluton type.

13 In the wndow the opens up select the requred soluton type for lab #1 t s Electrostatc for lab # t s Magnetostatc. Clc OK once you selected the correct opton for you.. Clc on the Draw lne button shown below n order to draw the requred structure geometry dependng on the problem you are currently tryng to smulate. Clc somewhere on the whte grd located across the center of the nterface n order to place a pont clc agan to place another pont and a lne wll be automatcally drawn between them. Play around wth ths functon untl you are comfortable usng t to draw dfferent shapes. It s also possble to use the rectangle and crcle functons n order to draw shapes. Famlarze yourself wth how these functons wor as well. You can zoom n and out by holdng the control ey and scrollng wth your mouse ths may help n the case where you need a fner grd sze. (Zoom n for a fner grd spacng). If a rng geometry s requred draw one crcle wth a radus equal to the radus of the rng s outsde edge. Then draw another crcle nsde the frst crcle wth a radus equal to the rng s nner edge as shown below.

14 Next select both crcles. You may do ths by holdng the control ey selectng the larger crcle then select the smaller nner crcle and releasng the control ey. Once ths s done rght clc on the smaller nner crcle then navgate and clc on the menu opton shown below. Clc the OK button on the wndow the opens. You should be left wth a rng as shown below.

15 3. Once the requred shapes are drawn accordng to the problem clc on a shape n order to select t. Once t s selected t wll change color and ts propertes wll be dsplayed n the propertes pane on the left sde of the nterface. Select the property box contanng the materal value and clc on Edt and a wndow wll open. In the Search by name box type n the name of the materal you would le to smulate select t from the lst below the box and clc OK. Do ths for each dfferent shape usng the requred materal. 4. Once the geometres are drawn and have had materals assgned to them draw a large rectangle around them and set the materal of ths rectangle to ar. You may also change the transparency of ths rectangle by changng the Transparent value located n the Propertes pane. Once ths rectangle s drawn press the E ey and then whle holdng the control ey select all four edges of the large rectangle you ust drew that encapsulates the rest of the shapes. Now rght clc anywhere on the whte grd and from the rght clc menu navgate to Assgn Boundary (Balloon) and clc on Charge (f you get a boundary error try carefully repeatng ths process). Set the Balloon type as Charge n the wndow that opens up and clc OK. You may now press the O ey to return to obect selecton mode (as opposed to edge selecton mode). It s also possble to hde or show the dfferent geometres that you have drawn by clcng on the eye con n the upper toolbar and selectng whch shapes you want to reman vsble n the wndow that opens. Ths s shown n the fgure below. Note that f a shape s not vsble t wll stll be ncluded wthn the smulaton.

16 5. In order to assgn voltages to materals rght clc on the materal that you wsh to assgn an exctaton (for example a voltage) navgate to Assgn Exctaton n the rght clc menu and clc on oltage. Set the voltage value requred and clc o. 6. erfy that the shapes drawn have the correct sze ratos. You should now change the scale by navgatng to the top Modeler menu and clcng on t then select Unts and clc t. On the wndow that appears chec the Rescale to new unts box and select cm or the proper unt for your drawng. Play around wth ths untl your unts are correct you may verfy f they are correct by usng the scale located at the bottom of the whte grd worspace nterface shown below n the example for problem 1. (The copper plate thcness s set to 1mm n ths fgure.)

17 7. In order to smulate ths proect you must frst add a soluton setup by navgatng to the followng menu shown below and clcng on Add Soluton Setup. Clc OK on the wndow that opens and then save your proect by pressng down on the control ey and S ey at the same tme. Choose a sutable locaton to save your proect f requred. 8. In order to smulate the proect navgate to the Proect menu and clc on Analyze All as shown below. You must repeat ths f you mae any changes.

18 9. To plot the results select the large ar rectangle and rght clc on Feld Overlays wthn the Proect Manager pane on the left sde of the nterface as shown below. Select the type of result you would le to plot and a wndow wll open. Clc Done on the wndow and your plot wll be vsble. 10. To fnd the capactance before analyzng the problem navgate to the proect manager pane on the left sde. Rght clc on Parameters and assgn a matrx. Select one sgnal lne and one sgnal ground (one for each source). Clc OK then run the smulaton. Once the smulaton s completed expand the Parameters tree rght clc on Matrx1 or the matrx you created and choose ew Soluton. The capactance unts are shown near the top-rght of the wndow and the value s shown n the large dsplay box n the bottom half of the wndow. 11. Explore the Proect Manager pane as t contans lots of useful nformaton. You can modfy feld overlays by rght clcng on Feld Overlays and clcng on Modfy Plot Attrbutes. Ths s useful for changng the resoluton color and scale of the legend. You may also see your exctatons and boundares along wth other parameters. Explore the Propertes pane for useful settngs too. You can rename shapes that you ve created to custom names for easer dentfcaton f needed. Ths bref tutoral on usng the ANSYS Maxwell Solver should be enough to get you started there s plenty of documentaton avalable on the nternet as well as bult nto the program tself.

19 5. Problems Problem 1: Feld n a Hollow Ths problem models a parallel plate capactor wth one plate dented away from the other as shown below. The top plate s at 1.5 and the bottom plate s at 0 source. The materal of both plates s copper and the delectrc s ar. Answer the followng questons for Problem. a) Plot the equpotental voltages and electrc feld lnes of your structure as n Problem 1. mars b) Consder the regon between the two plates. Why s the electrc feld dfferent n the hollow? mars Problem : Feld at a Rased Pont Ths problem models a parallel plate capactor n whch one plate s dented toward the other as shown below. The top plate s at 1.5 and the bottom plate s at 0. The materal of both plates s copper. The materal around the plates s ar. Answer the followng questons for Problem 1. (a) Plot the equpotental voltages and the electrc feld lnes of your structure together n one prntout or ndvdually. Modfy the scale of the plot to have 10 dvsons (Instead of the default 15). Don t forget to clearly nclude the legends. mars (b) Where s the locaton of the maxmum electrc feld strength? What s the value of the maxmum feld strength? Use the coloured electrc feld ntensty plot and the accompaned legend. Don t forget unts. mars

20 (c) Insulatng materals wll brea down or become conductng f the electrc feld strength exceeds the breadown strength of the materal. For ar the breadown strength s about 3 x 10 6 /m. If the gap s reduced to 1 mm estmate the maxmum voltage that could be appled to the top plate. Answer ths queston usng theory and nclude unts. You may use the smulator to chec the calculaton (Note: The smulator doesn t actually smulate the delectrc breadown). 1 mar Problem 3: Parallel Wre Transmsson Lne HF and UHF antennas are usually connected to T sets by transmsson lnes consstng of two parallel wres of fxed separaton as shown below. To desgn the transmsson lne we need to fnd the capactance per unt length between the wres. The capactance per unt length s gven analytcally by (unts F/m) C D cosh 1 ( ) a where s the dfference n potental between the two wres s the delectrc constant of the homogeneous materal surroundng the wres D s the center to center wre spacng and a s the radus of the wres as shown below. The delectrc constant of ar s 0 = F/m. For other materals we multply ths value by the relatve delectrc constant cosh 1 r of the materal (that s = 0 r (7) ). The functon s found usng the hyp button on any scentfc calculator. The obect of problem 3 s to fnd the capactance numercally and compare wth the theoretcal value. We wll assume that the radus of the wre s always 1 mm but wll allow for dfferent spacng between the wres. The wres have a dameter of D = 6 mm. The materal of both wres s copper one wre s at whle the other s at -. If we assume that the parallel wres can be estmated by two parallel plates then the capactance neglectng frngng can also be wrtten as (unts F) [] (pg. 96) r A Q C 0 l Q El (8) where A s the area of the plates Q s the charge on the plates l s the dstance between the plates 0 s the delectrc constant of ar and r s the relatve delectrc constant of the materal between the plates.

21 Ths relaton ndcates that the electrc feld s related to the delectrc propertes of the materal n between the plates. Answer the followng questons for Problem 3. a) Plot the equpotental voltages and electrc feld lnes of your structure. mars b) What do you notce about the drecton of the electrc feld at any pont n relaton to the equpotental lnes? 1 mar c) Specfy the regon at whch the electrc feld s maxmum and state the maxmum value. Use the legend to gude you. Theoretcally you wll fnd that the maxmum should not be one pont but several ponts. 3 mars d) Estmate the capactance per unt length of the transmsson lne usng the software (refer to pont 10 of the tutoral pg. 18). In our case we are usng two wres wth = ( ( )) = 4. Therefore U C must tae the where = becomes 1 4 U C fracton nto account n your fnal result. 3 mars where = 4. If you follow the nstructons exactly you e) Calculate the theoretcal value of the capactance per unt length as explaned n the ntroducton to Problem 3. Compare to the estmated value of d) and explan any dscrepancy. Remember that you are comparng dfferent methods of solvng for capactance: numercal and analytcal. 3 mars

22 Problem 4: Transmsson Lne wth Plastc Coatng Now modfy the structure n Problem 3 so that the wres are coated wth a plastc (delectrc) layer of radus.0 mm. The plastc materal s Teflon and when drawng the center of the plastc should be the same as the center of the copper wre. Read the prevous secton for how to draw a rng. Answer the followng questons for Problem 4. a) Plot the equpotental voltages and electrc feld lnes of your structure. mars b) State the maxmum value of the electrc feld and state why t s greater or less than the maxmum values found n Queston 3. mars c) Estmate the capactance per unt length of the transmsson lne usng the smulaton software. mars d) Is the capactance greater or less than the one estmated n Problem 3? Explan. 3 mars

23 Problem 5: Rectangular potental well The sde plates and bottom plate are connected and all at 0. The top plate s at 150. The materal around the plates s ar. Answer the followng questons for Problem 5. a) Plot the equpotental and electrc feld lnes of your structure. mars b) Compare results obtaned here wth those calculated n the pre-lab secton. mars References [1] accessed September 008. [] Edmnster J.A. Schaums Outlnes: Electromagnetcs second edton 1993.