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 simply and directly
resists the flow of electrons at all periods of time, the waveform for
the voltage drop across the resistor is exactly in phase with the
waveform for the current through it. 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
current is zero, the instantaneous voltage across the resistor is also
zero. Likewise, at the moment in time where the current through the
resistor is at its positive peak, the voltage across 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 inductor circuits
Inductors do not behave the same as
resistors. Whereas resistors simply oppose the flow of electrons through
them (by dropping a voltage directly proportional to the current),
inductors oppose changes in current through them, by dropping a
voltage directly proportional to the rate of change of current.
In accordance with Lenz's Law, this induced voltage is always of
such a polarity as to try to maintain current at its present value. That
is, if current is increasing in magnitude, the induced voltage will
"push against" the electron flow; if current is decreasing, the polarity
will reverse and "push with" the electron flow to oppose the decrease.
This opposition to current change is called reactance, rather
than resistance.
Expressed mathematically, the
relationship between the voltage dropped across the inductor and rate of
current change through the inductor is as such:
The expression di/dt is one from
calculus, meaning the rate of change of instantaneous current (i) over
time, in amps per second. The inductance (L) is in Henrys, and the
instantaneous voltage (e), of course, is in volts. Sometimes you will
find the rate of instantaneous voltage expressed as "v" instead of "e"
(v = L di/dt), but it means the exact same thing. To show what happens
with alternating current, let's analyze a simple inductor circuit:
If we were to plot the current and
voltage for this very simple circuit, it would look something like this:
Remember, the voltage dropped across an
inductor is a reaction against the change in current through it.
Therefore, the instantaneous voltage is zero whenever the instantaneous
current is at a peak (zero change, or level slope, on the current sine
wave), and the instantaneous voltage is at a peak wherever the
instantaneous current is at maximum change (the points of steepest slope
on the current 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 voltage wave seems to have a "head start" on
the current wave; the voltage "leads" the current, and the current
"lags" behind the voltage.
Things get even more interesting when we
plot the power for this circuit:
Because instantaneous power is the
product of the instantaneous voltage and the instantaneous current (p=ie),
the power equals zero whenever the instantaneous current or
voltage is zero. Whenever the instantaneous current and voltage are both
positive (above the line), the power is positive. As with the resistor
example, the power is also positive when the instantaneous current and
voltage are both negative (below the line). However, because the current
and voltage waves are 90o out of phase, there are times when
one is positive while the other is negative, resulting in equally
frequent occurrences of negative instantaneous power.
But what does negative power mean?
It means that the inductor is releasing power back to the circuit, while
a positive power means that it is absorbing power from the circuit.
Since the positive and negative power cycles are equal in magnitude and
duration over time, the inductor releases just as much power back to the
circuit as it absorbs over the span of a complete cycle. What this means
in a practical sense is that the reactance of an inductor dissipates a
net energy of zero, quite unlike the resistance of a resistor, which
dissipates energy in the form of heat. Mind you, this is for perfect
inductors only, which have no wire resistance.
An inductor's opposition to change in
current translates to an opposition to alternating current in general,
which is by definition always changing in instantaneous magnitude and
direction. This opposition to alternating current is similar to
resistance, but different in that it always results in a phase shift
between current and voltage, and it dissipates zero power. Because of
the differences, it has a different name: reactance. Reactance to
AC is expressed in ohms, just like resistance is, except that its
mathematical symbol is X instead of R. To be specific, reactance
associate with an inductor is usually symbolized by the capital letter X
with a letter L as a subscript, like this: XL.
Since inductors drop voltage in
proportion to the rate of current change, they will drop more voltage
for faster-changing currents, and less voltage for slower-changing
currents. What this means is that reactance in ohms for any inductor is
directly proportional to the frequency of the alternating current. The
exact formula for determining reactance is as follows:
If we expose a 10 mH inductor to
frequencies of 60, 120, and 2500 Hz, it will manifest the following
reactances:
For a 10 mH inductor:
Frequency (Hertz) Reactance (Ohms)
----------------------------------------
| 60 | 3.7699 |
|--------------------------------------|
| 120 | 7.5398 |
|--------------------------------------|
| 2500 | 157.0796 |
----------------------------------------
In the reactance equation, the term "2πf"
(everything on the right-hand side except the L) has a special meaning
unto itself. It is the number of radians per second that the alternating
current is "rotating" at, if you imagine one cycle of AC to represent a
full circle's rotation. A radian is a unit of angular
measurement: there are 2π radians in one full circle, just as there are
360o in a full circle. If the alternator producing the AC is
a double-pole unit, it will produce one cycle for every full turn of
shaft rotation, which is every 2π radians, or 360o. If this
constant of 2π is multiplied by frequency in Hertz (cycles per second),
the result will be a figure in radians per second, known as the
angular velocity of the AC system.
Angular velocity may be represented by
the expression 2πf, or it may be represented by its own symbol, the
lower-case Greek letter Omega, which appears similar to our Roman
lower-case "w": ω. Thus, the reactance formula XL = 2πfL
could also be written as XL = ωL.
It must be understood that this "angular
velocity" is an expression of how rapidly the AC waveforms are cycling,
a full cycle being equal to 2π radians. It is not necessarily
representative of the actual shaft speed of the alternator producing the
AC. If the alternator has more than two poles, the angular velocity will
be a multiple of the shaft speed. For this reason, ω is sometimes
expressed in units of electrical radians per second rather than
(plain) radians per second, so as to distinguish it from mechanical
motion.
Any way we express the angular velocity
of the system, it is apparent that it is directly proportional to
reactance in an inductor. As the frequency (or alternator shaft speed)
is increased in an AC system, an inductor will offer greater opposition
to the passage of current, and visa-versa. Alternating current in a
simple inductive circuit is equal to the voltage (in volts) divided by
the inductive 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). An example circuit is shown here:
However, we need to keep in mind that
voltage and current are not in phase here. As was shown earlier, the
voltage has a phase shift of +90o with respect to the
current. If we represent these phase angles of voltage and current
mathematically in the form of complex numbers, we find that an
inductor's opposition to current has a phase angle, too:
Mathematically, we say that the phase
angle of an inductor's opposition to current is 90o, meaning
that an inductor's opposition to current is a positive 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 rather than scalar quantities of resistance and
reactance.
- REVIEW:
- Inductive reactance
is the opposition that an inductor offers to alternating current due
to its phase-shifted storage and release of energy in its magnetic
field. Reactance is symbolized by the capital letter "X" and is
measured in ohms just like resistance (R).
- Inductive reactance can be calculated
using this formula: XL = 2πfL
- The angular velocity of an AC
circuit is another way of expressing its frequency, in units of
electrical radians per second instead of cycles per second. It is
symbolized by the lower-case Greek letter "omega," or ω.
- Inductive reactance increases
with increasing frequency. In other words, the higher the frequency,
the more it opposes the AC flow of electrons.
Series resistor-inductor circuits
In the previous section, we explored what
would happen in simple resistor-only and inductor-only AC circuits. Now
we will mix the two components together in series form and investigate
the effects.
Take this circuit as an example to work
with:
The resistor will offer 5 Ω of resistance
to AC current regardless of frequency, while the inductor will offer
3.7699 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
inductor's reactance is an imaginary number (3.7699 Ω ∠ 90o,
or 0 + j3.7699 Ω), the combined effect of the two components will be an
opposition to current equal to the complex sum of the two numbers. This
combined opposition will be a vector combination of resistance and
reactance. In order to express this opposition succinctly, we need a
more comprehensive term for opposition to current than either resistance
or reactance alone. This term is called 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 resistance is. Any resistance and any
reactance, separately or in combination (series/parallel), can be and
should be represented as a single impedance in an AC circuit.
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 inductive impedance are always 0o and
+90o, respectively, regardless of the given phase angles for
voltage or current).
As with the purely inductive circuit, the
current wave lags behind the voltage wave (of the source), although this
time the lag is not as great: only 37.016o as opposed to a
full 90o as was the case in the purely inductive circuit.
For the resistor and the inductor, the
phase relationships between voltage and current haven't changed. Across
voltage across the resistor is in phase (0o shift) with the
current through it; and the voltage across the inductor is +90o
out of phase with the current going through it. We can verify this
mathematically:
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 inductor has a
phase angle of 52.984o, while the current through the
inductor has a phase angle of -37.016o, a difference of
exactly 90o between the two. This tells us that E and I are
still 90o out of phase (for the inductor only).
We can also mathematically prove that
these complex values add together to make the total voltage, just as
Kirchhoff's Voltage Law would predict:
Let's check the validity of our
calculations with SPICE:
ac r-l circuit
v1 1 0 ac 10 sin
r1 1 2 5
l1 2 0 10m
.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 7.985E+00 6.020E+00 1.597E+00
freq vp(1,2) vp(2) ip(v1)
6.000E+01 -3.702E+01 5.298E+01 1.430E+02
Note that just as with DC circuits, SPICE
outputs current figures as though they were negative (180o
out of phase) with the supply voltage. Instead of a phase angle of
-37.016o, we get a current phase angle of 143o
(-37o + 180o). This is merely an idiosyncrasy of
SPICE and does not represent anything significant in the circuit
simulation itself. Note how both the resistor and inductor voltage phase
readings match our calculations (-37.02o and 52.98o,
respectively), just as we expected them to.
With all these figures to keep track of
for even such a simple circuit as this, it would be beneficial for us to
use the "table" method. Applying a table to this simple series
resistor-inductor circuit would proceed as such. First, draw up a table
for E/I/Z figures and insert all component values in these terms (in
other words, don't insert actual resistance or inductance values in Ohms
and Henrys, respectively, into the table; rather, convert them into
complex figures of impedance and write those in):
Although it isn't necessary, I find it
helpful to write both the rectangular and polar forms of each
quantity in the table. If you are using a calculator that has the
ability to perform complex arithmetic without the need for conversion
between rectangular and polar forms, then this extra documentation is
completely unnecessary. However, if you are forced to perform complex
arithmetic "longhand" (addition and subtraction in rectangular form, and
multiplication and division in polar form), writing each quantity in
both forms will be useful indeed.
Now that our "given" figures are inserted
into their respective locations in the table, we can proceed just as
with DC: determine the total impedance from the individual impedances.
Since this is a series circuit, we know that opposition to electron flow
(resistance or impedance) adds to form the total opposition:
Now that we know total voltage and total
impedance, we can apply Ohm's Law (I=E/Z) to determine total current:
Just as with DC, the total current in a
series AC circuit is shared equally by all components. This is still
true because in a series circuit there is only a single path for
electrons to flow, therefore the rate of their flow must uniform
throughout. Consequently, we can transfer the figures for current into
the columns for the resistor and inductor alike:
Now all that's left to figure is the
voltage drop across the resistor and inductor, respectively. This is
done through the use of Ohm's Law (E=IZ), applied vertically in each
column of the table:
And with that, our table is complete. The
exact same rules we applied in the analysis of DC circuits apply to AC
circuits as well, with the caveat that all quantities must be
represented and calculated in complex rather than scalar form. So long
as phase shift is properly represented in our calculations, there is no
fundamental difference in how we approach basic AC circuit analysis
versus DC.
Now is a good time to review the
relationship between these calculated figures and readings given by
actual instrument measurements of voltage and current. The figures here
that directly relate to real-life measurements are those in polar
notation, not rectangular! In other words, if you were to connect a
voltmeter across the resistor in this circuit, it would indicate
7.9847 volts, not 6.3756 (real rectangular) or 4.8071 (imaginary
rectangular) volts. To describe this in graphical terms, measurement
instruments simply tell you how long the vector is for that particular
quantity (voltage or current).
Rectangular notation, while convenient
for arithmetical addition and subtraction, is a more abstract form of
notation than polar in relation to real-world measurements. As I stated
before, I will indicate both polar and rectangular forms of each
quantity in my AC circuit tables simply for convenience of mathematical
calculation. This is not absolutely necessary, but may be helpful for
those following along without the benefit of an advanced calculator. If
we were restrict ourselves to the use of only one form of notation, the
best choice would be polar, because it is the only one that can be
directly correlated to real measurements.
- REVIEW:
- Impedance
is the total measure of opposition to electric current and is the
complex (vector) sum of ("real") resistance and ("imaginary")
reactance. It is symbolized by the letter "Z" and measured in ohms,
just like resistance (R) and reactance (X).
- 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
- A purely resistive impedance will
always have a phase angle of exactly 0o (ZR = R
Ω ∠ 0o).
- A purely inductive impedance will
always have a phase angle of exactly +90o (ZL =
XL Ω ∠ 90o).
- Ohm's Law for AC circuits: E = IZ ; I
= E/Z ; Z = E/I
- When resistors and inductors are mixed
together in 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.
- 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-inductor circuits
Let's take the same components for our
series example circuit and connect them in parallel:
Because the power source has the same
frequency as the series example circuit, and the resistor and inductor
both have the same values of resistance and inductance, respectively,
they must also have the same values of impedance. So, we can begin our
analysis table with the same "given" values:
The only difference in our analysis
technique this time is that we will apply the rules of parallel circuits
instead of the rules for series circuits. The approach is fundamentally
the same as for DC. We know that voltage is shared uniformly by all
components in a parallel circuit, so we can transfer the figure of total
voltage (10 volts ∠ 0o) to all components columns:
Now we can apply Ohm's Law (I=E/Z)
vertically to two columns of the table, calculating current through the
resistor and current through the inductor:
Just as with DC circuits, branch currents
in a parallel AC circuit add to form the total current (Kirchhoff's
Current Law still holds true for AC as it did for DC):
Finally, total impedance can be
calculated by using Ohm's Law (Z=E/I) vertically in the "Total" column.
Incidentally, parallel impedance can also be calculated by using a
reciprocal formula identical to that used in calculating parallel
resistances.
The only problem with using this formula
is that it typically involves a lot of calculator keystrokes to carry
out. And if you're determined to run through a formula like this
"longhand," be prepared for a very large amount of work! But, just as
with DC circuits, we often have multiple options in calculating the
quantities in our analysis tables, and this example is no different. No
matter which way you calculate total impedance (Ohm's Law or the
reciprocal formula), you 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 inductors 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.
Inductor quirks
In an ideal case, an inductor acts as a
purely reactive device. That is, its opposition to AC current is
strictly based on inductive reaction to changes in current, and not
electron friction as is the case with resistive components. However,
inductors are not quite so pure in their reactive behavior. To begin
with, they're made of wire, and we know that all wire possesses some
measurable amount of resistance (unless it's superconducting wire). This
built-in resistance acts as though it were connected in series with the
perfect inductance of the coil, like this:
Consequently, the impedance of any real
inductor will always be a complex combination of resistance and
inductive reactance.
Compounding this problem is something
called the skin effect, which is AC's tendency to flow through
the outer areas of a conductor's cross-section rather than through the
middle. When electrons flow in a single direction (DC), they use the
entire cross-sectional area of the conductor to move. Electrons
switching directions of flow, on the other hand, tend to avoid travel
through the very middle of a conductor, limiting the effective
cross-sectional area available. The skin effect becomes more pronounced
as frequency increases.
Also, the alternating magnetic field of
an inductor energized with AC may radiate off into space as part of an
electromagnetic wave, especially if the AC is of high frequency. This
radiated energy does not return to the inductor, and so it manifests
itself as resistance (power dissipation) in the circuit.
Added to the resistive losses of wire and
radiation, there are other effects at work in iron-core inductors which
manifest themselves as additional resistance between the leads. When an
inductor is energized with AC, the alternating magnetic fields produced
tend to induce circulating currents within the iron core known as
eddy currents. These electric currents in the iron core have to
overcome the electrical resistance offered by the iron, which is not as
good a conductor as copper. Eddy current losses are primarily
counteracted by dividing the iron core up into many thin sheets
(laminations), each one separated from the other by a thin layer of
electrically insulating varnish. With the cross-section of the core
divided up into many electrically isolated sections, current cannot
circulate within that cross-sectional area and there will be no (or very
little) resistive losses from that effect.
As you might have expected, eddy current
losses in metallic inductor cores manifest themselves in the form of
heat. The effect is more pronounced at higher frequencies, and can be so
extreme that it is sometimes exploited in manufacturing processes to
heat metal objects! In fact, this process of "inductive heating" is
often used in high-purity metal foundry operations, where metallic
elements and alloys must be heated in a vacuum environment to avoid
contamination by air, and thus where standard combustion heating
technology would be useless. It is a "non-contact" technology, the
heated substance not having to touch the coil(s) producing the magnetic
field.
In high-frequency service, eddy currents
can even develop within the cross-section of the wire itself,
contributing to additional resistive effects. To counteract this
tendency, special wire made of very fine, individually insulated strands
called Litz wire (short for Litzendraht) can be used. The
insulation separating strands from each other prevent eddy currents from
circulating through the whole wire's cross-sectional area.
Additionally, any magnetic hysteresis
that needs to be overcome with every reversal of the inductor's magnetic
field constitutes an expenditure of energy that manifests itself as
resistance in the circuit. Some core materials (such as ferrite) are
particularly notorious for their hysteretic effect. Counteracting this
effect is best done by means of proper core material selection and
limits on the peak magnetic field intensity generated with each cycle.
Altogether, the stray resistive
properties of a real inductor (wire resistance, radiation losses, eddy
currents, and hysteresis losses) are expressed under the single term of
"effective resistance:"
It is worthy to note that the skin effect
and radiation losses apply just as well to straight lengths of wire in
an AC circuit as they do a coiled wire. Usually their combined effect is
too small to notice, but at radio frequencies they can be quite large. A
radio transmitter antenna, for example, is designed with the express
purpose of dissipating the greatest amount of energy in the form of
electromagnetic radiation.
Effective resistance in an inductor can
be a serious consideration for the AC circuit designer. To help quantify
the relative amount of effective resistance in an inductor, another
value exists called the Q factor, or "quality factor" which is
calculated as follows:
The symbol "Q" has nothing to do with
electric charge (coulombs), which tends to be confusing. For some
reason, the Powers That Be decided to use the same letter of the
alphabet to denote a totally different quantity.
The higher the value for "Q," the "purer"
the inductor is. Because it's so easy to add additional resistance if
needed, a high-Q inductor is better than a low-Q inductor for design
purposes. An ideal inductor would have a Q of infinity, with zero
effective resistance.
Because inductive reactance (X) varies
with frequency, so will Q. However, since the resistive effects of
inductors (wire skin effect, radiation losses, eddy current, and
hysteresis) also vary with frequency, Q does not vary proportionally
with reactance. In order for a Q value to have precise meaning, it must
be specified at a particular test frequency.
Stray resistance isn't the only inductor
quirk we need to be aware of. Due to the fact that the multiple turns of
wire comprising inductors are separated from each other by an insulating
gap (air, varnish, or some other kind of electrical insulation), we have
the potential for capacitance to develop between turns. AC capacitance
will be explored in the next chapter, but it suffices to say at this
point that it behaves very differently from AC inductance, and therefore
further "taints" the reactive purity of real inductors.
More on the "skin effect"
As previously mentioned, the skin effect
is where alternating current tends to avoid travel through the center of
a solid conductor, limiting itself to conduction near the surface. This
effectively limits the cross-sectional conductor area available to carry
alternating electron flow, increasing the resistance of that conductor
above what it would normally be for direct current:
The electrical resistance of the
conductor with all its cross-sectional area in use is known as the "DC
resistance," the "AC resistance" of the same conductor referring to a
higher figure resulting from the skin effect. As you can see, at high
frequencies the AC current avoids travel through most of the conductor's
cross-sectional area. For the purpose of conducting current, the wire
might as well be hollow!
In some radio applications (antennas,
most notably) this effect is exploited. Since radio-frequency ("RF") AC
currents wouldn't travel through the middle of a conductor anyway, why
not just use hollow metal rods instead of solid metal wires and save
both weight and cost? Most antenna structures and RF power conductors
are made of hollow metal tubes for this reason.
In the following photograph you can see
some large inductors used in a 50 kW radio transmitting circuit. The
inductors are hollow copper tubes coated with silver, for excellent
conductivity at the "skin" of the tube:
The degree to which frequency affects the
effective resistance of a solid wire conductor is impacted by the gauge
of that wire. As a rule, large-gauge wires exhibit a more pronounced
skin effect (change in resistance from DC) than small-gauge wires at any
given frequency. The equation for approximating skin effect at high
frequencies (greater than 1 MHz) is as follows:
The following table gives approximate
values of "k" factor for various round wire sizes:
Gage size k factor
======================
4/0 ---------- 124.5
2/0 ---------- 99.0
1/0 ---------- 88.0
2 ------------ 69.8
4 ------------ 55.5
6 ------------ 47.9
8 ------------ 34.8
10 ----------- 27.6
14 ----------- 17.6
18 ----------- 10.9
22 ----------- 6.86
For example, a length of number 10-gauge
wire with a DC end-to-end resistance of 25 Ω would have an AC
(effective) resistance of 2.182 kΩ at a frequency of 10 MHz:
Please remember that this figure is
not impedance, and it does not consider any reactive effects,
inductive or capacitive. This is simply an estimated figure of pure
resistance for the conductor (that opposition to the AC flow of
electrons which does dissipate power in the form of heat),
corrected for skin effect. Reactance, and the combined effects of
reactance and resistance (impedance), are entirely different matters.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first.
Jim Palmer
(June 2001): Identified and offered correction for typographical error
in complex number calculation.
Jason Starck
(June 2000): HTML document formatting, which led to a much
better-looking second edition.
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