elen726 microwave measurements: theory & techniques · : δu =f 1 2q u fffffffffffas 2q u δ=1...
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ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
ELEN726 Microwave Measurements: Theory & Techniques
http://www.hfoscillators.com/apskhanna/teaching.html
Lecture 4Resonator Measurements
References:http://www.hfoscillators.com/apskhanna/scu/WI04_elen726/class4.html
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
• Review & Questions from Lecture 3. Lab organization.RF/Microwave Spectrum
• Resonator – definitions and applications.• Measurement parameters• One port vs. two port resonators• Definition & Measurement of:
– Loaded, Unloaded and External Q– Coupling Coefficent, VSWR–
• Types of resonators:– Transmission Line– Dielectric Resonators– FBAR
………..• Home Assignment
What will we try to cover today ?
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Frequency Spectrum
C = f.λ with c = 3 x 108 m/s
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Resonator Definition
•Microwave Resonator is a microwave junction in which electromagnetic energy can be excited for one or more specific frequencies.
•Different types of resonators comprised of conductors, dielectric material ferrite, acoustic material, crystal etc. are capable of resonating at frequencies related to their dimensions and material properties.
•Types of resonators: Lumped elements, transmission line, waveguide, coaxial, ceramic, quartz crystal, FBAR, SAW, YIG etc…..
•Important Characteristics: fo, QL, Qu, Qe, Zo, insertion loss, coupling coefficient β, tunability and temperature stability.
•Applications: Oscillators, Filters, Diplexers, Discriminators ……
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Resonator Characteristics• Resonant Frequency fo corresponds to the frequency where electric field
energy and magnetic field energy are equal. This condition is fulfilled at a number of frequencies corresponding to separate modes of resonances.
• Coupling coefficient β expresses the relationship between the intrinsic resonator characteristics and the external network to which the resonator is connected.
ZoZo
L
R
C
L
R
C
L
R
C
Around one resonant frequency the the equivalent circuit is reduced to a single resonant circuit.
ZoZo
L
R
C
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Quality Factor
Q = 2π W max
W d cycle*fffffffffffffffffffffffffffff
Qu = 2π ------------------------------------time averaged stored energyenergy lost in resonator / cycle
QL = 2π ------------------------------------time averaged stored energyenergy lost in the system / cycle
Qe = 2π ------------------------------------time averaged stored energyenergy lost in the external circuit / cycle
Unloaded Quality Factor
Loaded Quality Factor
External Quality Factor
Quality Factor Q represents the influence of resonator loss on its properties around fo and is expressed by:
For Detuned Short Qp =ωo
2Gpfffffffffffff dB
dωffffffffff g
and for DO Qs =ωo
2Rs
ffffffffff dXdωfffffffffff g
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Unloaded Q for Inductor & Capacitor
Qu =XRs
fffffff Inductor Unloaded Qu =ωLRs
fffffffffCapacitor Unloaded Qu =1
ωR Cfffffffffffffffff
When the loss is in series with the reactance:
When the loss is in parallel with the reactance:
Qu =Rp
XffffffffInductor Unloaded Qu =
Rp
ωLfffffffffand Capacitor UnloadedQu = ωRp C
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Quality Factor
W m t` a
=12fffLp I 2
W e t` a
=12fffC pV 2
Pd =12fffRpV 2
W d = Pd A t
ωres =1LCpwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwffffffffffffffffff
Q = 2π1 2+ C p V 2
1 2+ V 2 Rp tfffffffffffffffffffffffffffffffff
= ωres C p Rp
C
L RC
RpLp
W e t` a
=12fffLs I 2
Pd =12fffRs I 2
W d = Pd A t
ωres =1LCpwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwffffffffffffffffff
Q =ωresLs
Rs
fffffff
=1
ωres Cs Rs
ffffffffffffffffffffffffffffff
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Loaded Q of a terminated series resonant Circuit
From the reponse:
QL =f o
BWffffffffffff
From the Circuit:
QL =X
RTOTAL
fffffffffffffffffff
RTOTAL here = 100Ωb c
= 15.9
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Loaded Q of a terminated parallel Resonant Circuit
From the reponse:
QL =f o
BWffffffffffff
From the Circuit:
QL =RTOTAL
Xfffffffffffffffffff
RTOTAL here = 25Ωb c
QL = 15.9
Difference in series & parallel tuned circuits?
-Q-R-L
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Loaded Q and Group Delay
Group Delay td = @∂φ∂ωfffffffff
QL =ω td
2fffffffffff
Where phase & frequency are in radians
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
QL, QU & Insertion Loss of a Single Pole Resonator
Insertion Loss = 20 LogQU
Qu@QL
fffffffffffffffffffffffff
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Unloaded, Loaded and External Q
Qu =ωo CGo
fffffffffffffff QL =ωo C
GL + GS + GO
fffffffffffffffffffffffffffffffffffffffffff QE =ωO C
GL + GS
ffffffffffffffffffffffffff
LGoC
GS GL
For (Gs = GL)
Also Qe/Qu = Go / 2 GL or Go/GL = 2. Qe/Qu
η = PL /Pin = GL/(Go + GL) = 1/(1+Go/GL) =1/(1 + 2Qe/Qu)
Note: for high ηQu >> Qe
1QL
ffffffff= 1QU
fffffffff+ 1QE
fffffffff β =QU
QEffffffffff= 2GL
GO
ffffffffffffff= 2Y res
GO
ffffffffffffffff= 2 RO
Zres
fffffffffffff
QL =QU
1 + βffffffffffffffff
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Types of Resonators
• One Port Resonators:•Series Tuned vs. Parallel tuned•Smith Chart Representation•Q circles•Measurement of β and different Q’s
•Two Port Resonators:•Detuned Match•S-Parameters•Measurement of Coupling Coeff. & Q’s
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Detuned Short vs. Detuned Open
ZoL RC
Zo
L
R
C
λ/4
Detuned Short
Detuned Open
Zds = -------------β Zo 1 + j2Qoδ
where δ = (f-fo)/f& R = β Zo
Yds = ---- (1 + j2Qoδ )Yoβ
Zdo = ---------------------Zo(1 + j2Qoδ)β
Yd0 = -----------------β Yo(1 + j2Qoδ )
where R = Zo/ β
= β Zo
= Zo/ β
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Smith Chart Representation
R = 0R = 8
R = 1R = 0 R = 8R = 1
Detuned Short Detuned open
under coupled
near critical
over coupled
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Two Port Resonators
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Q CirclesZin
Zo
fffffffff= β1F j2Qu δfffffffffffffffffffffffffffffffff= β
1F j2QL 1 + βb c
δfffffffffffffffffffffffffffffffffffffffffffffffffffffffff= β
1F j2Qe βδffffffffffffffffffffffffffffffffffffff
ForQu : δu =F1
2Qu
fffffffffffas 2Qu δ = 1
SubstitutingZin
Zo
fffffffff= β1F jfffffffffffffff= β2
ffffFβ2ffff
R = X or G = B
ForQe : δe =F1
2Qe
fffffffffff
Zin e` a
Zoffffffffffffffff= β
1F jβffffffffffffffffffff
orY in e
` a
Yoffffffffffffffff= 1
βffffF j or B = 1
ForQL : δ l =F1
2QL
ffffffffffff
Y in L` a =
1βffffF j 1
βffff+ 1
f g
B = G + 1
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Linear Scale for the Q circle impedance locus
Z =βZo
1 + 2 jQo δffffffffffffffffffffffffffffffffff
Γ =Z@Zo
Z + ZO
fffffffffffffffffffff =
ZZO
ffffffffff@ 1
ZZo
ffffffff+ 1fffffffffffffffffffff=
β @ 1 + j2Qo δb c
β + 1 + j2Qo δb c
fffffffffffffffffffffffffffffffffffffffffffffffffff
Γ + 1 = 2ββ + 1 + j2Qo δfffffffffffffffffffffffffffffffffffffffffffff
Aβ + 1@ j2Qo δβ + 1@ j2Qo δffffffffffffffffffffffffffffffffffffffffffffff= K A β + 1
b c
@ j2Qo δ
φ = arg Γ + 1` a
= tan@ 12QO δβ + 1ffffffffffffffffff
or tanφ =2Qo δβ + 1fffffffffffffffff
This means that the intercept along the axis AB which is proportional to tan ϕ , is proportional to f (or δ )
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
FBAR ResonatorBulk Acoustic Wave Resonator
– Three-Layer StructureMetal - Aluminimum Nitride - Metal
– Acoustic Resonator– Piezoelectric Coupling to very High-Q Mechanical
Resonance– Integrable, Compact and Low Cost High Q
Resonator for applications from 0.5 to 10 GHz– Silicon-based– High Performance
40 mil x 40 mil x 5 mil
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
Technology Comparison at L Band
FBAR SAWs Ceramics
Frequency 500MHz-10GHz 30 MHz-2.5GHz 0.1 to 50 GHz
Size Semiconductor(acoustic)
Semiconductor(acoustic)
Quarter wave(electric)
Q >500 100s 1000s
Temp Coeff 20 to 30 ppm/C 35 to 80 ppm 0 to 5 ppm/C
Power Handling > 1 W < 1W >>1 W
ESD 200 - 500 V 200 V >1000 VHuman Body Model
IC IntegrationCompatibility
Moderate Difficult N/A
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
FBAR Equivalent CircuitA series tuned and parallel tuned composite resonator
High Q: >500High Acoustic Coupling: >6
Lm
Cm
Rm
Cp
Rseries
Gshunt
Parallel Resonancefp = (Lm Cm ) -1/2 ( 1 + Cm/Cp ) 1/2
Rp = Z2plate (Rm + 1/Gshunt ) -1
Series Resonancefs = (Lm Cm ) -1/2
Rs = Rseries + Rm
ELEN726 – Khanna - Spring 2007Santa Clara University, CA. USA
DR Filter Example
From: http://tmo.jpl.nasa.gov/tmo/progress_report/42-141/141J.pdf