hw04_prob (1)

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EECS 16B Designing Information Systems and Devices II UC Berkeley Fall 2023 Homework 4 This homework is due on Saturday, September 23, 2023 at 11:59PM. Self-grades and HW Resubmissions are due the following Saturday, Septem- ber 30, 2023 at 11:59PM. 1. Adapted from Hambley P6.27 Suppose you have a filter with transfer function H ( j ω ) = 1 1 + j ω ω c , with ω c = 400 rad s . The input signal of the filter with this transfer function is v in ( t ) = 1 + 2 cos ( 400 t + 30 ◦ ) + 3 cos 10 10 t (1) Find an expression for the output voltage (you may approximate). 1

EECS 16B Homework 4 2023-09-17 23:37:19-07:00 2. Hambley P6.33 Consider the circuit shown in Figure 1 . This circuit consists of a source having an internal resistance of R s , an RC lowpass filter, and a load resistance of R l . + − v s ( t ) R s R C + − v out ( t ) R l Figure 1: P6.33(a) + − v s ( t ) R s R C + − v out ( t ) R l Figure 2: P6.33(b) Show that the transfer function of this circuit is given by H ( j ω ) = e V out e V s = R l R s + R + R l × 1 1 + j ω ω c (2) in which the cutoff frequency ω c is given by ω c = 1 R t C where R t = R L ( R s + R ) R l + R s + R . Notice that R t is the parallel combination of R l and ( R s + R ) . (HINT: One way to make this problem easier is to rearrange the circuit as shown in Figure 2 and then to find the Thevenin equivalent for the source and resistances.) © UCB EECS 16B, Fall 2023. All Rights Reserved. This may not be publicly shared without explicit permission. 2

EECS 16B Homework 4 2023-09-17 23:37:19-07:00 3. Hambley P6.74 Derive an expression for the resonant frequency of the circuit shown in Figure 3 . We define the reso- nant frequency to be the frequency at which the impedance is purely real (no imaginary component). C L R Figure 3: P6.74 © UCB EECS 16B, Fall 2023. All Rights Reserved. This may not be publicly shared without explicit permission. 3

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EECS 16B Homework 4 2023-09-17 23:37:19-07:00 4. Circuit Design In this problem, you will find a circuit where several components have been left blank for you to fill in. Assume that the op-amp is ideal . A special note on op amps in frequency domain analysis: The op-amps you learned about in 16A can be used in exactly the same way for setting up differential equations and even Phasor analysis in 16B. Treat them as ideal op-amps and invoke the Golden Rules. You have at your disposal only one of each of the following components (not including R 1 ): (a) an open circuit (b) a short circuit R (c) a resistor (you choose from the values R = 1 k \YX , 10 k \YX , 20 k \YX ) C (d) a capacitor (you choose from the values C = 0.5 µF, 1 µF, 2 µF) Consider the circuit below. The labeled voltages e V in ( j ω ) and e V out ( j ω ) are the phasor representations of v in ( t ) and v out ( t ) respectively, where v in ( t ) has the form v in ( t ) = v 0 cos ( ω t + ϕ ) . The transfer function H ( j ω ) is defined as H ( j ω ) = e V out ( j ω ) e V in ( j ω ) . Z 1 Z 2 − + R 1 e V in ( j ω ) e V out ( j ω ) (a) Let Z 1 ( j ω ) and Z 2 ( j ω ) are the impedances of the boxes shown in the circuit diagram. Write the expression of the transfer function H ( j ω ) . (b) Let R 1 be 1 k \YX . We have to find Z 1 and Z 2 , such that the circuit’s transfer function H ( j ω ) has the following properties: • | H ( j0 ) | = 0. • | H ( j ª§¦ª ) | = 10. • | H ( j10 3 ) | = √ 50. Using the fact that the circuit is a high pass filter, infer the components (we will find values later) of Z 1 and Z 2 . Write the transfer function H ( j ω ) using these components. (HINT: Try method of elimination: figure out what Z 2 cannot be. Once you find what Z 2 is, what does Z 1 have to be for the circuit to be a filter?) (c) Now use the facts that | H ( j ª§¦ª ) | = 10 and R 1 = 1 k \YX to find the component value of Z 2 . (d) Finally use the fact that | H ( j10 3 ) | = √ 50 and the values of R 1 and Z 2 to find the component value of Z 1 . © UCB EECS 16B, Fall 2023. All Rights Reserved. This may not be publicly shared without explicit permission. 4

EECS 16B Homework 4 2023-09-17 23:37:19-07:00 5. Designing Filters In the lab, we will design various filter circuits using low-pass, high-pass, and band-pass filter ele- ments. In this problem, we will walk through the use cases of these filter elements. (a) First, you remember that you saw in lecture that you can build simple filters using a resistor and a capacitor. Design a simple first-order passive low-pass filter with the following specification using a 1 µF capacitor. (“Passive” means that the filter does not require any power supply to operate on the input signal. Passive components include resistors, capacitors, inductors, diodes, etc., while an example of an active component would be an op-amp). • Low-pass filter: cut-off frequency f c = 2400 Hz, ω c = 2 π · 2400 rad s . Hz can be interpreted as "cycles/sec", and rad s can be interpreted as "2 π radians/cycle". Recall that the cutoff-frequency of such a filter is just where the magnitude of the filter is 1 √ 2 of its peak value. Show your work to find the resistor value that creates this low-pass filter. Draw the schematic- level representation of your design. Please mark V in , V out , and the ground node(s) in your schematic. Round your results to two significant figures. (b) Now design a simple first-order passive high-pass filter with the following specification using a 1 µF capacitor. • High-pass filter: cut-off frequency f c = 100 Hz, ω c = 2 π · 100 rad s Show your work to find the resistor value that creates this high-pass filter. Draw the schematic- level representation of your design. Please mark V in , V out , and the ground node(s) in your schematic. Round your results to two significant figures. (c) You can try to build a bandpass filter by cascading the first-order low-pass and high-pass filters you designed in parts (a) and (b). To do this, you might be tempted to connect the V out node of your low-pass filter directly to the V in node of your high-pass filter. However, if you did this, just as you saw in 16A for voltage dividers, the purported high-pass filter would “load” the low-pass filter and you might get some potentially complicated mess instead of what you wanted. Show how you can use an ideal op-amp configured as a unity gain buffer to eliminate this loading effect to cascade the low-pass and high-pass filters, and write the resulting transfer function of the combined circuit. Draw the magnitude and phase transfer functions of the combined circuit (you can use Bode Plot approximations). What kind of filter is this? (d) Write down an expression for the time-domain output waveform V out ( t ) of this filter if the input voltage is V in ( t ) = 1 sin ( 1000 t ) V . Round your answer to 2 significant digits. Contributors: • Yen-Sheng Ho. • Sidney Buchbinder. • Ayan Biswas. • Druv Pai. • Antroy Roy Chowdhury. • Kyoungtae Lee. • Nikhil Shinde. © UCB EECS 16B, Fall 2023. All Rights Reserved. This may not be publicly shared without explicit permission. 5

EECS 16B Homework 4 2023-09-17 23:37:19-07:00 • Nathan Lambert. • Chancharik Mitra. • Nikhil Jain. © UCB EECS 16B, Fall 2023. All Rights Reserved. This may not be publicly shared without explicit permission. 6

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