AC capacitor circuits
Capacitors do not behave the same as
resistors. Whereas resistors allow a flow of electrons
through them directly proportional to the voltage drop,
capacitors oppose changes in voltage by drawing or
supplying current as they charge or discharge to the new
voltage level. The flow of electrons "through" a capacitor
is directly proportional to the rate of change of
voltage across the capacitor. This opposition to voltage
change is another form of reactance, but one that is
precisely opposite to the kind exhibited by inductors.
Expressed mathematically, the relationship
between the current "through" the capacitor and rate of
voltage change across the capacitor is as such:
The expression de/dt is one from
calculus, meaning the rate of change of instantaneous
voltage (e) over time, in volts per second. The capacitance
(C) is in Farads, and the instantaneous current (i), of
course, is in amps. Sometimes you will find the rate of
instantaneous voltage change over time expressed as dv/dt
instead of de/dt: using the lower-case letter "v" instead or
"e" to represent voltage, but it means the exact same thing.
To show what happens with alternating current, let's analyze
a simple capacitor circuit:
If we were to plot the current and voltage
for this very simple circuit, it would look something like
this:
Remember, the current through a capacitor is
a reaction against the change in voltage across it.
Therefore, the instantaneous current is zero whenever the
instantaneous voltage is at a peak (zero change, or level
slope, on the voltage sine wave), and the instantaneous
current is at a peak wherever the instantaneous voltage is
at maximum change (the points of steepest slope on the
voltage wave, where it crosses the zero line). This results
in a voltage wave that is -90o out of phase with
the current wave. Looking at the graph, the current wave
seems to have a "head start" on the voltage wave; the
current "leads" the voltage, and the voltage "lags" behind
the current.
As you might have guessed, the same unusual
power wave that we saw with the simple inductor circuit is
present in the simple capacitor circuit, too:
As with the simple inductor circuit, the 90
degree phase shift between voltage and current results in a
power wave that alternates equally between positive and
negative. This means that a capacitor does not dissipate
power as it reacts against changes in voltage; it merely
absorbs and releases power, alternately.
A capacitor's opposition to change in
voltage translates to an opposition to alternating voltage
in general, which is by definition always changing in
instantaneous magnitude and direction. For any given
magnitude of AC voltage at a given frequency, a capacitor of
given size will "conduct" a certain magnitude of AC current.
Just as the current through a resistor is a function of the
voltage across the resistor and the resistance offered by
the resistor, the AC current through a capacitor is a
function of the AC voltage across it, and the reactance
offered by the capacitor. As with inductors, the reactance
of a capacitor is expressed in ohms and symbolized by the
letter X (or XC to be more specific).
Since capacitors "conduct" current in
proportion to the rate of voltage change, they will pass
more current for faster-changing voltages (as they charge
and discharge to the same voltage peaks in less time), and
less current for slower-changing voltages. What this means
is that reactance in ohms for any capacitor is inversely
proportional to the frequency of the alternating current:
For a 100 uF capacitor:
Frequency (Hertz) Reactance (Ohms)
----------------------------------------
| 60 | 26.5258 |
|--------------------------------------|
| 120 | 13.2629 |
|--------------------------------------|
| 2500 | 0.6366 |
----------------------------------------
Please note that the relationship of
capacitive reactance to frequency is exactly opposite from
that of inductive reactance. Capacitive reactance (in ohms)
decreases with increasing AC frequency. Conversely,
inductive reactance (in ohms) increases with increasing AC
frequency. Inductors oppose faster changing currents by
producing greater voltage drops; capacitors oppose faster
changing voltage drops by allowing greater currents.
As with inductors, the reactance equation's
2πf term may be replaced by the lower-case Greek letter
Omega (ω), which is referred to as the angular velocity
of the AC circuit. Thus, the equation XC =
1/(2πfC) could also be written as XC = 1/(ωC),
with ω cast in units of radians per second.
Alternating current in a simple capacitive
circuit is equal to the voltage (in volts) divided by the
capacitive reactance (in ohms), just as either alternating
or direct current in a simple resistive circuit is equal to
the voltage (in volts) divided by the resistance (in ohms).
The following circuit illustrates this mathematical
relationship by example:
However, we need to keep in mind that
voltage and current are not in phase here. As was shown
earlier, the current has a phase shift of +90o
with respect to the voltage. If we represent these phase
angles of voltage and current mathematically, we can
calculate the phase angle of the inductor's reactive
opposition to current.
Mathematically, we say that the phase angle
of a capacitor's opposition to current is -90o,
meaning that a capacitor's opposition to current is a
negative imaginary quantity. This phase angle of reactive
opposition to current becomes critically important in
circuit analysis, especially for complex AC circuits where
reactance and resistance interact. It will prove beneficial
to represent any component's opposition to current in
terms of complex numbers, and not just scalar quantities of
resistance and reactance.
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REVIEW:
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Capacitive reactance is the
opposition that a capacitor offers to alternating current
due to its phase-shifted storage and release of energy in
its electric field. Reactance is symbolized by the capital
letter "X" and is measured in ohms just like resistance
(R).
-
Capacitive reactance can be calculated
using this formula: XC = 1/(2πfC)
-
Capacitive reactance decreases with
increasing frequency. In other words, the higher the
frequency, the less it opposes (the more it "conducts")
the AC flow of electrons.
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