2-The Selectivity Problem

8/3/2019 2-The Selectivity Problem

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Electronics for Telecommunications

2 Selectivity, matching and filtering RLC and crystal resonators

The maximum power transfer theorem

a c ng c rcu s: -s ape , -s ape an -s ape c rcu s

Filter properties and specifications in magnitude and phase

Filter Types: Butterworth, Chebychev and Bessel

General filter design procedure: the Insertion Loss Method (ILM)

Component selection: filter order and tables

Passive integrated components: resistors, capacitors and inductors

1 SAW filters basics


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Overview Electronics systems for telecommunications (especially transmitters and

receivers) rely on many passive components (also integrated)

Passive components are essential to design selective networks (i.e. analog

filters) and matching circuits.

At very high frequency distributed parameter circuits (i.e. transmission lines)

have to be used rather than lumped components. Using distributed or lumped

elements depends on the dimensions of the circuit considered. However, thedesign procedure is similar in both cases.

Given any passive network provided with reactive elements we refer to the

quality factor Q of the circuit at a frequency as:

d

c

P

EQ =

2

Dissipated power


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Parallel RLC resonator

( ) CjL

jRX

jX

jR

jY

+=+=11111

Admittance:

LC R

(Anti) Resonance freq. (XL=Xc):

C

L

0

0

1

=

LC

f2

10 =

21

Q-factor at 0: LRC

RIQ

002

21

20 ===

Bandwidth:f

fB dB0

3

12 ==

The higher Q, the narrower the bandwidth is for a given f0

dBf 3

|Z(f)|

dBf 3

f

3The impedance is maximum in f0


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Series RLC resonator

( )C

jLjRjXjXRjZ CL

1

+=++=Impedance:

LC Resonance frequency (XL=Xc):

CL

00

1

=

LCf

2

10 =

-L0 1==

RCR 0

Bandwidth:L

R

Q

fB

2

0 ==

Similarly to the previous case, the higher Q, the narrower B for a given f0

|Z(f)|

dBf 3 dBf 3

f0

4

The impedance is minimum in f0


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Other RLC resonatorsAny resonant circuit can be transformed into a standard parallel or series one in

in a limited frequency range

Example

C LS

RS

P P

Assume that Qs=Qp=Q and that the behavior of the two resonator must be the samearound 0

pp

ppss

LjR

RLjLjR

0

00

+=+ ( )11 2

20

+=

+= QRL

RR sp

psp These equations are

absolutely general, i.e.

they hold also in the case

s

s

p

p

R

L

L

RQ 0

0

==

+=

2

2 1

Q

QLL sp

of a serial to parallel

transformation and vice

versa around 0

5


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Crystal resonators 1

For f


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Crystal resonators 2 Usually C1


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High-frequency resonators

Microwave resonators for RF applications (100 MHz-100 GHz) can not beimplemented using traditional techniques because:

Traditional cr stal resonators are too thin frail devices

Circuit connections become transmission lines and lumped circuit

components such as R, L and C are impossible to design exactly because of, ,

Solutions

Resonant cavities (e.g. transmission lines or waveguides which are short-circuited at both ends) very used in high-end telecommunication equipment

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The maximum power transfer theorem

= To transfer the maximum amount of active s s s

ZL=R

L+jX

L

v t =V cos t

power from a source with a given internal

impedance to the load, the impedance of

the load must be the complex conjugate of

the source.

Proof: the avera e active ower rovided to the load is iven b :

( ) ( )[ ]222

LSLS

LSL

XXRR

RVPeff

+++=

=

The maximum of this quantity is achieved for:

2

0=LdP

2) SL RR =L

S

LR

Peff

4=

max

ax mum powerthat can betransferred to a load

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Types of matching networks Matching (i.e. ZI=ZL*) can be obtained inserting 2-port networks (e.g. T, or Lnetworks) made up of lumped circuit elements before the load ZL

L-shaped networks

jX1 jX2ZL

ZL

Case (a): |ZL|>|ZI| downward conversion Case (b): |ZL|


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L-shaped impedance network design - 1

L2

Upward transformation case (RL


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L-sha ed im edance network desi n - 2

L2Downward transformation case (RL>RI)

Cs

RIR

L

RI2

C1Rs

=R

R L +2 1Q

B1=j0C1

=

1+

Q

21

Q

s

Question: what are the values of C and L to assure a erfect matchin at

12

+

=

Q

RR LI

= 11'

1=I

L

R

RQ

QC1 =

PP LR 0102

+=

22

21

1

QL

020

0 ==L

CX SI

12

210

Q


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-shaped impedance network design

L

RI RL

L1 L2RZ

RI

QLeft QRight

LILI RRRRf ++=+== 110

1, 2 0within a given bandwidth B?

In case of matching

LZ RR = ZZZZRRRRB

ZI RR =

011

20

2

2

20 =

+

LQ

QCXZ

RZ

+

=

2

2

220

21

1

Right

Right

Q

QC

L

01

zLeftRQL =L

Right

R

QC

02 =

01

1

2

2

10

10 =

+

Q

QL

CXI

+

=2

120

11

1

LeftQL

C

13Both bandwidth and matching can be obtained

Left


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T-shaped impedance network design1

RI R

2

C

L1 L2

C

Z

RI

QLeft QRight

RRRRf

This is dual to the previous case

LILI

RightLeft

RRRRB

++=+==

RZ

QC=

RQ LRight=

RQL ILeft=

14

z00 0


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Example

RL= 50

RI = 5

0 = 2 1 GHz

Bd=25 MHz

Design the L-shaped and -shaped matching networks centered in 0meeting the wanted bandwidth constraint

1) L-shaped (downward transformer) Q=3 (fixed) C1=9.55 pF; L2=2.4 nH

but B=333 MHz. The bandwidth specifications are not met.

2)-shaped from Bd we obtain that Q=40 Rz=0.054 QRight=30.4 and

Left . 1 2 . . .

specifications are met, but the components can be hardly built using

integrated technologies: capacitances are too large and inductances too

15

.


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Introduction to filters

Analog filters must provide good matching (i.e. power transfer) over acertain bandwidth, while assuring adequate selectivity. Filters can be of

- -, . . , ,

Bandpass filter (BPF), Bandstop filter (BSF).

, .

active filter there can be amplification of the signal power in the passband

region, whereas passive filter do not provide any amplification.

Ref. Pictures from H. Khorramabadi

presentation, Analysis and Design of

VLSI Analog-to-Digital Interface

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n egra e c rcu s , er e ey,


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Filter frequency characteristics

17

. . , - -

circuits, UC Berkeley, 2009


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Group delay and phase distortion - 1

18


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Grou dela and hase distortion - 2

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An example of critical phase distortion effect

Intersymbol interference (ISI) is the impulse broadening in time resulting ininterference between successive TX samples

Ref. Pictures from H. Khorramabadi presentation,

- -

Linear phase filter Nonlinear phase filter

Interface Integrated circuits, UC Berkeley, 2009

ISI is larger in

this case dueto the tail in

the time

res onse

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The Insertion Loss Method ILM

The insertion loss method (ILM)

and synthesize a filter with aknown frequency response.

me o a so a ows er

performance to be improved in a

straightforward manner, at the

.

Phase information is totally

ignored. So the filter type must

There is a historical reason why phase

information is ignored. Original filter design

methods were developed for voice and

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human ear is insensitive to phase distortion.


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Power Loss Ratio

Ideally PLR should be equal to 1 in the

bandwidth and infinite out of bandwidth.

In real filters we have to set 1PLR1+k2 over

the bandwidth

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Filter t es

As a general rule, in the ILM, the PLR function is approximatedby suitable characteristic polynomials, i.e.

221 PkP +=

,

filters are: Maximally flat or Butterworth filters: Moderately linear phase

response, s ow cu -o , smoo a enua on n pass an .

Chebyshev filters: Bad phase response, rapid cut-off for similar

order, contains ripple in passband. May have impedancem sma c or even.

Bessel filters: Good phase response, linear. Very slow cut-off.Smooth amplitude response in passband.

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Characteristic Polynomial Functions

Low-pass case

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Butterworth Low-pass Filters

Maximally flat in the passband: all

derivatives of the transfer function

till order N in =0 are equal to 0

All poles on the unitary circle with

equal anglesRef. Pictures from H. Khorramabadi presentation,Analysis and Design of VLSI Analog-to-Digital

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Interface Integrated circuits, UC Berkeley, 2009


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Chebychev I Low-pass Filters

Equiripple in the passband Sharper transition bandwidth

Severe phase distortionRef. Pictures from H. Khorramabadi presentation,Analysis and Design of VLSI Analog-to-Digital

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po es on e pses ns e e un ary c rc eInterface Integrated circuits, UC Berkeley, 2009


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Bessel Low-pass Filters

Flat in the passband

Poor out-of-band attenuation

uas - near p ase

All poles outside the unitary circle

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Ref. Pictures from H. Khorramabadi presentation, Analysis and Design of

VLSI Analog-to-Digital Interface Integrated circuits, UC Berkeley, 2009


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Ex m l f P f r L -P Fil r - 1

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Examples ofPLR

for Low-Pass Filters - 2

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30


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The Low-Pass Protot e LPP

R

ZL

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General LPP design procedure

The LPP is the building block from which real filters can be constructed.

Various transformations may be used to convert it into a high-pass, band-

pass or other filter of arbitrary center frequency and bandwidth.

Let us consider the first circuit shown in the previous slide, and let refer to RIas the output resistance of the signal source and to ZL()=RL()+jXL() asthe input impedance of the filter.

( )( ) ( ) ( )( ) ( ) 222222 XRRXRR LILLLI =

+=

++=

By equating the Ncoefficients of the two polynomials in shown above for

( ) ( ) 44 RRRR LILI

. . , ,

gk of eitherLkorCk fork=1,,Nare determined.

The values of the coefficients gk

for a certain filter if both RIand R

Lare equal

o norma ze va ues are usua y a e see nex s e .

In order to determine the values if RL1, RI and Lk should be multiplied times

RL while Ckshould be divided by RL

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Tables examples for LPP design

Table for Chebychev LPP filters

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Impedance Denormalization and

Frequency Transformation of LPP - 1 , ,

the filter at the operating frequency can be scaled from unity to other values RL.

This is called impedance denormalization.

so, e er e cu o requency can e rans orme o o er requenc es an

unity or LPP parameters Lkand Ckcan be mapped to values Lkand Ckcorresponding to other filter types such as highpass, bandpass and bandstop.

To this purpose, we can use a new variable =f() so that:

( ) ( )[ ] fPkPkPLR2222 11 +=+= '

Examples

=' Lk

k R

L

L ='

1 LPF with cutoff fre uenc k

k

C

C =

'

c c Lc

c=' kRL

C1

='2) LPF to HPFL

kC

RL ='

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Impedance Denormalization and

Frequency Transformation of LPP - 2

=

20

2

12

1'3) LPF to BPFwhere

Lk

RL

=''

= 0

RL

LLk

k

'

Lk Ckk

k

CC =''

k0

Lkk

RLC

0

='

L0

where

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Note: many other transformations are possible (i.e. bandstop, notch,.)


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Summary of the design steps

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Exam le 1: Butterworth LPF

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Example 2: Chebychev LPF

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I t t d l i t


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Integrated planar resistors

Single square resistorResistorlength

=w

lR

w

l

zS

lR []=

==

3-square resistorMaterial Resistor cross-section

l

ep z x w w

R[] depends on the manufacturing process and

Large resistor

.

/square up to 100 /square)

w can e con ro e y e es gner w

high accuracy

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Large resistors are expensive in terms of area


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Integrated planar capacitors

Capacitor area

S

tt

==

constant(usually SiO2)

Dielectric

thickness

/t depends on the IC manufacturing process

wl can be controlled b the IC desi ner with hi h accurac

Large capacitors are expensive in terms of area. Area reduction canbe achieved usin various techni ues e. . throu h trench or stacked

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capacitors)


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Lateral flux ca acitors

Lateral flux capacitors are designed toexploit lateral electric fields.

The idea is to increase the capacitance

at no price in terms of planar area.

low series resistance and inductance

higher Q, better robustness to the etching

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process.


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Integrated monolithic inductors

4.7 nH The structure of a monolithic inductor consists of a spiral ofdifferent shape (rectangular, circular, hexagonal or octagonal)

35 / Total area of the metal track

.

square inductors L is approximately given by:

33 nH

( ) 41476171031

///.

wGwSL m

+

Total area of the

inductor Width of the metal trackDistance between coils

Performance of a monolithic inductor are limited by 3 mainparasitic effects:

Metal wire resistance (reduced using circular shape and by

removing the most internal 4 or 5 turns of the coil)

apac ve coup ng re uce y ncreas ng e s ance

between inductor and chip surface)

Ma netic cou lin reduced b usin a atterned round shield

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(PGS) between the inductor coil and the chip surface)


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Quality factor of integrated inductors

Q is measured at frequencies of operation (typically > 1GHz) Hard to have Q larger than 10 with CMOS technologies

Other substrates or techniques are required for better performance

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M lithi T f 1


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Monolithic Transformers 1

Monolithic transformers have been used

extensively in RF circuits. They usually

consist of 2 inductors combined in different

way.

For different transformer structures, the

coupling coefficient k, the turn ratio n, and

other parameters may vary considerably.

Depending on whether the lateral or vertical

magne c coup ng s use , rans ormers can

beplanarorstacked.

.Usually n is small and coupling kup to 0.7.

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M lithi T f 2


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Monolithic Transformers 2

In stackedtransformers, both vertical and lateral magnetic coupling is used

- = . ,

Cons: high-parasitic capacitance, poorer quality factor

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SAW fil b i 1


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SAW filter basics - 1 Surface Acoustic Wave filters are

based on interdigital trasducers (IDT)

arge y use or an ters ut

not on the same chip

Piezolectric substrate quartz,tourmaline, gallium phosphate,

lithium niobate, lithium tantalate and

piezoceramics

Etched Aluminium pattern on top

RF electric energy is transferred to

the cr stal and turns into vibrational

Vibration propagate along the surface

Longitudinal waves: ~6000 m/s

Transversal waves: ~3000 m/s

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Ref. Pictures from D. Morgan, Surface Acoustic Wave Filters with applications to electronic communications and signal

processing, Elsevier, 2007

SAW filt b i 2


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SAW filter basics - 2

Piezoelectric crystals have some ideal

properties for wave propagation:

Anisotropy

Low losses (if we have a pure crystal)

Abilit to ro a ate si nals atfrequencies larger than 1 GHz

Best coupling achieved when the

electrode width is in the order of/4

By properly designing the pattern of theoutput IDT only some frequencies are

behavior

Every stage may cause reflection and a

Linear phase response

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Ref. Pictures from D. Morgan, Surface Acoustic Wave Filters with applications to electronic communications and signal

processing, Elsevier, 2007

2-The Selectivity Problem

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