Capacitive Reactance
Complete treatment of this topic  SOURCE
REACTANCE AND IMPEDANCE  CAPACITIVE
AC resistor circuits
If we were to plot the current and voltage for a very simple AC circuit
consisting of a source and a resistor, it would look something like this:
Because the resistor allows an amount of current directly proportional to
the voltage across it at all periods of time, the waveform for the current is
exactly in phase with the waveform for the voltage. We can look at any point in
time along the horizontal axis of the plot and compare those values of current
and voltage with each other (any "snapshot" look at the values of a
wave are referred to as instantaneous values, meaning the values at that instant
in time). When the instantaneous value for voltage is zero, the instantaneous
current through the resistor is also zero. Likewise, at the moment in time where
the voltage across the resistor is at its positive peak, the current through the
resistor is also at its positive peak, and so on. At any given point in time
along the waves, Ohm's Law holds true for the instantaneous values of voltage
and current.
We can also calculate the power dissipated by this resistor, and plot those
values on the same graph:
Note that the power is never a negative value. When the current is positive
(above the line), the voltage is also positive, resulting in a power (p=ie) of a
positive value. Conversely, when the current is negative (below the line), the
voltage is also negative, which results in a positive value for power (a
negative number multiplied by a negative number equals a positive number). This
consistent "polarity" of power tells us that the resistor is always
dissipating power, taking it from the source and releasing it in the form of
heat energy. Whether the current is positive or negative, a resistor still
dissipates energy.
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 lowercase 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 90^{o} 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 X_{C}
to be more specific).
Since capacitors "conduct" current in proportion to the rate of
voltage change, they will pass more current for fasterchanging voltages (as
they charge and discharge to the same voltage peaks in less time), and less
current for slowerchanging 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 lowercase Greek letter Omega (ω), which is referred to as the angular
velocity of the AC circuit. Thus, the equation X_{C} = 1/(2πfC)
could also be written as X_{C} = 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 +90^{o}
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 90^{o}, 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.

REVIEW:
 
Capacitive reactance is the opposition that a capacitor offers to
alternating current due to its phaseshifted 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: X_{C}
= 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.

Series resistorcapacitor circuits
In the last section, we learned what would happen in simple resistoronly
and capacitoronly AC circuits. Now we will combine the two components together
in series form and investigate the effects.
Take this circuit as an example to analyze:
The resistor will offer 5 Ω of resistance to AC current regardless of
frequency, while the capacitor will offer 26.5258 Ω of reactance to AC
current at 60 Hz. Because the resistor's resistance is a real number (5 Ω
∠ 0^{o}, or 5 + j0 Ω), and the capacitor's reactance is an
imaginary number (26.5258 Ω ∠ 90^{o}, or 0  j26.5258 Ω),
the combined effect of the two components will be an opposition to current equal
to the complex sum of the two numbers. The term for this complex opposition to
current is impedance, its symbol is Z, and it is also expressed in the
unit of ohms, just like resistance and reactance. In the above example, the
total circuit impedance is:
Impedance is related to voltage and current just as you might expect, in a
manner similar to resistance in Ohm's Law:
In fact, this is a far more comprehensive form of Ohm's Law than what was
taught in DC electronics (E=IR), just as impedance is a far more comprehensive
expression of opposition to the flow of electrons than simple resistance is. Any
resistance and any reactance, separately or in combination (series/parallel),
can be and should be represented as a single impedance.
To calculate current in the above circuit, we first need to give a phase
angle reference for the voltage source, which is generally assumed to be zero.
(The phase angles of resistive and capacitive impedance are always 0^{o}
and 90^{o}, respectively, regardless of the given phase angles for
voltage or current).
As with the purely capacitive circuit, the current wave is leading the
voltage wave (of the source), although this time the difference is 79.325^{o}
instead of a full 90^{o}.
As we learned in the AC inductance chapter, the "table" method of
organizing circuit quantities is a very useful tool for AC analysis just as it
is for DC analysis. Let's place out known figures for this series circuit into a
table and continue the analysis using this tool:
Current in a series circuit is shared equally by all components, so the
figures placed in the "Total" column for current can be distributed to
all other columns as well:
Continuing with our analysis, we can apply Ohm's Law (E=IR) vertically to
determine voltage across the resistor and capacitor:
Notice how the voltage across the resistor has the exact same phase angle as
the current through it, telling us that E and I are in phase (for the resistor
only). The voltage across the capacitor has a phase angle of 10.675^{o},
exactly 90^{o} less than the phase angle of the circuit current.
This tells us that the capacitor's voltage and current are still 90^{o}
out of phase with each other.
Let's check our calculations with SPICE:
ac rc circuit
v1 1 0 ac 10 sin
r1 1 2 5
c1 2 0 100u
.ac lin 1 60 60
.print ac v(1,2) v(2,0) i(v1)
.print ac vp(1,2) vp(2,0) ip(v1)
.end
freq
v(1,2)
v(2)
i(v1)
6.000E+01 1.852E+00 9.827E+00
3.705E01
freq
vp(1,2) vp(2)
ip(v1)
6.000E+01 7.933E+01 1.067E+01 1.007E+02
Once again, SPICE confusingly prints the current phase angle at a value
equal to the real phase angle plus 180^{o} (or minus 180^{o}).
However, it's a simple matter to correct this figure and check to see if our
work is correct. In this case, the 100.7^{o} output by SPICE for
current phase angle equates to a positive 79.3^{o}, which does
correspond to our previously calculated figure of 79.325^{o}.
Again, it must be emphasized that the calculated figures corresponding to
reallife voltage and current measurements are those in polar form, not
rectangular form! For example, if we were to actually build this series
resistorcapacitor circuit and measure voltage across the resistor, our
voltmeter would indicate 1.8523 volts, not 343.11 millivolts (real
rectangular) or 1.8203 volts (imaginary rectangular). Real instruments connected
to real circuits provide indications corresponding to the vector length
(magnitude) of the calculated figures. While the rectangular form of complex
number notation is useful for performing addition and subtraction, it is a more
abstract form of notation than polar, which alone has direct correspondence to
true measurements.

REVIEW:
 
Impedance is the total measure of opposition to electric current
and is the complex (vector) sum of ("real") resistance and
("imaginary") reactance.
 
Impedances (Z) are managed just like resistances (R) in series circuit
analysis: series impedances add to form the total impedance. Just be sure to
perform all calculations in complex (not scalar) form! Z_{Total} = Z_{1}
+ Z_{2} + . . . Z_{n}
 
Please note that impedances always add in series, regardless of what
type of components comprise the impedances. That is, resistive impedance,
inductive impedance, and capacitive impedance are to be treated the same way
mathematically.
 
A purely resistive impedance will always have a phase angle of exactly 0^{o}
(Z_{R} = R Ω ∠ 0^{o}).
 
A purely capacitive impedance will always have a phase angle of exactly
90^{o} (Z_{C} = X_{C} Ω ∠ 90^{o}).
 
Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
 
When resistors and capacitors are mixed together in circuits, the total
impedance will have a phase angle somewhere between 0^{o} and 90^{o}.
 
Series AC circuits exhibit the same fundamental properties as series DC
circuits: current is uniform throughout the circuit, voltage drops add to
form the total voltage, and impedances add to form the total impedance.

Parallel resistorcapacitor circuits
Using the same value components in our series example circuit, we will
connect them in parallel and see what happens:
Because the power source has the same frequency as the series example
circuit, and the resistor and capacitor both have the same values of resistance
and capacitance, respectively, they must also have the same values of impedance.
So, we can begin our analysis table with the same "given" values:
This being a parallel circuit now, we know that voltage is shared equally by
all components, so we can place the figure for total voltage (10 volts ∠ 0^{o})
in all the columns:
Now we can apply Ohm's Law (I=E/Z) vertically to two columns in the table,
calculating current through the resistor and current through the capacitor:
Just as with DC circuits, branch currents in a parallel AC circuit add up to
form the total current (Kirchhoff's Current Law again):
Finally, total impedance can be calculated by using Ohm's Law (Z=E/I)
vertically in the "Total" column. As we saw in the AC inductance
chapter, parallel impedance can also be calculated by using a reciprocal formula
identical to that used in calculating parallel resistances. It is noteworthy to
mention that this parallel impedance rule holds true regardless of the kind of
impedances placed in parallel. In other words, it doesn't matter if we're
calculating a circuit composed of parallel resistors, parallel inductors,
parallel capacitors, or some combination thereof: in the form of impedances (Z),
all the terms are common and can be applied uniformly to the same formula. Once
again, the parallel impedance formula looks like this:
The only drawback to using this equation is the significant amount of work
required to work it out, especially without the assistance of a calculator
capable of manipulating complex quantities. Regardless of how we calculate total
impedance for our parallel circuit (either Ohm's Law or the reciprocal formula),
we will arrive at the same figure:

REVIEW:
 
Impedances (Z) are managed just like resistances (R) in parallel circuit
analysis: parallel impedances diminish to form the total impedance, using
the reciprocal formula. Just be sure to perform all calculations in complex
(not scalar) form! Z_{Total} = 1/(1/Z_{1} + 1/Z_{2}
+ . . . 1/Z_{n})
 
Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
 
When resistors and capacitors are mixed together in parallel circuits
(just as in series circuits), the total impedance will have a phase angle
somewhere between 0^{o} and 90^{o}. The circuit current
will have a phase angle somewhere between 0^{o} and +90^{o}.
 
Parallel AC circuits exhibit the same fundamental properties as parallel
DC circuits: voltage is uniform throughout the circuit, branch currents add
to form the total current, and impedances diminish (through the reciprocal
formula) to form the total impedance.

Capacitor quirks
As with inductors, the ideal capacitor is a purely reactive device,
containing absolutely zero resistive (power dissipative) effects. In the real
world, of course, nothing is so perfect. However, capacitors have the virtue of
generally being purer reactive components than inductors. It is a lot
easier to design and construct a capacitor with low internal series resistance
than it is to do the same with an inductor. The practical result of this is that
real capacitors typically have impedance phase angles more closely approaching
90^{o} (actually, 90^{o}) than inductors. Consequently, they
will tend to dissipate less power than an equivalent inductor.
Capacitors also tend to be smaller and lighter weight than their equivalent
inductor counterparts, and since their electric fields are almost totally
contained between their plates (unlike inductors, whose magnetic fields
naturally tend to extend beyond the dimensions of the core), they are less prone
to transmitting or receiving electromagnetic "noise" to/from other
components. For these reasons, circuit designers tend to favor capacitors over
inductors wherever a design permits either alternative.
Capacitors with significant resistive effects are said to be lossy,
in reference to their tendency to dissipate ("lose") power like a
resistor. The source of capacitor loss is usually the dielectric material rather
than any wire resistance, as wire length in a capacitor is very minimal.
Dielectric materials tend to react to changing electric fields by producing
heat. This heating effect represents a loss in power, and is equivalent to
resistance in the circuit. The effect is more pronounced at higher frequencies
and in fact can be so extreme that it is sometimes exploited in manufacturing
processes to heat insulating materials like plastic! The plastic object to be
heated is placed between two metal plates, connected to a source of
highfrequency AC voltage. Temperature is controlled by varying the voltage or
frequency of the source, and the plates never have to contact the object being
heated.
This effect is undesirable for capacitors where we expect the component to
behave as a purely reactive circuit element. One of the ways to mitigate
the effect of dielectric "loss" is to choose a dielectric material
less susceptible to the effect. Not all dielectric materials are equally "lossy."
A relative scale of dielectric loss from least to greatest is given here:
Vacuum  (Low Loss)
Air
Polystyrene
Mica
Glass
LowK
ceramic
Plastic film
(Mylar)
Paper
HighK
ceramic
Aluminum
oxide
Tantalum pentoxide  (High Loss)
Dielectric resistivity manifests itself both as a series and a parallel
resistance with the pure capacitance:
Fortunately, these stray resistances are usually of modest impact (low
series resistance and high parallel resistance), much less significant than the
stray resistances present in an average inductor.
Electrolytic capacitors, known for their relatively high capacitance and low
working voltage, are also known for their notorious lossiness, due to both the
characteristics of the microscopically thin dielectric film and the electrolyte
paste. Unless specially made for AC service, electrolytic capacitors should
never be used with AC unless it is mixed (biased) with a constant DC voltage
preventing the capacitor from ever being subjected to reverse voltage. Even
then, their resistive characteristics may be too severe a shortcoming for the
application anyway.
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