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.
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