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Capacitors do not have a stable "resistance" as
conductors do. However, there is a definite mathematical relationship
between voltage and current for a capacitor, as follows:
The lower-case letter "i" symbolizes
instantaneous current, which means the amount of current at a
specific point in time. This stands in contrast to constant current or
average current (capital letter "I") over an unspecified period of
time. The expression "dv/dt" is one borrowed from calculus, meaning
the instantaneous rate of voltage change over time, or the rate of
change of voltage (volts per second increase or decrease) at a
specific point in time, the same specific point in time that the
instantaneous current is referenced at. For whatever reason, the
letter v is usually used to represent instantaneous voltage
rather than the letter e. However, it would not be incorrect to
express the instantaneous voltage rate-of-change as "de/dt" instead.
In this equation we see something novel to our
experience thusfar with electric circuits: the variable of time.
When relating the quantities of voltage, current, and resistance to a
resistor, it doesn't matter if we're dealing with measurements taken
over an unspecified period of time (E=IR; V=IR), or at a specific
moment in time (e=ir; v=ir). The same basic formula holds true,
because time is irrelevant to voltage, current, and resistance in a
component like a resistor.
In a capacitor, however, time is an essential
variable, because current is related to how rapidly voltage
changes over time. To fully understand this, a few illustrations may
be necessary. Suppose we were to connect a capacitor to a
variable-voltage source, constructed with a potentiometer and a
battery:
If the potentiometer mechanism remains in a single
position (wiper is stationary), the voltmeter connected across the
capacitor will register a constant (unchanging) voltage, and the
ammeter will register 0 amps. In this scenario, the instantaneous rate
of voltage change (dv/dt) is equal to zero, because the voltage is
unchanging. The equation tells us that with 0 volts per second change
for a dv/dt, there must be zero instantaneous current (i). From a
physical perspective, with no change in voltage, there is no need for
any electron motion to add or subtract charge from the capacitor's
plates, and thus there will be no current.
Now, if the potentiometer wiper is moved slowly and
steadily in the "up" direction, a greater voltage will gradually be
imposed across the capacitor. Thus, the voltmeter indication will be
increasing at a slow rate:
If we assume that the potentiometer wiper is being
moved such that the rate of voltage increase across the
capacitor is steady (for example, voltage increasing at a constant
rate of 2 volts per second), the dv/dt term of the formula will be a
fixed value. According to the equation, this fixed value of dv/dt,
multiplied by the capacitor's capacitance in Farads (also fixed),
results in a fixed current of some magnitude. From a physical
perspective, an increasing voltage across the capacitor demands that
there be an increasing charge differential between the plates. Thus,
for a slow, steady voltage increase rate, there must be a slow, steady
rate of charge building in the capacitor, which equates to a slow,
steady flow rate of electrons, or current. In this scenario, the
capacitor is acting as a load, with electrons entering the
negative plate and exiting the positive, accumulating energy in the
electric field.
If the potentiometer is moved in the same
direction, but at a faster rate, the rate of voltage change (dv/dt)
will be greater and so will be the capacitor's current:
When mathematics students first study calculus,
they begin by exploring the concept of rates of change for
various mathematical functions. The derivative, which is the
first and most elementary calculus principle, is an expression of one
variable's rate of change in terms of another. Calculus students have
to learn this principle while studying abstract equations. You get to
learn this principle while studying something you can relate to:
electric circuits!
To put this relationship between voltage and
current in a capacitor in calculus terms, the current through a
capacitor is the derivative of the voltage across the capacitor
with respect to time. Or, stated in simpler terms, a capacitor's
current is directly proportional to how quickly the voltage across it
is changing. In this circuit where capacitor voltage is set by the
position of a rotary knob on a potentiometer, we can say that the
capacitor's current is directly proportional to how quickly we turn
the knob.
If we were to move the potentiometer's wiper in the
same direction as before ("up"), but at varying rates, we would obtain
graphs that looked like this:
Note how that at any given point in time, the
capacitor's current is proportional to the rate-of-change, or slope
of the capacitor's voltage plot. When the voltage plot line is rising
quickly (steep slope), the current will likewise be great. Where the
voltage plot has a mild slope, the current is small. At one place in
the voltage plot where it levels off (zero slope, representing a
period of time when the potentiometer wasn't moving), the current
falls to zero.
If we were to move the potentiometer wiper in the
"down" direction, the capacitor voltage would decrease rather
than increase. Again, the capacitor will react to this change of
voltage by producing a current, but this time the current will be in
the opposite direction. A decreasing capacitor voltage requires that
the charge differential between the capacitor's plates be reduced, and
the only way that can happen is if the electrons reverse their
direction of flow, the capacitor discharging rather than charging. In
this condition, with electrons exiting the negative plate and entering
the positive, the capacitor will act as a source, like a
battery, releasing its stored energy to the rest of the circuit.
Again, the amount of current through the capacitor
is directly proportional to the rate of voltage change across it. The
only difference between the effects of a decreasing voltage and
an increasing voltage is the direction of electron flow.
For the same rate of voltage change over time, either increasing or
decreasing, the current magnitude (amps) will be the same.
Mathematically, a decreasing voltage rate-of-change is expressed as a
negative dv/dt quantity. Following the formula i = C(dv/dt),
this will result in a current figure (i) that is likewise negative in
sign, indicating a direction of flow corresponding to discharge of the
capacitor.
There are three basic factors of capacitor
construction determining the amount of capacitance created. These
factors all dictate capacitance by affecting how much electric field
flux (relative difference of electrons between plates) will develop
for a given amount of electric field force (voltage between the two
plates):
PLATE AREA: All other factors being equal,
greater plate area gives greater capacitance; less plate area gives
less capacitance.
Explanation: Larger plate area results in
more field flux (charge collected on the plates) for a given field
force (voltage across the plates).
PLATE SPACING: All other factors being equal, further plate
spacing gives less capacitance; closer plate spacing gives greater
capacitance.
Explanation: Closer spacing results in a
greater field force (voltage across the capacitor divided by the
distance between the plates), which results in a greater field flux
(charge collected on the plates) for any given voltage applied across
the plates.
DIELECTRIC MATERIAL: All other factors being equal, greater
permittivity of the dielectric gives greater capacitance; less
permittivity of the dielectric gives less capacitance.
Explanation: Although it's complicated to
explain, some materials offer less opposition to field flux for a
given amount of field force. Materials with a greater permittivity
allow for more field flux (offer less opposition), and thus a greater
collected charge, for any given amount of field force (applied
voltage).
Relative" permittivity means the permittivity of a
material, relative to that of a pure vacuum. The greater the number,
the greater the permittivity of the material. Glass, for instance,
with a relative permittivity of 7, has seven times the permittivity of
a pure vacuum, and consequently will allow for the establishment of an
electric field flux seven times stronger than that of a vacuum, all
other factors being equal.
The following is a table listing the relative
permittivities (also known as the "dielectric constant") of various
common substances:
Material Relative permittivity (dielectric constant)
============================================================
Vacuum ------------------------- 1.0000
Air ---------------------------- 1.0006
PTFE, FEP ("Teflon") ----------- 2.0
Polypropylene ------------------ 2.20 to 2.28
ABS resin ---------------------- 2.4 to 3.2
Polystyrene -------------------- 2.45 to 4.0
Waxed paper -------------------- 2.5
Transformer oil ---------------- 2.5 to 4
Hard Rubber -------------------- 2.5 to 4.80
Wood (Oak) --------------------- 3.3
Silicones ---------------------- 3.4 to 4.3
Bakelite ----------------------- 3.5 to 6.0
Quartz, fused ------------------ 3.8
Wood (Maple) ------------------- 4.4
Glass -------------------------- 4.9 to 7.5
Castor oil --------------------- 5.0
Wood (Birch) ------------------- 5.2
Mica, muscovite ---------------- 5.0 to 8.7
Glass-bonded mica -------------- 6.3 to 9.3
Porcelain, Steatite ------------ 6.5
Alumina ------------------------ 8.0 to 10.0
Distilled water ---------------- 80.0
Barium-strontium-titanite ------ 7500
An approximation of capacitance for any pair of
separated conductors can be found with this formula:
A capacitor can be made variable rather than fixed
in value by varying any of the physical factors determining
capacitance. One relatively easy factor to vary in capacitor
construction is that of plate area, or more properly, the amount of
plate overlap.
The following photograph shows an example of a
variable capacitor using a set of interleaved metal plates and an air
gap as the dielectric material:
As the shaft is rotated, the degree to which the
sets of plates overlap each other will vary, changing the effective
area of the plates between which a concentrated electric field can be
established. This particular capacitor has a capacitance in the
picofarad range, and finds use in radio circuitry.
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