An electric pendulum
Capacitors store energy in the form of an
electric field, and electrically manifest that stored energy as a
potential: static voltage. Inductors store energy in the form of
a magnetic field, and electrically manifest that stored energy as a
kinetic motion of electrons: current. Capacitors and inductors
are flip-sides of the same reactive coin, storing and releasing energy
in complementary modes. When these two types of reactive components are
directly connected together, their complementary tendencies to store
energy will produce an unusual result.
If either the capacitor or inductor
starts out in a charged state, the two components will exchange energy
between them, back and forth, creating their own AC voltage and current
cycles. If we assume that both components are subjected to a sudden
application of voltage (say, from a momentarily connected battery), the
capacitor will very quickly charge and the inductor will oppose change
in current, leaving the capacitor in the charged state and the inductor
in the discharged state:
The capacitor will begin to discharge,
its voltage decreasing. Meanwhile, the inductor will begin to build up a
"charge" in the form of a magnetic field as current increases in the
circuit:
The inductor, still charging, will keep
electrons flowing in the circuit until the capacitor has been completely
discharged, leaving zero voltage across it:
The inductor will maintain current flow
even with no voltage applied. In fact, it will generate a voltage (like
a battery) in order to keep current in the same direction. The
capacitor, being the recipient of this current, will begin to accumulate
a charge in the opposite polarity as before:
When the inductor is finally depleted of
its energy reserve and the electrons come to a halt, the capacitor will
have reached full (voltage) charge in the opposite polarity as when it
started:
Now we're at a condition very similar to
where we started: the capacitor at full charge and zero current in the
circuit. The capacitor, as before, will begin to discharge through the
inductor, causing an increase in current (in the opposite direction as
before) and a decrease in voltage as it depletes its own energy reserve:
Eventually the capacitor will discharge
to zero volts, leaving the inductor fully charged with full current
through it:
The inductor, desiring to maintain
current in the same direction, will act like a source again, generating
a voltage like a battery to continue the flow. In doing so, the
capacitor will begin to charge up and the current will decrease in
magnitude:
Eventually the capacitor will become
fully charged again as the inductor expends all of its energy reserves
trying to maintain current. The voltage will once again be at its
positive peak and the current at zero. This completes one full cycle of
the energy exchange between the capacitor and inductor:
This oscillation will continue with
steadily decreasing amplitude due to power losses from stray resistances
in the circuit, until the process stops altogether. Overall, this
behavior is akin to that of a pendulum: as the pendulum mass swings back
and forth, there is a transformation of energy taking place from kinetic
(motion) to potential (height), in a similar fashion to the way energy
is transferred in the capacitor/inductor circuit back and forth in the
alternating forms of current (kinetic motion of electrons) and voltage
(potential electric energy).
At the peak height of each swing of a
pendulum, the mass briefly stops and switches directions. It is at this
point that potential energy (height) is at a maximum and kinetic energy
(motion) is at zero. As the mass swings back the other way, it passes
quickly through a point where the string is pointed straight down. At
this point, potential energy (height) is at zero and kinetic energy
(motion) is at maximum. Like the circuit, a pendulum's back-and-forth
oscillation will continue with a steadily dampened amplitude, the result
of air friction (resistance) dissipating energy. Also like the circuit,
the pendulum's position and velocity measurements trace two sine waves
(90 degrees out of phase) over time:
In physics, this kind of natural
sine-wave oscillation for a mechanical system is called Simple
Harmonic Motion (often abbreviated as "SHM"). The same underlying
principles govern both the oscillation of a capacitor/inductor circuit
and the action of a pendulum, hence the similarity in effect. It is an
interesting property of any pendulum that its periodic time is governed
by the length of the string holding the mass, and not the weight of the
mass itself. That is why a pendulum will keep swinging at the same
frequency as the oscillations decrease in amplitude. The oscillation
rate is independent of the amount of energy stored in it.
The same is true for the
capacitor/inductor circuit. The rate of oscillation is strictly
dependent on the sizes of the capacitor and inductor, not on the amount
of voltage (or current) at each respective peak in the waves. The
ability for such a circuit to store energy in the form of oscillating
voltage and current has earned it the name tank circuit. Its
property of maintaining a single, natural frequency regardless of how
much or little energy is actually being stored in it gives it special
significance in electric circuit design.
However, this tendency to oscillate, or
resonate, at a particular frequency is not limited to circuits
exclusively designed for that purpose. In fact, nearly any AC circuit
with a combination of capacitance and inductance (commonly called an "LC
circuit") will tend to manifest unusual effects when the AC power source
frequency approaches that natural frequency. This is true regardless of
the circuit's intended purpose.
If the power supply frequency for a
circuit exactly matches the natural frequency of the circuit's LC
combination, the circuit is said to be in a state of resonance.
The unusual effects will reach maximum in this condition of resonance.
For this reason, we need to be able to predict what the resonant
frequency will be for various combinations of L and C, and be aware of
what the effects of resonance are.
- REVIEW:
- A capacitor and inductor directly
connected together form something called a tank circuit, which
oscillates (or resonates) at one particular frequency. At that
frequency, energy is alternately shuffled between the capacitor and
the inductor in the form of alternating voltage and current 90 degrees
out of phase with each other.
- When the power supply frequency for an
AC circuit exactly matches that circuit's natural oscillation
frequency as set by the L and C components, a condition of
resonance will have been reached.
Simple parallel (tank circuit) resonance
A condition of resonance will be
experienced in a tank circuit when the reactances of the capacitor and
inductor are equal to each other. Because inductive reactance increases
with increasing frequency and capacitive reactance decreases with
increasing frequency, there will only be one frequency where these two
reactances will be equal.
In the above circuit, we have a 10 µF
capacitor and a 100 mH inductor. Since we know the equations for
determining the reactance of each at a given frequency, and we're
looking for that point where the two reactances are equal to each other,
we can set the two reactance formulae equal to each other and solve for
frequency algebraically:
So there we have it: a formula to tell us
the resonant frequency of a tank circuit, given the values of inductance
(L) in Henrys and capacitance (C) in Farads. Plugging in the values of L
and C in our example circuit, we arrive at a resonant frequency of
159.155 Hz.
What happens at resonance is quite
interesting. With capacitive and inductive reactances equal to each
other, the total impedance increases to infinity, meaning that the tank
circuit draws no current from the AC power source! We can calculate the
individual impedances of the 10 µF capacitor and the 100 mH inductor and
work through the parallel impedance formula to demonstrate this
mathematically:
As you might have guessed, I chose these
component values to give resonance impedances that were easy to work
with (100 Ω even). Now, we use the parallel impedance formula to see
what happens to total Z:
We can't divide any number by zero and
arrive at a meaningful result, but we can say that the result approaches
a value of infinity as the two parallel impedances get closer to
each other. What this means in practical terms is that, the total
impedance of a tank circuit is infinite (behaving as an open circuit)
at resonance. We can plot the consequences of this over a wide power
supply frequency range with a short SPICE simulation:
tank circuit frequency sweep
v1 1 0 ac 1 sin
c1 1 0 10u
* rbogus is necessary to eliminate a direct loop
* between v1 and l1, which SPICE can't handle
rbogus 1 2 1e-12
l1 2 0 100m
.ac lin 20 100 200
.plot ac i(v1)
.end
freq i(v1) 3.162E-04 1.000E-03 3.162E-03 1.0E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 9.632E-03 . . . . *
1.053E+02 8.506E-03 . . . . * .
1.105E+02 7.455E-03 . . . . * .
1.158E+02 6.470E-03 . . . . * .
1.211E+02 5.542E-03 . . . . * .
1.263E+02 4.663E-03 . . . . * .
1.316E+02 3.828E-03 . . . .* .
1.368E+02 3.033E-03 . . . *. .
1.421E+02 2.271E-03 . . . * . .
1.474E+02 1.540E-03 . . . * . .
1.526E+02 8.373E-04 . . * . . .
1.579E+02 1.590E-04 . * . . . .
1.632E+02 4.969E-04 . . * . . .
1.684E+02 1.132E-03 . . . * . .
1.737E+02 1.749E-03 . . . * . .
1.789E+02 2.350E-03 . . . * . .
1.842E+02 2.934E-03 . . . *. .
1.895E+02 3.505E-03 . . . .* .
1.947E+02 4.063E-03 . . . . * .
2.000E+02 4.609E-03 . . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The 1 pico-ohm (1 pΩ) resistor is placed
in this SPICE analysis to overcome a limitation of SPICE: namely, that
it cannot analyze a circuit containing a direct inductor-voltage source
loop. A very low resistance value was chosen so as to have minimal
effect on circuit behavior.
This SPICE simulation plots circuit
current over a frequency range of 100 to 200 Hz in twenty even steps
(100 and 200 Hz inclusive). Current magnitude on the graph increases
from left to right, while frequency increases from top to bottom. The
current in this circuit takes a sharp dip around the analysis point of
157.9 Hz, which is the closest analysis point to our predicted resonance
frequency of 159.155 Hz. It is at this point that total current from the
power source falls to zero.
Incidentally, the graph output produced
by this SPICE computer analysis is more generally known as a Bode
plot. Such graphs plot amplitude or phase shift on one axis and
frequency on the other. The steepness of a Bode plot curve characterizes
a circuit's "frequency response," or how sensitive it is to changes in
frequency.
- REVIEW:
- Resonance occurs when capacitive and
inductive reactances are equal to each other.
- For a tank circuit with no resistance
(R), resonant frequency can be calculated with the following formula:
-
- The total impedance of a parallel LC
circuit approaches infinity as the power supply frequency approaches
resonance.
- A Bode plot is a graph plotting
waveform amplitude or phase on one axis and frequency on the other.
Simple series resonance
A similar effect happens in series
inductive/capacitive circuits. When a state of resonance is reached
(capacitive and inductive reactances equal), the two impedances cancel
each other out and the total impedance drops to zero!
With the total series impedance equal to
0 Ω at the resonant frequency of 159.155 Hz, the result is a short
circuit across the AC power source at resonance. In the circuit
drawn above, this would not be good. I'll add a small resistor in series
along with the capacitor and the inductor to keep the maximum circuit
current somewhat limited, and perform another SPICE analysis over the
same range of frequencies:
series lc circuit
v1 1 0 ac 1 sin
r1 1 2 1
c1 2 3 10u
l1 3 0 100m
.ac lin 20 100 200
.plot ac i(v1)
.end
freq i(v1) 3.162E-02 1.000E-01 3.162E-01 1.0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.038E-02 * . . . .
1.053E+02 1.176E-02 . * . . . .
1.105E+02 1.341E-02 . * . . . .
1.158E+02 1.545E-02 . * . . . .
1.211E+02 1.804E-02 . * . . . .
1.263E+02 2.144E-02 . * . . . .
1.316E+02 2.611E-02 . * . . . .
1.368E+02 3.296E-02 . .* . . .
1.421E+02 4.399E-02 . . * . . .
1.474E+02 6.478E-02 . . * . . .
1.526E+02 1.186E-01 . . . * . .
1.579E+02 5.324E-01 . . . . * .
1.632E+02 1.973E-01 . . . * . .
1.684E+02 8.797E-02 . . * . . .
1.737E+02 5.707E-02 . . * . . .
1.789E+02 4.252E-02 . . * . . .
1.842E+02 3.406E-02 . .* . . .
1.895E+02 2.852E-02 . *. . . .
1.947E+02 2.461E-02 . * . . . .
2.000E+02 2.169E-02 . * . . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
As before, circuit current amplitude
increases from left to right, while frequency increases from top to
bottom. The peak is still seen to be at the plotted frequency point of
157.9 Hz, the closest analyzed point to our predicted resonance point of
159.155 Hz. This would suggest that our resonant frequency formula holds
as true for simple series LC circuits as it does for simple parallel LC
circuits, which is the case:
A word of caution is in order with series
LC resonant circuits: because of the high currents which may be present
in a series LC circuit at resonance, it is possible to produce
dangerously high voltage drops across the capacitor and the inductor, as
each component possesses significant impedance. We can edit the SPICE
netlist in the above example to include a plot of voltage across the
capacitor and inductor to demonstrate what happens:
series lc circuit
v1 1 0 ac 1 sin
r1 1 2 1
c1 2 3 10u
l1 3 0 100m
.ac lin 20 100 200
.plot ac i(v1) v(2,3) v(3)
.end
legend:
*: i(v1)
+: v(2,3)
=: v(3)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
(*)----------- 1.000E-02 3.162E-02 1.000E-01 0.3162 1
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
(+)----------- 1.000E+00 3.162E+00 1.000E+01 31.62 100
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
(=)----------- 1.000E-01 1.000E+00 1.000E+01 100 1000
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
freq i(v1)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.038E-02 * + = . . . .
1.053E+02 1.176E-02 . * + =. . . .
1.105E+02 1.341E-02 . * + = . . .
1.158E+02 1.545E-02 . * + .= . . .
1.211E+02 1.804E-02 . * + . = . . .
1.263E+02 2.144E-02 . * +. = . . .
1.316E+02 2.611E-02 . *+ = . . .
1.368E+02 3.296E-02 . .*+ = . . .
1.421E+02 4.399E-02 . . *+ = . . .
1.474E+02 6.478E-02 . . *+= . .
1.526E+02 1.186E-01 . . .=*+ . .
1.579E+02 5.324E-01 . . . = . x .
1.632E+02 1.973E-01 . . . = x . .
1.684E+02 8.797E-02 . . x = . .
1.737E+02 5.707E-02 . . +* = . . .
1.789E+02 4.252E-02 . . + * = . . .
1.842E+02 3.406E-02 . +.* = . . .
1.895E+02 2.852E-02 . + *. = . . .
1.947E+02 2.461E-02 . + * . = . . .
2.000E+02 2.169E-02 . + * . = . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
According to SPICE, voltage across the
capacitor and inductor (plotted with "+" and "=" symbols, respectively)
reach a peak somewhere between 100 and 1000 volts (marked by the "x"
where the graphs overlap)! This is quite impressive for a power supply
that only generates 1 volt. Needless to say, caution is in order when
experimenting with circuits such as this.
- REVIEW:
- The total impedance of a series LC
circuit approaches zero as the power supply frequency approaches
resonance.
- The same formula for determining
resonant frequency in a simple tank circuit applies to simple series
circuits as well.
- Extremely high voltages can be formed
across the individual components of series LC circuits at resonance,
due to high current flows and substantial individual component
impedances.
Applications of resonance
So far, the phenomenon of resonance
appears to be a useless curiosity, or at most a nuisance to be avoided
(especially if series resonance makes for a short-circuit across our AC
voltage source!). However, this is not the case. Resonance is a very
valuable property of reactive AC circuits, employed in a variety of
applications.
One use for resonance is to establish a
condition of stable frequency in circuits designed to produce AC
signals. Usually, a parallel (tank) circuit is used for this purpose,
with the capacitor and inductor directly connected together, exchanging
energy between each other. Just as a pendulum can be used to stabilize
the frequency of a clock mechanism's oscillations, so can a tank circuit
be used to stabilize the electrical frequency of an AC oscillator
circuit. As was noted before, the frequency set by the tank circuit is
solely dependent upon the values of L and C, and not on the magnitudes
of voltage or current present in the oscillations:
Another use for resonance is in
applications where the effects of greatly increased or decreased
impedance at a particular frequency is desired. A resonant circuit can
be used to "block" (present high impedance toward) a frequency or range
of frequencies, thus acting as a sort of frequency "filter" to strain
certain frequencies out of a mix of others. In fact, these particular
circuits are called filters, and their design constitutes a
discipline of study all by itself:
In essence, this is how analog radio
receiver tuner circuits work to filter, or select, one station frequency
out of the mix of different radio station frequency signals intercepted
by the antenna.
- REVIEW:
- Resonance can be employed to maintain
AC circuit oscillations at a constant frequency, just as a pendulum
can be used to maintain constant oscillation speed in a timekeeping
mechanism.
- Resonance can be exploited for its
impedance properties: either dramatically increasing or decreasing
impedance for certain frequencies. Circuits designed to screen certain
frequencies out of a mix of different frequencies are called
filters.
Resonance in series-parallel circuits
In simple reactive circuits with little
or no resistance, the effects of radically altered impedance will
manifest at the resonance frequency predicted by the equation given
earlier. In a parallel (tank) LC circuit, this means infinite impedance
at resonance. In a series LC circuit, it means zero impedance at
resonance:
However, as soon as significant levels of
resistance are introduced into most LC circuits, this simple calculation
for resonance becomes invalid. We'll take a look at several LC circuits
with added resistance, using the same values for capacitance and
inductance as before: 10 µF and 100 mH, respectively. According to our
simple equation, the resonant frequency should be 159.155 Hz. Watch,
though, where current reaches maximum or minimum in the following SPICE
analyses:
resonant circuit
v1 1 0 ac 1 sin
c1 1 0 10u
r1 1 2 100
l1 2 0 100m
.ac lin 20 100 200
.plot ac i(v1)
.end
freq i(v1) 7.079E-03 7.943E-03 8.913E-03
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 7.387E-03 . . * . .
1.053E+02 7.242E-03 . . * . .
1.105E+02 7.115E-03 . .* . .
1.158E+02 7.007E-03 . *. . .
1.211E+02 6.921E-03 . * . . .
1.263E+02 6.859E-03 . * . . .
1.316E+02 6.823E-03 . * . . .
1.368E+02 6.813E-03 . * . . .
1.421E+02 6.830E-03 . * . . .
1.474E+02 6.874E-03 . * . . .
1.526E+02 6.946E-03 . * . . .
1.579E+02 7.044E-03 . *. . .
1.632E+02 7.167E-03 . .* . .
1.684E+02 7.315E-03 . . * . .
1.737E+02 7.485E-03 . . * . .
1.789E+02 7.676E-03 . . * . .
1.842E+02 7.886E-03 . . *. .
1.895E+02 8.114E-03 . . . * .
1.947E+02 8.358E-03 . . . * .
2.000E+02 8.616E-03 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Minimum current at 136.8 Hz instead of 159.2 Hz!
Here, an extra resistor (Rbogus)
is necessary to prevent SPICE from encountering trouble in analysis.
SPICE can't handle an inductor connected directly in parallel with any
voltage source or any other inductor, so the addition of a series
resistor is necessary to "break up" the voltage source/inductor loop
that would otherwise be formed. This resistor is chosen to be a very
low value for minimum impact on the circuit's behavior.
resonant circuit
v1 1 0 ac 1 sin
r1 1 2 100
c1 2 0 10u
rbogus 1 3 1e-12
l1 3 0 100m
.ac lin 20 100 400
.plot ac i(v1)
.end
freq i(v1) 7.943E-03 1.000E-02 1.259E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.176E-02 . . . * .
1.158E+02 9.635E-03 . . * . .
1.316E+02 8.257E-03 . . * . .
1.474E+02 7.430E-03 . * . . .
1.632E+02 6.998E-03 . * . . .
1.789E+02 6.835E-03 . * . . .
1.947E+02 6.839E-03 . * . . .
2.105E+02 6.941E-03 . * . . .
2.263E+02 7.093E-03 . * . . .
2.421E+02 7.268E-03 . * . . .
2.579E+02 7.449E-03 . * . . .
2.737E+02 7.626E-03 . * . . .
2.895E+02 7.794E-03 . *. . .
3.053E+02 7.951E-03 . * . .
3.211E+02 8.096E-03 . .* . .
3.368E+02 8.230E-03 . . * . .
3.526E+02 8.352E-03 . . * . .
3.684E+02 8.464E-03 . . * . .
3.842E+02 8.567E-03 . . * . .
4.000E+02 8.660E-03 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Minimum current at roughly 180 Hz instead of 159.2 Hz!
Switching our attention to series LC
circuits, we experiment with placing significant resistances in parallel
with either L or C. In the following series circuit examples, a 1 Ω
resistor (R1) is placed in series with the inductor and
capacitor to limit total current at resonance. The "extra" resistance
inserted to influence resonant frequency effects is the 100 Ω resistor,
R2:
resonant circuit
v1 1 0 ac 1 sin
r1 1 2 1
c1 2 3 10u
l1 3 0 100m
r2 3 0 100
.ac lin 20 100 400
.plot ac i(v1)
.end
freq i(v1) 1.000E-02 1.259E-02 1.585E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 8.488E-03 . * . . .
1.158E+02 1.034E-02 . . * . .
1.316E+02 1.204E-02 . . * . .
1.474E+02 1.336E-02 . . . * .
1.632E+02 1.415E-02 . . . * .
1.789E+02 1.447E-02 . . . * .
1.947E+02 1.445E-02 . . . * .
2.105E+02 1.424E-02 . . . * .
2.263E+02 1.393E-02 . . . * .
2.421E+02 1.360E-02 . . . * .
2.579E+02 1.327E-02 . . . * .
2.737E+02 1.296E-02 . . . * .
2.895E+02 1.269E-02 . . * .
3.053E+02 1.244E-02 . . *. .
3.211E+02 1.222E-02 . . * . .
3.368E+02 1.202E-02 . . * . .
3.526E+02 1.185E-02 . . * . .
3.684E+02 1.169E-02 . . * . .
3.842E+02 1.155E-02 . . * . .
4.000E+02 1.143E-02 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Maximum current at roughly 178.9 Hz instead of 159.2 Hz!
And finally, a series LC circuit with the
significant resistance in parallel with the capacitor:
resonant circuit
v1 1 0 ac 1 sin
r1 1 2 1
c1 2 3 10u
r2 2 3 100
l1 3 0 100m
.ac lin 20 100 200
.plot ac i(v1)
.end
freq i(v1)
freq i(v1) 1.259E-02 1.413E-02 1.585E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.336E-02 . . * . .
1.053E+02 1.363E-02 . . * . .
1.105E+02 1.387E-02 . . * . .
1.158E+02 1.408E-02 . . * .
1.211E+02 1.426E-02 . . .* .
1.263E+02 1.439E-02 . . . * .
1.316E+02 1.447E-02 . . . * .
1.368E+02 1.450E-02 . . . * .
1.421E+02 1.447E-02 . . . * .
1.474E+02 1.438E-02 . . . * .
1.526E+02 1.424E-02 . . .* .
1.579E+02 1.405E-02 . . *. .
1.632E+02 1.382E-02 . . * . .
1.684E+02 1.355E-02 . . * . .
1.737E+02 1.325E-02 . . * . .
1.789E+02 1.293E-02 . . * . .
1.842E+02 1.259E-02 . * . .
1.895E+02 1.225E-02 . * . . .
1.947E+02 1.190E-02 . * . . .
2.000E+02 1.155E-02 . * . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Maximum current at 136.8 Hz instead of 159.2 Hz!
The tendency for added resistance to skew
the point at which impedance reaches a maximum or minimum in an LC
circuit is called antiresonance. The astute observer will notice
a pattern between the four SPICE examples given above, in terms of how
resistance affects the resonant peak of a circuit:
- Parallel ("tank") LC circuit:
- R in series with L: resonant frequency
shifted down
- R in series with C: resonant frequency
shifted up
- Series LC circuit:
- R in parallel with L: resonant
frequency shifted up
- R in parallel with C: resonant
frequency shifted down
Again, this illustrates the complementary
nature of capacitors and inductors: how resistance in series with one
creates an antiresonance effect equivalent to resistance in parallel
with the other. If you look even closer to the four SPICE examples
given, you'll see that the frequencies are shifted by the same amount,
and that the shape of the complementary graphs are mirror-images of each
other!
Antiresonance is an effect that resonant
circuit designers must be aware of. The equations for determining
antiresonance "shift" are complex, and will not be covered in this brief
lesson. It should suffice the beginning student of electronics to
understand that the effect exists, and what its general tendencies are.
Added resistance in an LC circuit is no
academic matter. While it is possible to manufacture capacitors with
negligible unwanted resistances, inductors are typically plagued with
substantial amounts of resistance due to the long lengths of wire used
in their construction. What is more, the resistance of wire tends to
increase as frequency goes up, due to a strange phenomenon known as the
skin effect where AC current tends to be excluded from travel
through the very center of a wire, thereby reducing the wire's effective
cross-sectional area. Thus, inductors not only have resistance, but
changing, frequency-dependent resistance at that.
As if the resistance of an inductor's
wire weren't enough to cause problems, we also have to contend with the
"core losses" of iron-core inductors, which manifest themselves as added
resistance in the circuit. Since iron is a conductor of electricity as
well as a conductor of magnetic flux, changing flux produced by
alternating current through the coil will tend to induce electric
currents in the core itself (eddy currents). This effect can be
thought of as though the iron core of the transformer were a sort of
secondary transformer coil powering a resistive load: the
less-than-perfect conductivity of the iron metal. This effects can be
minimized with laminated cores, good core design and high-grade
materials, but never completely eliminated.
One notable exception to the rule of
circuit resistance causing a resonant frequency shift is the case of
series resistor-inductor-capacitor ("RLC") circuits. So long as all
components are connected in series with each other, the resonant
frequency of the circuit will be unaffected by the resistance:
series rlc circuit
v1 1 0 ac 1 sin
r1 1 2 100
c1 2 3 10u
l1 3 0 100m
.ac lin 20 100 200
.plot ac i(v1)
.end
freq i(v1) 7.943E-03 8.913E-03 1.000E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 7.202E-03 * . . .
1.053E+02 7.617E-03 . * . . .
1.105E+02 8.017E-03 . .* . .
1.158E+02 8.396E-03 . . * . .
1.211E+02 8.747E-03 . . * . .
1.263E+02 9.063E-03 . . . * .
1.316E+02 9.339E-03 . . . * .
1.368E+02 9.570E-03 . . . * .
1.421E+02 9.752E-03 . . . * .
1.474E+02 9.883E-03 . . . *.
1.526E+02 9.965E-03 . . . .
1.579E+02 9.999E-03 . . . *
1.632E+02 9.988E-03 . . . *
1.684E+02 9.936E-03 . . . *.
1.737E+02 9.850E-03 . . . * .
1.789E+02 9.735E-03 . . . * .
1.842E+02 9.595E-03 . . . * .
1.895E+02 9.437E-03 . . . * .
1.947E+02 9.265E-03 . . . * .
2.000E+02 9.082E-03 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Maximum current at 159.2 Hz once again!
Note that the peak of the current graph
has not changed from the earlier series LC circuit (the one with the 1 Ω
token resistance in it), even though the resistance is now 100 times
greater. The only thing that has changed is the "sharpness" of the
curve. Obviously, this circuit does not resonate as strongly as one with
less series resistance (it is said to be "less selective"), but at least
it has the same natural frequency!
It is noteworthy that antiresonance has
the effect of dampening the oscillations of free-running LC circuits
such as tank circuits. In the beginning of this chapter we saw how a
capacitor and inductor connected directly together would act something
like a pendulum, exchanging voltage and current peaks just like a
pendulum exchanges kinetic and potential energy. In a perfect tank
circuit (no resistance), this oscillation would continue forever, just
as a frictionless pendulum would continue to swing at its resonant
frequency forever. But frictionless machines are difficult to find in
the real world, and so are lossless tank circuits. Energy lost through
resistance (or inductor core losses or radiated electromagnetic waves or
. . .) in a tank circuit will cause the oscillations to decay in
amplitude until they are no more. If enough energy losses are present in
a tank circuit, it will fail to resonate at all.
Antiresonance's dampening effect is more
than just a curiosity: it can be used quite effectively to eliminate
unwanted oscillations in circuits containing stray inductances
and/or capacitances, as almost all circuits do. Take note of the
following L/R time delay circuit:
The idea of this circuit is simple: to
"charge" the inductor when the switch is closed. The rate of inductor
charging will be set by the ratio L/R, which is the time constant of the
circuit in seconds. However, if you were to build such a circuit, you
might find unexpected oscillations (AC) of voltage across the inductor
when the switch is closed. Why is this? There's no capacitor in the
circuit, so how can we have resonant oscillation with just an inductor,
resistor, and battery?
All inductors contain a certain amount of
stray capacitance due to turn-to-turn and turn-to-core insulation gaps.
Also, the placement of circuit conductors may create stray capacitance.
While clean circuit layout is important in eliminating much of this
stray capacitance, there will always be some that you cannot eliminate.
If this causes resonant problems (unwanted AC oscillations), added
resistance may be a way to combat it. If resistor R is large enough, it
will cause a condition of antiresonance, dissipating enough energy to
prohibit the inductance and stray capacitance from sustaining
oscillations for very long.
Interestingly enough, the principle of
employing resistance to eliminate unwanted resonance is one frequently
used in the design of mechanical systems, where any moving object with
mass is a potential resonator. A very common application of this is the
use of shock absorbers in automobiles. Without shock absorbers, cars
would bounce wildly at their resonant frequency after hitting any bump
in the road. The shock absorber's job is to introduce a strong
antiresonant effect by dissipating energy hydraulically (in the same way
that a resistor dissipates energy electrically).
- REVIEW:
- Added resistance to an LC circuit can
cause a condition known as antiresonance, where the peak
impedance effects happen at frequencies other than that which gives
equal capacitive and inductive reactances.
- Unwanted resistances inherent in
real-world inductors can contribute greatly to conditions of
antiresonance. One source of such resistance is the skin effect,
caused by the exclusion of AC current from the center of conductors.
Another source is that of core losses in iron-core inductors.
- In a simple series LC circuit
containing resistance (an "RLC" circuit), resistance does not
produce antiresonance. Resonance still occurs when capacitive and
inductive reactances are equal.
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.
|
