Chapter 3: From lumped to distributed elements
Lumped elements
Parasitics limit the performance
Avoid shunt C, series L
Resistors
Ideal resistor value: R value pole and zero cancel out
Capacitors
Above self-resonant frequency the capacitor behaves as an inductor
Q-factor = (Below SFR)
Inductors
Resonators
Quality factor Q = omega0*av. Energy stored/av. Energy dissipated
Q = f0/3dB BW (relatively low due to losses in coil)
High tolerance on resonant frequency
At resonance: currents trough C and L cancel out and equal Q times the resistor current
Piezo-acoustic resonators
Quartz (SiO2) crystal
High Q, stability, small size, low cost
C1<C0, so slightly difference in resonance frequencies: f_series ≲ f_parallel
fP by L1 and (C1 series C0 )
Artificial lines
Repetitive ladder network
Input impedance of an infinite ladder network
does not change when a section is added
Artificial lines
Ladder network of reactive components
Behaves like real transmission lines (limited bandwidth)
Uses: Large broadband delays
Cutoff frequency: above ZIN is purely reactive, no longer a TL: make LC low
Group delay: calculate per section: make LC high
Termination
Use an extra half section!
m-derived half section
Gives even better results
Trade-off bandwidth ó delay
Bandwidth is traded for delay rather than for gain
Solution: traveling wave amplifier (ch. 6)
Transmission lines
Telegrapher equations
equals
Two waves propagating in opposite direction: incident and reflected wave
Terminated lossless transmission line
Gamma = 0: line is perfectly terminated
ð no reflections
Return loss (RL) dB = -20 log|Gamma|
Voltage standing wave ratio (VSWR)
= 1+|Gamma|/1-|Gamma|
Easy way to measure matching
Gamma vs. ZIN
Gamma tells everything about input impedance at distance l from the load
Gamma(l): rotates clockwise around the origin as l increases: lambda/2 => full circle
Two port models
Lossy transmission lines
In the conductor: skin effect arises: High Z0
In the dielectric (G): stray current between transmission lines:
ð Low Z0
Causes: attenuation and dispersion
ð Dispersion free when L/R = C/G
Stubs
Shorted: can replace an inductor
Open: can replace a capacitor
Higher Q, cheaper at high f
only at GHz
Quarter wavelength TL transformer
ó =>
The load impedance is inverted with a scale factor Short=>Open; L=>C; up or down transforming of Z2
Very dispersive
Broadband components
Port properties
Reciprocal lossless three-port cannot de matched at three ports
Lossless reciprocal four-ports with two matched and reciprocally decouples ports are matched and decoupled at the two other ports
Reciprocal lossless circuits with four matched ports always have two reciprocally decoupled ports; these circuits are called 4-ports couplers
Power dividers and combiners
Directional couplers
Separate left/right traveling waves
Insertion loss = P1/P2 (dB)
Coupling = P1/P3 (dB)
Isolation = P1/P4 (dB)
Directivity = P3/P4 = I-C, ability to isolate forward and backward waves
90° all limited bandwidth
180° Transformer 180° hybrid is not resonant, so it’s broadband
Transformers
Autotransformer (tapped coil) vs. Conventional transformer
Restrictions:
ð LF: tau0 = Lm/R needs to be increased -> N big
HF: tauc = RCp or Ls/R lowered -> N small
Transmission line transformer
Stray inductance (Ls) and winding capacitance (Cp) absorbed
Line must be short compared to the wavelength => both ends in phase
Impedance for maximum BW = ratio of in/output voltage to the current trough the line
Unbalanced - unbalanced
Impedance transformer
Balanced – unbalanced
Signal on both ports
1:4 balun with very high BW 1:4/9 balun for push-pull amplifiers
Two parallel amplifiers in anti-phase
Combining the output power of two
Linear amplifiers (e.g. class B)
Use a hybrid (transformer + resistor) combiner
Advantages:
ð Combining 2 amplifiers
o Power doubles
o OIP + 3dB
ð Using anti-phase amplifiers
o Even harmonics are suppressed
ð Extra component Rh
o One more degree of freedom
• equal PALOAD = R
Directional bridge
Accepts loss to extend bandwidth (to DC)
To separate transmitted/reflected waves
Surface acoustic wave devices
In contrast with Bulk acoustic waves
Central frequency between 10M-2GHz
Interdigital transducer (IDT)
Fundamental resonance frequency:
Lambda = IDT period
SAW transversal filter
Use two IDTs one for generating
and one for capturing
FIR filter
Greater loss
Constant pass band group delay
How to build
IDT is trapped between two Bragg reflector gratings => high Q