IN5240 Passive Filters Part 2: LC Ladder Filters · analog-to-digital converter (ADC) IN5240:...

Institutt for Informatikk

IN5240 Passive Filters Part 2:

LC Ladder Filters

Sumit Bagga* and Dag T. Wisland**

*Staff IC Design Engineer, Novelda AS**CTO, Novelda AS


Review Series & Parallel RLC

• Series à voltage magnification– Voltage across inductor, !" = $%" = &

' %" = (!

– w/ ( = )'

"* =

+,-. = )

+,./

• Parallel à current magnification– Current through inductor, $" = &

01= 2'

01= ($

– w/ ( = 3 *" =

'+,-

= 45RC

IN5240: Design of CMOS RF-Integrated Circuits, Dag T. Wisland and Sumit Bagga


Laplace Transform



Direct-RF Receiver Front-End

Receiver comprises high-pass filter (HPF) for interference rejection, impedance and noise-matched low-noise amplifier (LNA), and high-speed analog-to-digital converter (ADC)


LNAHPF

ADC


Filter Design Steps

• Choose filter type and order

• Realize a normalized lowpass prototype (LPP) filter

• Frequency transformation from LPP (to highpass,

bandpass, bandstop)

• Impedance and frequency scaling

• Impedance transformation

– "# ≠ "%

IN5240: Design of CMOS RF-Integrated

Circuits, Dag T. Wisland and Sumit Bagga


Specifications

• Insertion loss, bandwidth, passband-to-stopband transition and attenuation, group delay, termination/matching, overshoot and ringing in the impulse/step response

• Single-ended, single-ended to differential, DE to DE or DE to pseudo-DE

• RF design à high-Q reactive components à low-insertion loss, e.g., inductors, capacitors, varactors– Metal conductivity of Cu (5.8x107 S/m), distance to

substrate and substrate resistivity (e.g., 10 Ω-cm)



Comparison Table of the Classics



PLR

• Filter response is defined by the Power Loss Ratio (PLR) and is !"# $ = &',)

&*= +

+,|.(0)|2, where Γ($)is the input reflection coefficient and is 1/|6+7|7, assuming that the input and output are matched

• Let |Γ($)|7 be an even order 8(9), then !"# $ =1 + ;(02)

<(02), where = and A are real polynomials of the order 2



Butterworth or Maximally Flat Magnitude Filter

• All pole filter (!/# semi-circle LHP ) w/ no ripple (linear phase) in the passband w/ transfer function, $% &' =

)

*+,- …(*+,0)and squared magnitude of

$% &' , |$%(&')|3 =)

)4(5/56)7

• LPF=>? = 1 + B3('/'C)3D, w/' = 'C,=>? =

1 + B3,andifB = 1à -3dBpoint

• PLRvs'/'C à higherorderà higherstopbandattenuation



Chebyshev Type 1 and Type 2

• All pole filter w/ ripple in passband (Type1) or

stopband (Type 2); faster transition vs Butterworth

• Move poles of the normalized Butterworth àellipse of a unit circle ß product ℜ and ℑ pole

positions and constants, &' < 1 and &* < 1• LPF +,- = 1 + 01231 (5/57)13, where 231 is the

Nth-order Chebyshev polynomial, and oscillates

between ±1 in the passband

• Ripple is determined by 01IN5240: Design of CMOS RF-Integrated



Elliptic (or Cauer)

• Ripple in passband and stopband w/ rapid transition (narrow transition band) vs Butterworth and Chebyshev

• Pole-zero configuration comprises poles and zeros• Large phase distortions at edge of the passband à

non-linear phase characteristic• LPF !"# = 1 + '()*((,/,.)(0, where )* is

the Nth-order Elliptic function



Butterworth, Chebyshev and Elliptic

* M. J. Roberts - All Rights Reserved 8

Chebyshev, Elliptic and Bessel Filters


[Roberts, 2012]


RC, RL & LC

• J LC: no noise, no power dissipation, double pole (second-order) filters à higher stopband attenuation

• L LC: may resonate if not properly damped, more expensive than RC due to higher inductor cost àdie area

• @ RF: LC• @ BB: active filters as LC not feasible à large

inductor values



RC vs RL

A comparison of RC and RL networks.


[Storey, 1992]


LPP Ladder Normalized


EE133 – Winter 2002 Cookbook Filter Guide

3

Design a Normalized Low-Pass Filter using a Table:

Once that is done, we can now design a second order prototype filter for a source impedance of 1 ohm, a cut-off frequency of 1 rad/sec. As shown in the figure below, we use one of two equivalent ladder circuits. Note the way the element values are numbered, with g0 at the generator to gN+1 at the load.

How to read this chart: go= generator resistance or a generator conductance

gk= inductance for series inductors or a capacitance for shunt capacitors gN+1= load resistance if gN is a shunt C or a load conductance if gN is a series L A key point is that the components alternate between shunt and series. Note that during out prototyping, inductors are always series, capacitors are always shunt. The only difference is whether or not the first element is series or shunt.

[EE133, 2002]


Impedance and Frequency Scaling

• Normalized capacitors and inductors (!", !$,…!&) à denormalized by:

' = )*($,-.)0

& 1 = 2*0$,-.

• !" and !&4" denote source and load impedances and are equal to 1

• ', 1 and ! (normalized values) are obtained from a look-up table



LP à HP Transformation

• Inductors à capacitors and capacitors àinductors

• Transform LPF normalized component values ànormalized HPF values




Transformation from LPP


[Kim, EEE 194]


Frequency Transformation

LP normalized to un-normalized LP, high pass, band pass or band stop• LP to LP: ! → !/$%• LP to HP: ! → $%/!• LP to BP: ! → (!'+ $)$*)/(s($) − $*))• LP to BS: ! → s($) − $*))/(!'+ $)$*)



T and π Filter Networks (rfcec.com)



Transforming T to π and vice versa


• Pi: !"|!$|!%– !" = (!)!* + !)!, + !*!,)/!*– !$ = (!)!* + !)!, + !*!,)/!)– !% = (!)!* + !)!, + !*!,)/!,

• T: !)|!,|!*– !) = (!"!%)/(!" + !$ + !%)– !* = (!$!%)/(!" + !$ + !%)– !, = (!"!$)/(!" + !$ + !%)


1st-Order Filters

EE3

23-

Filte

rs

85


Institutt for InformatikkIN5240: Design of CMOS RF-Integrated Circuits, Dag T. Wisland and Sumit Bagga

1st-Order Filters contd.

EE323-Filters

86

IV.SECONDORDERFUNCTIONS

Thegeneralsecondorder(bi-quadratic)filtertransferfunctionisgiveby:

20

0201

22

sQ

s

asasa)Ts

ω+ω+

++=

whereω0andQdeterminethepolesaccordingto:20Q2

!21

Q4

11!jp,p 0 −±−=

WeareusuallyinterestedinthecaseofcomplexconjugatepolesobtainedforQ>0.5.

• Theradialdistanceofthepolesfromtheoriginisthepolefrequency(ω0).• Qiscalledthepolequalityfactor.• ThehigherthevalueoftheQ,thecloserthepolesaretothejωaxisandthemoreselective

(higherpeakandinitialroll-off)thefilterresponsebecomes.• AninfinitevalueofQlocatesthepolesonthejωaxisandcanyieldsustainedoscillations.


2nd-Order Filters


[analog.com]


2nd-Order Filters contd.

EE3

23-

Filte

rs

87

Fo

llow

ings

are

the

tran

sfer

func

tions

and

resp

onse

sof

var

ious

2nd

ord

erfi

lters

.



2nd-Order Filters contd.

EE3

23-

Filte

rs

88



E.g. Q-Boosted 5th-Order SHT HPF

Comprises capacitive elements C1, C2, V1 (varactor) and inductive elements L1, L2 and L3


L1 C2C1

L2 L3

RFo,+RFi,+

V1

L1C2C1

RFo,-RFi,-

VDD

M1

A B

C D

A

B

M2

A

M4A

C

M3C

D

I1 I2

V1


Q-Boosted 5th-Order SHT HPF contd.

-55-50-45-40-35-30-25-20-15-10-5 0

1 2 3 4 5 6 7 8 9 10

Forw

ard

Tran

s. C

oeff.

, S21

[dB]

Frequency [GHz]

HPF S21 with and without Q-Boosting

M-S21 w/ Q-BM-S21 w/o Q-B

-35

-30

-25

-20

-15

-10

-5

0

5 6 7 8 9 10

Inpu

t Ref

lect

ion

Coe

ffici

ent,

S 11

[dB]

Frequency [GHz]

HPF S11 with and without Q-Boosting

M-S11 w/ Q-BM-S11 w/o Q-B



E.g. Q-Boosted 5th-Order DHT HPF


M5RFi,+

A

M6

RFi,+

M8RFi,+

RFi,-

M7RFi,-

C

I3 I4

L1 C2C1

L2 L3

RFo,+RFi,+

V1

L1C2C1

RFo,-

VDD

A B

C D

V1

L4

RFi,-

L4

M1A

B

M2

A

M4A

C

M3C

D

I1 I2

2 parallel resonant tanks(L1-V1, L4-C1)

-60

-50

-40

-30

-20

-10

0

1 2 3 4 5 6 7 8 9 10

Forw

ard

Tran

s. C

oeff.

, S21

[dB]

Frequency [GHz]

SHT and DHT HPF Frequency Responses

T-DHTT-SHT


Transformers


[Pavan, 2006]


Transformer Leakage Model

!" = $%&'&'

&%( !) = $%&%

( !) = $%&%$ &'&%

!) = *+!)



Homework

• Design a 50 Ω doubly termination BPF w/ lower

and upper cut-off frequencies at 6 and 10 GHz,

respectively, and at least 20 dB stopband

attenuation at 500 MHz offset.

• Choose the order and type of the filter

appropriately.

• Hint: maximize Q of the passive components!




References


1. M. J. Roberts, Signals and Systems: Analysis Using Transform Methods & MATLAB, 2012

2. N. Storey, Electronics a System Approach, 19923. Cookbook Winter Guide, EE133, 2002

4. S. Pavan, Integrated circuit implementation for power and area efficient adaptive equalization, US 7142596 B2, 2006

5. E. Kim, EEE 194

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IN5240 Passive Filters Part 2: LC Ladder Filters · analog-to-digital converter (ADC) IN5240:...

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