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 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.
- REVIEW:
- 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.
Series resistor-capacitor circuits
In the last section, we learned what
would happen in simple resistor-only and capacitor-only 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 ∠ 0o, or 5 + j0 Ω), and the
capacitor's reactance is an imaginary number (26.5258 Ω ∠ -90o,
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 0o and
-90o, 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 79.325o instead of a full 90o.
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 -10.675o,
exactly 90o less than the phase angle of the circuit
current. This tells us that the capacitor's voltage and current are
still 90o out of phase with each other.
Let's check our calculations with SPICE:
ac r-c 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 3.705E-01
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 180o
(or minus 180o). However, it's a simple matter to correct
this figure and check to see if our work is correct. In this case, the
-100.7o output by SPICE for current phase angle equates to a
positive 79.3o, which does correspond to our previously
calculated figure of 79.325o.
Again, it must be emphasized that the
calculated figures corresponding to real-life voltage and current
measurements are those in polar form, not rectangular form! For
example, if we were to actually build this series resistor-capacitor
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! ZTotal = Z1 + Z2
+ . . . Zn
- 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 0o (ZR = R
Ω ∠ 0o).
- A purely capacitive impedance will
always have a phase angle of exactly -90o (ZC =
XC Ω ∠ -90o).
- 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 0o and -90o.
- 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 resistor-capacitor 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 ∠ 0o) 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!
ZTotal = 1/(1/Z1 + 1/Z2 + . . . 1/Zn)
- 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 0o
and -90o. The circuit current will have a phase angle
somewhere between 0o and +90o.
- 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 90o (actually, -90o) 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 high-frequency 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
Low-K ceramic
Plastic film (Mylar)
Paper
High-K 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.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first. See Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck
(June 2000): HTML document formatting, which led to a much
better-looking second edition.
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