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 factors. 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, sodaborosilicate  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