ideal capacitors

Chapter 2—Ideal Capacitors 2–1 IDEAL CAPACITORS Capacitors are used in almost every activity of electrical engineering, yet information on capacitor characteristics is scattered through a variety of textbooks, databooks, and manu-facturers literature. The following is an attempt to organize some of this information. A typical capacitor is a two-terminal device consisting of two conductors separated by a dielectric. When a voltage difference V ois applied to the conductors, a charge of + Qwill appear on one conductor and an equal and opposite charge −Qon the other conductor. The capacitance Cis defined as the ratio of the charge on one conductor to the potential difference. C=Q Vo(1) where Cis in farads, Qis in coulombs, and Vois in volts. Actually, one farad is a rather large capacitance, so capacitance values are usually expressed in terms of µF( 1 0−6F) or pF (10−12F). The total energy stored in a capacitor is WE=1 2/integraldisplay vol/epsilon1E2dv=1 2CV2 o=1 2QVo=1 2Q2 C(2) where WEis in joules, Eis the electric field in V/m, and /epsilon1is the permittivity. The integral expression shows that the energy stored in a capacitor with a fixed voltage difference acrossit increases as the permittivity of the material increases. The permittivity is usually expressed as the product of a relative permittivity /epsilon1 rand the permittivity of free space /epsilon1o. /epsilon1=/epsilon1r/epsilon1o (3) where /epsilon1o=8.854×10−12F/m( 4 ) The relative permittivity is unity for a vacuum and typically in the range of 2 to 6 for most dielectrics, as we shall discuss in more detail later. Capacitors are frequently used in series in a circuit, as shown in Fig. 1. There will be no actual charge transfer through the dielectric material. However, the electric fields will causea movement of charge within the series string. The battery supplies a positive charge to theleft plate of C 1. This positive charge attracts an equivalent negative charge on the right plate Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–2 ofC1. The movement of this negative charge leaves behind a positive charge of the same amount, which the electric field will force onto the left plate of C2. The process continues until each capacitor has the same charge Q=Qson its left plate. That is, Qs=Q1=Q2=Q3 (5) The total voltage across the series combination is V=V1+V2+V3 (6) and since V=Q C(7) Q Cs=Q C1+Q C2+Q C3(8) which can be solved for the series capacitance Cs. Cs=1 1 C1+1 C2+1 C3(9)    - - -  - rrC1 C2 C3V V1 V2 V3+++−−− Figure 1: Capacitors in Series A circuit of parallel capacitors is shown in Fig. 2. The voltage on each capacitor is the same and the amount of stored charge on each capacitor will be proportional to the individual capacitance values. It is not hard to show that the total parallel capacitance Cpis given by Cp=C1+C2+C3 (10) Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–3 C1 C2 C3 rr Figure 2: Capacitors in Parallel 1 Capacitance of Common Geometries The ratio of QtoVodepends on the geometrical arrangement of the conductors and on the electrical characteristics of the dielectric. The capacitance of a parallel plate capacitor, as illustrated in Fig. 3, is C=/epsilon1A d(11) where A is the area in m2anddis the separation between plates. This formula is accurate only when fringing can be neglected, that is, when dis small in comparison with A. AreaA /epsilon1 6 ?d Figure 3: A Parallel Plate Capacitor Later on, we will be interested in the capacitance of geometries where there are two different dielectrics. The simplest case is shown in Fig. 4. There is a layer of dielectric withrelative permittivity /epsilon1 r>1 of thickness x, and a second layer of air, with thickness y.T h e boundary between the two dielectrics can be considered a floating electrode. In fact, we canplace a conducting plate on the boundary without changing the results at all. We basicallyhave two capacitors in series. When we solve for the series capacitance, we get C=/epsilon1 oA y/epsilon1r (/epsilon1r+x/y)(12) Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–4 /epsilon1r /epsilon1o x y - -   Figure 4: Capacitor with Different Dielectrics Another important geometry is that of a coaxial cable of inner radius a, outer radius b, and length /lscript, which has capacitance C=2π/epsilon1/lscript ln(b/a)(13) The geometry for the coaxial cable is shown in Fig. 5. "!# j - 2b - /lscript 6 ? 2a Figure 5: Coaxial Cable The common 50 Ω coaxial cable 213/U (RG-8A/U) has a nominal capacitance of 29.5 pF/ft (96.8 pF/m). The small 75 Ω video cable 59B/U has a nominal capacitance of 21.0pF/ft (68.9 pF/m). Most other 50 and 75 Ω cables will have capacitance values very close tothese. Physically larger cables capable of carrying more power will have bandaincreased in the same proportion so the ratio b/aand the capacitance will remain the same as that of a smaller cable. Another geometry of great practical interest is the twin conductor transmission line, shown in Fig. 6a. This is composed of two conductors of radius r, with separation 2 hbetween conductor centers. The conductor-to-conductor capacitance C ccis given by Ccc=π/epsilon1/lscript ln[(h+√ h2−r2)/r]=π/epsilon1/lscript cosh−1(h/r)(14) If the two conductors have a small radius and are located far apart, the expression for capacitance becomes Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–5 Ccc=π/epsilon1/lscript ln(2h/r)(15) The error in the approximate expression is only 5.26% when h=2rand 1.16% when h=3rso the latter equation really has a wide range of usefulness. The conductor-to-plane capacitance Ccpbetween a cylindrical conductor of radius rand a conducting plane a distance hfrom the cylinder, as shown in Fig. 6b, is twice the value given by the previous two equations. Ccp=2π/epsilon1/lscript cosh−1(h/r)≈2π/epsilon1/lscript ln(2h/r)(16) This equation can be used to find the capacitance between two unequal conductors. We find the capacitance of each conductor to an imaginary ground plane, and then combine thetwo values for C cpusing the formula for capacitors in series. "!# "!# ? 6 2h Ccc"!# ? 6 hCcp (b) (a) Figure 6: Twin Conductor Transmission Line Another geometry of interest is that of a spherical capacitor of two concentric spheres with radiiaandb(b>a) as shown in Fig. 7. It is not practical to actually build capacitors this way, but the symmetry allows an exact formula for capacitance to be calculated easily. Thisis done in most introductory courses of electromagnetic theory. The capacitance is given by[4, Page 165] C=4π/epsilon1 1/a−1/b(17) If the outer sphere is made larger, the capacitance decreases, but does not go to zero. In the limit as b→∞,t h e isolated orisotropic capacitance of a sphere of radius abecomes C∞=4π/epsilon1a (18) Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–6 "!#  6 ?6 ?2b 2a Figure 7: Spherical Capacitor C∞gives us a lower bound for the capacitance of a spherical top loading element of a Tesla coil with respect to ground. One way of arriving at a reasonable estimate of the actualcapacitance is to start with the isotropic capacitance C ∞and add a correction term, as we shall see in the next section. We will also be interested is isotropic capacitances of shapes other than spheres. Some will be difficult to impossible to calculate analytically, so we will use the isotropic capacitance ofa sphere as a starting point for making an estimate. Rectangular boxes and short cylinders,for example, will have similar isotropic capacitances to a sphere. We just need to find anequivalent radius (or diameter) for these nonspherical shapes. Several different equivalentscould be used, such as a geometrical mean equivalent, an arithmetic mean equivalent, or theradius of a sphere with the same surface area as the other shape. It turns out that the simplestmethod, the arithmetic mean, does quite well [2]. Consider a rectangular box with orthogonaledge dimensions a,b,a n d c. Define an equivalent sphere diameter /lscript ewhere /lscripte=a+b+c 3(19) For other shapes we use the dimensions of the box in which the other shape can be placed. Making the change from radius to diameter, the isotropic capacitance is now C∞=2π/epsilon1/lscripte (20) This approach will give acceptable results in many cases. But, of course, there is no way of knowing the amount of error, or when some other approach would yield better results. It will get us in the right ballpark, however, and sometimes allow us to determine lower bounds of acceptable values obtained from other techniques. Suppose, for example, that we wanted the capacitance between two spheres separated by several diameters. We suspect that the parallel plate capacitor formula will not be veryaccurate and are unable to locate a better formula. What can we do? The lower bound for thecapacitance between two spheres is just half of C ∞, the isotropic capacitance for one sphere, a sc a nb ea r g u e df r o mF i g .8 . The dark point at the center of the figure is obviously ‘the’ point at infinity that math- ematicians love to talk about. If the capacitance of each sphere with respect to this ‘point’ Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–7 "!# "!# v∞C∞ C∞ Figure 8: Capacitance Between Two Spheres isC∞, then the capacitance between spheres has a lower bound of C∞/2, from the formula for two capacitors in series. Bringing the spheres closer together will increase the capacitancebut they cannot be separated far enough to reduce the capacitance below C ∞/2. 2 Toroid Capacitance An important emphasis of this book is the analysis of the Tesla coil. Among other things we will be interested in the capacitance of the top element (usually a ‘fat’ toroid) with respect to ground. We will also need the capacitance between adjacent turns (which look like ‘thin’toroids) and finally the capacitance of a turn with respect to ground. As usual, we will usethe published results as much as possible and leave the derivations to others. The dimensions of a toroid are shown in Fig. 9. .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................ .......................................................................................................................................................................... - ? -  6 ? w r D dtop view side view Figure 9: Toroid Dimensions There are other coordinate systems besides rectangular, cylindrical, and spherical in which variables can be separated and Laplace’s equation solved. One of these is toroidal coordinates. Moon and Spencer use this coordinate system to solve for the capacitance of an isolated toroid [3, Page 375] as Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–8 CMS=8a/epsilon1/bracketleftBigg Q−1/2(coshη0) P−1/2(coshη0)+2∞/summationdisplay n=1Qn−1/2(coshη0) Pn−1/2(coshη0)/bracketrightBigg (21) where PandQare Legendre functions of first and second order and aandη0will be discussed later. The subscript ‘MS’ refers to Moon and Spencer, to distinguish the capacitance obtainedfrom the value to be obtained from some empirical formulas later. Note that Eq. 21 is the corrected version. Godfrey Loudner [1] found that Moon and Spencer had an extra πin their expression (second equation from the top on p. 375), and also in Eq. 13.29 and Eq. 13.29a. There is also a typo on p. 373, second equation from the top,where V/2 should be replaced by V/π. Even the great ones can make a mistake! The Legendre functions can be written in many different forms, as integrals or infinite series, converging for arguments greater than unity or less than unity, and so on. The non-integer subscript ( n−1/2) adds another layer of complexity. Many math books do not mention the non-integer case, so one must be diligent in finding the correct expressions. The order of difficulty is much greater than for the sphere. Our mothers warned us that there would bedays like this, but let us proceed. Moon and Spencer give us an expression for Q n−1/2(coshη0) [3, Page 373] Qn−1/2(coshη0)=1 √ 2/integraldisplayπ 0cosnθ dθ /radicalbig (coshη0−cosθ)(22) For some reason, they do not give a similar expression for Pn−1/2(coshη0). However, they do give plots for both functions, which is convenient for checking computational results. Smythe [6, Page 159] gives an expression for Pin the form πPm n/prime(x)=(n/prime+1 ) (n/prime+2 )···(n/prime+m)/integraldisplayπ 0[x+/radicalbig x2−1cosθ]n/primecosmθ dθ (23) This expression is valid for x>0a n df o ra n y n/prime, including 1/2, 3/2, etc. We are only interested in the case m= 0. If we interpret the product of terms ahead of the integral sign as a factorial with zero entries (that is with a value of unity), the expression becomes Pn/prime(x)=1 π/integraldisplayπ 0[x+/radicalbig x2−1cosθ]n/primedθ (24) These expressions for PandQcan be numerically integrated using any scientific program- ming language (QuickBasic, etc.). We now return to our discussion of aandη0. These are coordinate values in toroidal coordinates, hence need to be translated into a more familiar coordinate system. The majortoroid radius wand the minor radius rare given by Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–9 w=acothη (25) r=a sinhη(26) Eliminating abetween the two equations yields w=rsinhηcothη=rcoshη (27) sow r=c o s h η=x (28) We then solve for ηas η=c o s h−1w r=l n/parenleftBigg w+√ w2−r2 r/parenrightBigg (29) and for aas a=rsinhη=r 2(/epsilon1η−/epsilon1−η) (30) This ‘exact’ formulation is of most interest to EM theorists and computer programmers. It will seem like overkill to most Tesla coil enthusiasts who just need to get in the right ballparkwith a capacitance estimate. For this reason empirical formulas have been developed whichyield an approximate value, adequate for most purposes, but obtained with much less effort.Empirical formulas for the capacitance of a toroid are given by [5] C S=1.8(D−d) ln(8(D−d)/d)(d/D < 0.25) (31) CS=0.37D+0.23d (d/D > 0.25) (32) where Dis the toroid major diameter, outside to outside, in cm, dis the toroid minor diameter in cm, and the capacitance is given in pF. Table 1 gives some isotropic capacitance values,both from the Moon and Spencer numerical integration and the empirical formulas. Thedeviation or error of C Swith respect to CMSis given in the last column in percent. We see that the empirical formulation agrees with Moon and Spencer to within 1% for the case of fat toroids, but gets progressively worse as the toroid gets thinner. The error is within5% for ddown to about 0.5 cm (4 gauge wire). Toroids this thin do not have the mechanical strength necessary to serve as top loading elements of a Tesla coil, so we can conclude thatthe empirical formulas are quite adequate for most purposes. Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–10 Table 1: Isotropic Capacitance of Toroids wr D d C MS CSerror meters cm pF pF % .3 .15 90 30 40.46 40.20 −0.63 .2 .1 60 20 26.97 26.80 -0 .63 .1 .05 30 10 13.49 13.40 −0.63 .2 .08 56 16 24.55 24.40 −0.62 .2 .06 52 12 22.02 21.93 −0.42 .2 .04 48 8 19.28 19.52 +1 .23 .2 .02 44 4 16.00 16.43 +2 .70 .2 .01 42 2 13.72 14.19 +3 .42 .2 .005 41 1 12.00 12.48 +4 .05 .2 .0025 40.5 .5 10.62 11.14 +4 .97 .2 .001 40.2 .2 9.09 9.76 +7 .36 3 Solenoid Capacitance (Medhurst) The isotropic capacitance of a sphere was given above as a simple formula. We looked at the theoretical formulas for capacitance of a toroid, but basically gave up and went to a simpler empirical version. After that learning experience, we will not even try to write exact equations for the isotropic capacitance of a cylinder. We will immediately write the empiricalequations as developed many years ago by a man named Medhurst. These will be expressedin several different versions, to meet different needs. The simplest expression for the isotropiccapacitance of a cylindrical coil of wire, with diameter Dand coil length /lscript,i s C M=HDpF (33) where Dis in cm, and His a multiplying factor that equals 0.51 for /lscript/D=2 ,0 . 8 1f o r /lscript/D = 5, and varies linearly between 0.51 and 0.81 for /lscript/Dbetween 2 and 5. Most coilers prefer values for /lscript/Dbetween 3.5 and 4.5, so this linear range is adequate for most purposes. An expression for Hthat works for /lscript/Dbetween 2 and 8 is H=0.100976/lscript D+0.30963 (34) Another expression for Hthat works for /lscript/Db e t w e e n1a n d8i s H=0.0005(/lscript D)4−0.0097(/lscript D)3+0.0648(/lscript D)2−0.0757(/lscript D)+0.4723 (35) Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–11 4 Tesla Coil Capacitance We now have expressions for the isotropic capacitance CSof a toroid and the isotropic ca- pacitance CMof a coil. The next step would be to set the toroid on top the coil and add the two capacitances to get an effective capacitance Ctcfor the Tesla coil. Unfortunately, this only works when the toroid is far away from the solenoid. As the toroid is brought near thecoil form, shielding occurs such that the effective capacitance is less than the sum of the twoisotropic capacitances. The Tesla coil capacitance might be written as C tc=CM+KCS (36) where K<1. A value of K=0.75 should result in a number for Ctcwithin 20% of the correct value for most Tesla coils. The resonant frequency is related to the square root of Ctcso a 20% error in capacitance results in only a 10% error in resonant frequency. Most readers probably feel disappointed here. We have gone to considerable effort and still come up short of an accurate formula for Ctc. Our effort is not entirely wasted because we can do ‘what if’ analyses relatively quickly. Questions about the effect of changing coildiameter, coil length, or toroid diameter can be answered with adequate accuracy. Someone might suggest using a modern digital capacitance meter to measure C tc.T h i s method would probably have greater error than the above formula, because the leads of thecapacitance meter have a similar capacitance value as C tc. Also the presence of the meter and a human body will change the capacitance. It is possible to calculate Ctcnumerically using Gauss’s Law. If one is careful about measuring and entering all the dimensions and the locations of grounded surfaces, one shouldget a value for C tcwell within 5% of the correct value. There are programs available in the Tesla coil community that do this. References [1] Loudner, Godfrey. Privat e communi cation, December 26, 2001. [2] Maruvada, P. Sarma and N. Hylt´ en-Cavallius, “Capacitance Calculations for Some Basic High Voltage Electrode Configurations”, IEEE Transactions on Power Apparatus andSystems, Vol. PAS-94, No. 5, September/October 1975, pp. 1708-1713. [3] Moon, Parry and Domina Eberle Spencer, Field Theory for Engineers ,D .V a nN o s t r a n d Company, Princeton, New Jersey, 1961. [4] Plonus, Martin A., Applied Electromagnetics , McGraw-Hill, New York, 1978. [5] Schoessow, Michael, TCBA News , Vol. 6, No. 2, April/May/June 1987, pp. 12-15. Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001 Chapter 2—Ideal Capacitors 2–12 [6] Smythe, William R., Static and Dynamic Electricity , Hemisphere Publishing Corporation, A member of the Taylor & Francis Group, New York, Third Edition, Revised Printing,1989. Solid State Tesla Coil by Dr. Gary L. Johnson December 27, 2001