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 !u = 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 = 1; 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; !u = 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) = !u=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 - - - - (c) (d) 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 1. 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(!t), 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 = 1). 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 = f1; 2g 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 = 1. 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 fil- ter with R = 100 kΩ that will accomplish this.