ELEN726 Microwave Measurements: Theory & Techniques · : δu =F 1 2Q u fffffffffffas 2Q u δ=1...

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


• 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 ?


Frequency Spectrum

C = f.λ with c = 3 x 108 m/s


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 ……


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


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


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


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


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


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


Loaded Q and Group Delay

Group Delay td = @∂φ∂ωfffffffff

QL =ω td

2fffffffffff

Where phase & frequency are in radians


QL, QU & Insertion Loss of a Single Pole Resonator

Insertion Loss = 20 LogQU

Qu@QL

fffffffffffffffffffffffff


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


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


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


Smith Chart Representation

R = 0R = 8

R = 1R = 0 R = 8R = 1

Detuned Short Detuned open

under coupled

near critical

over coupled


Two Port Resonators


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


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 )


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


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


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


DR Filter Example

From: http://tmo.jpl.nasa.gov/tmo/progress_report/42-141/141J.pdf

ELEN726 Microwave Measurements: Theory & Techniques · : δu =F 1 2Q u fffffffffffas 2Q u δ=1...

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