RLC RESONATORS
Resonators trap energy:
R
C
V
Also:
+
I
L
G
L
- C
Parallel RLC resonator
Series RLC resonator
terminated
TEM lines,
waveguides
Circuit equations, series resonator:
L di + Ri + 1 ∫ i dt = 0 ⇒ jωL I + R I + I = 0
dt
C
jωC
R 1 ⎤
⎡
2
(jω)
+
(jω)
+
I = 0 ⇒ (jω – s1)(jω – s2) = 0*
⎢⎣
L LC ⎥⎦
s1,2 = - R ± j 1 - ( R )2
2L
LC
2L
ω’
i(t) = R e { Ioe
jω't
}e
-
Note: s2 = s1*
ω' =
1 for R → 0
LC
R
t
2L
2
* Let jω = si; recall: as + bs + c = 0
(-b± b2 -4ac )
si =
2a
L14-1
RLC RESONATOR WAVEFORMS
i(t)
Series resonator current i(t):
i(t) = R e { I oe jω't } e
-
R
t
2L
= Iocos(ω't + φ)e
Energy w(t):
1
wm (t) = Li2 ∝ cos2 (ω't) e
2
1
-
-
R
t
2L
Io e-(R/2L)t = Io e-αt
R
t
L
0
R
t
L
w e (t) = Cv 2 ∝ sin2 (ω't) e
2
Stored energy
wTo
wT = wTo e-(R/L)t
wemax = wmmax
= wm(t) + we(t)
L
⇒ vmax = imax
C
we(t)
wTo/e
0
~Q radians,
Q/ω’ seconds
wm(t)
t
t
Let φ = 0
e-2αt ≅ e-ω’t/Q
= e-(R/L)t
Q ≅ ω’/2α = Lω’/R
≅ L/C / R
(series resonance)
Q ≅ ω’/2α = Cω’/G
≅ C/L / G
(parallel)
L14-2
COUPLING TO RLC RESONATORS
Thevenin and Norton Equivalent Sources:
I(ω)
RTh
VTh
C
Pd(ω)
V
RL
+
L
INo
Thevenin equivalent source
GNo
G
- C
Norton equivalent
L
Δω
1
1/2
ω
ωo
Resonance
Power dissipated Pd(f) in R = RL+RTh:
| VTh |2
| VTh |2
1
1
Pd (ω) =
R=
R
Dominant factor near ωo
2 | Z |2
2 | R + jωL + 1 |2
j ωC
−2
| VTh |2 Rω2
1
R
1
R
(ω −
=
− j )(ω +
−j )
2
2L
2L
LC
LC
2L
Half-power frequencies: ω ≅ ωο ± R/2L = ωο ± α, where ωο = 1/ LC
so: Δω = 2α = ωo/Q
and
Q = ωo/Δω
L14-3
RESONATOR Q
General derivation of Q (all resonators):
wT ≅ wToe-ω’t/Q
(total stored energy [J])
Pd = -dwT/dt ≅ (ω’/Q)wT (power dissipated [W])
Q ≅ ω’wT/Pd *
(resonator Q [“radians” is dimensionless])
Internal, external, and loaded Q (QI, QE, QL):
QI = ω’wT/PdI
(PdI is power dissipated internally, in R)
QE = ω’wT/PdE (PdE is power dissipated externally, in RTh)
QL = ω’wT/PdT (PdT is the total power dissipated, in R and RTh)
R
RTh
VTh
C
L
+
RTh
VTh
G
- C
PdT = PdI + PdE ⇒ QL-1 = QI-1 + QE-1
QL ≈ ωo/Δω
Perfect Match: QI = QE
*IEEE definition: Q = ωowT/Pd
L
Pd(ω)
1
Δω
ωo
1/2
ω
L14-4
MATCHING TO RESONATORS
Transmission line feed:
Zo
VTh
R
Zo
Γ
C
L
Zo
VTh
Zo
+
Γ
R
L
- C
2
At ωo:
R − Zo
2
|Γ| =
R + Zo
=0
if matched, R = Zo
= 1/9 if R = Zo/2 or 2Zo
Pd(ω)
1
Behavior away from resonance:
Series resonance:
Open circuit
Parallel resonance:
Short circuit
Δω
1/2
ωo
0
= 1
L/C
ω
L14-5
EXAMPLE #1 – CELL PHONE FILTER
Bandpass filter specifications:
Looks like a short circuit far from ωo
At ωo: reflect 1/9 of the incident power and let Γ < 0
ωo = 5 × 109 and Δω = 5 × 107
Zo = 100-ohm line
Zo
Γ
Zo
VTh
Filter solution:
+
R
L
- C
Parallel resonators look like short circuits far from ωo
1+ Γ
Z
=
Z
2
o
|Γ| = 1/9 and Γ < 0 ⇒ Γ = -1/3 at ωo.
1 − Γ ⇒ R = 50Ω
LC = 1/ωo = (5x109)-1
QL = R’/ L/C (parallel) ⇒ L/C = R’/QL = 33/100 = 0.33 (R’ = R // Zo)
L=
LC L/C = (5x109)-1 x 0.33 = 6.67 x 10-12 [Hy]
C = LC / L/C = (5x109)-1/0.33 = 6 x 10-10 [F]
Small, hard to
build, use TEM?
L14-6
EXAMPLE #2 – BAND-STOP FILTER
Filter specifications:
Far from ωo the load is matched (signal goes to amplifier R)
At ωo reflect all incident power; let Γ = -1 (short circuit)
ωo = 5 × 106
Δω = 5 × 104 (rejected band, notch filter) ⇒ Q = 100
Zo = 100-ohm line
Zo
L
Zo
R
VTh
C
Filter solution:
Lossless series resonators look like short circuits at ωo
R = Zo = 100Ω
⇒
|Γ|2 = 0 at ω far from ωo
LC = 1/ωo = (5x106)-1
QL = L/C /RL (series)
⇒
L/C = RLQL = 50x100 = 5000
L = LC L/C = (5x106)-1 x 5000 = 10-3 [Hy]
C = LC / L/C = (5x106)-1/5000 = 4 x 10-11 [F]
L14-7
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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