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- 1. FINAL THESIS PROJECT FFiinniittee eelleemmeenntt mmooddeelliinngg ooff tthhee mmaaggnneettiicc ffiieelldd ooff gguuiittaarr ppiicckkuuppss Martín Martínez July 2003
- 2. Akustik Labor FINAL THESIS PROJECT Title: Finite element modeling of the magnetic field of guitar pickups Author: Martín Martínez Villar Date of finish: 24. July 2.003 Corrector: Prof. Dr. Manfred Zollner (FH Regensburg) English revisor: Prof. Dr. Manuel Villar (Universidad de Granada) Prof. Dr. David Villar (Universidad de Granada) Prof. Andrés Calavia (C.P. Escolapios Soria)
- 3. For my family, mainly my parents; for this opportunity; this is yours. THANKS FOR ALL…
- 4. Opening i Opening: The title of this work is “Finite element modeling of the magnetic field of guitar pickups”, and what does it means? It means that I will try to analyze the basic pieces of a magnetic pickup with the help of this powerful software called ANSYS, and step by step I will increase the difficulty of the experiment. ANSYS/Emag is an electromagnetic field simulation product designed for static and low- frequency electromagnetics, electrostatics, current conduction, circuit simulation, and coupled circuit-electromagnetic simulation. A typical ANSYS analysis has three distinct steps: • Build the model. • Apply loads and obtain the solution. • Review the results. First of all, when you build the model, you model and mesh it, and this is the reason because it’s called finite element modelling; you mesh your model in many little elements which will be analyzed one by one in the solution process, in the next step you apply loads to the previous created model; and next, you obtain the solution to your model. After that, there is another step called postproccessing where you have the possibility of review the results obtained in the solution. It’s very important that the results obtained in this process be compared with real results, because ANSYS is no more that a simulation program. But sometimes, this software is more effective than some theoretical equations, which need approximations. Due to this, we have been taking measures in each experiment for be sure that the results are correct. However, we live in a three-dimensional world, and one of the great values of electromagnetics is that makes one familiar with 3-D concepts. Another important concept is the maximum number of elements supported by ANSYS license; that limits the accuracy of the measure, mainly in 3D models, where there is an enormous number of elements. The solution process is another concept of great moment, because it limits the time of operation. In last examples of 3D models, we have been waiting for a nonlinear solution more or less 3- 4 hours; and I don’t contemplate the times where the solution was not the expected. This study consists on a Static Magnetic Scalar Analysis (not dynamic). I have to told you that this work has not been done yet, ‘cause it’s a very big project carried out by Mr. Zollner, so it can be considered a beginning, and perhaps some day we have a close solution, but now it is so far away; so if we found this goal, it will be a great present to this unexpected world. Anyway, the work has meant a new concept in my knowledge; both computer programming, language practice & electromagnetism.
- 5. Gratefulness ii Gratefulness: All my Spanish, even not Spanish Erasmus friends that have been bearing me in my bad times; specially Luis, my big friend in Regensburg. I want thanks to Prof. Dr. Manfred Zollner, who offered me a unexpected project, which mixed two very significant aspects of my life: science and music, and has been teaching & helping me in each moment. He is the reason for which I have learned all this concepts of guitar pickups, although our English communication was not the perfect one! Thanks to all my dear people in Spain that have been worried of me in my stay in Germany, to all my Spanish teachers and partners of the “Universidad Politécnica de Valencia”, and for you, that spend some time reading this work. I hope you spend a nice time reading it; and I hope that you can learn something about this… Regensburg, July 2.003 Martín Martínez
- 6. Table of contents iii Table of contents Opening i Gratefulness ii Table of contents iii Glossary iv 1 Prelude 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Main idea of a simple pickup . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Magnetic flux and ferromagnetic materials …………………………….. 2 1.2.2 Varying magnetic field ………………………………………………….. 2 1.2.3 Faraday’s Law and induction of current by a magnetic field …………… 3 1.2.4 Other types of pickups …………………………………….. …………… 4 1.3 Issue representation & Prodecure. . . . . . . . . . . . . . . . . . . . 5 2 Self- inductance of a simple coil 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Planning of the problem . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Solution for the problem . . . . . . . . . . . . . . . . . . . . . . . 10 3 Mutual inductance of two coils 14 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Planning of the problem . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Solution for the problem . . . . . . . . . . . . . . . . . . . . . . . 18 4 Magnets 22 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Planning of the problem . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Solution for the problem . . . . . . . . . . . . . . . . . . . . . . . 25 5 Models with magnet and string 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Planning of the problem . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Solution for the problem . . . . . . . . . . . . . . . . . . . . . . . 33 6 ANSYS EXAMPLES AND COMMERCIAL PICKUPS 38 7 BIBLIOGRAPHY 40
- 7. Glossary iv Glossary: A Ampere A area, 2 m B, B magnetic flux density, 2−= m WbT d distance, m D diameter, m E, E electric field density, 2− m C emf electromotive force, V H, H magnetic field, 1− m A I, I current, A L inductance, H l, L length, m M, M magnetization, 1− m A N number of turns ∧ n unit vector normal to a surface r radius, m S,s distance, also surface area, 2 m V voltage, V Z impedance, Ω µ permeability, 1− m H rµ relative permeability θ angle ω angular frequency φ magnetic flux, gaussWb 4 10= rµ permeability of vacuum: 7 104 − ⋅⋅π π pi 3.141592
- 8. 1 Prelude 1 1 Prelude 1.1 Motivation The transducer of an electric guitar is the pickup, and this consists of a coil of wire called bobbin and a permanent magnet placed underneath each guitar string, that has six poles pieces sticking out above the bobbin, each corresponding to one string on a guitar. Picture 1.1: Single-Coil Pickup – Top view The pickup itself is placed in the body of the guitar and aligned so that each pole piece is seated directly underneath its corresponding string. As it means, this pickup is used to turn the oscillations of a guitar string into a changing magnetic flux. They work on the same principal as electrical generators --Whenever there is relative motion between magnetic flux lines and an electrical conductor (copper wire), an electrical signal will be generated in the conductor--. The strength of the signal generated will depend on how fast the relative motion is, on how much on the conductor “cut” the magnetic flux lines, and of course, on how strong those flux lines are. The magnet in the pickup creates a magnetic field, which is disturbed by the vibration of the ferromagnetic guitar string located above it, causing the flux through the bobbin to be altered. In the bobbin, the change in magnetic flux is opposed by the induction of an alternating current because the bobbin itself is good conductor. The oscillatory motion of the string is the reason that there exists an alternating current rather that a direct current in the bobbin. Picture 1.2: Single- Coil Pickup- Side view End of winding End of winding Bobbin (insulated wire coil) Permanent magnet Guitar string Pole of permanent magnet Poles of the magnet Coil Guitar strings
- 9. 1 Prelude 2 1.2 Main idea 1.2.1 Magnetic Flux and Ferromagnetic Materials The magnetic flux is defined as the amount of magnetic field that passes through a surface, and can be changed in two ways; by changing the strength of the magnetic field, or by changing the size of the area of the area. Since the area of the bobbin is fixed, the only way to change the flux is to change the magnetic field. Picture 1.3: Magnetic Flux through a surface The movement of the guitar string that is made of steel (a ferromagnetic material), changes the magnetic field of the pickup. Ferromagnetic materials have no inherent magnetic field, but greatly enhance any field in which they are placed, due to the µ of this materials is very high (from 2 10 to 6 10 times 0µ ). 1.2.2 Varying magnetic field Ferromagnetic materials can be used to enhance magnetic fields. Solenoids use this principle to create larger magnetic fields for the same amount of energy. Since a solenoid’s magnetic field depends only on the permeability constant, the current running through the wire, and the number of turns per unit length, the cheapest way to increase the magnetic field is to change the permeability constant. This is similar to the interaction between a guitar pickup and a ferromagnetic string. Before the guitar is played, while the string is at rest, the magnetic pickup induces a magnetic flux in the guitar string. This effectively increases the length of the magnet, and thereby increases the magnetic flux through the bobbin. When the string is plucked, the magnetic flux through the bobbin changes as a
- 10. 1 Prelude 3 function of the distance from the head of the pickup. When the distance is smaller, the effective magnetic field is larger, and the flux is greater, and when the distance is greater, the flux is less than usual. The rate of change of the flux depends on the movement of the string. When a guitar is plucked, it moves in an oscillatory manner. The rate of oscillation depends on the material, length, and tension of the string. 1.2.3 Faraday’s Law and Induction of Current by a Magnetic Field The next task of the guitar pickup is to convert the magnetic flux into a current through the bobbin. This procedure is described by Faraday’s Law; which states that a time varying magnetic flux will procedure a voltage within a metallic ring. The simplest approach to visualizing this is to imagine a magnetic material passing through the center of a loop of wire. The magnet-string interaction produces a magnetic flux that is dependent upon time. The magnetic flux varies as a sinusoid, as does the electric field produced in the bobbin. “ Whenever a magnetic force increases or decreases, it produces electricity; the faster it increases or decreases, the more electricity it produces.” - Michael Faraday. Using both Faraday’s Law and Maxwell Equations we arrive to this expression: ( )02/cos ϕπωω ++⋅⋅⋅⋅= tNABE This is the electric field caused by the magnetic flux occurring in the coil due to the vibrating string. As can be seen from this equation, the magnitude of the electric field is directly proportional to the frequency of the stimulus. Additionally, the frequency with which the string vibrates is also the frequency of the electric field. Picture 1.4: Single coil pickup
- 11. 1 Prelude 4 1.2.4 Other types of electric guitar pickups An inherent problem with all pickups coils is that they are susceptible to electro- magnetic interference that can be caused by a.m. radios, lights or house wiring. In the case of electrical guitars, having one’s guitar plugged in right to any sort of amplifier would cause the pickups to hum and maybe even begin to feedback, a very undesirable trait especially in the middle of a song. To fix this, Gibson invented hum- canceling pickups, called “humbuckers”; this consists of two coils of wire, which are wired in series but out of phase with each other, and two sets of magnet pole pieces with opposite polarities. However, the currents generated by the vibrating strings are actually duplicated instead of cancelled. Picture 1.5: Humbucker pickup
- 12. 1 Prelude 5 1.4 Issue Representation & Procedure Once we have made this first introduction, I want to explain you how we will explain each example made. First of all, there is a little theoretical introduction, in which are explained all the equations and parameters that will be necessary to be known. In a second step, there is a explanation of the problem; in which there will be all data for the issue, and finally, there is a third or solution step where there is shown the solution for the problem, made with ANSYS. If you need to know how each example is made, there is a CD where are sited all the examples in txt files. Each one represents the commands needed by ANSYS, but you need to have installed ANSYS in your computer to run them. At the end of the procedure, there is a commentary where is explained the possible complications, a comparison between real and simulated results and possible next step if the accuracy is not the expected. Made this prelude, we start the work with the most simple and important thing (both magnet) in a pickup; the coil.
- 13. 2 Self-Inductance of a simple coil 6 2 Self-Inductance of a simple coil 2.1 Introduction The electromagnetic inductance is the production of electrical currents by magnetic fields variables with time. The discovery of Faraday and Henry introduced such symmetry inside electromagnetic’s world. Oersted discovered that a electrical current induces (produces) a magnetic field; and this multiplied the number of experiments searching new relations between electricity and magnetism. Faraday named induced currents these electrical currents produced by magnetic fields. At the same time, a magnetic field can induce a electrical current, but in this case there must be a variation of the field. On the other hand, the impedance (total resistance) of an inductor depends on the inductance of the coil and on the frequency. Any coil of wire is an inductor, and its impedance varies with the frequency; the higher the frequency, the higher the impedance. When you add more turns of wire to a coil, you are increasing the inductance and thus altering the frequency response. LfjLjwZ ⋅⋅=⋅= π2 (1.1) If the current I that travels on a cylindrical coil of length L and section 2 rA ⋅=π , being r the radium of the coil, with a number of turns N ; is created a magnetic field in the coil. If rl >> , the field B is uniform: l N IB ⋅⋅= µ (1.2) where µ is the permeability of the medium. The magnetic flux through the solenoid is given by: l N AIBAN 2 ⋅⋅⋅=⋅⋅= µφ (1.3) When the flux depends on time, a voltage is induced between the ends of the coil: dt dI L dt dI l N AI dt d Vind ⋅−=⋅⋅⋅⋅−=−= 2 µ φ (1.4)
- 14. 2 Self-Inductance of a simple coil 7 where: l N rL 2 2 ⋅⋅⋅= πµ (1.5) is the coefficient of autoinductance of the coil and depends only on the number of turns and the size of the coil. It consists in a induction of the own coil on itself. Picture 2.1: Simple coil The aim of this chapter is the study of this parameter, and to know how it depends on the parameters like µ and N .
- 15. 2 Self-Inductance of a simple coil 8 2.2 Planning of the problem The self-inductance of a coil depends on the µ , characteristic for each material (relatively permeability of cupper approximately 1 0µµ ≅c ), its dimensions (diameter and section area) and of course the number of loops of wire). For a first approximation we made some quadrangular section- coils of determined characteristics, which are shown in the table below: Coil Area ( 2mm ) Radium ( mm ) Number of turns Diameter ( mm ) 1 1 x 1 2.3 100 0.08 2 1 x 1 2.3 25 0.2 Table 1: Characteristics of both coils. The equation (1.5) permits us calculate the inductance know all the parameters of the coil. But due to the small solenoid that we are using, we must apply a correction (end correction for small coils) to fix the result: Wheeler correction: Dll ⋅+→ 45.0 (1.6) so finally, the self-inductance results: Dl N rL ⋅+ ⋅⋅⋅= 45.0 2 2 πµ (1.7) Picture 2.2: Cross-section of the coil l D
- 16. 2 Self-Inductance of a simple coil 9 ( ) HL µπ 35.67 )106.245.0101( 100 106.4102567.1 33 2 236 1 = ⋅⋅+⋅ ⋅⋅⋅⋅⋅= −− −− ( ) HL µπ 46.4 )106.245.0101( 25 106.4102567.1 33 2 236 2 = ⋅⋅+⋅ ⋅⋅⋅⋅⋅= −− −− Picture 2.3: Cross-section of both coils To compare these results with real ones, we made two coils of similar diameter section; a 100-turned one of 0.08mm for copper diameter (100 turns of 0.08mm copper and covered coil approximately 1mm) and a second coil of 0.2mm diameter (25 turns of 0.2mm approximately 1mm, too) and then, once finished the measuring, it resulted: HL µ771 = HL µ6.42 = so we conclude the explanation with the next data- table: Coil Theoretical ( Hµ ) Real( Hµ ) (%)ε 1 67.35 77 14.32 2 4.46 4.6 3.14 Table 2: Comparison between theoretical and real calculation. 0.08 mm . 10 x 10 = 100 turns 1 0.2 mm . 5 x 5 = 25turns 2
- 17. 2 Self-Inductance of a simple coil 10 2.3 Solution for the problem The solution for this case using ANSYS could be considered easy at a first moment. The analysis is considered 2-D Static Magnetic, and to be easy, I chose an axisymmetric model. So we started to make the commands program using two element types: PLANE 13: 2-D Coupled- Field solid. INFIN110: 2-D Infinite solid. Picture 2.4: Detail of area plotting Initially, there are defined two semi-circles, which determine the measure field; the outer one defines far field conditions (it will be made of INFIN110 element type). In the center, is located the coil. There are defined two materials, first for air ( 11 =rµ ) and second for copper ( 12 =rµ ). Once the modeling process is finished, and material proprieties are given for each area, we start to mesh. I preferred to mesh line by line because in this example there are not a very high number of lines, so it is easier to make it in this way. Picture 2.5: Element plotting coil Air INFIN110 Air PLANE13
- 18. 2 Self-Inductance of a simple coil 11 TYPE NUMBER OF ELEMENTS ELEMENT TYPE NUMBER 1 1975 PLANE13 2 60 INFIN110 Table 3: Number of elements used in Coil1.txt. As the number of elements generated in the meshing process is very important (more in 3D models than in two); I have preferred to include in each program the number of elements obtained. As more elements we have, more time we will spend in the solution step. Finally, is created a component with areas of material two, named coil, and the loads were applied. The only loads needed are the current density applied to the coil and a infinite flag in the outer circle. Finally, we solve the model. In General PostProcessing, we can review the results obtained. There are some important functions called Macros, which help us to calculate important parameters. One of them is PLD2D; it makes a plot of the flux lines of the model. As we see in the figure, the lines do not come out to the INF110. That is very important because our model must complete far field conditions. In this POST step there is another macro called SRCS, which calculates terminal parameters for a stranded coil in a linear static analysis. The expression for this macro is the following: • SRCS, NTURN, CURR, FREQ, PSYM, CSYM where: NTURN: Number of turns in the coil winding. CURR: Current per turn applied to the coil. FREQ: Harmonic frequency of the coil current (Defaults to 1) PSYM: Planar symmetry factor. When we use symmetry around X-Axis. CSYM: Only for 3-D analysis. The following terminal parameters are calculated: Energy input to the coil, terminal inductance, terminal voltage, and flux linkages. So the output SRCS command for the 100 turned coil is: _______________ Calculated coil terminal parameters __________________ Planar symmetry factor = 1, Circumferential symmetry factor = 1. Energy: Joules, Winding Inductance: Henries, VLTG: volts, Current per turn: Amps, Flux linkage: Webers, Frequency: Hertz. Energy input= 3.391810372E-05, Winding inductance= 6.783620744E-05. Current per turn= 1, Flux linkages= 6.783620744E-05. Terminal Voltage = 4.262274619E-04. Parameters defined for the coil: WIN (energy input), INDL (Winding inductance), IWIND (Current per turn), FLNK (Flux linkage), VLTG (Terminal voltage) ______________________________________________________________________
- 19. 2 Self-Inductance of a simple coil 12 We make the same for the 25 turned-coil, obtaining: _______________ Calculated coil terminal parameters __________________ Planar symmetry factor = 1, Circumferential symmetry factor = 1. Energy: Joules, Winding Inductance: Henries, VLTG: volts, Current per turn: Amps, Flux linkage: Webers, Frequency: Hertz. Energy input= 3.391810372E-05, Winding inductance= 4.239762965E-06. Current per turn= 4, Flux linkages= 1.695905186E-05. Terminal Voltage = 1.065568655E-04. Parameters defined for the coil: WIN (energy input), INDL (Winding inductance), IWIND (Current per turn), FLNK (Flux linkage), VLTG (Terminal voltage) ______________________________________________________________________ Picture 2.6: Magnetic Flux lines for a single coil Coil Theoretical ( Hµ ) ANSYS( Hµ ) Real( Hµ ) 1 67.35 67.83 77 2 4.46 4.24 4.6 Table 4: Comparison between results. It is clear that the theoretical and ANSYS results are similar (there is a smaller error that 5%), but real measures go away from this, perhaps due to parasite contributions of external noises (lights, A.M. radios…), or because the dimensions of the test- coil are no similar as theoretical one (it´s impossible to get a perfect coil when you have to make manually one, since you have to glue them, and there are air gaps). We must know that for these small results, we may take attention because a little error can cause a big deviation on the result.
- 20. 2 Self-Inductance of a simple coil 13 We tried to correct this error by changing the dimensions of the ANSYS coils: Picture 2.7: New dimensions for the test- coils Finally, for those dimensions, the final results were the next: Coil ANSYS( Hµ ) Real( Hµ ) 1 62.26 77 2 3.9 4.6 Table 5: Comparison between new results. This time the results are worse than before, but at the end, the accuracy is not so bad. Otherwise, for next measuring, we corrected the dimensions of Ansys coils. 1.5mm 1.7mm
- 21. 3 Mutual inductance of two coils 14 3 Mutual inductance of two coils 3.1 Introduction Up to now we have been discussing magnetic fields in free space, or in air. In air the distances apart of the molecules are relatively large, and the influence of the air molecules on the magnetic fields is very small. Suppose that we put a sample of material into a magnetic field. How is the field changed and what are the new field equations? How do describe what happens to the material? These questions are discussed in this chapter. The figure 3.1 shows two long solenoids, one wound on top of the other. The length of each solenoid is l , and the common radius is r . The bottom (primary) coil has 1N turns per unit length and carries a current 1I which produces a field 1B and a total flux 2φ through the top (secondary) coil. When 1I changes, 2φ changes and a e.m.f. 2V is induced in the secondary given by: −=⋅= 2 2 2 s dt d dlEV φ (3.1) In this formula, the secondary loop 2s has to be traversed in the direction which makes ⋅= 2 12 s dSBφ positive, in a manner consistent with the discussion of Lenz´s Law (chapter 1). We then have: dt dI MV 1 122 ⋅−= (3.2) with: l rNN M 2 210 12 ⋅⋅⋅⋅ = πµ (3.3) where 2N is the number of turns per unit length on the top coil. For 1I increasing so that dtdI /1 is positive, 2V is negative and the induced current which is trying to maintain 2φ constant is in the direction shown on Figure 3.1:
- 22. 3 Mutual inductance of two coils 15 Picture 3.1: A mutual inductance made by winding one long solenoid in top of another If we reverse the roles of the top and bottom coils by letting a changing current 2I in the top coil induce a e.m.f. 1V on the bottom coil it can be seen that: dt dI MV 2 211 ⋅−= , (3.4) where l rNN M 2 210 21 ⋅⋅⋅⋅ = πµ (3.5) The quantities 12M and 21M thus have the same value and can bee given by the symbol M is called the mutual inductance of the two coils, and depends only on their construction and geometry. M is equal to the flux through one due to unit current in the other if the two coils are in air, where the field produced is proportional to current. This parameter can be put in function of the self inductance of both coils: ( )212121 LLkMMM ⋅⋅=== (3.6) where the dimensionless number k, called the coefficient of coupling, is less than or equal to unity. This phenomenon constitutes the principle of electrical transformer, an instrument that permits to rise or to reduce alternate voltage. A transformer consists mainly of two coils rounded to an iron core. The first coil, in which we apply the AC voltage is called primary and the second coil, where appears the transformed voltage is called secondary. The f.e.m. in primary and secondary coils is shown below: t N Λ Λ ⋅−= φ ε 11 t N Λ Λ ⋅−= φ ε 22 (3.7) 1I 2I
- 23. 3 Mutual inductance of two coils 16 The presence of the iron core avoids the dispersion of the magnetic flux, so we can accept that is the same for each case. Combining the previous equations it results: 2 2 1 1 NN εε = , (3.9) that is the equation for an ideal transformer. 3.2 Planning of the problem In the same way as a transformer, a pickup can include more than one coil (humbuckers), and its interesting to know the mutual inductance between them. Now, it is considered a system with two simple coils, with the first dimensions that we supposed ( mmmm 11 × ) and separated mm14 each other. Picture 3.2: Model of the two coils of the problem with and without magnet 1a 2a l d Coil 1: 1a 2a l d 100=N turnmAI /1= mml 14= 21 1 amma == mmd 6.4= r mmr 2.4= Magnet: µs mms 15=
- 24. 3 Mutual inductance of two coils 17 Then, it will be defined a path in the inactive coil to calculate the magnetic flux that crosses the second coil; originated by the first coil. As can be seen, the relative permeability of the medium is 1, due to we are not using a ferromagnetic material. But, what is the matter when we collocate a magnetic material inside? As it is logical, the magnetic flux through the path will increase depending on the value of the permeability of the material. It is important to know the dependence of magnetic flux and distance from the excited coil. Taking advantage of the results obtained in last chapter, now we will calculate the mutual inductance between two coils sited in the same measure field. The results obtained will depend on the ferromagnetic material and size, so it is important to understand how it depends. For this reason there will be included some graphs that can help us understanding it.
- 25. 3 Mutual inductance of two coils 18 3.3 Solution for the problem The mutual inductance of the two coils, according to the equation 3.5 results: ( ) H l rNN MM µ ππµ 4.17 103 103.225100104 3 23722 210 1221 = ⋅ ⋅⋅⋅⋅⋅⋅ = ⋅⋅⋅⋅ == − −− Now, we will see how the flux density varies depending on the distance and the proprieties of the material between them. First, the medium is the air, so 0µµ = In the modeling process we create the two coils as we told in the previous chapter. After meshing, and loading the model, we solve it. Picture 3.3 shows the flux lines created by the upper coil. Picture 3.3: Plot of flux lines The magnetic flux that crosses the downer coil is: Wbc 12 2 105464.1 − ⋅=φ Now, in twice of measure the flux crossing the coil, we will define a path with the aim of compare results depending on distance and permeability of the medium.
- 26. 3 Mutual inductance of two coils 19 Picture 3.4: Path defined In the next graph is shown the flux density variation along the defined path when the medium is the air 0µµ = , which can be seen below the graph. Table 3.1: Flux values for the defined path Picture 3.5: Flux variation Flux (nWb)Flux (nWb)),( 11 yx ),( 22 yx
- 27. 3 Mutual inductance of two coils 20 If we put a magnetic material inside, as it is normal, the flux increases due to the bigger µ of the material. We have tried for µ =5. µ =1 µ =2 µ =5 Flux in nWb vs. Distance in mm. Picture 3.6: Comparison for µ =1, µ =2 and µ =5 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Distance (mm) Flux(nWb)
- 28. 3 Mutual inductance of two coils 21 Here are shown both flux lines plots for µ =1, µ =2 and µ =5 Picture 3.7: Flux lines for µ =1, µ =2 and µ =5 µ =1 µ =2 µ =5 449,9647 559,6930 706,8504 530,7358 690,6601 919,6008 316,0158 468,1927 717,5316 166,9999 278,3069 490,1813 95,3650 175,5187 350,0643 57,8657 115,9750 258,5388 37,0524 79,6877 195,8641 24,7861 56,4536 151,1502 17,2320 41,0408 118,3066 12,4148 30,5367 93,6736 9,2252 23,1442 74,7216 7,0692 17,8128 59,8305 5,6190 13,8798 47,8536 4,7010 10,8760 37,8469 4,3072 8,4730 29,0204 4,3210 6,3415 20,2009 3,8216 5,2389 15,1911 Table 3.2: Magnetic flux for µ =1, µ =2, and µ =5 At this point, we tried to find the maximum magnetic flux originated by a determined permeability, and we gave a value of 10.000, getting the next maximum flux value. We will discuss this parameter in the next chapter. Picture 3.8: Comparison for the different µ values Picture 3.9: Flux lines for µ =10.000 Flux in the center 0 200 400 600 800 1000 1200 1400 1600 Flux in nWb Reihe1 530,7358 Reihe2 690,6601 Reihe3 919,6008 Reihe4 1427,7498 1
- 29. 4 Magnets 22 4 Magnets 4.1 Introduction What we have learned in Chap. 1 and 2 about loops and solenoids we can now be applied to magnetic materials. We learnt that the magnetic effects of these materials originate in atomically smart current loops and that when great numbers of these in an iron bar have their magnetic moments aligned in a uniform manner, the bar is magnetically equivalent (externally) to an air-filled solenoid of equal sheet current density. The magnetic vectors B, H, and M are explained in this chapter. After a discussion of the boundary relations for magnetic fields, the rest of the chapter deals with the behavior of ferromagnetic materials, such as alnico, iron, and calculations of their parameters for different applications. If a horizontal bar magnet is freely suspended, as in a compass, it turns into the earth´s magnetic field, so that one end points north. This end is called north pole of the magnet. The other end is its south pole (equal strength but opposite polarity). All magnetized bodies have both a north and a south pole, and they cannot be isolated. If a bar magnet is placed on a wooden table and covered with a sheet of paper, iron fillings sprinkled on the sheet align themselves along the magnetic lines of the magnet. Picture 4.1: Magnetic field lines around a magnetized iron rod.
- 30. 4 Magnets 23 All materials show some magnetic effects. In many substances the effects are so weak that the materials are often considered non ferromagnetic. In general materials can be classified according to their magnetic behavior into diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, and superparamagnetic. In diamagnetic materials, magnetic effects are weak. In a few materials, such as iron, nickel, and cobalt, a special phenomenon occurs which greatly facilitates the alignment process. In these substances, called ferromagnetic, there is a quantum effect known as “exchange coupling” between adjacent atoms in the crystal lattice of the material which locks their magnetic moments into a rigid parallel configuration over regions, called domains, which contain many atoms. However, at temperatures above a critical value, known as the Curie temperature, the exchange coupling disappears and the material reverts to an ordinary paramagnetic type. The relative permeability of ferromagnetic materials varies over a wide range for different applied fields. It also depends on the previous history of the specimen. However, the maximum relative permeability is a relatively definite quantity for a particular ferromagnetic material although in different materials the maximum may occur at different values of the applied field. Substance Group type Relative permeability, rµ Silver Diamagnetic 0.99998 Copper Diamagnetic 0.999991 Cobalt Ferromagnetic 250 Nickel Ferromagnetic 600 Iron (0.2 impurity) Ferromagnetic 5000 Superalloy (5 Mo, 79 Ni) Ferromagnetic 1.000.000 Table 4.1: Relative permeability for a number of substances.
- 31. 4 Magnets 24 4.2 Planning of the problem The aim of this chapter is to find out the permeability of the magnet that we are using. We have chosen in previous chapters values for this permeability, but we did not know with accuracy the value for it. After that, we will try to calculate in the same way the permeability of the string that we will use in the next chapter. Picture 4.2: Air-core solenoid with B and H fields, also B, M and H along the solenoid axis. Picture 4.3: Flux lines of the test magnet
- 32. 4 Magnets 25 4.3 Solution for the problem At this moment, we have supposed a determined value for the permeability of the magnet, so in the looking for the way to find the permeability of the test magnets, we made more measures with the different test magnets and coils, and trying with ANSYS, we finally found this value. Now we will explain how. In this first example, we put the coil on the centre of the magnet: . We stat with 2=rµ , because the permeability of the test magnet is more than air´s one. CASE 1: - Diameter of the coil: 0.2 mm - N= 25 turns - L MEASURED= 8.2 Hµ - Diameter of the magnet: 5 mm Here I expose the values obtained for the supposed permeability for the magnet: HLr µµ 67.62 == HLr µµ 99,73 == %6.16166.1 67.6 99.7 = difference between both values. CASE 2: - Diameter of the coil: 0.08 mm - N= 100 turns - L MEASURED= 116 Hµ - Diameter of the magnet: 4.25 mm Here I expose the values obtained for the supposed permeability for the magnet: HLr µµ 24.972 == HLr µµ 97.1113 == %4.15154.1 24.97 97.111 = difference between both values. same difference between permeability values for each case.
- 33. 4 Magnets 26 The value for the real measurements is a bit greater than the results obtained for 3=rµ , so let´s try with 5.3=rµ . No, we assign this permeability to the magnet in each case: CASE 1: HLr µµ 40.1185.3 == CASE 2: HLr µµ 269.1275.3 == Those results are valid for us, due to the difference with obtained can be attributed to possible interferences in the measure of parasites contributions. . Now, we take the two coils and we collocate them as the next figure. Picture 4.3: Upside is sited the 25 turned coil, and downside, the 100 turned one. The excited coil is the upper one. After that we defined a path to calculate the flux in the second coil resulting: ! "#! "#! "#! "# 3.35 mm 2 mm PATH DEFINED
- 34. 4 Magnets 27 We choose a frequency of 1000 hz; the voltage in the path defined can be calculated: VuNu µπφω 231061633.310010002 11 ~~ =⋅⋅⋅⋅⋅=→⋅⋅= − Now, we insert the magnet with the last value of permeability; as it is logical, the flux will increase; and this increment will help us finding the inductance. $ $$ $$ $$ $ ! "#! "#! "#! "# VuNu µπφω 56108.9291310010002 11 ~~ =⋅⋅⋅⋅⋅=→⋅⋅= − With these two results, and another more on the center of the magnet, a comparison is presented between measured values and Ansys ones where the results obtained for different positions of the coils in the magnet length are compared : On The top On the center % difference ANSYS VALUES 56 66 18% MEASURE 1 42 55 31% MEASURE 2 48 61 27% MEASURE 3 51 64 25% Table 4.2: Comparison between Ansys and measured voltages The error was reduced when we improved the accuracy of the real measurement. Picture 4.4: Magnetic Flux Density distribution on the magnet for 5.3=rµ
- 35. 4 Magnets 28 In the end, we arrived at the conclusion that the relative permeability of the test magnet´s material is approximately 3.5, and for this value, and a coercive force, the maximum flux density is 0.18 T on the centre of the magnet. In the same way, we also wanted to know an approximate value for the permeability of a guitar string, and I wrote a program to calculate it. Below is shown the model used to make it: Picture 4.4: Model for measuring the rµ for the guitar string. The calculated values for the inductance of the coil were: HL stringwithout µ9_ = HL stringwith µ90_ = The piece used as the magnet, is made of little pieces of guitar string of 70mm each one. Now, as before, we suppose a value for the rµ of the string, and start calculate inductance values: 1. HLstringno sstringrairrcoilr µµµµ 312.5)_(1,1,1 ___ =→→=== 2. HLsstringrairrcoilr µµµµ 561.5550,1,1 ___ =→=== 3. HLsstringrairrcoilr µµµµ 65.77100,1,1 ___ =→=== 4. HLsstringrairrcoilr µµµµ 34.90150,1,1 ___ =→=== 5. HLsstringrairrcoilr µµµµ 58.98200,1,1 ___ =→=== The value for the measure obtained in case 4 is very similar to the real one, so we can finally say that the relative permeability of the string is approximately 150. 70mm 0.33mm 0.1mm N=233 turns 14mm GUITAR STRING COIL cu φ =1.8mm
- 36. 5 Models with magnet and string 29 5 Models with magnet and string 5.1 Introduction Ferromagnetic materials exhibit strong magnetic effects and are the most important magnetic substances. The permeability of these materials is not a constant but is a function both of the applied field and of the previous magnetic history of the specimen. In those substances, as already explained, the atomic dipoles tend to align in the same direction over regions, or domains, containing many atoms. Thus, a domain acts like a small, but not atomically small, bar magnet. With further increase of the field more domains change over, each as an individual unit, until when all the domains are in the same direction, magnetic saturation is reached. If the majority of the domains retain their directions after the applied field is removed, the specimen is said to be permanently magnetized. Heat and mechanical shock tend to return the crystal to the original unmagnetized state, and if the temperature is raised sufficiently high, the domains themselves are demagnetized and the substance changes from ferromagnetic to paramagnetic. From iron this transition temperature, or Curie point is 770ºC. The residual magnetism is so weak compared to the ferromagnetic case that the material is usually considered to be unmagnetized. Magnetization, which appears only in the presence of an applied field may be spoken of as induced magnetization, as distinguished from permanent magnetization, which is present in the absence of an applied field. The permeability µ of a substance is given by: r H B µµµ ⋅== 0 (5.1) To illustrate the relation of B to H, a graph showing B as a function of H is used. The line or curve showing B as a function of H on such a BH chart is called magnetization curve. Picture 5.1: Hysteresis loop showing path of B as H is changed.
- 37. 5 Models with magnet and string 30 Starting with an unmagnetized iron specimen, let us trace what happens to the flux density B as we change the applied field H. Starting at the origin (at 1), B follows the initial magnetization curve as H is increased to a value mH where the curve flattens off and saturation is reached (at 2). Now on reducing H to zero, B does not go to zero but has a residual flux density or remanence rB (at 3). If now we reverse H, by reversing the battery polarity, and increase H negatively, B comes to zero at a negative field cH− called the coercive force (at 4). As H is increased still more in the negative direction, the specimen becomes magnetized further with negative polarity, the magnetization at first being easy and then hard as saturation is reached when the field equals mH− (at 5). Bringing the applied field H to zero again leaves a residual magnetization with flux density rB− (at 6). Reversing H and increasing it in the positive direction, B comes to zero at a positive field (or coervice force) cH (at 7). With further increase in H the specimen reaches saturation with the original polarity. When the field equals mH+ it completes (back at 2) out “tour” of what is called a hysteresis loop. “For a given specimen, no points can be reached on the BH diagram outside the saturation hysteresis loop, but any point inside can”. In many applications permanent magnets play an important part. In dealing with permanent magnets, the section of the hysteresis loop in the second quadrant of the BH diagram is of particular interest. If the loop is a saturation or major hysteresis loop, the section in the second quadrant is called the demagnetization curve. This curve is characteristic for a given magnetic material. The intercept of the curve with the B axis is the maximum possible residual flux density rB , or the retentivity, for a material, and the intercept with the H axis is the maximum coercive force, or the coercivity. It is usually desirable that the permanent magnet materials have a high retentivity, but it is also important that the coercivity be large so that the magnet will not be easily demagnetized. Picture 5.2: Demagnetization and BH product curve for Alnico V.
- 38. 5 Models with magnet and string 31 5.2 Planning of the problem The last stage of my work was to find out the skill of the real magnet and the string used by us. Picture 5.3: Alnico V test magnet. As we anticipated in Chapter 4, ferromagnetic materials are better for us, due to their proprieties. Until this point, we have not applied the “real” proprieties to this magnet. Now, we can assign the proprieties to our Alnico 5 magnet: Coercive force: mAHc /352.49= H [A/m] 0 1592 7164 9552 17512 25472 33432 41392 49352 62884 B[T] 0 0.34 0.93 0.99 1.07 1.117 1.156 1.19 1.22 1.25 Table 5.1: Alnico 5 B-H table Picture 5.2: BH- graph of Alnico 2, 3 and 5.
- 39. 5 Models with magnet and string 32 … on the other way, the proprieties to the used string are presented: H [A/m] 0 400 1400 2400 3400 4400 5400 6400 7400 7900 B[T] 0 0.52 0.8 0.88 0.94 0.98 1.01 1.03 1.05 1.06 Table 5.1: String B-H table Picture 5.3: BH and NB graphs of string and its table in ANSYS. Picture 5.4: BH- graph of guitar string So in the solution we will analyse the results obtained for ANSYS programs, the distribution of magnetic flux density depending on the distance and vector plots which will help us understanding the skill of the pickup.
- 40. 5 Models with magnet and string 33 5.3 Solution for the problem He will show both 2-D and 3-D ANSYS solutions for magnet and string of the pickup, and we will try to explain the differences between them. - The first difference is the number of elements in both models. In our 2-D example we used 26.714 elements of 32.000 available, and for 3-D we used 30.000 approximately. - The second difference is the accuracy; as logical, for the same number of elements, is better the result of the 2-D model, but there are more options to appreciate in the 3-D model. - The third, and a very important difference is the time of the solution process. In a non- linear analysis of the 3-D model, you can spend 4 hours in solution process time, because ANSYS has to find a convergence solution for all the elements of the model. You can select one of a lot of ANSYS solvers if you want to decrease the solution time, but the solution could be not the expected. Characteristics of the models: 2D MODEL 3D MODEL MAXIMUM FLUX DENSITY 0.8 T 1 T NUMBER OF ELEMENTS 26.614 23000 SOLUTION TIME 5 min 3-4 hours Table 5.2: Main differences between models
- 41. 5 Models with magnet and string 34 2D MODEL SOLUTION: Picture 5.5: Magnetic flux density Picture 5.6: Magnetic flux lines
- 42. 5 Models with magnet and string 35 Picture 5.7: Detail of magnetic flux vector plot Looking at the obtained results, we can say that our model has a good accuracy: The most relevant results: - The maximum of the flux density on the string is not just above the magnet, but it is situated 0.4 mm from the centre, and this value is a bit more than the middle of the maximum flux density on the magnet. - The flux lines do not cross the string, but rather they enter and travel through the string.
- 43. 5 Models with magnet and string 36 3D MODEL SOLUTION: In the 3D model, the steps in an analysis are the same as 2D model, but we must be care because we work with volumes, so one coordinate more increases the difficulty of the program. Element types used: - SOLID98: Tetrahedral Coupled- Field Solid - INFIN111: 3-D Infinite Solid Picture 5.8: Magnetic flux density The most relevant results: - Due to the low accuracy of this model, we can not distinguish the transitions on the flux density, but the maximum flux density is more or less the same as 2D model. If we want to improve the model, we could give more elements to the magnet, due to there is a bad meshing on it. However, if we need to work in 3D model, we need more available elements; and also the solution time will be bigger. It is a difficult compromise.
- 44. 5 Models with magnet and string 37 . PATH DEFINITIONS: Picture 5.8: Paths defined to measure the magnetic flux density Picture 5.9: Total Flux density for each defined path 1. TB 78022.0max = 2. TB 4155.0max = → maximum flux density on string 3. TB 7922.0max = → maximum flux density on magnet 2mm Path 1 Path 2 Path 3
- 45. 6 ANSYS EXAMPLES 38 6 ANSYS EXAMPLES With this example of the model with magnet and string in 2 Dimensional Static Analisys, we finish the work. You can find more of these examples in the assistant CD, where the programs with ANSYS commands made by me are located and orderly by chapter !**************************************************************************************** ! Description: Magnet and string in a 2D model ! Models: PLANE13 INF110 ! Ansys: 7.0 ! Date: 15.07.03 ! Author: Martin Martinez !**************************************************************************************** ! Input data ***************************************************************************************** r1=5e-2 !m ! Radium of outter circle r2=4e-2 !m ! Radium of inner circle r=4.8e-3 !m ! Diam. of the magnet l=15e-3 !m ! length of the magnet ! /TITLE, 2D MAGNET AND STRING ! Title to the work /prep7 ! Preprocessor ANTYPE,STATIC,NEW ! Static new analisys /PNUM,AREA,1 ! Plot Area, turn on numbers/colors RECTNG,0,r/2,-l/2,l/2 ! Magnet RECTNG,0,2e-2,10e-3,10.4e-3 ! String (Diam=.2mm) cyl4,0,0,r1,-90,0,90 ! External circle (INF110) cyl4,0,0,r2,-90,0,90 ! Internal circle (PLANE13) AOVLAP,all ! Overlap all areas NUMCMP,area ! Compress out unused area numbers ! ! Material models mp,murx,1,1 ! Material 1 for air TB,BH,2 ! B-H Curve definition (mat 2) TBPT,,1592,.34 TBPT,,7164,.93 TBPT,,9552,.99 TBPT,,17512,1.07 TBPT,,25472,1.117 TBPT,,33432,1.156 TBPT,,41392,1.19 TBPT,,49352,1.22 TBPT,,62884,1.25 TBPLOT,BH,2 ! Material 2 for magnet MP,MGYY,2,49352 TB,BH,3 ! B-H Curve definition: String TBPT,,400,.52 TBPT,,1400,.8 TBPT,,2400,.88 TBPT,,3400,.94 TBPT,,4400,.98 TBPT,,5400,1.01 TBPT,,6400,1.03 TBPT,,7400,1.05 TBPT,,7900,1.06 TBPLOT,BH,3 ! Two kind of elements, PLANE13: 2-D Coupled-Field Solid and INFIN110: 2-D Infinite Solid ! Element Type et,1,13,,,1 ! Element type 1: PLANE13 and behaviour to axisymmetric et,2,110,,,1 ! Element type 2: INFIN110 and behaviour to axisymmettic ! ! Material attributes
- 46. 6 ANSYS EXAMPLES 39 ASEL,S,AREA,,3 ! Assign attributes to magnet AATT,2,1,1 ASEL,S,AREA,,1 ! Assign attributes to magnet AATT,3,1,1 ASEL,S,AREA,,4 ! Assign attributes to air AATT,1,1,1 ASEL,S,AREA,,2 ! Assign attributes to air AATT,1,1,2 ! In this case, element type 2 INF110 APLOT ! Plot areas ALLSEL,ALL ! ! MESHING PROCESS LESIZE,7,,,200 LESIZE,5,,,200 LESIZE,6,,,10 LESIZE,8,,,10 AMESH,1 ! Magnet ! LESIZE,3,,,10 LESIZE,1,,,10 LESIZE,18,,,50 LESIZE,17,,,50 LESIZE,2,,,100 AMESH,3 ! String ! LESIZE,12,,,100 ! 100 divisions on line LESIZE,9,,,100 ! 100 divisions on line LESIZE,16,,,1 LESIZE,15,,,1 ! LESIZE,19,,,100,3 LESIZE,21,,,70,3 LESIZE,20,,,70 !ASEL,s,area,,5 ! Select area !MSHAPE,0,2D ! Free meshing !MSHKEY,0 AMESH,4 ! Mesh area ! /PNUM,MAT,2 ESEL,S,MAT,,2 CM,MAGNET,ELEM ! Component -> magnet ALLSEL,all ! /PNUM,MAT,2 ESEL,S,MAT,,3 CM,STRING,ELEM ! Component -> string ALLSEL,all ! DEFINE LOADS AND BOUNDARY CONDITIONS LSEL,s,,,9 ! Select line9 to make a flag SFL,all,INF ! Apply boundary condition ALLSEL,all EPLOT,all ! NSEL,ALL ! Use frontal solver FINISH ! Finish of preprocessor ! **************************** obtain solution ***************************************** /SOLU MAGSOLV FINISH ! *************************** retrieve results ***************************************** /POST1 PATH,Bsum,2,,600 ! Magnetic Flux Density PPATH,1,,0,10.2e-3,0 PPATH,2,,2e-2,10.2e-3,0 PDEF,Bsum,B,SUM /AXLAB,sum,Magnetic Flux Density (T) PLPATH,Bsum ! Path to calculate the maximum flux density on the string
- 47. 7 BIBLIOGRAPHY 40 7 BIBLIOGRAPHY [1] John D. Kraus: Electromagnetics, Fourth edition. Mc. Graw Hill, 1.992. [2] I.S. Grant – W.R. Phillips: Electromagnegnetism, Second edition. Manchester Physics Series, 1.990. [3] B.H. Flowers and E. Mensoza: Properties of Matter. Manchester Physics Series, 1.987. [4] Halliday & Resnick: Fundamentals of physics, third edition. Library of Congress Cataloging, 1988. [5] P.A. Tipler: Physics for scientists and engineers World Publishers, 1995. [6] J. Ewing: Experimental researches in magnetism Phil. Trans. Roy. Soc. London, 1985. [7] ANSYS 5.5 & 7.0 Documentation Ansys Corp. [8] Magnetfabrik Bonn Permanent magnets [9] Engineering Methods Inc: http://www.engmeth.com/ [10] Gary´s Guitars: http://www.6strings.com/ [11] Gibson Official Webpage: http://www.gibson.com/ [12] Fender Official Webpage: http://www.fender.com/ [13] Ansys Emag Webpage: http://www.ansys.com/
- 48. Magnetic Flux Density, 2mm above Magnets SINGLE COIL PICKUPS. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Telecaster Bridge (level), with plate FluxdensityinmT mm 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Telecaster Bridge (level), without plate FluxdensityinmT mm Telecaster (level, with plate) Telecaster (level, without plate) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Jazzmaster Bridge FluxdensityinmT mm 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 0 5 10 15 20 25 30 35 40 45 50 DiMarzio SDS1 F lu x d e n sity in m T mm Jazzmaster Bridge DiMarzio SDS1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Stratocaster 70s (staggered) FluxdensityinmT mm 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Gibson P90 FluxdensityinmT mm Stratocaster 70s (staggered) Gibson P90
- 49. HUMBUCKERS (Two coils): 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Gretsch Filtertron FluxdensityinmT mm 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 0 5 10 15 20 25 30 35 40 45 50 Squier FluxdensityinmT mm Gretsch Filtertron Squier Humbucker 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 0 5 10 15 20 25 30 35 40 45 50 ES335 Bridge FluxdensityinmT mm 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 10 15 20 25 30 35 40 45 50 ES335 Neck FluxdensityindB mm Gibson PAF Bridge Gibson PAF Neck Preliminary draft! Zollner, Manfred: The Physics of Electric Guitars; Fachhochschule Regensburg, 2003. (to be published) In those pictures is shown the relation between flux density and distance of most commercial pickups. Each curve represents a magnet of the pickup, so in a graph we consider all the magnets of its.
- 50. Erklärung 1. Mir ist bekannt, dass die Diplomarbiet als Prüfungsleistung in das Eigentum des Freistaats Bayern übergeht. Hiermit erkläre ich mein Einverständnis, dass die Fachhochschule Regensburg diese Prüfungsleistung die Studenten der Fachhochschule Regensburg einsehen lassen darf, und dass sie die Abschlussarbeit unter Nennung meines Names als Urheber veröffentlichen darf. 2. Ich erkläre hiermit, dass ich diese Diplomarbeit selbständig verfasst, noch nicht anderweitig für andere Prüfungszwecke vorgelegt, keine anderen als die angegeben Quellen und Hilfsmittel benützt sowie wörtliche und sinngemässe Zitate als solche gekennzeichnet habe. Regensburg, den 24.07.03 Martín Martínez Villar Unterschrift Diese Erklärungen sind mit der Diplomarbeit (eingeheftet ) abzugeben.

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