The Inverting Amplifier
Why Do You Need To Know About Inverting Amplifiers?
Analysis Of The Inverting Amplifier
Connecting The Inverting Amplifier
Testing The Circuit
What If Questions
Other Possibilities
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The Inverter - A Simple Operational Amplifier Circuit

        Operational amplifiers can be used to perform mathematical operations on voltage signals such as inversion, addition, subtraction, integration, differentiation, and multiplication by a constant.  You need to understand how to figure out what an operational amplifier circuit does.  We will start with a simple circuit so that we can examine a method that will permit you to figure out how these circuits work and then you will have a more general method you can use for more complex circuits.  So, you have two goals in this section.

   Given an inverting amplifier op-amp circuit with resistor values,
    Be able to compute the gain of the circuit.
   Given an operational amplifier op-amp circuit similar to the inverting amplifier,
Be able to compute the output voltage of the circuit in terms of the signals and electrical elements in the circuit.

Analyzing the Inverting Amplifier

        The circuit we will examine is shown below.  This circuit amplifies a voltage by a factor (- Ro / R1).  What is important in this circuit is that it amplifies by almost  exactly (-Ro /R1) so that the gain of the circuit can be controlled precisely by controlling the resistor values precisely.  The gain of the circuit will not depend upon paramters of the "Op-Amp".  We'll analyze this circuit to get a mathematical prediction of how it works.

Note:  The power supply connections are not shown.  For the operational amplifier to function properly you will need to supply power to it.  Click here for some more information on power supplies for operational amplifiers.

        Also, note that the input labelled "+" is referred to as the non-inverting input, and the input labelled "-" is referred to as the inverting input.  The operational amplifier amplifies the difference between the non-inverting input and the inverting input.  In other words, the output of the operational amplifier is:

Because the operational amplifier amplifies the difference between the two voltages - rather than a single voltage - the operational amplifier is a differential amplifier.

        The operational amplifier has a long history.  It first appeared in vacuum tube manifestations in the 1950s.  These amplifiers were heavy, expensive and prone to failure.  They usually required a power supply that most normal people could not lift.  The amplifiers cost about a hundred dollars, and the supplies several hundred in an era when a hundred dollars was worth much more than today's value.  In the 1960s transistorized versions of operational amplifiers came on the market, and in the middle to late 1960s the first integrated circuit operational amplifiers came on the market.  (Although different technologies were used at different times, the amplifiers had characteristics in common that marked them as operational amplifiers even though built in widely different forms!)

        In this era of team engineering it is interesting to note that one individual, Robert Widlar, was responsible for the development of the integrated circuit operational amplifier in a form that led to wide acceptance.  Today, thanks to Widlar and many others operational amplifiers are available for less than 25 cents and the operational amplifier is probably the most widely used analog integrated circuit.

        Widlar is known not just for his creativity, but also for his zany antics.  Of course, those two characteristics might just go together.  When National Semiconductor went into a period of austerity, and stopped mowing the grass, Bob Widlar brought a sheep and turned it loose to graze.

Analysis Of The Inverting Amplifier

        To analyze the inverting amplifier, we start by making an assumption that the output voltage, Vout, is some "reasonable" value - a value somewhere between the values of the positive and negative power supply voltages.  (Click here for more information on output voltage limits.)  That may well seem like an odd place to start, but we can begin to look at the consequences of making that assumption.

        For example, we might have an output voltage of ten (10) volts. We can figure out what input voltage caused that output voltage of ten volts.  If the gain of the operational amplifier is 100,000, then the input difference, (V+ - V-), must be 10/100,000 or .00001 volts.  That's 100 microvolts, and it's pretty darn small.

        Notice what happens here.  This assumption that the difference between the inverting and noninverting input voltages is just a consequence of the very high gain of the operational amplifier.  It's not special to this circuit.  It's a general idea that we can make use of in other amplifier circuits.

        For all practical purposes that voltage is close enough to zero that we will call it zero when we calculate how the circuit behaves.  We know it isn't zero, but it has such a small value that it will not affect any of our calculations.  You'll need to remember the logic here.

        And, note the following:         For all practical purposes that voltage is close enough to zero that we will call it zero when we calculate how the circuit behaves.  We know it isn't zero, but it has such a small value that it will not affect any of our calculations.

        Since the difference between the operational amplifier input voltages are practially zero and the internal input resistance is very large, we can  make the assumption that the current flowing into the amplifier through either of the input terminals is so small as to be negligible.  Most of the time that's a good assumption because:

Here is a modified circuit diagram that shows the input resistance of the operational amplifier.  You can visualize the input resistance as a resistor connected between the input terminals of the operational amplifier.

        Let's take a minute to summarize the few assumptions we have made so far.

        Assuming that the input difference is small, we can write KCL at the inverting node:  (Notice the little red dot at the inverting node in the circuit diagram.)  (Note also, that we have defined two voltages, V1 and Vout that are both measured with respect to the ground.)

        Here's the KCL equation using the assumption that the voltage at the amplifier input - at the input node - is zero.

I1 + I0 = 0

Technically, we can write KCL in terms of all the voltages involved (taking V+ and V- as the voltages - with respect to ground - at the "+" and "-" terminals respectively).  Doing that we obtain:

( V1 - V- )/ R1 + ( Vout - V- )/ R0 = 0

        However, since we assume that there is no voltage difference between V+ and V- , we can replace V- with V+ and we have the inverting input terminal connected to ground, so V- = 0.  That means we get:

 V1 / R1 + Vout / R0 = 0

After all is said and done, we can solve for the output voltage, and doing that we find:

Vout  = - V1 R0 / R1

There are two things to note about this expression for the output voltage.

And we reiterate the conditions/assumptions under which this result is true.         The result above - for the output voltage - is the result we wanted.  It gives the output voltage in terms of the input voltage and the two resistors in the circuit.  Amazingly, no property of the OpAmp shows up in this expression although the presence of the OpAmp in the circuit is what makes it work the way it does.  The op-amp doesn't show up in the final expression for the output, but the circuit wouldn't work without the op-amp.  Finally, you might want to read the note on gain - especially since it relates to this circuit.
Problems & Questions

Q1.   In this circuit, what is the expression for the ratio of output voltage to input voltage (Vout/V1)?

P1.   In this circuit, you want a gain of ten.  Actually, you are going to have to settle for a gain of -10 because you can't get rid of the minus sign.  Still, if R1 is 5 kW, what is the value you need to use for R0?  Give your answer in ohms.

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P2.   In this circuit, you have it set up for a gain of -10.  The input voltage is .24v.  What is the output voltage?

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P3.   For the same conditions as in Problem 2, the input is changed to -.35 volts.  What is the output voltage now?

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P4.   Assuming that you have an inverting amplifier with  R1 = 10,000W, determine a value for Ro that will produce an amplifier with a gain of 3.3.

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P5.   Assuming that you have an inverting amplifier with Ro = 4,700W, determine a value for R1 that will produce an amplifer with a gain of -5.

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        Let's summarize what we did to solve for the output of the inverting amplifier.  Each step we took was simple, but they combine to give us a powerful method that we can use when we examine other operational amplifer circuits, so it will be worthwhile to review the process to be sure that we understand the approach.         We need to be more specific about what we mean when we say the output is "reasonable".  Whenever you use an operational amplifier you need to use two power supplies.  Often you use a +12v and a -12v supply, although +/-15v is also common.  Whatever the power supply, the output of the operational amplifier is limited by the supply you use.  Usually the limit is within a volt or so of the power supply voltage, so if the supply voltage is +/-12v, you might be able to drive the op-amp up to 10.8v (or something like that) and down to -10.8v.  (And, it's not always symmetrical so you need to be careful.)  If your circuit tries to make the op-amp output voltage 17.3 v in that situation, you aren't going to see that voltage.  You'll get 10.8 instead.  When that happens, you say that the op-amp is saturated.  Otherwise, when everything is copasetic, and the amplifier is not saturated, you say that the op-amp is operating in the linear range.

        When the op-amp is operating in the linear range, then there is an expression for the output voltage in terms of the gain.

Vout  = Gain*( V+- V- )

where:         The gain of a 741 operational amplifier is typically well over 100,000.  So, if the output is limited to something like ten (10) or eleven (11) volts, the input difference, V+ -V-, can't be more than about 100mv (microvolts).


P6.   An operational amplifier has a gain of 250,000.  The output voltage is 3.75 v.  Calculate the difference between the input voltages and the inverting and non-inverting pins on the op-amp.

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            The inverting amplifier is an important - and often used - operational amplifier circuit.  It can also be viewed as a prototype circuit that is a starting point for more complex and even more interesting circuits.  You need to be sure that you remember the assumptions and analysis techniques you use to figure out the output voltage in this circuit.  Here is a summary.
Wiring The Op-Amp Inverter Circuit

       Next, we are going to show you how an inverter circuit is connected.  Please read through the steps carefully and we will show you how the components look as you insert them into a circuit board.  Before we do that, you need to understand how the chip is wired internally.  Here is the pin-out for a typical 741 op-amp in a DIP (Dual In-line Package).

        Finally, if you want to, you may proceed directly to the section on testing the circuit.  Click here if you want to go to the section on testing the circuit.  Otherwise, you should be ready to go.  Check all the connections made in the first four steps.  When you are sure you have it correct, then you can turn on the power supplies and begin testing your circuit.  Below there's a photo of a completed circuit, and a hotlink to take you to the section on testing the amplifier.

Testing Your Circuit

        Here's the circuit again.  We'll step through what you need to do to check that you have the circuit wired correctly.

Some Practical Considerations

        Whenever you use an operational amplifier, the power supply voltages limit the output voltage.  Let's go back to the inverter circuit we considered in this section.  Let's assume that we have an inverter with a gain of -2.  (That is, Ro/R1 = 2.)  If V1 = 5.0 volts, we would expect the output voltage to be -10. volts.  That's probably OK.  However. if V1 = 10.0 volts, then we might expect the output voltage to be -20. volts.  But, the output voltage can't be -20 volts.  If you have power supply voltages of  +/-12, it can only go as low as -10.5 or so, so that's what the output will be, -10.5 volts.

        We have to conclude that the output voltage is always limited by the supply voltages, and try as we might, we can't make the output voltage go outside the limits set by the supply voltages.  If we were to build an inverter circuit, with a gain of -1, then a plot of output versus input has to look like the one below.  Without power supply limitations we would expect a straight line with a slope of -1, and we would not expect the saturation characteristic found below for the plot of output against input.

        The net result is that whenever the input voltage is such that it would drive the output voltage beyond the rails, the output voltage gets clipped (does not reach a value higher than the saturation value!), and never reaches the desired value.  This produces distortion in the output voltage when you try to amplify voice or music signals, for example distortion that can be heard.  You'll have to have earphones or speakers connected to your sound card to hear this.

Notice the difference in the two sounds.  Even though the two sounds have just about the same amplitude the clipped sound is harsher than the pure sine signal.  That's the effect of the "hard limiting" - the clipping - of the saturation in an operational amplifier.

What If Questions

        The operational amplifier is a versatile circuit element, and is used in many different ways.  There are many ways that the op-amp is used, and many of them involve variations on the basic inverter circuit.  Now that you've examined the inverter circuit pretty exhaustively, we can start to look at other possibilities.  First, let's consider what some of those possibilities might be.  Here's the circuit again.  Think about what could be changed.

Problems & Questions

Q2.   Which devices can be used in place of the input and feedback resistors?

    Click here if you would like to see an interesting possibility with a feedback capacitor.

        There are other possibilities for changes that could be made in the inverting circuit.  Did you think of using two input resistors like this circuit?

        This is actually a version of a widely used circuit, and it's important enought that you should understand how it works.  This circuit is interesting because it has two (!) inputs and the output is going to depend upon both of those inputs now.  We're going to ask you to try to analyze this circuit and determine exactly how the output depends upon those two inputs.

        You will need to plan how you will analyze this circuit.  How can you analyze the circuit?  Where do you start?  It will be helpful to check what you did when you analyzed the inverter circuit.  There were some important assumptions you made.  They're shown again below.  Will they help you this time?  Are they true in this circuit?

  • The output voltage, Vout, is within the value between the positive and negative voltage supply.  It's a "reasonable" value.
  • The input difference, (V+ - V- ) is small enough that we can consider the value to be approximately zero.  This is due to large gain of the amplifier - the infinite gain assumption.  We will assume that the input voltage difference is zero.
  • Since we will treat the input difference as zero, and assume input resistance (the resistance between the non-inverting and inverting inputs) is infinte, then the current flowing through both of the inputs of the amplifier will also be so small that it is negligible.  We will assume that no current enters the input terminals of the op-amp.
  •         If these assumptions are true, it will help us a lot because then we can assume that the input voltage (at the node with the red dot in the circuit diagram) is zero.  Then you can write KCL at the node with the red dot.  Maybe that red dot is Rudolph's nose, and he's bringing you a present for the holiday season!

            At this point it is up to you to finish the analysis of this circuit.  You should plan what you're going to do.  Here's an approach we recommend.

            On the other hand, you can get more insight by clicking here.

            Finally, there is one other "What if?" question that many people raise.  Here's the question:

            If you examine this circuit carefully, it is still possible to miss the fact that the two inputs have been reversed.  The inverting input is grounded, and the feedback, through Ro, is to the non-inverting input.  That's a change from what we have been doing.

            Does this make a difference?  Yes, it makes a major difference.  This circuit has positive feedback, and that may mean the circuit is unstable.  It could oscillate or it could hang up at a saturation limit.  There's even an outside chance that it will work, but it probably won't.  That's a much more advanced topic, and you can click here if you want to examine why the circuit is unstable, but you may need to work through material on circuits, Laplace transforms, linear systems, and more, to understand the argument.

            One last point is that even our original circuit - with the input polarities correct - might not work.  It all depends on the frequency response of the operational amplifier, and there are some special-purpose operational amplifiers that don't work in some simple circuits.  The author has had that experience, and it's tough to figure out what is going wrong.  So, if you connect an op-amp circuit and it doesn't work as you predicted, you probably didn't connect it correctly, but there's a small, non-zero, chance that you did everything right and it's just not going to work.  However, for the circuits we discuss, using a 741 style op-amp will almost always result in a circuit that works - if you connect it correctly.


            At this point you've looked at one operational amplifier circuit and done a little thinking about how you could make it into something else.  That's a good start on operational amplifiers.  You've learned a little about how to analyze operational amplifier circuits and the kinds of assumptions you often, but not always, make when you work with those circuits.  You have the basic knowledge you need to go on.  Here's hoping that you continue to have fun in this area.  Build the circuits.  If you overheat a 741 or two it's no big deal.  Learn and have fun.

    Problems & Questions

    Q3.   What kind of an amplifier is an operational amplifier?

    Q4.   What is a typical gain for an operational amplifier like a 741?

    Q5.   Who was the individual most responsible for the development of the integrated circuit operational amplifier?

    Q6.   What are typical power supply voltages for a 741?

    Q7.   What are typical limits for the output voltage of a 741 when operated with +12 and -12 volt supplies?

    P7.   You have an inverting amplifer with R1 =   Ro = 2,700W, determine the output voltage when the input voltage is +5 volts.

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    P8.   You have an inverting amplifer with R1 =   Ro = 2,700W, determine the output voltage when the input voltage is -5 volts. 

    Enter your answer in the box below, then click the button to submit your answer.

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