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By: Neil E.
Cotter
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Filters
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RLC filters
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LC resonance
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Tool: At
a any particular frequency, ωs, a circuit consisting of
only L's and C's is equivalent to a single L or C.
Lemma: We
may place an appropriate L or C in parallel or series with a circuit consisting
of only L's and C's to create a resonance at any particular frequency, ωs.
Tool: Given
an L in series with a C with resonant frequency ωo, at any particular frequency, ωs, the L and C are
equivalent to:
1) A single
C if ωs < ωo
2) A wire if
ωs = ωo
3) A single
L if ωs > ωo
Tool: Given
an L in parallel with a C with resonant frequency wo, at any particular frequency, ws, the L and C are
equivalent to:
1) A single
L if ωs < ωo
2) An open
circuit if ωs = ωo
3) A single
C if ωs > ωo
Comment: A circuit consisting of
only L's and C's looks like a single L or C at one frequency, ωs, because all the impedances are purely
imaginary. Thus, the impedance of the entire circuit, ztot, is purely imaginary.
If ztot is positive imaginary,
then ztot = jωsL for some L.
If ztot is negative imaginary,
then ztot =
–j/(ωsC) for some C.
Tool: Summary
of LC behavior:
LC
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ω = 0
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ω < ωo
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ω = ωo
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ω > ωo
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ω → ∞
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series
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– – (open)
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–jX
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––– (wire)
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jX
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– – (open)
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parallel
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––– (wire)
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jX
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– – (open)
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–jX
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––– (wire)
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