Overview of Passive Components
– Size
– Reliability
– Tolerance
Usage of Passives for RFICs
• Low Vdd design
• High Z( j!L)
• Generate Re( Zin)
Usage of inductor for RFICs
• Tuned circuit
• High Z
Y = G + jωC + 1
ω →∞ : Y → ∞ as
C dominates (open Bhaskar Banerjee, EERF 6330,
Sp‘2013, UTD 10
Parallel RLC Network
ω0L
RESONANCE
Parallel RLC Network
√
LC
Parallel RLC Network
– At !0, ‘L’ and ‘C’ cancel each other
– However, the currents in the branches ‘L’ and ‘C’ are very
large!
– The currents in ‘L’ and ‘C’ cancel each other
R L CIin Vout
Q-Factor
• Most fundamental definition - expression is
independent of ‘what’ stores/dissipates the energy and ‘how’
-Dimensionless
pkR
2 CR2I 2 pk
Q-Factor of the RLC Network at Resonance
At resonance frequency !0, let the peak current be Ipk. The avg.
power dissipated is only across R, given by,
The total energy stored is the sum of the energies stored in C and
L - which goes back and forth between the two.
2 I 2 pkR
Q-Factor of the RLC Network at Resonance
Q = R
L
C
has dimensions of Resistance and is called the ‘Characteristic
Impedance’ (Z 0) of the network
At resonance, the inductive and capacitive impedances:
|Z L| = |Z C | = ω0L = L
√
Z =
Q-Factor of the RLC Network at Resonance
Note: as R→∞, Q→ (LC network)
Branch Currents at resonance:
Current flowing in the inductive and capacitive branches is
Q times as large as the net current!
Bandwidth and Q
Q-factor of non-ideal L and C
Z = R + j X
Q-factor of non-ideal L and C
!
Series RLC Network
Generalized RLC Network
C LP RP
Generalized RLC Network
Equating Re and Im parts:
RS = ω 2 L 2
P RP
Series/Parallel Transformation
RLC Network as Impedance Transformers
• For maximum power transfer: ZL = ZS*.
• Hence we need to TRANSFORM impedances to maximize POWER
GAIN.
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RLC Network as Impedance Transformers
• Resonate the Network at the required frequency (0).
• C and LS cancel each other.
25
RLC Network as Impedance Transformers
• Resonate the Network at the required frequency (0).
• CS and L cancel each other.
26
RLC Network as Impedance Transformers
• “L-Matching Network”
Generalized L-Matching Network
• One of X1 and X2 is an inductor, the other a
capacitor.
• Transform R1 to R2 and R2 to R1.
• Note: R1 > R2.
Generalized L-Matching Network
• Two degrees of freedom: L and C
• When the Z-transform ratio AND the frequency 0 are fixed
=> Q is fixed!
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BW = 2π × 25× 106 Hz and ω0 = 2π × 109 rad/s
Q = ω0
BW = 40
Q = ω0L
Example
• Frequency: 1 GHz, R1 = 50 ", R2 = 5 ", BW = 25
MHz.
• L-match:
30
-Matching
31
!
-Matching
32
• Additional degree of freedom: Q can be tuned!
C1 C2
-Matching
33
!
• If the required Q of the network is known: R INT can be
calculated.
• Use a graphing calculator or iteration to find R INT.
• For iteration a good starting point is:
• Once RINT is known rest can be easily calculated.
T-Network
• Similar to the # Network, provides another degree of
freedom.
• Useful when the source/termination parasitics are inductive so
that can be absorbed into the network.
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r
RINT
R1
− 1 +
r
RINT
R2
− 1
ω0RINT
L1 =
QleftR1
ω0
L2 =
QleftR2
ω0
T-Network
Tapped Capacitor/Inductor Resonator
36
BW = 2π × 25× 106 Hz and ω0 = 2π × 109 rad/s
Q = ω0
BW = 40
Example
• Frequency: 1 GHz, R1 = 50 ", R2 = 5 ", BW = 25
MHz.
• #-match:
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