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 �F 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
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.
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