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
=
3. Thecircuit in Fig. 2 is
Ic =
4. Thecircuit in Fig. 3 is
vo
vi
=
RL
+
-vi
+
-
vo
Figure1:
1mA
Ic
1x 1x
Figure2:
5. Thecircuit in Fig. 4 is
6. Thecircuit in Fig. 5 is
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:
E4215: Analog Filter Synthesis and Design: HW1
Nagendra Krishnapura ([email protected])
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
E4215: Analog Filter Synthesis and Design: HW2
Nagendra Krishnapura ([email protected])
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:
E4215: Analog Filter Synthesis and Design: HW3
Nagendra Krishnapura ([email protected])
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:
E4215: Analog Filter Synthesis and Design: HW4
Nagendra Krishnapura ([email protected])
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.
E4215: Analog Filter Synthesis and Design: HW5
Nagendra Krishnapura ([email protected])
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)?
E4215: Analog Filter Synthesis and Design: HW6
Nagendra Krishnapura ([email protected])
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.
E4215: Analog Filter Synthesis and Design: HW7
Nagendra Krishnapura ([email protected])
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.
E4215: Analog Filter Synthesis and Design: HW8
Nagendra Krishnapura ([email protected])
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.
E4215: Analog Filter Synthesis and Design: HW9
Nagendra Krishnapura ([email protected])
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
E4215: Analog Filter Synthesis and Design: HW10
Nagendra Krishnapura ([email protected])
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
Nagendra Krishnapura ([email protected])
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.