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.
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REVIEW:
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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.
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