E4215: Analog Filter Synthesis and Design: HW0 · 2006. 8. 21. · E4215: Analog Filter Synthesis and Design: HW0 Nagendra Krishnapura([email protected]) due on 21 Jan. 2003 This

E4215: Analog Filter Synthesis and Design: HW0

Nagendra Krishnapura ([email protected])

due on 21 Jan. 2003

ThisassignmenthasZERO credit anddoesnot con-

tributeto thefinal grade. Its purposeis to gaugeyour

familiarity of prerequisitetopics.

1. Checkthetermsthatareunfamiliar to you:

• Laplacetransform

• Impulseresponse

• Frequency response

• Transferfunction

• Bodeplot

• Operationalamplifier

• Bipolar transistor

• MOStransistor

• Smallsignalequivalentcircuit

• Commondrainamplifier

• Loopgain

• Gainmargin

• Phasemargin

2. Thecircuit in Fig. 1 is

vo

vi

=


Ic =


vo

vi

=

RL

+

-vi

+

-

vo

Figure1:

1mA

Ic

1x 1x

Figure2:



Vx =

Vy =

7. Transferfunctionof thecircuit in Fig. 6:

Vo(s)

Vi(s)=

1

2

1KΩ 1KΩ

2mA

+

-Vi

+

-Vo

v1 v2

Figure3:

+

-vi +

-

vo

Figure4:

8. In Fig. 7

Vo =

9. Transferfunctionof thecircuit in Fig. 8:

Vo(s)

Vi(s)=

10. In Fig. 9:

vo

vi

=

−

++

-

+

-

+

- ideal opamp

1KΩ

2KΩ

1V VyVx

Figure5:

+

-

+

-

R

CVi Vo

Figure6:

3Vcos(ωt)

2KΩ1KΩ

+

-Vo

+

-

Figure7:

+

-

+

-

RC

Vi Vo

L

Figure8:

+

-

+

-vovi

gm rds RL

Figure9:



due on 28 Jan. 2003

+

-Vi(s)

+

-Vo(s)

R C

+

-

v i(t)=

1Vco

s(t/R

C)

+

-vo(t)

R C

+

-Vi(s)

+

-Vo(s)

R/2 2C

+

-

+

-vo(t)

v i(t)=

1Vco

s(t/R

C)

R/2 2C

(a)

(c) (d)

(b)ii(t) ii(t)

+ -v(t)

i(t)

(e)

Figure 1:

1. (5 pts.) For the circuits in Fig. 1(a) and

Fig. 1(b), evaluate the transfer function H(s) =

Vo(s)/Vi(s), and the impulse response h(t) cor-

responding to H(s). Approximately sketch

the magnitude and phase of H(s) (Bode Plot).

What is the difference between the two circuits?

2. (5 pts.) In the circuits in Fig. 1(c) and Fig. 1(d),

evaluate the current ii(t) through the input volt-

age source. Evaluate the average power dis-

sipated in the voltage source and the resistor.

What is the difference between the two circuits?

Note: Average power dissipated in an element

with a voltage v(t) across it and a current i(t)

through it (see Fig. 1(e)) is given by

P =1

T

∫T

0

v(t)i(t)dt

3. (5 pts.) Write the expressions for the transfer

function H(s) = Vo(s)/Vi(s) for the circuits in

+

-Vi(s)

R1 C1

(a)

+

-

R2 C2

+

-vx vx

+

-Vi(s)

R1 C1

(b)

+

-

R2

C2

+

-vx vx

+

-Vo(s)

-Vo(s)+

T

0V

1V

(c)

vi(t)

R2/4

Figure 2:

Fig. 2(a) and Fig. 2(b). Sketch the Bode plots

assuming R1C1 = 4R2C2.

4. (5 pts.) The circuit in Fig. 2(b) is driven by a

pulse with an amplitude 1V and lasting T sec-

onds (Fig. 2(c)). Assuming T = R1C1, sketch

the intermediate voltage vx(t). Sketch the out-

put voltage vo(t) assuming that R2C2 = R1C1.

1



due on 4 Feb. 2003

For theopamps,usetheappropriatemodelbasedon

theparameters provided. i.e. if nothingis given,as-

sumean ideal opampwith infinite gain; if the unity

gain frequencyis given,usethe integrator model;if

the dc gain and the unity gain frequencyare given,

usethefirstordermodeletc.Thisholdsfor all future

assignments.

+

-

?

?

R

R2R

gmvovo

+

-

vi

v1

+−

Figure1:

1. (2 pts.) [Fig. 1, gm = 4/R] Assign the cor-

rectsignsto theopampsuchthatit hasnegative

feedbackatdc.

2. (2 pts.) [Fig. 1, gm = 4/R] Assumingthat the

opamphasa transferfunction A(s) = ωu/s,

determinethe transfer functions Vo(s)/Vi(s),

V1(s)/Vi(s).

3. (4 pts.) [Fig. 1, gm = 4/R] Determinetheloop

gainT (s) aroundthis feedbackloop. Assuming

that the opamphasa dc gain Ao = 100 anda

unity gain frequency ωu = 1 Grad/s1, draw the

Bode plot (magnitudeand phase)of loop gain

T (s) andopampgainA(s).

4. (6 pts.) Assume gm = 1 mS, R1 =

900 kΩ, R2 = 100 kΩ, RL = ∞, Ao = 1000.

For thecircuits in Fig. 2(a)andFig. 2(b), eval-

uatethegain Vo/Vi andthefeedbackloop gain

T. Repeat,assumingRL = 1 MΩ.

5. (6 pts.) Assume gm = 1 mS, R1 =

900 kΩ, R2 = 100 kΩ, CL = 10 pF, Ao =

1000, ωu = 100 Mrad/s2. For the circuits

in Fig. 2(c) and Fig. 2(d), evaluatethe trans-

fer functionVo(s)/Vi(s) andthefeedbackloop

gain T(s). Write the transferfunctionsin the

standardfirst order form andcomparethe two

results.Repeat,assumingCL = 20 pF.

1giga radians/second; giga=109

2mega radians/second

1

2

+- −

+

+

- -

+

-

+

-

+

Vi Vo Vi Vo

R2

RL

+- −

+

+

- -

+

-

+

Vi Vo Vi

R1

R2

R1

RL

-

+

Vo

R2

R1

R2

R1

CL

CL

gm = 1mS

gm = 1mS Ao = 1000, ωu = 100Mrad/s

Ao = 1000

(a) (b)

(c) (d)

Figure2:



due on 11 Feb. 2003

In addition to the problems here, problems 1, 2, 3

from HW2 are also due on 11 Feb. 2003.

−

+

+

-Vin1

Vo+

-

1kΩ

1kΩ

1kΩ

opampwith offset

−

+

+

-Vin2

1kΩ

1kΩ

1kΩ

opampwith offset

1kΩ

1kΩ

−

+

+

-Vin1

1kΩ

1kΩ

1kΩ

opampwith offset

Vo+

-

+

-Vo2

+

-Vo1

(a)

(b)

Figure1:

1. (9 pts.) The opampsin Fig. 1 have an input

referredoffset voltage Vos, but are otherwise

ideal(A0 = ∞). For Fig. 1(a), derive the ex-

pressionrelatingtheoutputVo to theinputVin1

andtheoffsetVos. Draw thedc transfercharac-

teristicsVo vs.Vin1 includingtheeffectof offset

assumingthatVos > 0. Show theinput referred

offset and the outputoffset of the amplifier in

Fig. 1(a) on this plot. (Hint: In a circuit with

multiple inputs,try usingsuperposition).

If the standarddeviation of Vos is σ = 5 mV,

what is the standarddeviation of the input re-

ferredoffsetandtheoutputoffsetof theampli-

fier in Fig. 1(a).

What is the net outputoffset(in the outputVo)

of thecircuit in Fig.1(b)?(Hint: Usetheresults

relatedto Fig. 1(a) to determineVo1 and Vo2.

RelateVo to Vo1 andVo2)

2. (5 pts.) In Fig.2(a),determineVp,max, themax-

imum value of Vp such that the output vo(t)

is sinusoidal. The opamphasthe characteris-

tic shown in Fig. 2(b)(Theslopeof thevertical

part is ∞. Sketchvo(t) whenVp = Vp,max/2

andwhenVp = 2Vp,max

3. (3 pts.) In Fig. 3, vo = f(vi) = vi + a2v2

i +

a3v3

i . If vi = Vp cos(ωt), expressvo(t) asasum

of sinusoids.Find the ratio of the 2nd and3rd

harmonicamplitudesto thatof thefundamental.

If a2 = 10−3 V−1, a3 = 10−3 V−2, find thein-

put peakVp suchthat the secondharmonicis

60dB below thefundamental.Repeattheexer-

1

2

−

+

1kΩ

2kΩ

+

-vin=Vpcos(ωt) +

-vo

1V

-1V

vid

vout

(a)

(b)

Figure2:

f(vo)vi vo

Figure3:

cisefor thethird harmonic.

4. (3 pts.) Assumingideal transconductors1, de-

rive expressionsrelating Vo to Vi in Fig. 4(a)

andto Vi1 andVi2 in Fig. 4(b).

Repeatfor Fig. 4(a)assumingthatthetranscon-

ductorgmx hasanoutputresistancerox andin-

put and output capacitancesCix, Cox. x =

1, 2 for thetwo transconductorsin Fig. 4(a).

1voltage controlled current source

+-

+-

+-

+-

+-

-Vi

+

-Vi1

+

-Vi2

+

(a)

(b)

C

gm2

gm1

gm2

gm1

gm3

Figure4:



due on 18 Feb. 2003

first-orderfilter

RS

RLvs(t)vo(t)

+

-

Figure1:

1. Initially, assumeRS = 0, RL = ∞. Fig. 1

shows a first orderfilter whoseinput is thesum

of two sinusoidsvs(t) = 1V cos(1 Mrad/st) +

1V cos(1000 Mrad/st). The higher frequency

sinusoidshouldbeattenuatedby 40dB andthe

lower frequency sinusoidshouldbe attenuated

aslittle aspossible.

(2 pts.) Determinethe transferfunction of the

filter. Draw the schematicof a passive RC fil-

ter with R = 100 kΩ thatwill accomplishthis.

Whatis theattenuation(in dB) of thelower fre-

quency sinusoid?

(3 pts.) In the previously designedfilter, if

R and C can have variations of ±10%, (a)

What are the maximumand minimum values

of the pole frequency? What is the percent-

agevariationfrom thenominalvalue?(b) What

is the worst case(smallest)attenuationof the

higherfrequency signal?(c) What is theworst

case(largest)attenuationof thelowerfrequency

signal?

(1 pt.) Reevaluatethe transferfunction with

RS = 10 kΩ, RL = ∞. How wouldyourestore

thetransferfunctionto theoriginal?Reevaluate

thetransferfunctionwith RS = 0, RL = 1 MΩ.

How would you restorethepole to theoriginal

value?

(1 pt.) With RS = 10 kΩ, RL = 1 MΩ, choose

R, C suchthat thepoleof thefilter is thesame

asoriginally determined.What is the transfer

function?Determinetheattenuationof thetwo

sinusoids.

1

R

C

1

R

C

1

R

C

1 2 N

vo(t)+

-vs(t)+

-

Figure2:

2. (3 pts.) Fig. 2 shows a cascadeof N identi-

cal bufferedfirst orderfilter sections.vs(t) =

1V cos(1 Mrad/st) + 1V cos(10 Mrad/st). Us-

ing simpleBodePlots,determinethe smallest

N requiredto reducethehigherfrequency sig-

nal by 80dB while leaving thelower frequency

signalunchanged.Whatis thevalueof thepole

of eachsection?Now, usingthe transferfunc-

tion of the filter so obtained,find the actual

attenuationof the two signals. RecomputeN

andthepoleof thefilter if the lower frequency

shouldbeattenuatedby ≤ 3 dB andthehigher

frequency by≥ 80 dB.

3. (3pts.)Designanaccouplingstagebetweenthe

1

2

stage 1 stage 2

Ci = 1pF

ac coupling

dc bias=1V

+−1V

Figure3:

two stagesshown in Fig. 3. The secondstage

hasan input capacitanceCi = 1 pF. a) Theat-

tenuationfor very high frequencies(ω → ∞)

shouldbelessthan1dB, b) Theattenuationfor

10Mrad/sshouldbe lessthan4dB, c) The ca-

pacitorusedin thecircuit shouldbeminimized.

d) Thedc biasprovidedto the2ndstageshould

be1V (A 1V dcsourceis availableto you.).

4. (1 pt.) Designa filter with thetransferfunction

−k/(1 + s/p1), with k = 10, p1 = 10 Mrad/s.

Draw the schematicwith ideal opamps—use

C = 1 pF.

(2 pts.) Determinethedc gainAo andtheunity

gain frequency ωu of theopampsuchthateach

of thesenonidealities(actingby itself) changes

thepoleof thefilter by lessthan2.5%.

(2 pts.)Draw theBodeplot of theloopgain for

thefilter youdesigned.Useanintegratormodel

for theopampwith ωu determinedpreviously.

(2 pts.) Redesignthe filter (useideal opamps)

assumingthat the largest resistor allowed is

10 kΩ.



due on 25 Feb. 2003

+

-Vs Vo

L

C

Rs RL

+

-

Figure1:

1. (2 pts.) DetermineVo(s)/Vi(s) for the filter

in Fig. 1. For a given Rs, determineRL such

thatQ is maximum.What is themaximumQ?

Whatis ωp underthiscondition?

L

C

C

R

R

+

-Vi Vo

+

-

+

-Vi Vo

+

-

(a)

(b)

Figure2:

2. (2 pts.) What is the bandwidthof the circuit

in Fig. 2(a)? If you were allowed to placea

seriesinductor L as in Fig. 2(b), what value

would you choosefor it to maximizetheband-

width without introducingpeakingin themag-

nitude response?What is the resultingband-

width? Sketch the frequency responsesof the

two circuits.

Rs C=1pF L

Vs

C=1pFL

Vo

+

-

Vo

+

-

R

Rs

Vs

R

(a)

(b)

Figure3:

3. (4 pts.) For eachof Fig. 3(a)andFig. 3(b), (a)

AssumingRs = 0 determineL andR so that

a bandpassfilter with ωp/2π = 5 GHz1 anda -

3dBbandwidthof 1GHzis realized.(b) If vs(t)

is a 1V sinusoidat 5 GHz, what is the current

flowing throughthe input source?(c) What is

the valueof Rs, the sourceresistance,that re-

sultsin a10%deviation in Q?

4. (5 pts.) In Fig.4 considertwo casesR1 = R2 =

R andR1 = 2R, R2 = R/2.

For eachof these,(a) FindV1(s)/Vi(s) Is there

a difference? (b) EvaluateVk(s)/Vi(s), k =

2, 3 Is therea difference?What is the max-

imum of |Vk(jω)/Vi(jω)|? (c) The input is a

sinusoidvi(t) = Vip cos(ωt) whereω can be1This means that ωp = 2π × 5 Grad/s

1

2

−

+

−

+

−

+

R

5R

R

CC

R

V1 V2

V3

Vi

OPA1

OPA3

OPA2

R2

R1

Figure4:

anything. If the opampshave a swing limit of

1V, what is the largestVip that canbe applied

while maintainingall the opampsin the linear

region?

5. (3 pts.) (a) Designa secondordergm-C But-

terworth filter with dc gain=1 and-3dB band-

width=1MHz. Assumethat the smallestgm is

10µS. Give the transferfunction and all the

componentvaluesin thegm-C filter schematic.

(4 pts.) (b) Using the above filter asthe basis,

designa lowpassnotch filter with dc gain=10

and a notch at√

10 MHz. Use the voltage

summingtechnique.Give the transferfunction

andall thecomponentvaluesin thegm-C filter

schematic.What is the high frequency gain of

this filter? What is the attenuationof the filter

at 1MHz w.r.t. dc? Has the -3dB bandwidth

increasedor decreasedcomparedto thefilter in

(a)?



due on 4 Mar. 2003

In addition to the problems here, problem #5 from

HW5 is also due on 4 Mar. 2003

1. (1+3+3pts.)Repeatthedesignin problem#5 of

HW5 usingopampsandfeedforward technique.

Use10pF capacitors.

(a)DesigntheButterworth lowpassfilter.

(b) Obtain the lowpassnotchtransferfunction

at theoutputV11.

(c) Obtain the lowpassnotch transferfunction

at theoutputV2.

+-

+-

+-

+-

+-

gm

gm

gm/Q

CC

V2V1

Vi1

Vi2

gm1i

gm2i

Figure1:

2. (2 pts.) In Fig. 1, Determinethe transferfunc-

tionsfrom Vi1 andVi2 to voltagesV1 andV2.

1outputof OPA1; in thehandout“Transferfunctionsrealiz-

ablein a biquad”.

Vi+-

R L=1H Cgm

Figure2:

3. (1+2+2+2+1+3pts.) (a) Designa 1H inductor

usingtransconductorsanda 100pF capacitor.

(b) Derive the (passive) equivalent circuit of

the previously designedinductorif the capaci-

tor hada1MΩ resistoracrossit.

(c) Designan RLC bandpassfilter with ωp =

100 krad/sand Q = 10 using a 1H inductor.

The gain at the resonantfrequency shouldbe

10. Usethetopologyin Fig. 2.

(d) Replacetheinductorwith theequivalentcir-

cuit obtainedin (b) andre-evaluatethetransfer

functionVo(s)/Vi(s) What,if any, is thedevia-

tion from theintendeddesignin (c).

(e)How wouldyouchangethedesignto restore

theQ to 10? You cannot remove the1MΩ re-

sistorwhich is acrossthecapacitor.

(f) Simulate(i) thecircuit in Fig. 2, (ii) thecir-

cuit with theinductorreplacedby theactive in-

ductor2, and(iii) the repairedcircuit from (e).2usethecircuit with transconductorsandcapacitors,not the

equivalentobtainedin (b); Includethe1MΩ resistoracrossthe

1

2

Submitthemagnitudeandthephaseresponses;

overlaytheresponsesof thethreecircuits.

100pFcapacitor.



due on 25 Mar. 2003

For 1-5, give theschematicof thepassive filter with

all theelementvalues.For 1-3, give thethetransfer

functionin thenormalizedform which is

b0 + b1(s/Q/ωn) + b2(s/ωn)2

1 + s/Q/ωn + (s/ωn)2

whereωn is aconvenientnormalizingfrequency. For

3-5, give the expressionfor the frequency transfor-

mationalongwith the numericalvaluesfor the pa-

rametersin the transformation. For 6, give the fi-

nalschematicandexplainverybriefly thepurposeof

eachfeedforwardcomponent1 .

1. (1 pt.) Design a secondorder passive low-

C1

C2

L

R

+

-Vi

+

-Vo

Figure1:

passRLC notchfilter with Q = 1/√

2, ωp =

1 Mrad/s and a transmissionzero at ωz =

10 Mrad/s. Use the topology in Fig. 1 with

C1 + C2 = 10 nF. What is the attenuationin

dB at1Mrad/s? Call thisAp.

2. (2 pts.) Scalethe filter in (1) so that it uses

R = 1Ω and hasa notch at 10rad/s. What1Elementsfrom theinput to variousopamps.

is the frequency Ωp at which theattenuationis

Ap? What is the smallestfrequency2 at which

theattenuationis As = −20 dB?Call thisΩs.

3. (3 pts.) Transformthe prototypein (2) to a

passive RLC highpassfilter with anattenuation

Ap (determinedin (1)) at 10Mrad/sanda ter-

mination impedance10kΩ. What is the fre-

quency of the notch in this filter? Draw the

schematicreplacingthe inductorswith capaci-

tively terminatedgyratorswhosegyrationresis-

tanceis 10kΩ.

4. (4 pts.)Transformtheprototypein (2) to a pas-

sive RLC bandpassfilter whoseattenuationis

Ap atωp1 = 10 Mrad/sandωp2 = 12.1 Mrad/s.

The terminationimpedanceshould be 10kΩ.

What are the “stopband” edgesωs1 and ωs2

wherethe attenuationis As? What is the gain

of thefilter at 11Mrad/s? If oneof thenotches

of thefilter is at 4.7Mrad/s, whereis theother

notch?

5. (4 pts.) Transformthe prototypein (2) to a

passive RLC bandstopfilter whoseattenuation

is at leastAs in the range81 Mrad/s ≤ ω ≤100 Mrad/s. Use a terminationimpedanceof

1kΩ. What arethe “passband”edgesωp1 and

2You can calculatethis analytically-you’ll get a 2nd order

equationin Ω2; or determineit usingsimulation-besureto usea

sufficiently smallfrequency step.

1

2

ωp2 wherethe attenuationis Ap? What is the

filter’s attenuationat90Mrad/s?

6. (2 pts.) Realizeananopamp-RCversionof the

highpassfilter in (3). UsetheTow-Thomasbi-

quadwith feedforward techniqueto realizethe

zerosat the output of the first opamp. Use

R = 10 kΩ in theresonatorcore.

7. (1 pt.) Realizea bandpassfilter whoseatten-

uation is Ap at fp1 = 10 MHz and fp2 =

12.1 MHz. (Hint: You don’t have to go through

thewholesynthesisagain. Usethe resultfrom

(4)).

8. (2 pt.) Simulatethemagnituderesponseof the

passive circuitsin 1, 3(not thepartwith thegy-

rator),4, 5. (Plot all 4 magnituderesponsesin

4 subwindows of thesameplot for submission.

Useappropriaterangesfor x andy axesto show

all points of interest). In each,mark the fre-

quency of thenotch(es).

9. (1 pt.) Simulate the magnituderesponseof

the opamp-RCfilter in 6. For the opampsuse

ideal voltage controlled voltage sourceswith

gain=106.



due on 8 Apr. 2003

R=QC=1/ωp L=1/ωp

IC IR IL

+-

+-

+-

ViVi

tran

scon

duct

ance

=1S

b2IC

b1IR

b0IL

+

-

Vo

+

-

R=1C=1/ωp

IC IR

+-

+-

ViVi

tran

scon

duct

ance

=1S

b1IC

b0IR

+

-

Vo+

-

(a) (b)

"bilinear" "biquad"

Figure1:

1. (a) (5 pts.) ComputethetransferfunctionsVo/Vi in termsof theparameters(Q, ωp, b0, b1, b2) for

thecircuitsin Fig. 1(a,b).

(b) Turn thesecircuits into parameterizedsubcircuits“bilinear” and“biquad” in cadence1 with the

requiredparameters.You canthenusethesesubcircuitsto realizeidealcascaderealizationsof

any transferfunction.

0dB

-1dB

-40dB

2MH

z

4MH

z

0dB

-1dB

-40dB

1 ra

d/s

2 ra

d/s

(a) (b)

Figure2:

2. Youarerequiredto realizeafilter thatmeetsthespecificationsshown in Fig.2(a).Youaregiven(Table1)1In cadence,to realizeacurrentcontrolledvoltagesource,youalsoneedto havea0V voltagesourcethroughwhich thedesired

currentis flowing. Seetheexamplesubcircuit“lpf ” in thelibrary “E4215 examples”.

1

2

thepolesandzerosof 4 types(ExcludingBessel)of filters which satisfytheprototypespecifications

in Fig. 2(b).

(a) (4 pts.) Tabulate the order, the resonantfrequencies,the quality factorsof the poles,andthe

locationof transmissionzeros(if present)of thedifferenttypesof filters thatsatisfythespecs.

in Fig. 2(a).

(b) (7 pts.) Usingtheparameterizedsubcircuitsfor thebilinearandthebiquadraticfilters,simulate

thefour filters(usingthecascadestructure)in cadence.Usetherulesof cascadingdiscussedin

theclass.You do not have to submittheschematics.Clearlystatetheorderof cascadeandthe

polezeropairing.

Plottheirmagnitudeandphaseresponses2, andthegroupdelay(for this,youcanusethefunction

“groupDelay”in thecalculatorin cadence).

(c) (4 pts.) For eachfilter, determinethemaximumtransferfunctionmagnitudefrom the input to

eachof thestage(first or secondorder)outputs.If eachoutputwerelimited to 1V, what is the

maximuminputvoltagethatcouldbeappliedto eachwithouthaving distortion?

(d) (4 pts.)Simulatethetransferfunctionof theBesselfilter prototype(lastcolumnof Table1) using

thesametechniqueasabove. If thisfilter werescaledsuchthatit hadanattenuationAs = 40 dB

at 4MHz (the stopbandedge),what would be its attenuationat the passbandedge(2 MHz)?3

Doesit meetthespecsin Fig. 2(a)?

(e) (4 pts.) For eachof the 4 filters that satisfiesthe specsin Fig. 2(a), list the maximumquality

factorof the biquadstagesused,the maximumresonantfrequency, andthe maximumgroup

delayvariationin thepassband(< 2 MHz).

(2 pts.) Repeat3 for the Besselfilter. To find its maximumresonantfrequency, calculatethe

maximumresonantfrequency in theprototypeandmultiply it by thescalingfactordetermined

above.

Table1: Prototypezerosandpoles

Butterworth Chebyshev Inverse Chebyshev Elliptic Bessel

poles poles zeros poles zeros poles poles

−1.1031 ± j0.2194 −0.0895 ± j0.9901 ±j3.0671 −0.2811 ± j1.1013 ±j3.5251 −0.3643 ± j0.4786 −0.3868 ± j1.0991

−0.9351 ± j0.6248 −0.2342 ± j0.6119 ±j1.8956 −0.9461 ± j0.8751 ±j1.6095 −0.1053 ± j0.9937 −0.6127 ± j0.8548

−0.6248 ± j0.9351 −0.2895 −1.4202 −0.7547 ± j0.6319

−0.2194 ± j1.1031 −0.8453 ± j0.4179

−0.8964 ± j0.2080

−0.9129

2Plot the magnituderesponsesof the 4 filters in the sameplot; samefor the phaseresponseand the groupdelay. Plot the

magnituderesponse(in dB) twice—onceshowing thewholepictureandoncezoomedin on thepassband.Usesensiblescalesso

that thedetailsof the responsecanbe seen.e.g. with notches,the responsegoesdown to −∞dB andthe default scalemay be

totally unsuitable.3You don’t needto rescalethefilter andsimulate.Youshouldbeableto answerthis by lookingat theprototyperesponse.



due on 15 Apr. 2003

Designandsimulatethefollowing active versionsof theInverseChebyshev filter (scaledto a 2MHz pass-

band)givenin HW8. Startwith all resistorsof 10 kΩ or all gm of 100µS.

Scalethecircuit to have equalmaximain theac responseof all opamp/gm outputs.Submittheschematic

with all the componentvaluesandthe magnituderesponseplots beforeandafter scaling. Plot the output

magnitudesof all theoutputsin agivenfilter on thesameplot.

1. (10pts.)Cascadeof opamp-RCbiquadstages—zerosusingfeedforward.

2. (10pts.)gm-C ladderfilter.

Table1: Inversechebyshev prototypezerosandpoles:passbandcorner= 1rad/s

InverseChebyshev

zeros poles poleresonantfrequency polequality factor

±j3.0671 −0.2811 ± j1.1013 1.1366 2.0218

±j1.8956 −0.9461 ± j0.8751 1.2887 1.4202

−1.4202 n/a n/a

+

-

0.43518F

0.085068F

1.5592F

0.27046F

0.28141F

1.2496H 1.0290H

Vi Vo

-

+

1Ω 1Ω

Figure1: Inversechebyshev doublyterminatedladderprototypewith polesandzerosshown in Table1

1



due on 29 Apr. 2003

vi+α3vi3 1/kk

vi+α3vi3

(a)

(b)

Vi=Vpcos(ωt)VoΣ

vn

Σ

vn

Vi=Vpcos(ωt)Vo

Figure1:

1. (4+2+3pts.)Fig. 1 shows ablock thathasthird orderdistortionandanoutputnoisevn (rmsvolts). It

couldrepresenta filter or any othercircuit thathasdistortionandnoise.Theinput is a sinusoidwith

apeakVp.

(a) In Fig. 1(a,b) calculatethe following quantitiesat the output: Peakvalueof the fundamental

sinusoid,amplitudeof thethird harmonic,rmsoutputnoise,ratio of thethird harmonicpeakto

thefundamentalpeak,ratio of rmsnoiseto rmsfundamental.Neglectthecontribution from the

v3

i termwhile calculatingtheoutputfundamentalamplitude.

(b) How doesk affectthenoise/signal1 anddistortion/signalratios?Whatwouldyoudowith k to (a)

minimizenoise/signal,(b) distortion/signal?Giveaverybrief intuitive explanation.Computek

suchthatnoise/signalanddistortion/signalratiosareequal.

(c) If α3 = 0.002 V−2, vn =√

2mV, rms, Vp = 1 V, calculatek for equalnoise/signalanddistor-

tion/signalratios. With thesenumericalvalues,calculatethenoise/signalanddistortion/signal

ratiosin Fig. 1(a,b). How do thetwo circuitscompare?

2. (2+3+2+1+2pts.)C = 1/2π nF, R = 1 kΩ, L = 10/2π µH.

(a) Calculatetheoutputnoisevoltageof thecircuit in Fig. 2(a).

(b) Simulatethe noise in Fig. 2(a). To computethe meansquarednoise, integrate the spectral

densityfrom i) 1/10 the -3dB bandwidthto 10 timesthe -3dB bandwidth,andii) 1/100 the

-3dB bandwidthto 100timesthe-3dB bandwidth.How differentarethetwo values?1“signal” implicitly means“desiredsignal”, in this casethefundamental.

1

2

C C LR R

+

-

+

-

+

-

+

-Vi Vo

Vi Vo

(a) (b)

Figure2:

(c) Simulatethenoisein Fig.2(a).Tocomputethemeansquarednoise,integratethespectraldensity

in therangef0 ± 10fB wheref0 is thecenterfrequency andfB is the -3dB bandwidthof the

bandpassfilter.

(d) SetL = 0.1/2π µH andrepeattheprevioussimulation.

(e) Comparethe noisein the threecasesabove. What is the bandwidthof the circuit in the three

cases?Doesthevalueof themeansquarednoisemake sense,consideringthat it is thespectral

densityintegratedoveracertainbandwidth?

3. (1+4+4+2pts.)Theinput referrednoisevoltageof a transconductorgm is γ4kT/gm.

gm,OPA

R

R

+

-Vi +

-Vo

in,R

in,R

+−vn,gm

-+

-+

-+

+

-Vi

-Vo

+

gmgm

(a) (b)

in,gmin,gm

Figure3:

(a) Calculategm,OPA in Fig. 3(a) if theloop gainhasto be100(HW2 hadproblemsrelatedto the

useof a transconductorasanopamp).

(b) Calculate2 thenoisespectraldensityattheoutputin Fig.3(a,b) in termsof kT, gm, gm,OPA, R, γ.

(c) In theexpressionfor Fig. 3(a)substituethevalueof gm,OPA calculatedin (i). In theexpression

for Fig. 3(b) substitutegm = 1/R. What can you say aboutthe relative valuesof noisein

Fig. 3(a)andFig. 3(b) assuminge.g. γ = 5. Thecomparisonis typically true for opamp-RC

andgm-C filters.

(d) If Vi = Vp cos(ωt) whatis thepeakcurrentdrivenby eachactive componentin Fig. 3(a)?

2It is easiestif yourepresentthenoiseof differentcomponentsasshown. While analyzingFig. 3(a),youcanassumeanopamp

with infinite gain.

E4215: Analog Filter Synthesis and Design: Project

Equalizer for 1 Gb/s data


due on 6 May 2003

1 Description

+−

channel(lowpass)

Equalizinginput output

clk(1 GHz)

-1V

1V

Σ

100mV clock

transmitterAB

feedthrough

filter

2.5V

Figure1: Transmitter, channel,andtheequalizer

Digital dataat fs = 1 Gb/sfrom a transmitter(Fig. 1) passesthrougha lowpasschannelwhich attenuates

someof thehigh frequenciesof thesignal.Additionally, someof theclock at fs = 1 GHz leaksto thedata

output.

Your job is to designanequalizingfilter to boostthehigh frequenciesof thesignalaroundfs/2 = 500 MHz

andfilter theclock feedthroughatfs = 1 GHz. Thefilter is requiredto have a linearphase.

• For linearphase,startwith aseventhorderBesselfilter with a -3dB bandwidthof fs/2 = 500 MHz.

• Add a pair of complex conjugatezerosanda pair of equalandoppositerealzerosto obtaina +3dB

boostatfs/2 = 500 MHz and10dB attenuationatfs = 1 GHz.

• Youcanuseany topologythatstrikesyourfancy: opamp-RCor gm-C; ladderor cascade;singleended

or differential.

• Thetotal capacitanceusedin your filter mustbe2.xxpF wherexx arethelast2 digits of your social

securitynumber.

• Thedc gainof thefilter mustbe0dB.

1

2

2 Project submission

1. Giveacleardescriptionof thefollowing in your report.

• Prototypelowpassfilter design;computationof zerosto gettheboostatfs/2 andattenuationat

fs.

• Detaileddesignof thefilter atthedesiredfrequency with all theresistor/gm andcapacitorvalues.

• Scalingthefilter to have equalmaximain theacresponseat all opamp/gmoutputs.Scalingthe

filter to usea total capacitanceof 2.xxpF.

• A completeschematicwith all thecomponentvalues.Useasensiblehierarchysothatthedesign

is understandable.

2. Beforetheduedate(6 May 2003,5pm)e-mailmeyourcadencelibrary paththatcontainstheproject,

andthenameof thetopmostcell in your hierarchy.

3. Submitthefollowing simulationresults.

• Frequency response:magnituderesponseat the filter’s outputshowing the gain boostat fs/2

andtheattenuationatfs; groupdelayresponse;plot with overlaidmagnituderesponseatall the

opamp/gmoutputs.

• Transient:Show theresponseof thefilter for asingle1V pulsewhosedurationis 1/fs = 1 ns.

You will begiventhewaveformsof thebit streamsat A andB. Simulatethefilter with its input

being the sumof the channeloutputand the clock feedthrough(100mV sinusoidat 1 GHz).

Show the outputsof the transmitterandthe channel(waveformswill be given to you) andthe

outputof thefilter. Briefly describewhatyourfilter hasdoneto thesignal.

• Noise: Show thenoisespectraldensityat theoutput. Computethe integratednoiseuptofs =

1 GHz. Calculatetheoutputsignalto noiseratio,assumingthata1V sinusoidat low frequencies

is appliedto thefilter.

• Power dissipation:Plot the frequency responsemagnitude(with an input magnitudeof 1V) of

theoutputcurrentsof theeachof theopamp/gm. Tabulatethemaximumof eachof thecurrent

magnitudesover frequency. Thesewill bethelargestcurrentsdrawn from eachopamp/gm.

If you areusingopamps,take the largestof theseandmultiply by 8. This will be the current

drawn peropamp.Multiply by thenumberof opampsto arrive at the total currentdissipation.

This meansthatyouareusingidenticalopampswhich arecapableof driving thelargestcurrent

demandedin thiscircuit. This is acommonsituationin filter design.

If you areusinggms,multiply thelargestcurrentdrawn from eachgm, by 8 andsumtheresult

to obtainthetotal currentdissipation.Notethatyou cannotin generaluseidenticalgm s asthe

transferfunctiondependson thevalueof thegm s.

Computethepower dissipationassumingthatthesupplyvoltageis 2.5V.

3

3 Simulation/modeling

• You can generatea voltagesourcewith an arbitrary waveform using the voltagesourcevpwlf in

the library analogLib. You needto specifya file that hasthe voltagevaluesat certaintime points.

/u2/nagi/courses/E4215/project/tx output.dat and/u2/nagi/courses/E4215/project/channel output.dat

have thetransmitterandthechanneloutputsrespectively.

• Model theclock feedthroughusinga 100mV sinusoidatfs = 1 GHz in serieswith theinput voltage

source.

• Usethesubcircuitsin Fig.2 tomodelgm sandopamps.Youcanmaketheseintosubcircuits(parameterized

if necessary)andusethem. The1GΩ resistorsarethereto provide dc pathsto groundandsuppress

warningsfrom thesimulatoraboutfloatingnodes.They will not affect theoperationof thecircuit if

you have calculatedthe componentvaluesin your circuit correctly). The resistorin serieswith the

negative input of thecells is for modelingthenoiseof theopamps/gm s. They too will not affect the

operationof yourcircuit asthecurrentflowing throughthemis negligible.

+

-v gmv

+

- 4/gm

1GΩ 1GΩ

(noise)

+

-v

gmv+

-

+

-

4/gm

1GΩ0.5GΩ

(noise)gmv

0.5GΩ

+

-v

+

-

1GΩ

(noise)

+-

1000v

+

-v

+

-

+

-

1GΩ

(noise)

+-

+-

(a) (b)

(c) (d)

in

in

out

out

out

out

in

in

25Ω25Ω

500v

500v

Figure2: (a)Singleendedgm, (b) Differentialgm, (c) Singleendedopamp,(d) Differentialopamp.

4 Timeline

Thereare4.5 weeksto the projectdeadline.Budget2 weeksfor designand2 weeksfor simulationand

writing the report. The designcan be startedwith what you have learnedin the classso far. For the

prototypefilter you canconsultA. I. Zverev, Handbook of Filter Synthesis, Wiley, New York, 1967,which

is anoncirculatingreferencein theEngineeringlibrary.

Documents

E4215: Analog Filter Synthesis and Design: HW0 · 2006. 8. 21. · E4215: Analog Filter Synthesis and Design: HW0 Nagendra Krishnapura([email protected]) due on 21 Jan. 2003 This