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.
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