Πανεπιστήμιο Πατρών
Τμήμα Φσσικής
Εργαστήριο Ηλεκτρονικής
T.K. 26504, Ρίο Πατρών
Telecom’s electronics
PASSIVE RLC NETWORKS
S. Vlassis
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Electronics Laboratory
Contents
1. Quality Factor
2. Bandwidth and Quality Factor
3. RC and RL networks
4. Maximum power transfer theorem
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Passive RLC networks
•
•
•
•
•
Match or modify impedances
Cancel transistor parasitics
Provide high gain at high frequencies
Filter-out unwanted signals
more
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Parallel RLC network (or tank)
Admittance = Y
Impedance = Z
YR=G=1/R ,conductance
YC=1/Zc=jωC
YL=1/ZL=1/jωL
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Resonator
At resonant frequency ωο
Υ=1/R
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Quality Factor, Q
Dimensionless
Very fundamental
What stores or dissipates energy
Q applies both to resonant or non resonant networks
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Q of parallel RLC tank
At resonant ωο
L
 ZO
C
Zo characteristic impedance of the network at ωo
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Electronics Laboratory
Q of parallel RLC tank
At resonant frequency
Z C  Z L  o L 

Q
1
LC
R
R
 Q  RoC 
ZC
L/C
Z C  Z L , o
Q
1
L
L
 ZO
C
LC

R
1

Z L o L
R
Q
Z L ,C
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Branch current at resonance
of parallel RLC tank
At resonant frequency
Z L  ZC
I in R  V
I L  IC 
V
Z

I in R
o L
 Q I in
The branch currents are Q times as larger as the peak current
e.g. for Q=1000 Iin=1uA  IR=1A and |IL|=|IC|=1mA
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Electronics Laboratory
BANDWIDTH & Q of parallel RLC tank
ω=ωο+Δω

1 
jC

2
Y  G  j C 
2o    
G
C 


Y  G  j 2C

  
Y
-3dB
R
For ω=ωο-Δω
Y  G  j 2C
2C
w
wo
Δw=1/2RC
BW=1/RC
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Electronics Laboratory
BANDWIDTH & Q of parallel RLC tank
The bandwidth for a RLC network at resonant is
1
BW 
RC
Useful formulas
BW
o
1
LC



RCo
RC
L/C 1

R
Q
For large Q  narrow bandwidth
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Electronics Laboratory
SERIES RLC tanks
R
Iin
C
L
1 2
Etotal  LI pk
2
1 2
Eavg  I pk R
2
At resonant
o 
1
LC
Z L  ZC
o L
1
QS 

R
o RC
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SERIES RLC tanks
Useful formulas
Z L  Z C  QS R
•Large R  small Q
•Large R  Only R (LC effects negligible), only power dissipation
QS 
Z L ,C
R
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Electronics Laboratory
Branch current at resonance
of series RLC tank
R
C
Iin
At resonant
o 
V
1
LC
L
I in R  V
VL  VC  I in Z L ,C
1
 I in o L  I in
oC
VL  VC  Qs Vin
At resonance |VL|=|VC| are Q times larger than V
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Other resonant RLC tanks
Near to resonant we need
Ls
C
C
Rp
Rs
If parallel and series tanks are equivalent then
Parallel Qp and series Qs must be equal
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Other resonant RLC tanks
Near to resonant we need
Ls
C
C
Rp
Rs
using
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Other resonant RLC tanks
Near to resonant we need
Cs
L
L
Rp
Cp
Rs
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Other resonant RLC tanks
• Universal equations
For large Q 
R p  Q 2 Rs
X p  Xs
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Other resonant RLC tanks
Parallel tank
Series tank
1. The single pair of universal formulas is used to novert any “impure” RLC
tank into a purely parallel or series one that is straightforward to analyze.
2. The tank’s equivalences hold only over a narrow range of frequencies
centered about wo
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RL conversion
Ls
Rp
Lp
Rs
Qs 
o Ls
Rs
Qp 
Rp
o L p
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RC conversion
Cs
Rp
Cp
Rs
1
Qs 
oCs Rs
Qp  R poC p
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Maximum power transfer theorem
Power to load
•For a given fixed source impedance Zs what load impedance ZL maximizes
the power delivered to the load
•The power delivered to the load impedance is entirely due to RL since
reactive elements do not dissipate power.
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Maximum power transfer theorem
Maximum power : XL=-XS and RL=RS simultaneously are valid
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Electronics Laboratory