More than you ever wanted to know - SOURCE
REACTANCE AND IMPEDANCE -- INDUCTIVE
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
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
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
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)
| 7.5398 |
| 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 =
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
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.
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
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
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
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
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)
6.000E+01 7.985E+00 6.020E+00
6.000E+01 -3.702E+01 5.298E+01
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
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.
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
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
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:
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.
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
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
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
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.
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