[Note: various equations denoted by "e." are not yet avail., but coming
soon.]
1. Fundamentals of Capacitance
We are all familiar with the capacitance effect of the human body. The phenomenon
of building up a static charge when walking on a carpet illustrates the chargestoring
abilities of human body capacitance. Many electronic devices, such as touchscreen
displays, touchsensitive switches, and computer touch pads make use of this
property. In terms of electrical hazards, the capacitance is a property which
must be considered, along with inductance and resistance, in evaluating the
effect of electrical currents on the human body. Capacitors also present a
significant electrical safety hazard in their ability to store charge. Systems
and equipment which use capacitors must be designed in such a way that this
stored charge does not present a hazard to workers.
Capacitance is a property where an insulator opposes a change in the amount
of electrical potential, due to the electric field produced by the potential.
Since the human body is relatively small compared with the rate of change of
power frequency potentials, the capacitance effect is relatively small.
Nonetheless, it can have a significant effect at higher frequencies. However,
devices which have a large capacitance may be hazardous owing to the stored
charge which they may contain.
Electrical energy is stored in the electric field, and in monopoles (electrons
and ions) and dipoles (polarized molecules) within the material, all of which
must be factored into the total relationship between voltage and current.
The dielectric losses may be considered as a bulk resistance and analyzed
as previously discussed for resistors. For the initial discussion, the resistance
will be assumed to be infinite, so that the capacitive effects may be isolated
and analyzed separately. A real object, such as a body part, contains resistive,
capacitive, and inductive components.
Capacitors as with resistors are not only a material property, but can be
electronic components or power equipment. Capacitors are used in electronics
for a wide variety of purposes. A few examples are shown in FIG 1. Capacitors
may be used for blocking the flow of DC and only permitting the flow of AC;
this application is called a coupling capacitor. A coupling capacitor might
be applied in series with the input of an audio amplifier. Capacitors are widely
used for filtering out AC components of DC voltages, where they are connected
in shunt with the DC source. Bypass capacitors are small capacitors for filtering
out high frequencies. Filter capacitors are large capacitors for smoothing
out low frequencies, widely used in power supplies. Often, a large and a small
capacitor may be applied in parallel. Capacitors are used in a wide variety
of filter circuits, such as low pass, high pass, bandpass, and many more elaborate
designs.
The uses of capacitors in electronics are virtually unlimited, and there are
thousands of types available. Pictured here in FIG 1a is the ubiquitous ceramic
disk capacitor, which generally has a low capacitance value, usually in the
picofarad range, (this one has a capacitance of 47 pF) and can have a voltage
rating from 100 V up to several kilovolts. Capacitors made of a roll of foil
separated by an insulating material are used for higher capacitance values.
FIG 1b is a 0.1 µF, 600 VDC polyester film capacitor encapsulated in a hard
resin case.
Earlier versions of this type of capacitor were made with paper insulation
and dipped in wax for protection against moisture. When large capacitance values
are needed, electrolytic capacitors are normally used as shown in Figure 2.1(e)
and FIG 1c. These are polarized, and can carry a high capacitance value (10's
to 1000's of µF) at voltages from low DC values to hundreds of volts, in a
small package. Tantalum capacitors, FIG 1d, are widely used in computer circuits
owing to their small size and low series inductance. They are available in
both bypass and filter capacitor forms.
FIG. 1 Capacitors used in electronic applications. (a) Ceramic disk capacitor.
(b) Film capacitor. (c) Electrolytic capacitor. (d) Tantalum capacitors.
In terms of hazards to personnel, capacitors with a high capacitance value
present the greatest danger due to stored charges. Even at low voltages, a
discharge can produce a high current flow. Unless there is a parallel resistance
to discharge the capacitor, the stored charge may last for a long time after
the equipment is deenergized. Since the stored energy is as the voltage increases,
the amount of capacitance which presents a danger decreases. Capacitors should
always be discharged through a resistor, not by applying a short circuit, in
order to avoid the dangers of arcing.
Capacitors used in electric power applications are typified by the rectangular
cans with one or two bushings. The stack rack configuration FIG 2a is often
used in substations. The stack rack configuration allows parallel and series
combinations to be constructed, typically for application in substations. Capacitors
may also be mounted on poles for applications along distribution lines, FIG
2b. Power capacitors are generally constructed of rolls of aluminum foil separated
by polyethylene or similar plastic film, and then impregnated in oil. The rectangular
cans at high voltage will contain just such rolled capacitors. At lower voltages,
they may contain an array of smaller capacitors interconnected to make up the
required value. Power capacitors are generally specified in kVAr and voltage,
rather than in µF, although both values may be given.
FIG 2 (a) Stack rack capacitors used in electric power systems. (b) Distribution
capacitor bank with switches.
These capacitors are typically used for power factor correction and harmonic
filters. When capacitors are used on power distribution lines, they also provide
the function of voltage support. Capacitor banks are often switched due to
changes in load, power factor, or voltage.
Banks can be switched as a whole, or in steps.
Power capacitors are used in series with long, high voltage AC transmission
lines to reduce the inductance effect. Smaller capacitors, called surge capacitors,
are used to protect motors, generators, and transformers against high voltage
transients. In power conversion systems, such as variable frequency drives
(VFDs) for motor speed control, capacitors are used to store energy on an internal
DC bus.
There are a variety of power electronic devices which provide continuous variation
in capacitance at a faster speed than is possible with switching. These include
the STATCOM or static compensator, the SVC or static VAR controller, the TCSR
or thyristorcontrolled shunt reactor which controls capacitance by varying
a parallel reactor. Power capacitors are required to have internal discharge
resistors (IEEE, 2002c), which discharge a capacitor of 600 V or less in no
more than 1 min, and of greater than 600 V in no more than 5 min. The internal
discharge resistor is not a substitute for discharging the capacitor through
an external resistor before performing work on the bank.
2. Capacitance and Permittivity
The capacitance of the cylindrical geometry shown in FIG 3, is defined as…
eqn.1 where A = area of one of two identical conductive plates (m^2)
= distance between the surfaces of the two plates (m)
? = electrical permittivity of the material between the plates in farads/meter
(F/m).
? = product of two components, ? = ?0?r.
They are:
?0 = permittivity of free space ˜ 10^9/36p F/m
?r = relative permittivity of the material (unitless).
FIG 3 Capacitive energy storage.
The energy that is stored in a capacitor can be analyzed by drawing the Poynting
surface, , around one of the end plates. The energy will enter one side of
the plate, and exit the other side into the dielectric. Similarly, at the other
plate, the energy will exit around the other conductor. Since the conductance,
s, is zero, and the capacitor is lossless, the circuit model in FIG 4 is very
simple.
eqn.2
If the surface is taken around the dielectric, …
eqn.3
FIG 4 Circuit model of a lossless capacitive circuit.
In a pure capacitive volume, there is infinite resistivity, or zero conductivity,
and thus no current flow. However, the electromagnetic energy must transition
from one plate to another, so a vector quantity called the displacement current,
D, is postulated:
eqn.4
The relative permittivity of some common materials is listed in Table 1, and
of some common body parts in Table 2.
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Table 1 Relative Permittivity (?r) of Some Common Materials (at Room Temperature)
Material ?r
Copper 1.0
Vacuum 1.0
Teflon 2.1
Paraffin wax 2.2
Polyethylene 2.2
Plexiglas 3.4
Borosilicate glass 4.0
Ruby mica (muscovite) 5.4
Polyvinyl chloride 6.6
Pure water, sea water, fresh water 80
====
Table 2 Relative Permittivity (?r) of Some Body Parts (at Power Frequency)
Body Parts ?r
Skin (dry)corneum 283
Bone >3800
Fat 1.5 × 105
Muscle (perpendicular) 3.2 × 105
Lung 4.5 × 105
Liver 8.5 × 105
Muscle (parallel) 1.1 × 106
====
The displacement current is the subject of one of the most famous equations
in physics, Maxwell's displacement current correction of Ampère's law, which
is the cornerstone of Maxwell's equations.
eqn.5
This demonstrates that while the conventional current flow produces a magnetic
field, the change in displacement current also produces a magnetic field. It
is changing displacement which causes energy flow through a capacitor. A capacitor
is normally thought as providing energy storage in an electric field. This
is true in the static case. In the dynamic case, it provides energy flow between
two conducting materials, similar to a radio transmission. Each plate of the
capacitor may be considered as an antenna.
When the dielectric materials contain polarized molecules (dipoles), the polarization
may be caused by field cancellation from the capacitor plates. In addition,
charged particles (ions) may be present within the material.
eqn.6
Thus the relative permittivity relates the polarizability to the electric
field strength:
eqn.7 where
κ = electric susceptibility.
3. Capacitance in Electrical Circuits
Discrete capacitors are widely used in electrical and electronic circuits,
and often present safety hazards in their own right. In a simple DC circuit,
capacitors can be charged through a resistor (FIG 5) at an exponential rate.
The circuit equations are
e.8
Solving the differential equation:
e.9
The result is the familiar capacitor charging exponential, as shown in FIG
6. Similarly, the discharge circuit is shown in FIG 7 and its curve is shown
in FIG 8.
FIG 5 Capacitor charged from a DC source.
FIG 6 Voltage on 1.0 µF capacitor charged through a 100 Ohm resistor to 1 kV
with t = 100 ms.
FIG 7 Discharge of capacitor charged from a DC source.
FIG 8 Voltage on 1.0 µF capacitor discharged from 1 kV through a 100 Ohm resistor
with t = 100 ms.
The equation for the discharge is calculated similarly as for the charging,
and results in e.10 With an inductor in place of the resistor, the familiar
series resonant circuit is constructed (FIG 9). The resonant frequency is
e.11 while the voltage across the capacitor can be expressed as e.12…
...as shown in FIG 10.
FIG 9 Oscillatory LC circuit.
FIG 10 Voltage on 1.0 µF capacitor in oscillatory LC circuit with L = 20
mH and ?0 =
5.010 × 107.
The switching of capacitive loads, as in FIG 11, causes transient oscillations
with high di/dt.
FIG 11 Series RLC circuit.
The voltage source is sinusoidal or e.13 where the peak voltage is:
e14
Using the definitions of the voltages across resistors, inductors, and capacitors,
we have the classical differential equation for a series RLC circuit.
The inductor and capacitor are energy storage devices. The current through
the inductor is zero with no voltage applied. Since the current in an inductor
cannot change instantaneously, the current is zero both immediately before
and after time t = 0.
e.15
The voltage across the capacitor is zero if it is discharged before the start
of the switching operation. Since the voltage in a capacitor cannot change
instantaneously, the voltage is zero both immediately before and after time.
e.16
Using the definitions of the voltages across resistors, inductors, and capacitors,
we have the classical differential equation for a series RLC circuit:
e.17
This equation can be solved by a number of means, giving a generic solution
of e.18 where e.19
e.20
The form of the solution is determined by the relation
between and...
If , the system is described as being overdamped. The solution to the differential
equation becomes: …
If , the system is described as being critically damped. The solution to the
differential equation becomes: .
If , the system is described as being underdamped. The solution to the differential
equation becomes or more conveniently, Supposing that the values of R, L, and
C are then the resonant frequency is Then, because …. The system is underdamped.
Initial voltages are and… At time , the loop equation becomes…
At time the underdamped solution equation is Since or else there would not
be a solution, and , it follows that .
The derivative of the underdamped solution equation is:
Thus, A plot of the solution equation is shown in FIG 12.
FIG 12 Oscillatory current transient in series RLC circuit with L = 12 µH,
C = 0.08 µF, and R = 14 Ohm.
4. Capacitance of Body Parts
Capacitance in the human body is primarily present between the layers of skin
(0.020.06 µF/cm^2), and between the body as a whole and the earth.
4.1 ExampleSkin Capacitance
Skin capacitance resides primarily in the dried layer of dead cells called
the corneum, which is the normal surface layer. The thickness of the corneum
is in the range 1020 µm, and it typically consists of approximately 200 layers
of cell membranes, which would each have a thickness of 0.05 µm. The capacitance
of a single membrane is in the range of 25 µF/cm^2.
This allows the relative permittivity to be calculated:
Using the data from the skin with an electrode, the area , and depth , assuming.
This is in the accepted range of 0.020.06 µF/ cm^2. At 60 Hz, the capacitive
reactance is The equivalent circuit of the skin can then be completed ( FIG
13). For AC current, the capacitance will dominate RP, resulting in lower skin
impedance.
FIG 13 Example of skin impedance.
4.2 ExampleCapacitance of Trunk and Limb
The trunk of the body from the previous example has a capacitance of At 60
Hz, the capacitive reactance is ...
The limb of the body from the previous example has a capacitance of flesh
and bone. The capacitance of the bone is…
The capacitance of the flesh is The total capacitance of the limb is…
At 60 Hz, the capacitive reactance is…
5. Electrical Hazards of Capacitance
Power capacitors are used in electric power applications mainly for power
factor correction and harmonic filters.
Filter capacitors for DC power supplies act to reduce the remaining AC components
in a pulsating DC voltage after the rectification process is completed. These
are usually large capacitors in the hundreds to thousands of microfarads, at
voltages from the single digits up to the kilovolt range. Filter capacitors
cause several electrical hazards. The first, and most obvious, is the risk
of electrical shock from a capacitor charged to a high voltage with a high
stored energy. A filter capacitor will charge to the peak value of the sinusoidal
voltage being rectified. For example, if the power transformer for a piece
of vacuumtube electronic equipment has a 500 centertapped secondary, such
that , the fullwave rectified voltage will be However, the filter capacitor
will charge to the peak value of voltage If the capacitor has a size of 100
µF, then the charge on the capacitor is The energy stored in the capacitor
is This may provide a noticeable shock. The second hazard is of high discharge
currents if such a capacitor is accidentally shorted, such as by an uninsulated
screwdriver or other tool.
DC Link capacitors for AC motor drives act to store energy from the AC system
after it has been rectified, in order that the inverter has a constant voltage
source. These are usually large capacitors in the hundreds to thousands of
microfarads, at voltages from the hundreds of volts up to the kilovolt range.
DC Link capacitors can cause several electrical hazards.
The first, and most obvious, is the risk of electrical shock from a capacitor
charged to a high voltage with a high stored energy. A filter capacitor will
charge to the peak value of the sinusoidal voltage being rectified. For example,
if the AC drive is powered by a 480 VAC system, and rectified by a threephase
bridge, the rectified voltage will be However, the filter capacitor will charge
to the peak value of voltage If the capacitor has a size of 4500 µF, then the
charge on the capacitor is The energy stored in the capacitor is This is more
than sufficient to provide a fatal discharge if touched, and a severe arc flash
if accidentally shorted.
6. Capacitance of Cables
The capacitance of a cable is a distributed quantity, and the longer the cable,
the larger the capacitance (Sutherland, 2012). When cables are deenergized,
they hold a charge for a long period of time unless properly discharged. It
is important to know the capacitance of a cable and the voltage at which it
was operated in establishing the proper safety precautions when working on
it, even if deenergized.
Capacitance, , is calculated as e.21 where
= radius of the shield
= radius of the center conductor
?0 = 8.857 × 1012 F/m, and
?r = (unitless) is the relative permittivity (dielectric constant) of the
insulation.
The dimensions and are shown in FIG 14. Values of for some typical cableinsulating
materials are shown in Table 3 . More detailed simulations will also include
the permittivity and conductivity of the semiconducting shields.
FIG 14 Shielded cable cross section showing radius of the wire and shield.
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Table 3 Dielectric Constants of Some Cable Insulation Materials
Material  Description  Maximum Service; Dielectric Constant; Temperature
(°C) PVC Polyvinyl chloride 105 3.4
XLPE Crosslinked polyethylene 90 5.0
MVXLPE Medium voltage crosslinked polyethylene 90 2.3
EPR Ethylene propylene rubber 90 2.5
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