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Analog Signal Processing Problems

Module by: Don Johnson

Summary: Problems Dealing with Analog Signal Processing

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.
Circuit Acircuit6.png
Subfigure 1.1
Circuit Bcircuit7.png
Subfigure 1.2
Figure 1
Problem 2:

Equivalent Resistance

For each of the following circuits, find the equivalent resistance using series and parallel combination rules.
circuit acircuit8a.png
Subfigure 2.1
circuit bcircuit8b.png
Subfigure 2.2
circuit ccircuit8c.png
Subfigure 2.3
circuit dcircuit8d.png
Subfigure 2.4
Figure 2
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.
circuit9.png
Figure 3
  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 of voltage divider rules to calculate the indicated circuit variables in Figure 4.
circuit acircuit10a.png
Subfigure 4.1
circuit ccircuit10b.png
Subfigure 4.2
circuit bcircuit10c.png
Subfigure 4.3
Figure 4
Problem 5:

Thévenin and Mayer-Norton Equivalents

Find the Thévenin and Mayer-Norton equivalent circuits for the following circuits.
circuit acircuit11a.png
Subfigure 5.1
circuit bcircuit11b.png
Subfigure 5.2
circuit ccircuit11c.png
Subfigure 5.3
Figure 5
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 .
circuit17.png
Figure 6
Problem 7:

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.
circuit acircuit18a.png
Subfigure 7.1
circuit bcircuit18b.png
Subfigure 7.2
circuit ccircuit18c.png
Subfigure 7.3
circuit dcircuit18d.png
Subfigure 7.4
Figure 7
Problem 8:

Using Impedances

Find the differential equation relating the indicated variable to the source(s) using impedances for each circuit shown in Figure 8.
circuit acircuit19a.png
Subfigure 8.1
circuit bcircuit19b.png
Subfigure 8.2
circuit ccircuit19c.png
Subfigure 8.3
circuit dcircuit19d.png
Subfigure 8.4
Figure 8
Problem 9:

Transfer Functions

In the following circuit, the voltage source equals v in t=10sint2 v in t 10 t 2 .
circuit21.png
Figure 9
  1. Find the transfer function between the source and the indicated output voltage.
  2. For the given source, find the output voltage.
Problem 10:

A Simple Circuit

You are given this simple circuit.
circuit33.png
Figure 10
  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 11:

Circuit Design

circuit23.png
Figure 11
  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 8 Ω for frequencies above 1 kHz, the frequency at which the transfer function is maximum. Find element values that satisfy this criterion.
Problem 12:

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,0≤t≤T: 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 13:

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 14:

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 12).
circuit31.png
Figure 12
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 15:

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 Sxt-τ=yt-τ 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 16:

Long and Sleepless Nights

Sammy went to lab after a long, sleepless night, and constructed the circuit shown in Figure 13.
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.
circuit1.png
Figure 13
Problem 17:

A Testing Circuit

The simple circuit here was given on a test.
circuit34.png
Figure 14
When the votlage source is 5sint 5 t , the current it=2cost-arctan2-π4 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 18:

Mystery Circuit

You are given a circuit that has two terminals for attaching circuit elements.
circuit27.png
Figure 15
When you attach a voltage source equaling sint t to the terminals, the current through the source equals 4sint+π4-2sin4t 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 19:

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 32sint-π4 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 20:

Find the Load Impedance

The depicted circuit has a transfer function between the output voltage and the source equal to Hf=-8π2f2-8π2f2+4+ⅈ6πf H f -8 2 f 2 -8 2 f 2 4 6 f .
circuit24.png
Figure 16
  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 21:

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.
circuit28a.png
Figure 17
  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.
circuit28b.png
Figure 18
Problem 22:

An Interesting Circuit

circuit29.png
Figure 19
  1. For the circuit shown in Figure 19, 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 23:

A Circuit

You are given the depicted circuit.
circuit30.png
Figure 20
  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 24:

An Interesting and Useful Circuit

The depicted circuit has interesting properties, which are exploited in high-performance oscilloscopes.
circuit20.png
Figure 21
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 25:

A Circuit Problem

You are given the depicted circuit.
circuit2.png
Figure 22
  1. Find the differential equation relating the output voltage to the source.
  2. What is the impedance “seen” by the capacitor?
Problem 26:

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 23).
circuit32.png
Figure 23
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 27:

Dependent Sources

Find the voltage v out v out in each of the depicted circuits.
circuit acircuit21a.png
Subfigure 24.1
circuit bcircuit21b.png
Subfigure 24.2
Figure 24
Problem 28:

Transfer Functions and Circuits

You are given the depicted network.
circuit26.png
Figure 25
  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 29:

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 30:

Operational Amplifiers

Find the transfer function between the source voltage(s) and the indicated output voltage for the circuits shown in Figure 26.
op-amp aopamp1.png
Subfigure 26.1
op-amp bopamp2.png
Subfigure 26.2
op-amp copamp3.png
Subfigure 26.3
op-amp dopamp15.png
Subfigure 26.4
Figure 26
Problem 31:

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.
opamp4.png
Figure 27
  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 28 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?
opamp5.png
Figure 28
Problem 32:

Operational Amplifiers

Consider the depicted circuit.
opamp6.png
Figure 29
  1. Find the transfer function relating the voltage v out t v out t to the source.
  2. In particular, R 1 =530Ω R 1 530 Ω , C 1 =1μF C 1 1 μF , R 2 =5.3kΩ R 2 5.3 kΩ , C 2 =0.01μF C 2 0.01 μF , and R 3 = R 4 =5.3kΩ R 3 R 4 5.3 kΩ . Characterize the resulting transfer function and determine what use this circuit might have.
Problem 33:

Designing a Bandpass Filter

We want to design a bandpass filter that has transfer the function Hf=10ⅈ2πfⅈf f l +1ⅈf 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 34:

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).
opamp21.png
Figure 30
  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 35:

Active Filter

Find the transfer function of the depicted active filter.
opamp17.png
Figure 31
Problem 36:

This is a filter?

You are given a circuit.
opamp18.png
Figure 32
  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 37:

Optical Receivers

In your optical telephone, the receiver circuit had the form shown.
opamp7.png
Figure 33
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.
opamp8.png
Figure 34
Problem 38:

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.
opamp19.png