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Problems on Analog Signal Processing

Module by: Don Johnson. E-mail the author

Summary: Problems Dealing with Analog Signal Processing

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Problem Set

Problem 1

Solving Simple Circuits

  1. Write the set of equations that govern Circuit A's behavior.
  2. Solve these equations for i 1 i 1 : In other words, express this current in terms of element and source values by eliminating non-source voltages and currents.
  3. For Circuit B, find the value for R L R L that results in a current of 5 A passing through it.
Figure 1
Circuit A
Circuit A (circuit6.png)
Circuit B
Circuit B (circuit7.png)

Problem 2

Equivalent Resistance

For each of the following circuits, find the equivalent resistance using series and parallel combination rules.

Figure 2
circuit a
circuit a (circuit8a.png)
circuit b
circuit b (circuit8b.png)
circuit c
circuit c (circuit8c.png)
circuit d
circuit d (circuit8d.png)

Calculate the conductance seen at the terminals for circuit (c) in terms of each element's conductance. Compare this equivalent conductance formula with the equivalent resistance formula you found for circuit (b). How is the circuit (c) derived from circuit (b)?

Problem 3

Superposition Principle

One of the most important consequences of circuit laws is the Superposition Principle: The current or voltage defined for any element equals the sum of the currents or voltages produced in the element by the independent sources. This Principle has important consequences in simplifying the calculation of ciruit variables in multiple source circuits.

Figure 3
Figure 3 (circuit9.png)
  1. For the depicted circuit, find the indicated current using any technique you like (you should use the simplest).
  2. You should have found that the current ii is a linear combination of the two source values: i= C 1 v in + C 2 i in i C 1 v in C 2 i in . This result means that we can think of the current as a superposition of two components, each of which is due to a source. We can find each component by setting the other sources to zero. Thus, to find the voltage source component, you can set the current source to zero (an open circuit) and use the usual tricks. To find the current source component, you would set the coltage wource to zero (a short circuit) and find the resulting current. Calculate the total current ii using the Superposition Principle. Is applying the Superposition Principle easier than the technique you used in part (1)?

Problem 4

Current and Voltage Divider

Use current of voltage divider rules to calculate the indicate circuit variable.

Figure 4
circuit a
circuit a (circuit10a.png)
circuit c
circuit c (circuit10b.png)
circuit b
circuit b (circuit10c.png)

Problem 5

Thévenin and Mayer-Norton Equivalents

Find the Thévenin and Mayer-Norton equivalent circuits for the following circuits.

Figure 5
circuit a
circuit a (circuit11a.png)
circuit b
circuit b (circuit11b.png)
circuit c
circuit c (circuit11c.png)

Problem 6

Detective Work

In the depicted circuit, the circuit N 1 N 1 has the v-i relation v 1 =3 i 1 +7 v 1 3 i 1 7 when i s =2 i s 2 .

  1. Find the Thévenin equivalent circuit for circuit N 2 N 2 .
  2. With i s =2 i s 2 , determine RR such that i 1 =-1 i 1 -1 .

Figure 6
Figure 6 (circuit17.png)

Problem 7

Transfer Functions

Find the transfer function relating the complex amplitudes of the indicated variable and the source. Plot the magnitude and phase of the transfer function.

Figure 7
circuit a
circuit a (circuit18a.png)
circuit b
circuit b (circuit18b.png)
circuit c
circuit c (circuit18c.png)
circuit d
circuit d (circuit18d.png)

Problem 8

Using Impedances

Find the differential equation relating the indicated variable to the source(s) using impedances.

Figure 8
circuit a
circuit a (circuit19a.png)
circuit b
circuit b (circuit19b.png)
circuit c
circuit c (circuit19c.png)
circuit d
circuit d (circuit19d.png)

Problem 9

Transfer Functions

In the following circuit, the voltage source equals vint=10sint2 vin t 10 t 2 .

Figure 9
Figure 9 (circuit21.png)
  1. Find the transfer function between the source and the indicated output voltage.
  2. For the given source, find the output voltage.

Problem 10

Circuit Design

Figure 10
Figure 10 (circuit23.png)
  1. Find the transfer function between the input and the output voltages.
  2. At what frequency does the transfer function have a phase shift of zero? What is the circuit's gain at this frequency?
  3. Specifications demand that this circuit have an output impedance (its equivalent impedance) less than 8 Ω for frequencies above 1 kHz, the frequency at which the transfer function is maximum. Find element values that satisfy this criterion.

Problem 11

Sharing a Channel

Two transmitter-receiver pairs want to share the same digital communications channel. The transmitter signals will be added together by the channel. Receiver design is greatly simplified if first we remove the unwanted transmission (as much as possible). Each transmitter signal has the form t ,0tT:xit=Asin2πfit t 0 t T xi t A 2 fi t where the amplitude is either zero or AAand each transmitter uses its own frequency fi fi . Each frequency is harmonically related to the bit interval duration T T, where the transmitter 1 uses the the frequency 1T 1 T . The datarate is 10Mbps.

  1. Draw a block diagram that expresses this communication scenario.
  2. Find circuits that the receivers could employ to separate unwanted transmissions. Assume the received signal is a voltage and the output is to be a voltage as well.
  3. Find the second transmitter's frequency so that the receivers can suppress the unwanted transmission by at least a factor of ten.

Problem 12

Circuit Detective Work

In the lab, the open-circuit voltage measured across an unknown circuit's terminals equals sint t . When a 1 Ω resistor is place across the terminals, a voltage of 12sint+π4 1 2 t 4 appears.

  1. What is the Thévenin equivalent circuit?
  2. What voltage will appear if we place a 1 F capacitor across the terminals?

Problem 13

Linear, Time-Invariant Systems

For a system to be completely characterized by a transfer function, it needs not only be linear, but also to be time-invariant. A system is said to be time-invariant if delaying the input delays the output by the same amount. Mathematically, if Sxt=yt S x t y t , meaning yt y t is the output of a system S S when xt x t is the input, S S is the time-invariant if Sxtτ=ytτ S x t τ y t τ for all delays ττ and all inputs xt x t . Note that both linear and nonlinear systems have this property. For example, a system that squares its input is time-invariant.

  1. Show that if a circuit has fixed circuit elements (their values don't change over time), its input-output relationship is time-invariant. Hint: Consider the differential equation that describes a circuit's input-output relationship. What is its general form? Examine the derivative(s) of delayed signals.
  2. Show that impedances cannot characterize time-varying circuit elements (R, L, and C). Consequently, show that linear, time-varying systems do not have a transfer function.
  3. Determine the linearity and time-invariance of the following. Find the transfer function of the linear, time-invariant (LTI) one(s).
    1. diode
    2. yt=xtsin2πf0t y t x t 2 f0 t
    3. yt=xtτ0 y t x t τ0
    4. yt=xt+Nt y t x t N t

Problem 14

Long and Sleepless Nights

Sammy went to lab after a long, sleepless night, and constructed the following circuit.

Figure 11
Figure 11 (circuit1.png)

He cannot remember what the circuit, represented by the impedance ZZ, was. Clearly, this forgotten circuit is important as the output is the current passing through it.

  1. What is the Thévenin equivalent circuit seen by the impedance?
  2. In searching his notes, Sammy finds that the circuit is to realize the transfer function Hf=1i10πf+2 H f 1 10 f 2 Find the impedance ZZ as well as values for the other circuit elements.

Problem 15

Mystery Circuit

You are given a circuit that has two terminals for attaching circuit elements.

Figure 12
Figure 12 (circuit27.png)

When you attach a voltage source equaling sint t to the terminals, the current through the source equals 4sint+π42sin4t 4 t 4 2 4 t . When no source is attached (open-circuited terminals), the voltage across the terminals has the form Asin4t+φ A 4 t φ .

  1. What will the terminal current be when you replace the source by a short circuit?
  2. What are AA and φφ?
  3. If you were to build a circuit that was identical (from the viewpoint of the terminals) to the given one, what would your circuit be?

Problem 16

Mystery Circuit

Sammy must determine as much as he can about a mystery circuit by attaching elements to the terminal and measuring the resulting voltage. When he attaches a 1 Ω resistor to the circuit's terminals, he measures the voltage across the terminals to be 3sint 3 t . When he attaches a 1 F capacitor across the terminals, the voltage is now 3×2sintπ4 3 2 t 4 .

  1. What voltage should he measure when he attaches nothing to the mystery circuit?
  2. What voltage should Sammy measure if he doubled the size of the capacitor to 2 F and attached it to the circuit?

Problem 17

Find the Load Impedance

The depicted circuit has a transfer function between the output voltage and the source equal to Hf=-8π2f28π2f2+4+i6πf H f -8 2 f 2 -8 2 f 2 4 6 f .

Figure 13
Figure 13 (circuit24.png)
  1. Sketch the magnitude and phase of the transfer function.
  2. At what frequency does the phase equal π2 2 ?
  3. Find a circuit that corresponds to this load impedance. Is your answer unique? If so, show it to be so; if not, give another example.

Problem 18

Analog "Hum" Rejection

"Hum" refers to corruption from wall socket power that frequently sneaks into circuits. "Hum" gets its name because it sounds like a persistent humming sound. We want to find a circuit that will remove hum from any signal. A Rice engineer suggests using a simple voltage divider circuit consisting of two series impedances.

Figure 14
Figure 14 (circuit28a.png)
  1. The impedance Z1 Z1 is a resistor. The Rice engineer must decide between two circuits for the impedance Z2 Z2. Which of these will work (see figure below)?
  2. Picking one circuit that works, choose circuit element values that will remove hum.
  3. Sketch the magnitude of the resulting frequency response.
Figure 15
Figure 15 (circuit28b.png)

Problem 19

An Interesting Circuit

Figure 16
Figure 16 (circuit29.png)
  1. Find the circuit's transfer function.
  2. What is the output voltage when the input has the form iin=5sin2000πt iin 5 2000 t ?

Problem 20

A Circuit

You are given the following circuit:

Figure 17
Figure 17 (circuit30.png)
  1. What is the transfer function between the source adn the output voltage?
  2. What will the voltage be when the source equals sint t ?
  3. Many function generators produce a constant offset in addition to a sinusoid. If the source equals 1+sint 1 t , what is the output voltage?

Problem 21

An Interesting and Useful Circuit

The following circuit has interesting properties, which are exploited in high-performance oscilloscopes.

Figure 18
Figure 18 (circuit20.png)

The portion of the circuit labeled "Oscilloscope" represents the scope's input impedance. R2=1 MΩ R2 1 MΩ and C2=30 pF C2 30 pF (note the label under the channel 1 input in the lab's oscilloscopes). A probe is a device to attach an oscilloscope to a circuit, and it has the indicated circuit inside it.

  1. Suppose for a moment that the probe is merely a wire and that the oscilloscope is attached to a circuit that has a resistive Thévenin equivalent impedance. What would be the effect of the oscilloscope's input impedance on measured voltages?
  2. Using the node method, find the transfer fucntion relating the indicated voltage to the source when the probe is used.
  3. Plot the magnitude and phase of this transfer function when R1=9 R1 9 and C1=2 pF C1 2 pF .
  4. For a particular relationship among the element values, the transfer function is quite simple. Find that relationship and describe what is so special about it.
  5. The arrow through C1 C1 indicates that its value can be varied. Select the value for this capacitor to make the special relationship balid. What is the impedance seen by the circuit being measured for this special value?

Problem 22

A Circuit Problem

You are given the following circuit.

Figure 19
Figure 19 (circuit2.png)
  1. Find the differential equation relating the output voltage to the source.
  2. What is the impedance "seen" by the capacitor?

Problem 23

Find the voltage vout vout in each of the following circuits.

Figure 20
circuit a
circuit a (circuit21a.png)
circuit b
circuit b (circuit21b.png)

Problem 24

Transfer Functions and Circuits

You are given the depicted network.

Figure 21
Figure 21 (circuit26.png)
  1. Find the transfer function between Vin Vin and Vout Vout.
  2. Sketch the magnitude and phase of your transfer function. Label important frequency, amplitude and phase values.
  3. Find vout t vout t when vint=sint2+π4 vin t t 2 4 .

Problem 25

Fun in the Lab

You are given an unopenable box that has two terminals sticking out. You assume the box contains a circuit. You measure the voltage sint+π4 t 4 across the terminals when nothing is connected to them and the current 2cost 2 t when you place a wire across the terminals.

  1. Find a circuit that has these characteristics.
  2. You attach a 1 H inductor across the terminals. What voltage do you measure?

Problem 26

Operational Amplifiers

Find the transfer function between the source voltage(s) and the indicated output voltage.

Figure 22
op-amp a
op-amp a (opamp1.png)
op-amp b
op-amp b (opamp2.png)
op-amp c
op-amp c (opamp3.png)
op-amp d
op-amp d (opamp15.png)

Problem 27

Why Op-Amps are Useful

The following circuit of a cascade of op-amp circuits illustrate the reason why op-amp realizations of transfer functions are so useful.

Figure 23
Figure 23 (opamp4.png)
  1. Find the transfer function relating the complex amplitude of the voltage vout t vout t to the source. Show that this transfer function equals the product of each stage's transfer function.
  2. What is the load impedance appearing across the first op-amp's output?
  3. The following illustrates that sometimes "designs" can go wrong (see below). Find the transfer function for this op-amp circuit, and then show that it can't work! Why can't it?
Figure 24
Figure 24 (opamp5.png)

Problem 28

Operational Amplifiers

Consider the following circuit.

Figure 25
Figure 25 (opamp6.png)
  1. Find the transfer function relating the voltage vout t vout t to the source.
  2. In particular, R1=530 R1 530 , C1=1μF C1 1 μF , R2=5.3KΩ R2 5.3 KΩ , C2=0.01μF C2 0.01 μF , and R3=R4=5.3KΩ R3 R4 5.3 KΩ . Characterize the resulting transfer function and determine what use this circuit might have.

Problem 29

Designing a Bandpass Filter

We want to design a bandpass filter that has transfer the function Hf=102πf(ffl+1)(iffh+1) H f 10 2 f f fl 1 f fh 1 Here, fl fl is the cutoff frequency of the low-frequency edge of the passband and fh fh is the cutoff frequency of the high-frequency edge. we want fl=1 kHz fl 1 kHz and fh=10 kHz fh 10 kHz .

  1. Plot the magnitude and phase of this frequency response. Label important amplitude and phase values and the frequency response. Label important amplitude and phase values and the frequencies at which they occur.
  2. Design a bandpass filter that meets these specifications. Specify component values.

Problem 30

Active Filter

Find the transfer function of the depicted active filter.

Figure 26
Figure 26 (opamp17.png)

Problem 31

This is a filter?

You are given the following circuit.

Figure 27
Figure 27 (opamp18.png)
  1. What is this circuit's transfer function? Plot the magnitude and phase.
  2. If the input signal is the sinusoid sin2πf0t 2 f0 t , what will the output be when f0 f0 is larger than the filter's "cutoff frequency"?

Problem 32

Optical Receivers

In you optical telephone, the receiver circuit had the following form:

Figure 28
Figure 28 (opamp7.png)

This circuit served as a transducer, converting light energy into a voltage vout vout. The photodiode acts as a current source, producing a current proportional to the light intesity falling upon it. As is often the case in this crucial stage, the signals are small and noise can be a problem. Thus, the op-amp stage serves to boost the signal and to filter out-of-band noise.

  1. Find the transfer function relating light intensity to vout vout.
  2. What should the circuit realizing the feedback impedance Zf Zf be so that the transducer acts as a 5 kHz lowpass filter?
  3. A clever engineer suggests an alternative circuit to accomplish the same task. Determine whether the idea works or not. If it does, find the impedance Zin Zin that accomplishes the lowpass filtering task. If not, show why it does not work.

Figure 29
Figure 29 (opamp8.png)

Problem 33

Reverse Engineering

The following circuit has been developed by the TBBG Electronics design group. They are trying to keep its use secret; we, representing RU Electronics, have discovered the schematic and want to figure out the intended application. Assume the diode is ideal.

Figure 30
Figure 30 (opamp19.png)
  1. Assuning the diode is a short-circuit (it has been removed from the circuit), what is the circuit's transfer function?
  2. With the diode in place, what is the circuit's output when the input voltage is sin2π f 0 t 2 f 0 t ?
  3. What function might this circuit have?

Problem 34

Rectifying Amplifier

The following circuit has been designed by a clever engineer. For positive voltages, the diode acts like a short circuit and for negative voltages like an open circuit.

Figure 31
Figure 31 (opamp16.png)
  1. How is the output related to the input?
  2. Does this circuit have a transfer function? If so, find it; if not, why not?

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