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Analog Signal Processing Problems

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 RL RL that results in a current of 5 A passing through it.
  4. What is the power dissipated by the load resistor RL RL in this case?
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 voltage source 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 or voltage divider rules to calculate the indicated circuit variables in Figure 4.

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

Cartesian to Polar Conversion

Convert the following expressions into polar form. Plot their location in the complex plane.

  1. 1+-32 1 -3 2
  2. 3+i4 3 4
  3. 2i632+i63 2 6 3 2 6 3
  4. (4i3)(1+i12) 4 3 1 1 2
  5. 3eiπ+4eiπ2 3 4 2
  6. (3+i)2×2e(iπ4) 3 2 2 4
  7. 31+i3π 3 1 3

Problem 8

The Complex Plane

The complex variable z z is related to the real variable u u according to z=1+eiu z 1 u

  • Sketch the contour of values zz takes on in the complex plane.
  • What are the maximum and minimum values attainable by |z|z?
  • Sketch the contour the rational function z1z+1 z 1 z 1 traces in the complex plane.

Problem 9

Cool Curves

In the following expressions, the variable xx runs from zero to infinity. What geometric shapes do the following trace in the complex plane?

  1. eix x
  2. 1+eix 1 x
  3. exeix x x
  4. eix+ei(x+π4) x x 4

Problem 10

Trigonometric Identities and Complex Exponentials

Show the following trigonometric identities using complex exponentials. In many cases, they were derived using this approach.

  1. sin2u=2sinucosu 2 u 2 u u
  2. cos2u=1+cos2u2 u 2 1 2u 2
  3. cos2u+sin2u=1 u 2 u 2 1
  4. ddusinu=cosu u u u

Problem 11

Transfer Functions

Find the transfer function relating the complex amplitudes of the indicated variable and the source shown in Figure 7. 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 12

Using Impedances

Find the differential equation relating the indicated variable to the source(s) using impedances for each circuit shown in Figure 8.

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 13

Transfer Functions

In the following circuit, the voltage source equals v in t=10sint2 v in 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 14

A Simple Circuit

You are given this simple circuit.

Figure 10
Figure 10 (circuit33.png)
  1. What is the transfer function between the source and the indicated output current?
  2. If the output current is measured to be cos2t 2 t , what was the source?

Problem 15

Circuit Design

Figure 11
Figure 11 (circuit23.png)
  1. Find the transfer function between the input and the output voltages for the circuits shown in Figure 11.
  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 8Ω for frequencies above 1 kHz, the frequency at which the transfer function is maximum. Find element values that satisfy this criterion.

Problem 16

Power Transmission

The network shown in (Reference) represents a simple power transmission system. The generator produces 60 Hz and is modeled by a simple Thévenin equivalent. The transmission line consists of a long length of copper wire and can be accurately described as a 50Ω resistor.

  1. Determine the load current RL RL and the average power the generator must produce so that the load receives 1,000 watts of average power. Why does the generator need to generate more than 1,000 watts of average power to meet this requirement?
  2. Suppose the load is changed to that shown in (Reference). Now how much power must the generator produce to meet the same power requirement? Why is it more than it had to produce to meet the requirement for the resistive load?
  3. The load can be compensated to have a unity power factor (see exercise) so that the voltage and current are in phase for maximum power efficiency. The compensation technique is to place a circuit in parallel to the load circuit. What element works and what is its value?
  4. With this compensated circuit, how much power must the generator produce to deliver 1,000  average power to the load?

Figure 12
Simple power transmission system
Simple power transmission system (circuit37a.png)
Modified load circuit
Modified load circuit (circuit37b.png)

Problem 17

Optimal Power Transmission

The following figure shows a general model for power transmission. The power generator is represented by a Thévinin equivalent and the load by a simple impedance. In most applications, the source components are fixed while there is some latitude in choosing the load.

  1. Suppose we wanted the maximize "voltage transmission:" make the voltage across the load as large as possible. What choice of load impedance creates the largest load voltage? What is the largest load voltage?
  2. If we wanted the maximum current to pass through the load, what would we choose the load impedance to be? What is this largest current?
  3. What choice for the load impedance maximizes the average power dissipated in the load? What is most power the generator can deliver?
One way to maximize a function of a complex variable is to write the expression in terms of the variable's real and imaginary parts, evaluate derivatives with respect to each, set both derivatives to zero and solve the two equations simultaneously.

Figure 13
Figure 13 (circuit38.png)

Problem 18

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: x i t=Asin2π f i t t 0 t T x i t A 2 f i t where the amplitude is either zero or AA and each transmitter uses its own frequency f i f i . 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 19

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 20

More Circuit Detective Work

The left terminal pair of a two terminal-pair circuit is attached to a testing circuit. The test source v in t v in t equals sint t (Figure 14).

Figure 14
Figure 14 (circuit31.png)

We make the following measurements.

  • With nothing attached to the terminals on the right, the voltage vt v t equals 12cost+π4 1 2 t 4 .
  • When a wire is placed across the terminals on the right, the current it i t was sint t .
  1. What is the impedance “seen” from the terminals on the right?
  2. Find the voltage vt v t if a current source is attached to the terminals on the right so that it=sint i t t .

Problem 21

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 τ τ 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=xtsin2π f 0 t y t x t 2 f 0 t
    3. yt=xt τ 0 y t x t τ 0
    4. yt=xt+Nt y t x t N t

Problem 22

Long and Sleepless Nights

Sammy went to lab after a long, sleepless night, and constructed the circuit shown in Figure 15.

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.

Figure 15
Figure 15 (circuit1.png)

Problem 23

A Testing Circuit

The simple circuit here was given on a test.

Figure 16
Figure 16 (circuit34.png)

When the votlage source is 5sint 5 t , the current it=2costarctan2π4 i t 2 t 2 4 .

  1. What is voltage v out t v out t ?
  2. What is the impedance ZZ at the frequency of the source?

Problem 24

Mystery Circuit

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

Figure 17
Figure 17 (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. 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?
  3. For your circuit, what are AA and φφ?

Problem 25

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 26

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 18
Figure 18 (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 27

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 19
Figure 19 (circuit28a.png)
  1. The impedance Z 1 Z 1 is a resistor. The Rice engineer must decide between two circuits for the impedance Z 2 Z 2 . Which of these will work?
  2. Picking one circuit that works, choose circuit element values that will remove hum.
  3. Sketch the magnitude of the resulting frequency response.
Figure 20
Figure 20 (circuit28b.png)

Problem 28

An Interesting Circuit

Figure 21
Figure 21 (circuit29.png)
  1. For the circuit shown in Figure 21, find the transfer function.
  2. What is the output voltage when the input has the form i in =5sin2000πt i in 5 2000 t ?

Problem 29

A Circuit

You are given the depicted circuit.

Figure 22
Figure 22 (circuit30.png)
  1. What is the transfer function between the source and 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 30

An Interesting and Useful Circuit

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

Figure 23
Figure 23 (circuit20.png)

The portion of the circuit labeled "Oscilloscope" represents the scope's input impedance. R 2 =1MΩ R 2 1 MΩ and C 2 =30pF C 2 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 function relating the indicated voltage to the source when the probe is used.
  3. Plot the magnitude and phase of this transfer function when R 1 =9MΩ R 1 9 MΩ and C 1 =2pF C 1 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 C 1 C 1 indicates that its value can be varied. Select the value for this capacitor to make the special relationship valid. What is the impedance seen by the circuit being measured for this special value?

Problem 31

A Circuit Problem

You are given the depicted circuit.

Figure 24
Figure 24 (circuit2.png)
  1. Find the differential equation relating the output voltage to the source.
  2. What is the impedance “seen” by the capacitor?

Problem 32

Analog Computers

Because the differential equations arising in circuits resemble those that describe mechanical motion, we can use circuit models to describe mechanical systems. An ELEC 241 student wants to understand the suspension system on his car. Without a suspension, the car's body moves in concert with the bumps in the raod. A well-designed suspension system will smooth out bumpy roads, reducing the car's vertical motion. If the bumps are very gradual (think of a hill as a large but very gradual bump), the car's vertical motion should follow that of the road. The student wants to find a simple circuit that will model the car's motion. He is trying to decide between two circuit models (Figure 25).

Figure 25
Figure 25 (circuit32.png)

Here, road and car displacements are represented by the voltages v road t v road t and v car t v car t , respectively.

  1. Which circuit would you pick? Why?
  2. For the circuit you picked, what will be the amplitude of the car's motion if the road has a displacement given by v road t=1+sin2t v road t 1 2 t ?

Problem 33

Dependent Sources

Find the voltage v out v out in each of the depicted circuits.

Figure 26
circuit a
circuit a (circuit21a.png)
circuit b
circuit b (circuit21b.png)

Problem 34

Transfer Functions and Circuits

You are given the depicted network.

Figure 27
Figure 27 (circuit26.png)
  1. Find the transfer function between V in V in and V out V out .
  2. Sketch the magnitude and phase of your transfer function. Label important frequency, amplitude and phase values.
  3. Find v out t v out t when v in t=sint2+π4 v in t t 2 4 .

Problem 35

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 36

Operational Amplifiers

Find the transfer function between the source voltage(s) and the indicated output voltage for the circuits shown in Figure 28.

Figure 28
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 37

Why Op-Amps are Useful

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

Figure 29
Figure 29 (opamp4.png)
  1. Find the transfer function relating the complex amplitude of the voltage v out t v out 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. Figure 30 illustrates that sometimes “designs” can go wrong. Find the transfer function for this op-amp circuit, and then show that it can't work! Why can't it?
Figure 30
Figure 30 (opamp5.png)

Problem 38

Operational Amplifiers

Consider the depicted circuit.

Figure 31
Figure 31 (opamp6.png)
  1. Find the transfer function relating the voltage v out t v out t to the source.
  2. In particular, R 1 =530 R 1 530 , C 1 =1μF C 1 1 μF , R 2 =5.3 R 2 5.3 , C 2 =0.01μF C 2 0.01 μF , and R 3 = R 4 =5.3 R 3 R 4 5.3 . Characterize the resulting transfer function and determine what use this circuit might have.

Problem 39

Designing a Bandpass Filter

We want to design a bandpass filter that has transfer the function Hf=10i2πf(if f l +1)(if f h +1) H f 10 2 f f f l 1 f f h 1 Here, f l f l is the cutoff frequency of the low-frequency edge of the passband and f h f h is the cutoff frequency of the high-frequency edge. We want f l =1kHz f l 1 kHz and f h =10kHz f h 10 kHz .

  1. Plot the magnitude and phase of this 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 40

Pre-emphasis or De-emphasis?

In audio applications, prior to analog-to-digital conversion signals are passed through what is known as a pre-emphasis circuit that leaves the low frequencies alone but provides increasing gain at increasingly higher frequencies beyond some frequency f 0 f 0 . De-emphasis circuits do the opposite and are applied after digital-to-analog conversion. After pre-emphasis, digitization, conversion back to analog and de-emphasis, the signal's spectrum should be what it was.

The op-amp circuit here has been designed for pre-emphasis or de-emphasis (Samantha can't recall which).

Figure 32
Figure 32 (opamp21.png)
  1. Is this a pre-emphasis or de-emphasis circuit? Find the frequency f 0 f 0 that defines the transition from low to high frequencies.
  2. What is the circuit's output when the input voltage is sin2πft 2 f t , with f=4kHz f 4 kHz ?
  3. What circuit could perform the opposite function to your answer for the first part?

Problem 41

Active Filter

Find the transfer function of the depicted active filter.

Figure 33
Figure 33 (opamp17.png)

Problem 42

This is a filter?

You are given a circuit.

Figure 34
Figure 34 (opamp18.png)
  1. What is this circuit's transfer function? Plot the magnitude and phase.
  2. If the input signal is the sinusoid sin2π f 0 t 2 f 0 t , what will the output be when f 0 f 0 is larger than the filter's “cutoff frequency”?

Problem 43

Optical Receivers

In your optical telephone, the receiver circuit had the form shown.

Figure 35
Figure 35 (opamp7.png)

This circuit served as a transducer, converting light energy into a voltage v out v out . 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 v out v out .
  2. What should the circuit realizing the feedback impedance Z f Z f 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 Z in Z in that accomplishes the lowpass filtering task. If not, show why it does not work.

Figure 36
Figure 36 (opamp8.png)

Problem 44

Reverse Engineering

The depicted 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 37
Figure 37 (opamp19.png)
  1. Assuming 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?

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