E4215: Analog Filter Synthesis and Design: HW0

Nagendra Krishnapura ([email protected])

due on 21 Jan. 2003

This assignment has ZERO credit and does not con- tribute to the final grade. Its purpose is to gauge your RL familiarity of prerequisite topics. + 1. Check the terms that are unfamiliar to you: v • Laplace transform + o v • Impulse response - i - • Frequency response • Transfer function Figure 1: • Bode plot • Operational amplifier 1mA • Bipolar transistor I • MOS transistor c • Small signal equivalent circuit 1x 1x • Common drain amplifier • Loop gain • Gain margin • Phase margin Figure 2: 2. The circuit in Fig. 1 is 5. The circuit in Fig. 4 is v o = vi 6. The circuit in Fig. 5 is 3. The circuit in Fig. 2 is Vx =

Ic = Vy =

4. The circuit in Fig. 3 is 7. Transfer function of the circuit in Fig. 6:

vo V s = o( ) vi = Vi(s)

1 2

Ω Ω 1K 1K 2KΩ

v1 v2 + 1KΩ V o − - + + + + 1V Vx + V Vi y - ideal opamp - - -

2mA

Figure 5: Figure 3: R + + Vi C Vo - -

+ vi + -

vo Figure 6:

1KΩ 2KΩ - + + ω 3Vcos( t) Vo Figure 4: - -

8. In Fig. 7 Figure 7:

Vo = L C R 9. Transfer function of the circuit in Fig. 8: + + Vi Vo V (s) - - o = Vi(s)

10. In Fig. 9: Figure 8: v o = vi gm rds RL + + v v - i - o

Figure 9: E4215: Analog Filter Synthesis and Design: HW1

Nagendra Krishnapura ([email protected])

due on 28 Jan. 2003

R C R/2 2C R C R C + + + + 1 1 2 2

Vi(s) Vo(s) Vi(s) Vo(s) + + + - - - - + Vi(s) vx - vx Vo(s) (a) (b) - - -

ii(t) ii(t) R C R/2 2C (a) + + + + R2 vo(t) vo(t) - - - - (t)=1Vcos(t/RC) (t)=1Vcos(t/RC) i i R1 C1 C2 R /4 v v 2 (c) (d) + + + i(t) + Vi(s) vx - vx Vo(s) - - - + v(t) - (e) (b)

1V Figure 1: vi(t) 1. (5 pts.) For the circuits in Fig. 1(a) and 0V Fig. 1(b), evaluate the transfer function H(s) = T (c) Vo(s)/Vi(s), and the impulse response h(t) cor- responding to H(s). Approximately sketch Figure 2: the magnitude and phase of H(s) (Bode Plot). What is the difference between the two circuits? Fig. 2(a) and Fig. 2(b). Sketch the Bode plots assuming R1C1 = 4R2C2. 2. (5 pts.) In the circuits in Fig. 1(c) and Fig. 1(d), evaluate the current ii(t) through the input volt- 4. (5 pts.) The circuit in Fig. 2(b) is driven by a age source. Evaluate the average power dis- pulse with an amplitude 1V and lasting T sec- sipated in the voltage source and the resistor. onds (Fig. 2(c)). Assuming T = R1C1, sketch What is the difference between the two circuits? the intermediate voltage vx(t). Sketch the out- Note: Average power dissipated in an element put voltage vo(t) assuming that R2C2 = R1C1. with a voltage v(t) across it and a current i(t) through it (see Fig. 1(e)) is given by 1 T P = v(t)i(t)dt T Z0 3. (5 pts.) Write the expressions for the transfer function H(s) = Vo(s)/Vi(s) for the circuits in

1 E4215: Analog Filter Synthesis and Design: HW2

Nagendra Krishnapura ([email protected])

due on 4 Feb. 2003

For the opamps, use the appropriate model based on unity gain frequency ωu = 1 Grad/s1, draw the the parameters provided. i.e. if nothing is given, as- Bode plot (magnitude and phase) of loop gain sume an ideal opamp with inﬁnite gain; if the unity T (s) and op amp gain A(s). gain frequency is given, use the integrator model; if 4. (6 pts.) Assume gm = 1 mS, R1 = the dc gain and the unity gain frequency are given, 900 kΩ, R2 = 100 kΩ, RL = ∞, Ao = 1000. use the ﬁrst order model etc. This holds for all future For the circuits in Fig. 2(a) and Fig. 2(b), eval- assignments. uate the gain Vo/Vi and the feedback loop gain T. Repeat, assuming RL = 1 MΩ.

m 1 2R 5. (6 pts.) Assume g = 1 mS, R = R 900 kΩ, R2 = 100 kΩ, CL = 10 pF, Ao = v1 1000, ωu = 100 Mrad/s2. For the circuits R ? in Fig. 2(c) and Fig. 2(d), evaluate the trans- + g v v + v + m o fer function Vo(s)/Vi(s) and the feedback loop i − ? o - - gain T(s). Write the transfer functions in the standard ﬁrst order form and compare the two results. Repeat, assuming CL = 20 pF. Figure 1:

1. (2 pts.) [Fig. 1, gm = 4/R] Assign the cor- rect signs to the opamp such that it has negative feedback at dc.

2. (2 pts.) [Fig. 1, gm = 4/R] Assuming that the opamp has a transfer function A(s) = ωu/s, determine the transfer functions Vo(s)/Vi(s), V1(s)/Vi(s).

3. (4 pts.) [Fig. 1, gm = 4/R] Determine the loop gain T (s) around this feedback loop. Assuming

that the opamp has a dc gain Ao = 100 and a 1giga radians/second; giga=109 2mega radians/second

1 2

gm = 1mS Ao = 1000

+ + + + - + − + 2 2 R R V V V V i o L i o L R R 1 1 R R - - - -

(a) (b)

ω gm = 1mS Ao = 1000, u = 100Mrad/s

+ + + + - + − + 2 2 R R

Vi Vo Vi Vo L L C C 1 1 R R - - - -

Figure 2: 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 pression relating the output Vo to the input Vin1 from HW2 are also due on 11 Feb. 2003. and the offset Vos. Draw the dc transfer charac-

teristics Vo vs. Vin1 including the effect of offset 1kΩ assuming that Vos > 0. Show the input referred offset and the output offset of the ampliﬁer in 1kΩ − Fig. 1(a) on this plot. (Hint: In a circuit with + 1kΩ multiple inputs, try using superposition). Vin1 + + - Vo opamp - If the standard deviation of Vos is σ = 5 mV, with offset what is the standard deviation of the input re- (a) ferred offset and the output offset of the ampli- ﬁer in Fig. 1(a). 1kΩ What is the net output offset (in the output Vo) of the circuit in Fig. 1(b)? (Hint: Use the results 1kΩ − + related to Fig. 1(a) to determine Vo1 and Vo2. Ω Vin1 1k + + Relate to 1 and 2) - Vo1 1kΩ Vo Vo Vo opamp - with offset 2. (5 pts.) In Fig. 2(a), determine Vp,max, the max-

Ω imum value of Vp such that the output vo(t) 1k + Vo is sinusoidal. The opamp has the characteris- - tic shown in Fig. 2(b) (The slope of the vertical 1kΩ − 1kΩ + part is ∞. Sketch vo(t) when Vp = Vp,max/2 1kΩ Vin2 + + and when - Vo2 Vp = 2Vp,max opamp - with offset 2 3. (3 pts.) In Fig. 3, vo = f(vi) = vi + a2vi + (b) 3 a3vi . If vi = Vp cos(ωt), express vo(t) as a sum of sinusoids. Find the ratio of the 2nd and 3rd Figure 1: harmonic amplitudes to that of the fundamental. −3 −1 −3 −2 1. (9 pts.) The opamps in Fig. 1 have an input If a2 = 10 V , a3 = 10 V , ﬁnd the in-

referred offset voltage Vos, but are otherwise put peak Vp such that the second harmonic is ideal (A0 = ∞). For Fig. 1(a), derive the ex- 60 dB below the fundamental. Repeat the exer-

1 2

2kΩ

1kΩ − + v =V cos(ωt) in p + + - vo - (a)

vout 1V gm2 + vid + - V - i - -1V + g (b) m1

Figure 2: (a) gm3 - + + vi vo Vi1 f(vo) -

gm2 Figure 3: + + - V - i2 - + cise for the third harmonic. C gm1 4. (3 pts.) Assuming ideal transconductors1, de- (b) rive expressions relating Vo to Vi in Fig. 4(a)

and to Vi1 and Vi2 in Fig. 4(b). Figure 4: Repeat for Fig. 4(a) assuming that the transcon-

ductor gmx has an output resistance rox and in-

put and output capacitances Cix, Cox. x = {1, 2} for the two transconductors in Fig. 4(a).

1voltage controlled current source E4215: Analog Filter Synthesis and Design: HW4

Nagendra Krishnapura ([email protected])

due on 18 Feb. 2003

RS the transfer function to the original? Reevaluate first-order + the transfer function with RS = 0, RL = 1 MΩ. RL vo(t) vs(t) filter - How would you restore the pole to the original value?

Figure 1: (1 pt.) With RS = 10 kΩ, RL = 1 MΩ, choose R, C such that the pole of the filter is the same 1. Initially, assume RS = 0, RL = ∞. Fig. 1 as originally determined. What is the transfer shows a first order filter whose input is the sum function? Determine the attenuation of the two of two sinusoids vs(t) = 1V cos(1 Mrad/s t) + sinusoids. 1V cos(1000 Mrad/s t). The higher frequency 1 2 N sinusoid should be attenuated by 40 dB and the R R R + 1 1 1 + lower frequency sinusoid should be attenuated vs(t) vo(t) - C C C - as little as possible.

(2 pts.) Determine the transfer function of the Figure 2: filter. Draw the schematic of a passive RC filter with R = 100 kΩ that will accomplish this. 2. (3 pts.) Fig. 2 shows a cascade of N identi- What is the attenuation (in dB) of the lower fre- cal buffered first order filter sections. vs(t) = quency sinusoid? 1V cos(1 Mrad/s t) + 1V cos(10 Mrad/s t). Us- (3 pts.) In the previously designed filter, if ing simple Bode Plots, determine the smallest R and C can have variations of 10%, (a) N required to reduce the higher frequency sig- What are the maximum and minimum values nal by 80 dB while leaving the lower frequency of the pole frequency? What is the percent- signal unchanged. What is the value of the pole age variation from the nominal value? (b) What of each section? Now, using the transfer func- is the worst case (smallest) attenuation of the tion of the filter so obtained, find the actual higher frequency signal? (c) What is the worst attenuation of the two signals. Recompute N case (largest) attenuation of the lower frequency and the pole of the filter if the lower frequency signal? should be attenuated by ≤ 3 dB and the higher frequency by ≥ 80 dB. (1 pt.) Reevaluate the transfer function with RS = 10 kΩ, RL = ∞. How would you restore 3. (3 pts.) Design an ac coupling stage between the

1 2

+ 1V −

dc bias=1V

stage 1 ac coupling stage 2

Ci = 1pF

Figure 3:

two stages shown in Fig. 3. The second stage has an input capacitance Ci = 1 pF. a) The attenuation for very high frequencies (ω → ∞) should be less than 1 dB, b) The attenuation for 10 Mrad/s should be less than 4 dB, c) The capacitor used in the circuit should be minimized. d) The dc bias provided to the 2nd stage should be 1 V (A 1 V dc source is available to you.).

4. (1 pt.) Design a ﬁlter with the transfer function −k/(1 + s/p1), with k = 10, p1 = 10 Mrad/s. Draw the schematic with ideal opamps—use C = 1 pF.

(2 pts.) Determine the dc gain Ao and the unity gain frequency ωu of the opamp such that each of these nonidealities (acting by itself) changes the pole of the filter by less than 2.5%. (2 pts.) Draw the Bode plot of the loop gain for the filter you designed. Use an integrator model for the opamp with ωu determined previously. (2 pts.) Redesign the filter (use ideal opamps) assuming that the largest resistor allowed is 10 kΩ. E4215: Analog Filter Synthesis and Design: HW5

Nagendra Krishnapura ([email protected])

due on 25 Feb. 2003

Rs L RL two circuits. + +

Vs C Vo - - Rs C=1pF L R +

Vs Vo - Figure 1: (a)

1. (2 pts.) Determine Vo(s)/Vi(s) for the ﬁlter Rs R L C=1pF in Fig. 1. For a given R , determine R such s L + that Q is maximum. What is the maximum Q? Vs Vo - What is ωp under this condition? (b) R Figure 3: + +

Vi C Vo - - 3. (4 pts.) For each of Fig. 3(a) and Fig. 3(b), (a)

Assuming Rs = 0 determine L and R so that (a) 1 a bandpass ﬁlter with ωp/2π = 5 GHz and a - L R 3dB bandwidth of 1 GHz is realized. (b) If vs(t) + + is a 1V sinusoid at 5 GHz, what is the current Vi C Vo - - ﬂowing through the input source? (c) What is the value of Rs, the source resistance, that re- (b) sults in a 10% deviation in Q?

Figure 2: 4. (5 pts.) In Fig. 4 consider two cases R1 = R2 = R and R1 = 2R, R2 = R/2. 2. (2 pts.) What is the bandwidth of the circuit For each of these, (a) Find V1(s)/Vi(s) Is there in Fig. 2(a)? If you were allowed to place a a difference? (b) Evaluate Vk(s)/Vi(s), k = series inductor L as in Fig. 2(b), what value 2, 3 Is there a difference? What is the max- would you choose for it to maximize the band- { } imum of Vk(jω)/Vi(jω) ? (c) The input is a width without introducing peaking in the mag- | | sinusoid vi(t) = Vip cos(ωt) where ω can be nitude response? What is the resulting band- 1 This means that ωp = 2π × 5 Grad/s width? Sketch the frequency responses of the

1 2

− R R1 V3 + 5R OPA3 C C

R V − i R2 − + V1 + V2 OPA1 OPA2

Figure 4:

anything. If the opamps have a swing limit of

1V, what is the largest Vip that can be applied while maintaining all the opamps in the linear region?

5. (3 pts.) (a) Design a second order gm-C But- terworth ﬁlter with dc gain=1 and -3dB band-

width=1 MHz. Assume that the smallest gm is 10 µS. Give the transfer function and all the

component values in the gm-C filter schematic. (4 pts.) (b) Using the above filter as the basis, design a lowpass notch filter with dc gain=10 and a notch at √10 MHz. Use the voltage summing technique. Give the transfer function

and all the component values in the gm-C filter schematic. What is the high frequency gain of this filter? What is the attenuation of the filter at 1 MHz w.r.t. dc? Has the -3dB bandwidth increased or decreased compared to the filter in (a)? E4215: Analog Filter Synthesis and Design: HW6

Nagendra Krishnapura ([email protected])

due on 4 Mar. 2003

g In addition to the problems here, problem #5 from Vi m + R L=1H C HW5 is also due on 4 Mar. 2003 - 1. (1+3+3 pts.)Repeat the design in problem #5 of HW5 using opamps and feedforward technique. Use 10 pF capacitors. (a) Design the Butterworth lowpass ﬁlter. Figure 2: (b) Obtain the lowpass notch transfer function 3. (1+2+2+2+1+3 pts.) (a) Design a 1 H inductor at the output V11. using transconductors and a 100 pF capacitor. (c) Obtain the lowpass notch transfer function (b) Derive the (passive) equivalent circuit of at the output V2. the previously designed inductor if the capaci-

gm tor had a 1 MΩ resistor across it. - + (c) Design an RLC bandpass ﬁlter with ωp = 100 krad/s and Q = 10 using a 1 H inductor. V gm1i i1 gm The gain at the resonant frequency should be + V1 V2 - g /Q + 10. Use the topology in Fig. 2. m - - C (d) Replace the inductor with the equivalent cir- + C cuit obtained in (b) and re-evaluate the transfer

gm2i function Vo(s)/Vi(s) What, if any, is the devia- Vi2 + tion from the intended design in (c). - (e) How would you change the design to restore the Q to 10? You cannot remove the 1 MΩ resistor which is across the capacitor. Figure 1: (f) Simulate (i) the circuit in Fig. 2, (ii) the cir- 2. (2 pts.) In Fig. 1, Determine the transfer func- cuit with the inductor replaced by the active in- 2 tions from Vi1 and Vi2 to voltages V1 and V2. ductor , and (iii) the repaired circuit from (e). 2use the circuit with transconductors and capacitors, not the 1output of OPA1; in the handout “Transfer functions realiz- equivalent obtained in (b); Include the 1 MΩ resistor across the able in a biquad”.

1 2

Submit the magnitude and the phase responses; overlay the responses of the three circuits.

100 pF capacitor. E4215: Analog Filter Synthesis and Design: HW7

Nagendra Krishnapura ([email protected])

due on 25 Mar. 2003

For 1-5, give the schematic of the passive filter with is the frequency Ωp at which the attenuation is 2 all the element values. For 1-3, give the the transfer Ap? What is the smallest frequency at which function in the normalized form which is the attenuation is As = 20 dB? Call this Ωs. − 2 b0 + b1(s/Q/ωn) + b2(s/ωn) 2 3. (3 pts.) Transform the prototype in (2) to a 1 + s/Q/ωn + (s/ωn) passive RLC highpass filter with an attenuation where ωn is a convenient normalizing frequency. For Ap (determined in (1)) at 10 Mrad/s and a ter- 3-5, give the expression for the frequency transfor- mination impedance 10 kΩ. What is the fre- mation along with the numerical values for the pa- quency of the notch in this filter? Draw the rameters in the transformation. For 6, give the fi- schematic replacing the inductors with capaci- nal schematic and explain very briefly the purpose of tively terminated gyrators whose gyration resis- each feedforward component1. tance is 10 kΩ. 1. (1 pt.) Design a second order passive low- 4. (4 pts.) Transform the prototype in (2) to a pas-

C1 sive RLC bandpass ﬁlter whose attenuation is

Ap at ωp1 = 10 Mrad/s and ωp2 = 12.1 Mrad/s. C2 R The termination impedance should be 10 kΩ. + + L What are the “stopband” edges ωs1 and ωs2 Vi Vo - - where the attenuation is As? What is the gain of the ﬁlter at 11 Mrad/s? If one of the notches of the ﬁlter is at 4.7 Mrad/s, where is the other Figure 1: notch?

pass RLC notch ﬁlter with Q = 1/√2, ωp = 5. (4 pts.) Transform the prototype in (2) to a

1 Mrad/s and a transmission zero at ωz = passive RLC bandstop ﬁlter whose attenuation

10 Mrad/s. Use the topology in Fig. 1 with is at least As in the range 81 Mrad/s ω ≤ ≤ C1 + C2 = 10 nF. What is the attenuation in 100 Mrad/s. Use a termination impedance of

dB at 1 Mrad/s? Call this Ap. 1 kΩ. What are the “passband” edges ωp1 and

2 nd 2. (2 pts.) Scale the ﬁlter in (1) so that it uses You can calculate this analytically-you’ll get a 2 order equation in Ω2; or determine it using simulation-be sure to use a R = 1 Ω and has a notch at 10 rad/s. What sufﬁciently small frequency step. 1Elements from the input to various opamps.

1 2

ωp2 where the attenuation is Ap? What is the ﬁlter’s attenuation at 90 Mrad/s?

6. (2 pts.) Realize an an opamp-RC version of the highpass ﬁlter in (3). Use the Tow-Thomas biquad with feedforward technique to realize the zeros at the output of the ﬁrst opamp. Use R = 10 kΩ in the resonator core.

7. (1 pt.) Realize a bandpass ﬁlter whose atten-

uation is Ap at fp1 = 10 MHz and fp2 = 12.1 MHz. (Hint: You don’t have to go through the whole synthesis again. Use the result from (4)).

8. (2 pt.) Simulate the magnitude response of the passive circuits in 1, 3 (not the part with the gy- rator), 4, 5. (Plot all 4 magnitude responses in 4 subwindows of the same plot for submission. Use appropriate ranges for x and y axes to show all points of interest). In each, mark the frequency of the notch(es).

9. (1 pt.) Simulate the magnitude response of the opamp-RC ﬁlter in 6. For the opamps use ideal voltage controlled voltage sources with gain=106. E4215: Analog Filter Synthesis and Design: HW8

Nagendra Krishnapura ([email protected])

due on 8 Apr. 2003

+ + - b2IC + + + b1IC b1IR V transconductance=1S ω - transconductance=1S ω ω - o C=1/ p R=1 C=1/ p R=Q L=1/ p + Vo + Vi + Vi + Vi - b0IR Vi - b0IL - - - - IC IR IC IR IL

(a) (b) "bilinear" "biquad"

Figure 1:

1. (a) (5 pts.) Compute the transfer functions Vo/Vi in terms of the parameters (Q, ωp, b0, b1, b2) for the circuits in Fig. 1(a, b). (b) Turn these circuits into parameterized subcircuits “bilinear” and “biquad” in cadence1 with the required parameters. You can then use these subcircuits to realize ideal cascade realizations of any transfer function.

0dB 0dB -1dB -1dB

-40dB -40dB 2MHz 4MHz 2 rad/s 1 rad/s (a) (b)

Figure 2:

2. You are required to realize a filter that meets the specifications shown in Fig. 2(a). You are given (Table 1) 1In cadence, to realize a current controlled voltage source, you also need to have a 0 V voltage source through which the desired current is flowing. See the example subcircuit “lpf” in the library “E4215 examples”.

1 2

the poles and zeros of 4 types (Excluding Bessel) of ﬁlters which satisfy the prototype speciﬁcations in Fig. 2(b).

(a) (4 pts.) Tabulate the order, the resonant frequencies, the quality factors of the poles, and the location of transmission zeros (if present) of the different types of filters that satisfy the specs. in Fig. 2(a). (b) (7 pts.) Using the parameterized subcircuits for the bilinear and the biquadratic filters, simulate the four filters (using the cascade structure) in cadence. Use the rules of cascading discussed in the class. You do not have to submit the schematics. Clearly state the order of cascade and the pole zero pairing. Plot their magnitude and phase responses2, and the group delay (for this, you can use the function “groupDelay” in the calculator in cadence). (c) (4 pts.) For each filter, determine the maximum transfer function magnitude from the input to each of the stage (first or second order) outputs. If each output were limited to 1 V, what is the maximum input voltage that could be applied to each without having distortion? (d) (4 pts.) Simulate the transfer function of the Bessel filter prototype (last column of Table 1) using

the same technique as above. If this filter were scaled such that it had an attenuation As = 40 dB at 4 MHz (the stopband edge), what would be its attenuation at the passband edge (2 MHz)?3 Does it meet the specs in Fig. 2(a)? (e) (4 pts.) For each of the 4 filters that satisfies the specs in Fig. 2(a), list the maximum quality factor of the biquad stages used, the maximum resonant frequency, and the maximum group delay variation in the passband (< 2 MHz). (2 pts.) Repeat3 for the Bessel filter. To find its maximum resonant frequency, calculate the maximum resonant frequency in the prototype and multiply it by the scaling factor determined above.

Table 1: Prototype zeros and poles

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 magnitude responses of the 4 ﬁlters in the same plot; same for the phase response and the group delay. Plot the magnitude response (in dB) twice—once showing the whole picture and once zoomed in on the passband. Use sensible scales so that the details of the response can be seen. e.g. with notches, the response goes down to −∞ dB and the default scale may be totally unsuitable. 3You don’t need to rescale the ﬁlter and simulate. You should be able to answer this by looking at the prototype response. E4215: Analog Filter Synthesis and Design: HW9

Nagendra Krishnapura ([email protected])

due on 15 Apr. 2003

Design and simulate the following active versions of the Inverse Chebyshev ﬁlter (scaled to a 2 MHz passband) given in HW8. Start with all resistors of 10 kΩ or all gm of 100 µS. Scale the circuit to have equal maxima in the ac response of all opamp/gm outputs. Submit the schematic with all the component values and the magnitude response plots before and after scaling. Plot the output magnitudes of all the outputs in a given ﬁlter on the same plot.

1. (10 pts.) Cascade of opamp-RC biquad stages—zeros using feedforward.

2. (10 pts.) gm-C ladder ﬁlter.

Table 1: Inverse chebyshev prototype zeros and poles: passband corner = 1 rad/s

Inverse Chebyshev zeros poles pole resonant frequency pole quality 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

1.2496H 1.0290H 1Ω 1Ω

+ + Vi 0.085068F 0.27046F Vo - -

0.43518F 1.5592F 0.28141F

Figure 1: Inverse chebyshev doubly terminated ladder prototype with poles and zeros shown in Table 1

1 E4215: Analog Filter Synthesis and Design: HW10

Nagendra Krishnapura ([email protected])

due on 29 Apr. 2003

V =V cos(ωt) i p α 3 Σ vi+ 3vi Vo

(a) vn

V =V cos(ωt) i p α 3 Σ k vi+ 3vi 1/k Vo

(b)

Figure 1:

1. (4+2+3 pts.) Fig. 1 shows a block that has third order distortion and an output noise vn (rms volts). It could represent a ﬁlter or any other circuit that has distortion and noise. The input is a sinusoid with

a peak Vp.

(a) In Fig. 1(a, b) calculate the following quantities at the output: Peak value of the fundamental sinusoid, amplitude of the third harmonic, rms output noise, ratio of the third harmonic peak to the fundamental peak, ratio of rms noise to rms fundamental. Neglect the contribution from the 3 vi term while calculating the output fundamental amplitude. (b) How does k affect the noise/signal1 and distortion/signal ratios? What would you do with k to (a) minimize noise/signal, (b) distortion/signal? Give a very brief intuitive explanation. Compute k such that noise/signal and distortion/signal ratios are equal. −2 (c) If α3 = 0.002 V , vn = √2mV, rms, Vp = 1 V, calculate k for equal noise/signal and distortion/signal ratios. With these numerical values, calculate the noise/signal and distortion/signal ratios in Fig. 1(a, b). How do the two circuits compare?

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

(a) Calculate the output noise voltage of the circuit in Fig. 2(a). (b) Simulate the noise in Fig. 2(a). To compute the mean squared noise, integrate the spectral density from i) 1/10 the -3 dB bandwidth to 10 times the -3 dB bandwidth, and ii) 1/100 the -3 dB bandwidth to 100 times the -3 dB bandwidth. How different are the two values? 1“signal” implicitly means “desired signal”, in this case the fundamental.

1 2

R C R C L + + + + V V Vi Vo i- - o - -

(a) (b)

Figure 2:

in the range f0 10fB where f0 is the center frequency and fB is the -3 dB bandwidth of the bandpass ﬁlter. (d) Set L = 0.1/2π µH and repeat the previous simulation. (e) Compare the noise in the three cases above. What is the bandwidth of the circuit in the three cases? Does the value of the mean squared noise make sense, considering that it is the spectral density integrated over a certain bandwidth?

3. (1+4+4+2 pts.) The input referred noise voltage of a transconductor gm is γ4kT/gm.

in,R

in,R R

R g gm - m + + - Vi + + V - V i - + gm,OPA o - + − - + + vn,gm Vo in,gm - in,gm

(a) (b)

Figure 3:

(a) Calculate gm,OP A in Fig. 3(a) if the loop gain has to be 100 (HW2 had problems related to the use of a transconductor as an opamp). 2 (b) Calculate the noise spectral density at the output in Fig. 3(a, b) in terms of kT, gm, gm,OP A, R, γ.

for Fig. 3(b) substitute gm = 1/R. What can you say about the relative values of noise in Fig. 3(a) and Fig. 3(b) assuming e.g. γ = 5. The comparison is typically true for opamp-RC

and gm-C ﬁlters.

(d) If Vi = Vp cos(ωt) what is the peak current driven by each active component in Fig. 3(a)?

2It is easiest if you represent the noise of different components as shown. While analyzing Fig. 3(a), you can assume an opamp 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

2.5V

1V -1V B channel A input Equalizing output transmitter Σ + (lowpass) filter −

100mV clock clk(1 GHz) feedthrough

Figure 1: Transmitter, channel, and the equalizer

Digital data at fs = 1 Gb/s from a transmitter (Fig. 1) passes through a lowpass channel which attenuates some of the high frequencies of the signal. Additionally, some of the clock at fs = 1 GHz leaks to the data output. Your job is to design an equalizing filter to boost the high frequencies of the signal around fs/2 = 500 MHz and filter the clock feedthrough at fs = 1 GHz. The filter is required to have a linear phase.

• For linear phase, start with a seventh order Bessel ﬁlter with a -3 dB bandwidth of fs/2 = 500 MHz.

• Add a pair of complex conjugate zeros and a pair of equal and opposite real zeros to obtain a +3 dB boost at fs/2 = 500 MHz and 10 dB attenuation at fs = 1 GHz.

• You can use any topology that strikes your fancy: opamp-RC or gm-C; ladder or cascade; single ended or differential.

• The total capacitance used in your ﬁlter must be 2.xx pF where xx are the last 2 digits of your social security number.

• The dc gain of the ﬁlter must be 0 dB.

1 2

2 Project submission

1. Give a clear description of the following in your report.

• Prototype lowpass ﬁlter design; computation of zeros to get the boost at fs/2 and attenuation at fs.

• Detailed design of the ﬁlter at the desired frequency with all the resistor/gm and capacitor values.

• Scaling the ﬁlter to have equal maxima in the ac response at all opamp/gmoutputs. Scaling the ﬁlter to use a total capacitance of 2.xx pF. • A complete schematic with all the component values. Use a sensible hierarchy so that the design is understandable.

2. Before the due date (6 May 2003, 5pm) e-mail me your cadence library path that contains the project, and the name of the topmost cell in your hierarchy.

3. Submit the following simulation results.

• Frequency response: magnitude response at the ﬁlter’s output showing the gain boost at fs/2 and the attenuation at fs; group delay response; plot with overlaid magnitude response at all the opamp/gmoutputs.

• Transient: Show the response of the filter for a single 1 V pulse whose duration is 1/fs = 1 ns. You will be given the waveforms of the bit streams at A and B. Simulate the filter with its input being the sum of the channel output and the clock feedthrough (100 mV sinusoid at 1 GHz). Show the outputs of the transmitter and the channel (waveforms will be given to you) and the output of the filter. Briefly describe what your filter has done to the signal.

• Noise: Show the noise spectral density at the output. Compute the integrated noise upto fs = 1 GHz. Calculate the output signal to noise ratio, assuming that a 1 V sinusoid at low frequencies is applied to the ﬁlter. • Power dissipation: Plot the frequency response magnitude (with an input magnitude of 1 V) of the output currents of the each of the opamp/gm. Tabulate the maximum of each of the current magnitudes over frequency. These will be the largest currents drawn from each opamp/gm. If you are using opamps, take the largest of these and multiply by 8. This will be the current drawn per opamp. Multiply by the number of opamps to arrive at the total current dissipation. This means that you are using identical opamps which are capable of driving the largest current demanded in this circuit. This is a common situation in ﬁlter design.

If you are using gms, multiply the largest current drawn from each gm, by 8 and sum the result to obtain the total current dissipation. Note that you cannot in general use identical gm s as the transfer function depends on the value of the gm s. Compute the power dissipation assuming that the supply voltage is 2.5 V. 3

3 Simulation/modeling

• You can generate a voltage source with an arbitrary waveform using the voltage source vpwlf in the library analogLib. You need to specify a ﬁle that has the voltage values at certain time points. /u2/nagi/courses/E4215/project/tx output.dat and /u2/nagi/courses/E4215/project/channel output.dat have the transmitter and the channel outputs respectively.

• Model the clock feedthrough using a 100 mV sinusoid at fs = 1 GHz in series with the input voltage source.

• Use the subcircuits in Fig. 2 to model gm s and opamps. You can make these into subcircuits (parameterized if necessary) and use them. The 1 GΩ resistors are there to provide dc paths to ground and suppress warnings from the simulator about ﬂoating nodes. They will not affect the operation of the circuit if you have calculated the component values in your circuit correctly). The resistor in series with the negative input of the cells is for modeling the noise of the opamps/gm s. They too will not affect the operation of your circuit as the current ﬂowing through them is negligible.

0.5GΩ 1GΩ + + gmv 1GΩ 1GΩ + + in v out + out - 4/gm - in v gmv gmv - 4/gm - (noise) - (noise) 0.5GΩ

(a) (b)

1GΩ + + + 500v Ω + - 1G v + + out in 25Ω out in v + - - + 500v Ω - 25 - - (noise) - - (noise) 1000v

Figure 2: (a) Single ended gm, (b) Differential gm, (c) Single ended opamp, (d) Differential opamp.

4 Timeline

There are 4.5 weeks to the project deadline. Budget 2 weeks for design and 2 weeks for simulation and writing the report. The design can be started with what you have learned in the class so far. For the prototype ﬁlter you can consult A. I. Zverev, Handbook of Filter Synthesis, Wiley, New York, 1967, which is a non circulating reference in the Engineering library.