Designing Information Devices and Systems II Spring 2017 Murat Arcak and Michel Maharbiz Homework 4. This homework is due February 22, 2017, at 17:00.

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1 EECS 16B Designing Information Devices and Systems II Spring 2017 Murat Arcak and Michel Maharbiz Homework 4 This homework is due February 22, 2017, at 17: Bandpass Consider the series bandpass below where Ṽ s and V o are phasor voltages: 20mH 0.5µF Ṽs 5Ω V o V (a) What is the transfer function, H(ω)= o V, of this circuit? s (b) What is ω 0 of this? (c) What is ω c1 and ω c2 of this? Hint: H(ω c1 ) = H(ω c2 ) = 1 2 (d) What is the bandwidth, B, of this? (e) What is the Q of this? 2. Bode plots (a) Transfer Functions to Bode Plots Figure 1: Reproduced with permission from Ulaby, Maharbiz, Furse. Circuits. Third Edition. (b) Bandstop EECS 16B, Spring 2017, Homework 4 1

2 Figure 2: Reproduced with permission from Ulaby, Maharbiz, Furse. Circuits. Third Edition. (c) Bandpass Figure 3: Reproduced with permission from Ulaby, Maharbiz, Furse. Circuits. Third Edition. 3. Ring oscillator Figure 4 shows a ring oscillator circuit with three inverters. These inverters are modeled as non-ideal op-amps using a general, non-ideal, op-amp model. Remember, our golden rules don t apply for the models below. Each op-amp acts as an inverter with gain. The voltage inputs terminals are considered open circuits. R out = 10kΩ, C o1 = C o2 = C o3 = 1pF, and K 1 = K 2 = K 3 = 2 EECS 16B, Spring 2017, Homework 4 2

3 V in1 v 1 V in2 v 2 V in3 v 3 R out R out R out C o1 C o2 C o3 K 1V in1 K 2V in2 K 3V in3 Figure 4: Ring Oscillator Modeled with Non-Ideal Op-Amps (a) First, lets look at the first op-amp in the chain. For the circuit in figure 5, find the transfer function for v 1 v0. V in1 v 1 v 0 R out C o1 K 1V in1 Figure 5: First Op-Amp in Ring Oscillator (b) Now, let s look at three of these op-amps cascaded together as seen in figure 6. What is the transfer function for v 3 v0? (Hint: since the input of each op-amp is an open circuit, the overall transfer function can be represented as the individual transfer functions of each amplifier cascaded together.) V in1 v 1 V in2 v 2 V in3 v 3 v 0 R out R out R out C o1 C o2 C o3 K 1V in1 K 2V in2 K 3V in3 Figure 6: Ring Oscillator without Feedback EECS 16B, Spring 2017, Homework 4 3

4 (c) Draw the bode plots for the magnitude and phase of v 3 (d) At what frequency is the phase of v 3 v0 equal to 2π? What is the magnitude of v 3 v0 at that frequency? How does v 3 compare to v 0 at this frequency? An interesting consequence of this result is that this system will have a sustained oscillation when placed in feedback. 4. Redo Problem 1 on the midterm (a) (b) (c) (d) 5. Redo Problem 2 on the midterm (a) (b) (c) (d) 6. Redo Problem 3 on the midterm (a) (b) 7. Redo Problem 4 on the midterm (a) (b) (c) (d) (e) 8. Redo Problem 5 on the midterm (a) (b) v0. EECS 16B, Spring 2017, Homework 4 4

5 EE 16B Midterm 1, February 15, 2017 Name: SID #: Discussion Section and TA: Discussion Section and TA: Lab Section and TA: Name of left neighbor: Name of right neighbor: Important Instructions: Show your work. An answer without explanation is not acceptable and does not guarantee any credit. Only the front pages will be scanned and graded. Back pages won't be scanned; you can use them as scratch paper. Do not remove pages, as this disrupts the scanning. Instead, cross the parts that you don't want us to grade. PROBLEM MAX

6 Well, Diotallevi and I are planning a reform in higher education. A School of Comparative Irrelevance, where useless or impossible courses are given. The school's aim is to turn out scholars capable of endlessly increasing the number of unnecessary subjects. Umberto Eco, Foucault's Pendulum Problem 1 Warm up (20 points) a) Consider the following circuit. Zeq is the impedance looking into the circuit from the left, as shown. Provide an expression for Zeq. Zeq =

7 b) If this impedance is driven by a sinusoidal source at frequency, ω [rad/s], for what ω is Zeq =? ω = c) What logic function does the following circuit perform?

8 d) Consider the following four circuits. For each, we define the voltage transfer function, Hv(ω) = Vout/Vin. With respect to HV(ω), circle what class of frequency response each circuit performs. Lowpass Highpass Bandpass Bandstop Lowpass Highpass Bandpass Bandstop Lowpass Highpass Bandpass Bandstop Lowpass Highpass Bandpass Bandstop

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10 You can tell you ve found a really interesting question when nobody wants you to answer it. James S.A. Corey, Nemesis Games Problem 2 (20 points) Consider the circuit below. Assume an ideal op amp. a) Find an expression that relates the derivative of vout (dvout/dt) to the input voltage (vin) and/or its derivative (dvin/dt). ddvv oooooo dddd = b) Now given that Cs = 1 nf, Cf = 5 nf, Cp = 1 nf, vcf(t<0) = vcs(t<0) = 0 and vin(t 0) = 5*t [volts], provide an expression for v out (t) for t 0.

11 Consider now the different circuit below. Assume an ideal op amp. c) Provide a symbolic expression for v out (t) for t 0. v out (t)= d) Assume vin(t 0) = 5*t [volts] and vcf(t<0) = 0. What is the value of vout(t) at t=1 s? v out (t)=

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13 Problem 3 (15 points) One should never mistake pattern for meaning. Iain Banks, The Hydrogen Sonata The following circuit is part of a near field communication system. A realistic voltage source (Vs, Rs) is connected through a switch onto a three component circuit. The inductor represents an antenna; the voltage across it modulates how much energy is radiated away from the system. The switch alternates continuously between position A and position B; it has been doing this since t = -. It spends π microseconds at each position. We want the voltage on the inductor, VL, to follow the curve plotted below. Specifically, we want to fulfill the following condition. Condition: The inductor voltage should oscillate 5 times during period when the switch is in position A. Plot of VL as a function of time with switch positions labeled. Note the units of time (10-6 seconds)!

14 a) If R and L is non-zero and known, provide an expression for C such that the above condition is met. (Reminder: the condition is that the inductor voltage should oscillate 5 times during period when the switch is in position A.) C = b) Unfortunately, a colleague tells you that R ; if L and C are known, provide an expression for R such that the above condition is met. (Reminder: the condition is that the inductor voltage should oscillate 5 times during period when the switch is in position A.) R =

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16 Chang Tzu tells us of a persevering man who after three laborious years mastered the art of dragonslaying. For the rest of his days, he had not a single opportunity to test his skills. Jorge Luis Borges, The Book of Imaginary Beings Problem 4 (30 points) Consider the circuit below. a) What is i(0)? Hint. What is the current flowing through L1 before the switch opens? Consequently, what is the current flowing through L2? b) What is di/dt (0)? c) What is the relationship between the voltages across L1 and R1?

17 d) Use KCL on Node A and the relationship derived above to arrive at a differential equation of the form, where i(t) is the current going through L2. e) Let R1 = R2 = R and L1 = L2 = L. Recall that the above differential equation can be reshaped into the follow linear algebra problem: What is the A matrix and what are its eigenvalues? f) Will this circuit exhibit any oscillations?

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19 Problem 5 (15 points) I am Groot. - Groot, Guardians of the Galaxy Consider the circuit below. a) Given an input voltage, v1(t), which is a sinusoid at frequency ω, and phasors corresponding to the input and output voltages, V1 and V2, find an expression for V2/V1. VV 2 VV 1 =

20 b) If v1(t) = cos(ωt) where ω = 10 6 rad/s and L = 1 µh, R = 1 Ω, and C = 0.5 µf, solve for v2(t). v2(t) =

21 Contributors: Brian Kilberg. EECS 16B, Spring 2017, Homework 4 21