LRC Circuits and
Resonance
Physics 116A 2009
D. Pellett
Brief Outline
Series LRC Circuit and Resonance
i
Z
(looking into
circuit)
Vout
= iR
1
Z = jωL +
+R
jωC
Vin
Vin
i =
=
Z
R + j[ωL −
Vin
=
1
R + jX
ωC ]
•
•
Current magnitude is maximum when reactance X = 0
•
Circuit above in resonance when ωL −
!
!
1
1
1
f
=
ωR = LC , R
2π
LC
Circuit with at least one capacitor and one inductor is in resonance
when the imaginary part of its impedance (or admittance) equals 0
•
1
ωC
=0
Q and Bandwidth
•
•
Refer to Q definition in text
•
Consider transfer function for series LRC network with
output taken across the resistor
The Q derivation for a series LRC circuit was done in class
on Friday
•
Gives band pass filter
Bandwidth for Series LRC Band Pass Filter
•
Find half-power points ω1, ω2 for band pass filter
•
•
Write H(jω) in terms of ωR and
1
Q=
R
!
L
ωR L
1
=
=
C
R
ωR RC
Half power when |H(jω)| = 1/√2
H(jω) ≡
Vout
R
=
Vin
R + j[ωL −
1
ωC ]
=
Half power when
1 + jQ[ ωωR −
ωR
ω ]
!(denominator)
√
(1, 1) length = 2
ωR
ω ±
ω − ωR 2 = 0
Q
2
Solutions (ω > 0):
1
!(denominator)
(1, −1)
!"
#2
ωR
1
ω1 = −
+ ωR
+1
2Q
2Q
!"
#2
ωR
1
ω2 =
+ ωR
+1
2Q
2Q
ωR
Bandwidth ≡ ω2 − ω1 =
Q
Resonance for Parallel LRC Circuit
•
See text: parallel R, L, C driven by current source i(t)
•
At resonance, Y has imaginary part = 0
!
!
•
•
Again, ωR =
•
Also read about high-Q coils. This can be used to model a
high-Q coil with series resistance Rs in parallel with C as a
parallel RLC circuit with a resistance-free L and effective
parallel resistance L/RsC
1
LC
, fR =
1
2π
1
LC
Q defined as before now gives (for parallel resonance)