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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
8
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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
9
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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
16
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
19
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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
20
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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
21
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
22
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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.
23
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Characteristic Polynomial Functions
Low-pass case
24
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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
25
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
26
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
27
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
28
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Examples ofPLR
for Low-Pass Filters - 2
29
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30
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The Low-Pass Protot e LPP
R
ZL
31
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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
32
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Tables examples for LPP design
Table for Chebychev LPP filters
33
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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 ='
34
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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
35
Note: many other transformations are possible (i.e. bandstop, notch,.)
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Summary of the design steps
36
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Exam le 1: Butterworth LPF
37
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Example 2: Chebychev LPF
38
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
39
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
40
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
41
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
42
(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
43
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.
44
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
45
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
46
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
47
Ref. Pictures from D. Morgan, Surface Acoustic Wave Filters with applications to electronic communications and signal
processing, Elsevier, 2007
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