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.
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