Chapter 3—Lossy Capacitors

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Chapter 3—Lossy Capacitors 3–1 LOSSY CAPACITORS 1 Dielectric Loss Capacitors are used for a wide variety of purposes and are made of many different materials in many different styles. For purposes of discussion we will consider three broad types, that is, capacitors made for ac, dc, and pulse applications. The ac case is the most general since ac capacitors will work (or at least survive) in dc and pulse applications, where the reverse may not be true. It is important to consider the losses i

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  Chapter 3—Lossy Capacitors 3–1 LOSSY CAPACITORS1 Dielectric Loss Capacitors are used for a wide variety of purposes and are made of many different materialsin many different styles. For purposes of discussion we will consider three broad types, thatis, capacitors made for ac, dc, and pulse applications. The ac case is the most general sinceac capacitors will work (or at least survive) in dc and pulse applications, where the reversemay not be true.It is important to consider the losses in ac capacitors. All dielectrics (except vacuum) havetwo types of losses. One is a conduction loss, representing the flow of actual charge throughthe dielectric. The other is a dielectric loss due to movement or rotation of the atoms ormolecules in an alternating electric field. Dielectric losses in water are the reason for food anddrink getting hot in a microwave oven.One way of describing dielectric losses is to consider the permittivity as a complex number,defined as  =   −  j  = |  | e −  jδ (1)where   = ac capacitivity   = dielectric loss factor δ = dielectric loss angleCapacitance is a complex number C  ∗ in this definition, becoming the expected real number C  as the losses go to zero. That is, we define C  ∗ = C  −  jC   (2)One reason for defining a complex capacitance is that we can use the complex value in anyequation derived for a real capacitance in a sinusoidal application, and get the correct phaseshifts and power losses by applying the usual rules of circuit theory. This means that most of our analyses are already done, and we do not need to start over just because we now have alossy capacitor.Equation 1 expresses the complex permittivity in two ways, as real and imaginary or asmagnitude and phase. The magnitude and phase notation is rarely used. Instead, peopleSolid State Tesla Coil by Dr. Gary L. Johnson December 10, 2001  Chapter 3—Lossy Capacitors 3–2usually express the complex permittivity by   and tan δ , wheretan δ =     (3)where tan δ is called either the loss tangent  or the dissipation factor  DF.The real part of the permittivity is defined as   =  r  o (4)where  r is the dielectric constant  and  o is the permittivity of free space.Dielectric properties of several different materials are given in Table 1 [4, 5]. Some of these materials are used for capacitors, while others may be present in oscillators or otherdevices where dielectric losses may affect circuit performance. The dielectric constant andthe dissipation factor are given at two frequencies, 60 Hz and 1 MHz. The righthand columnof Table 1 gives the approximate breakdown voltage of the material in V/mil, where 1 mil= 0.001 inch. This would be for thin layers where voids and impurities in the dielectrics arenot a factor. Breakdown usually destroys a capacitor, so capacitors must be designed with asubstantial safety factor.It can be seen that most materials have dielectric constants between one and ten. Oneexception is barium titanate with a dielectric constant greater than 1000. It also has relativelyhigh losses which keep it from being more widely used than it is.We see that polyethylene, polypropylene, and polystyrene all have small dissipation fac-tors. They also have other desirable properties and are widely used for capacitors. For highpower, high voltage, and high frequency applications, such as an antenna capacitor in an AMbroadcast station, the ruby mica seems to be the best.Each of the materials in Table 1 has its own advantages and disadvantages when used ina capacitor. The ideal dielectric would have a high dielectric constant, like barium titanate, alow dissipation factor, like polystyrene, a high breakdown voltage, like mylar, a low cost, likealuminum oxide, and be easily fabricated into capacitors. It would also be perfectly stable,so the capacitance would not vary with temperature or voltage. No such dielectric has beendiscovered so we must apply engineering judgment in each situation, and select the capacitortype that will meet all the requirements and at least cost.Capacitors used for ac must be unpolarized  so they can handle full voltage reversals. Theyalso need to have a lower dissipation factor than capacitors used as dc filter capacitors, forexample. One important application of ac capacitors is in tuning electronic equipment. Thesecapacitors must have high stability with time and temperature, so the tuned frequency doesnot drift beyond some specified amount.Another category of ac capacitor is the motor run or power factor correcting capacitor.These are used on motors and other devices operating at 60 Hz and at voltages up to 480V or more. They are usually much larger than capacitors used for tuning electronic circuits,Solid State Tesla Coil by Dr. Gary L. Johnson December 10, 2001  Chapter 3—Lossy Capacitors 3–3and are not sold by electronics supply houses. One has to ask for motor run capacitors at anelectrical supply house like Graingers. These also work nicely as dc filter capacitors if voltageshigher than allowed by conventional dc filter capacitors are required.The term power factor  PF may also be defined for ac capacitors. It is given by theexpressionPF = cos θ (5)where θ is the angle between the current flowing through the capacitor and the voltage acrossit.The capacitive reactance for the sinusoidal case can be defined as X  C  =1 ωC  (6)where ω = 2 πf  rad/sec, and f  is in Hz.In a lossless capacitor,   = 0, and the current leads the voltage by exactly 90 o . If    isgreater than zero, then the current has a component in phase with the voltage.cos θ =     (   ) 2 + (   ) 2 (7)For a good dielectric,      , socos θ ≈     = tan δ (8)Therefore, the term power factor  is often used interchangeably with the terms loss tangent  or dissipation factor  , even though they are only approximately equal to each other.We can define the apparent power flow into a parallel plate capacitor as S  = V I  = V  2 −  jX  C  = jV  2 ωC  ∗ = jV  2 ωAd (   −  j  ) = V  2 ωAd r  o (  j + DF) (9)By analogy, the apparent power flow into any arbitrary capacitor is S  = P  + jQ = V  2 ωC  (  j + DF) (10)Solid State Tesla Coil by Dr. Gary L. Johnson December 10, 2001  Chapter 3—Lossy Capacitors 3–4Table 1: Dielectric Constant  r , Dissipation Factor DF and Breakdown Strength V  b of selectedmaterials.Material  r  r DF DF V  b 60 Hz 10 6 Hz 60 Hz 10 6 Hz V/milAir 1.000585 1.000585 - - 75Aluminum oxide - 8.80 - 0.00033 300Barium titanate 1250 1143 0.056 0.0105 50Carbon tetrachloride 2.17 2.17 0.007 < 0.00004 -Castor oil 3.7 3.7 - - 300Glass, soda-borosilicate - 4.84 - 0.0036 -Heavy Soderon 3.39 3.39 0.0168 0.0283 -Lucite 3.3 3.3 - - 500Mica, glass bonded - 7.39 - 0.0013 1600Mica, glass, titanium dioxide - 9.0 - 0.0026 -Mica, ruby 5.4 5.4 0.005 0.0003 -Mylar 2.5 2.5 - - 5000Nylon 3.88 3.33 0.014 0.026 -Paraffin 2.25 2.25 - - 250Plexiglas 3.4 2.76 0.06 0.014 -Polycarbonate 2.7 2.7 - - 7000Polyethylene 2.26 2.26 < 0.0002 < 0.0002 4500Polypropylene 2.25 2.25 < 0.0005 < 0.0005 9600Polystyrene 2.56 2.56 < 0.00005 0.00007 500Polysulfone 3.1 3.1 - - 8000Polytetrafluoroethylene(teflon) 2.1 2.1 < 0.0005 < 0.0002 1500Polyvinyl chloride (PVC) 3.2 2.88 0.0115 0.016Quartz 3.78 3.78 0.0009 0.0001 500Tantalum oxide 2.0 - - - 100Transformer oil 2.2 - - - 250Vaseline 2.16 2.16 0.0004 < 0.0001 -Solid State Tesla Coil by Dr. Gary L. Johnson December 10, 2001
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