08/22/04

ECE 1002

Final Project

Operational Amplifiers: the Integrator Circuit

Camtasia Tutorial: Parts 1 through 4 (25,534 KB)

Camtasia Tutorial: Parts 5 through 8 (40,418 KB)

Note: Viewing the equations in this document with Internet Explorer requires the installation of MathPlayer software.

The name operational amplifier dates from the days when the main application of such circuits was performing integration and differentiation (calculus) for numerical simulation in analog computers, predecessors of contemporary digital computers. The operations of differentiation and integration were closely identified by the engineers of that day with the operational calculus introduced by the pioneering electrical engineer, Oliver Heaviside . Hence the name operational amplifier , or op amp , for short. (Subsequently, Heaviside's operational calculus was understood in terms of Laplace transforms, which you will study later.)

An integrator circuit provides, in real time, an output signal that is the time integral of the input signal. Among many other applications, integrators today form the basis for extracting information from digital signals corrupted by noise, such as voices in digital phones. The application of operational amplifiers is far from limited to integrators, however. For instance, designers can make amplifiers with any particular gain they choose with a particular operational amplifier chip simply by adjusting the ratio of a couple of resistors connected externally to the chip.

This example suggests how an operational amplifier chip, which includes the overwhelming majority of the circuit components (approaching 100 transistors and resistors on the chip) in the total circuit, can be mass produced and then customized for many specific purposes (realizing an amplifier with a particular gain, or an integrator, as examples) by the addition of a few external components.

The versatility of the op amp creates a demand that justifies mass production at low cost. When the application of operational amplifiers was limited mainly to synthesizing integrators in analog computers, the cost of a single op amp was several hundred dollars. Now, because we realize their versatility and employ them in many diverse applications, demand is large enough to drive the price for some common types, such as the type 741 that you will use in this project, below one dollar.

In this project, you will see how to build, analyze and test an integrator circuit. The heart of the integrator you build will be a type 741 operational amplifier (or op amp ) IC (integrated circuit) chip, represented by the triangle in the following circuit.

To accomplish to Final Project, you should know a few things about op amps and capacitors:

v o u t ( t )     =   A ( v + ( t )   −   v − ( t ) )

where v + ( t ) is the voltage applied to the non-inverting input (indicated by "+" in the figure above) of the op amp and v - ( t ) is the voltage applied to the inverting input (indicated by "-" in the figure above) of the op amp. The names of these inputs reflect the algebraic signs of their contributions to the output voltage of the op amp. The gain of op amps is large, typically 10,000 or more. Thus, if the voltage difference at the input terminals is 0.0001 V and the gain A = 10000 , then the output voltage of the op amp is ( 0.0001V )( 10000 ) = 1V .

i C ( t ) =   C d v C d t

where C is the capacitance, a constant, of the ideal capacitor. Note that the relationship between the voltage and current for an ideal capacitor involves calculus and hence is more complicated than the relationship between the voltage and current for an ideal resistor, which is given by a purely algebraic equation, Ohm's Law. This complication does not affect the validity of KVL and KCL, which, as long as the frequency is not too high, hold for circuits regardless of what kinds of elements (inductors, transistors, ...) the circuits contain.

Record and submit all your work for this Final Project in a Word document named final.doc placed in a folder named final on your hard disk. The first lines of this document should include the course number, the words Final Project, your name, your netID, and your e-mail address.

  1. (10 points) Relate the output voltage, v out ( t ), of the amplifier in the circuit above to the input voltage, v in ( t ), which, as the input voltage, we consider given.

    To begin the solution process, first write 6 equations in terms of the unknown voltages and currents v out ( t ) , v R ( t ) , i R ( t ) , v C ( t ) , i C ( t ) , v - ( t ). (Note that in the particular circuit above, the non-inverting input of the op amp is connected directly to ground so that v + ( t ) is zero.) Write 1 equation from KCL involving currents i R ( t ) and i C ( t ) at node X , 1 from KVL involving voltages v in ( t ), v R ( t ) , and v - ( t ), 1 from KVL involving voltages v - ( t ), v C ( t ) , and v out ( t ), 1 equation from Ohm's Law, 1 equation from the relationship between the voltage and current in the capacitor, and, finally, 1 equation from the fact that the output voltage of the op amp is the difference between the two input voltages multiplied by the gain, A of the op amp: v out ( t ) = A [ v + ( t ) - v - ( t )] = - A v - ( t ) since v + ( t ) is always zero. Combine these 6 equations to show that:

    d v o u t ( t ) d t   =   −   1 R   C ( 1 + 1 A )   v i n ( t ) −   1 A ( 1 + 1 A ) R C   v o u t ( t )

    Use the Equation Editor Equation Editor in Word to write the equations and the steps in your solution in the document final.doc .

    During part 1 you should have inserted into the document final.doc in a directory named final on your hard disk:

  2. (10 points) In ideal op amps, the gain A approaches infinity. Take the limit as A approaches infinity to show that, with an ideal op amp in the circuit above, the equation you just found simplifies to:

    d v o u t ( t ) d t   =   −   1 R   C   v i n ( t )

    Use the equation editor in Word to record your work in final.doc . In practical op amps, the gain, A , is much larger than 1 . For a 741 op amp A is at least 10,000. As a consequence, this expression for the gain holds with good accuracy when a 741 op amp is used in the circuit above rather than an ideal op amp.

    During part 2, you should have inserted into the document final.doc the mathematical steps, together with explanatory text, that lead from the equation given in part 1 to the equation given in part 2.

  3. (10 points) Note that the last equation shows that the derivative of the output voltage from the circuit above is directly proportional to the input voltage to the circuit. That means that the output voltage must be proportional to the integral of the input. Starting with the equation from part 2 , use indefinite integration to show that:

    v o u t ( t )   =   −   1 R   C   ∫ v i n ( t ) d t + K

    where K is a constant of integration. The constant K corresponds to a dc voltage that often is zero and is usually not important.

    The negative sign before the integral indicates that the output of the integrator is "upside down," so this integrator is an example of an inverting integrator. If inversion is a problem in a particular application, we can feed the output of the inverting integrator into an inverting amplifier and make the output right side up. Use the equation editor in Word to record your work in final.doc .

    The desire to make this last equation hold true drove the original development of operational amplifiers. It is ironic that just at the time that integrated circuit technology allowed manufacture of inexpensive op amps with good quality, their original application as integrators in analog computers largely disappeared as digital computers replaced analog computers. Not only other applications for integrators soon emerged, however, but also countless other applications of op amps in amplifiers and noise filters generated enough demand to ensure the eventual commercial availability of a variety of op amp chips at reasonable costs.

    During part 3, you should have inserted into the document final.doc the mathematical steps, together with explanatory text, that lead from the equation given in part 2 to the equation given in part 3.

  4. (30 points) Construct the circuit you designed by soldering the components into a General-Purpose IC PC Board (RadioShack Cat. No. 276-150) , or a similar board. Life will be simpler for you if you use an 8-Pin Low-Profile Socket (RadioShack Cat. No. 276-1995) instead of soldering the 741 op amp chip directly into the PC board. Use the circuit diagram shown below, which includes 3 more resistors than the circuit we considered above.

    Here is a possible configuration on the printed circuit board ( top view , bottom view ).

    The circuit and component values were chosen to accommodate periodic inputs, such as sinusoids and square waves, that have zero average value and contain frequencies of a few thousand Hz . The 100,000 Ohm resistor connected across the capacitor is chosen large enough to have little effect on the operation of the circuit other than to discharge the capacitor slowly in case any charge should accumulate on it. In effect, this resistor maintains a value of zero for K , the constant of integration that appeared in an earlier equation for the output voltage of the integrator.

    The 10,000 Ohm resistor across the output terminals, which in fact has little effect on the operation of the circuit, represents the effect of some circuit that would draw current from the output of the integrator when it is used in a practical application.

    The 10,000 Ohm resistor inserted between the non-inverting (+) input to the op amp and ground roughly matches the resistance at the inverting input so that any non-zero leakage currents at the input terminals of practical op amps produce approximately equal dc voltages at the inverting and non-inverting inputs which, because the op amp is a differential amplifier, cancel each other out as far as any effect on the output voltage is concerned.

    This circuit diagram also includes the 9V batteries that are necessary for the 741 chip to operate, but were not shown, for simplicity, in the earlier diagram.

    During part 4, you should have constructed the circuit in the diagram shown in part 4. Connect the 9V batteries only during testing. Do not submit them.

  5. (10 points) Use the Capture part of OrCAD PSpice ® to draw and then simulate this circuit. In a folder final on your hard drive, create an OrCAD PSpice simulation project named finalsine in which the sinusoidal input voltage has a frequency of 2000Hz and an amplitude of 1V . Run the simulation for times between 0 and 10ms so that you can see the first 20 cycles of the output waveform.

    Include in the Word document screen shots of the schematic and of a screen from OrCAD PSpice ® that shows simultaneously the input voltage and the simulated output voltage. Notice that during startup, the capacitor becomes charged and that the 100,000 Ohm resistor requires a few cycles to drain off this initial charge. A smaller value for this resistor would drain off the charge more quickly, but could interfere more with the basic operation of the integrator circuit. For reading values from the simulation, you may wish to change the OrCAD PSpice ® settings to display only the last few cycles of the simulation in more detail.

    Also include in the Word document a comparison of the relative amplitudes and phases of the output voltage with the input voltage and explain how these quantities are consistent with the output voltage being proportional to the integral of the input voltage. To compare the relative amplitude and phase of the input and output of the integrator, use the fact that if

    v i n ( t ) = M sin ⁡ ( 2 π f   t )

    where M is the amplitude of the sinusoid and f is its frequency in Hertz, then

    − ∫ v i n ( t ) d t = M 2 π f cos ⁡ ( 2 π f   t ) = M 2 π f sin ⁡ ( 2 π f   t + π 2 )

    so that for the case K=0 ,

    v o u t ( t )   =   −   1 R   C   ∫ v i n ( t ) d t = 1 R   C   M 2 π f sin ⁡ ( 2 π f   t + π 2 )

    That is, the inverted integral of a sinusoid is a sinusoid with the same frequency, with a phase that leads the original sinusoid by 90° , and with amplitude relative to the input of

      output   amplitude input   amplitude   = 1 R   C   M 2 π f / M =   1 2 π R C f

    Use the equation editor to include your calculations in the Word file. From the Capture part of OrCAD PSpice ® , save the project finalsine in the folder named final on your hard disk.

    During part 5 you should have inserted into the document final.doc :

    During part 5, you should have saved the project finalsine in the directory final on your hard disk.

  6. (10 points) Repeat part 5 for a 2000Hz square wave with amplitude of 1V by creating a new OrCAD PSpice project, finalsquare based on the project finalsine , in the folder final on your hard drive. Do not include another screenshot of the revised schematic, however.

    The output waveform should be triangular whenever the input waveform is a square wave. In the Word file, discuss why.

    Also, compare the simultaneous input and output waveforms and discuss the degree to which they are consistent with the output voltage being proportional to the integral of the input voltage for this particular waveform. Specifically, derive the result

    v o u t ( t )   =   −   1 R   C   ∫ v i n ( t ) d t = { − 1 R   C t ,             when   v i n ( t ) = + 1 V + 1 R   C t ,             when   v i n ( t ) = − 1 V

    and use it to calculate the theoretically expected slopes for the output waveform and compare these values with ones that you read from the simulated output waveform. (The units of the product RC turn out to be seconds , so the slopes ± 1V/RC of the triangular waveform have units of V/s .) Use the equation editor to include your calculations in the Word file.

    From the Capture part of OrCAD PSpice ® , save the project finalsquare in the folder named final on your hard disk.

    During part 6 you should have inserted into the document final.doc :

    During part 6, you should have saved the OrCAD PSpice project, finalsquare , in the directory final on your hard disk.

  7. (10 points) To begin investigating the performance of your integrator circuit experimentally open one instance of Cool Edit and generate a 2000Hz sinusoidal signal 1sec or so in duration. You need NOT save this signal to disk to play it and use it as an input to the integrator. You will connect the output of your sound card to the input of your integrator and play this sinusoidal file to generate an input signal for the integrator. Next, open a second instance of Cool Edit for recording the output signal from the integrator.

    With your audio cables, connect the microphone input of your laptop to the output of the integrator, and connect the headphone output of your laptop to the input of the integrator. Specifically, connect the appropriate red alligator clips to the input and output of your integrator and both black alligator clips to ground. Start playing the sinusoidal signal from the first instance of Cool Edit and, while it is playing, use the second instance of Cool Edit to record the output voltage. If necessary, adjust the playback volume from your laptop so that the maximum amplitude of output voltage is roughly 75% of the display range in the Cool Edit window. If the output voltage is 100% of the display or larger, the sound card may become overloaded and thus record a distorted version of the true output voltage. Save this WAV file as finalVoutsine.wav in the folder final on your hard disk. Into the Word file final.doc , insert a screen shot of the Cool Edit screen that shows the recorded output voltage of the integrator. Then zoom in so that only a few cycles of the output waveform are displayed and include a screenshot of this view, as well.

    Next, record the input to your integrator. Be careful not change the playback volume from that you used to record the output of the integrator. Move the red alligator clip that connects to the microphone input from the output terminal of the amplifier to the input terminal of the amplifier. (The black alligator clip, connected to ground, can remain in the same place.) Then record the input voltage as you play the sinusoidal signal again. Save this WAV file as finalVinsine.wav in the folder final on your hard disk. In the Word file final.doc , insert a screen shot of the Cool Edit screen that shows the recording of the input voltage to the integrator and verify that it is a sinusoid of the proper frequency.

    In the Word file, include a discussion that compares the ratio of the measured input and output sinusoidal amplitudes with the values that you obtained in part 5 from simulation and from analytical calculation. (Because many laptops do not record in stereo, we cannot record simultaneously the input and output to the integrator in separate stereo channels and thus we cannot compare the relative phases of the input and output signals. That could be done with a desktop computer that can record in stereo, although that would require an additional cable with alligator clips.)

    During part 7 you should have inserted into the document final.doc :

    During part 7, you should have saved the Cool Edit file that shows the output waveform of your circuit when the input is a 2000Hz sine wave as finalVoutsine.wav in the directory final on your hard disk.

    During part 7, you should have saved the Cool Edit file that shows the input waveform to your circuit as finalVinsine.wav in the directory final on your hard disk.

  8. (10 points) Repeat part 7 with a 2000Hz square wave as the input to the integrator. Save the input and output WAV files as finalVinsquare.wav and as finalVoutsquare.wav in the folder final on your hard disk. Include screenshots of the recorded output and input of the integrator, as well as zoomed screenshots that show only a few cycles of the output and input waveforms. In the Word file, compare the experimental input and output waveforms with those that resulted from simulation in part 6 .

    During part 8 you should have inserted into the document final.doc :

    During part 8, you should have saved the Cool Edit file that shows the output waveform of your circuit when the input is a 2000Hz square wave as finalVoutsquare.wav in the directory final on your hard disk.

    During part 8, you should have saved the Cool Edit file that shows the input waveform to your circuit as finalVinsquare.wav in the directory final on your hard disk.

For the final project, use ZIP utility software, such as WinZip , to prepare a ZIP file named final.zip that includes all files that you have placed in the directory final on your hard disk. Specifically, final.zip should include the following files:

After you have made the ZIP file, go to the homework submission page, http://www.ece.msstate.edu/courses/homework/submit.html , and submit final.zip .

Submit your circuit in the Ziploc ® bag provided. Write your name, NetID, and e-mail address on the bag's label.