Op-Amps (Operational Amplifiers) Explained In Detail

Introduction

 In many electronic circuits, the signals (voltage differences) that are generated and manipulated are very small.  Therefore, amplification is often essential.  When playing a CD, for instance, the signals generated in the CD player are quite small and will not adequately drive a speaker system.  The signals from the CD player are therefore passed into the stereo amplifier (which often comes as a tuner/amplifier combination in modern home stereo systems).  The heart of the stereo amplifier is the operational amplifier, or op-amp, which takes low level voltage signals as inputs and produces large output voltages that vary linearly with the input voltage.

Operational Amplifiers

The op-amp is a simple example of an integrated circuit.  The common 411 op-amp used in this laboratory contains 24 transistors on a single silicon chip.  Many integrated circuits are much larger:  a computer's microprocessor can contain several million separate elements.  Each transistor is a three terminal semiconductor device that controls a large current with a small one.  If you later study electronics, you will learn about transistors.  In this course, we will omit that stage and show how the functioning of op-amp circuits can be understood without knowing anything about the individual transistors of which op-amps are composed.  You need only understand a few basic principles (explained below) and Kirchoff's circuit laws.  Our reason for doing this lab is to show you how practical problems can be solved using electronic devices.  These days, most scientists solve practical instrumentation problems using op-amps and other integrated circuits rather than discrete components.

                     

An op-amp has three main terminals.  The circuit symbol for an op-amp is shown in figure 1.  The V_- input is called the inverting input, the V₊ input is called the non-inverting input, and V_out is the output voltage.  All voltages are measured relative to the ground line of the power supply for the op-amp.  All op-amps need a power supply in order to provide the amplification, since without a voltage higher than the input voltages it would be impossible to produce amplification.  Generally the power supply is provided by connections at +15 V and -15 V to the op-amp.  (Note that by convention, these power supply connections are not shown on the circuit symbol for the op-amp.  However, you must always connect them up in the lab.)  The supply voltages determine the maximum output voltage range of the op-amp, and if V_out reaches one of the supply voltages the op-amp is said to be in "saturation".  This situation is to be avoided since if the op-amp is in saturation, its output cannot be varying linearly with the inputs.

Figure 1:  Op-amp inputs and output

The op-amp will amplify both AC and DC signals, although there is a high frequency f_3dB (analogous to the f_3dB or “cutoff frequency of a low-pass filter) determined by the type of op-amp; frequencies beyond this value will be amplified less and less as the frequency increases. We describe the signal amplifying properties of the op-amp by giving its gain, and the ratio between the output signal and the input signal.  In the so-called "open loop" configuration shown in figure 1, the output voltage is given by

   

   

where the open loop voltage gain A₀ characterizes the op-amp.  Note that the voltage difference between the inputs is amplified and not the voltage between an input and ground.  If you add 5 volts to both inputs, this does not affect the output at all!  Equation 1 makes it clear why V_- is called the "inverting" input; it contributes negatively to the output signal.

        The input impedance of an op-amp is typically 10⁶ W although it can be as high as 10¹² W in some models.  The output impedance is usually very small.  The gain A₀ is extraordinarily high, typically 10⁶  at low frequencies, so that an op-amp hooked up solely with two inputs and its supplies would almost certainly be in saturation (a voltage difference of only 15 microvolts between the inputs would be sufficient to cause saturation).  By using "feedback" (see below), this high gain can be controlled and made useful. 

Feedback

        Because the op-amp has such a huge open loop gain A₀, it is always used with a feedback network that controls the inputs by returning some voltage from the output to the input.   This reduces the effective gain, but it also causes the amplification to be nearly independent of frequency up to much higher frequencies than the open loop f_3dB mentioned above.

        The term feedback refers to configurations in which a fraction of the output voltage is returned (it is "fed back") to one of the inputs (see figure 2).  Thus, the output V_out depends upon itself, as well as the input to the circuit, V_in.  (If you find this idea confusing, you are not alone.  The US Patent Office refused to grant its inventor a patent for this extremely important engineering concept because it didn't believe his idea would work!)  Feedback can be positive (returned to the non-inverting input) or negative (returned to the inverting input), but negative feedback is used primarily in analog circuits because it yields stable, controllable outputs, and we will concentrate on it.  For positive feedback, an increasing output V_out

                        Negative feedback--stable                         Positive feedback--unstable

Figure 2:  Feedback

drives the inputs even further positive, resulting in a still more positive V_out.  (A similar argument can be made that once V_out swings negative, it will result in a large negative swing.)  As a result, the device will always be in saturation.  While this can be useful for some purposes (e.g. for making oscillators and in digital circuits), we shall concentrate on "negative feedback" here.  To make the idea more comprehensible, we will first consider some non-electronic examples of feedback.  

 EXAMPLES OF FEEDBACK

 Steam engines were equipped with devices called governors to make sure their pressure did not exceed a safe level.  The governor consisted of a valve connected to an array of spinning weights.  The valve opened wider as the weights spun more quickly.  Steam from a vent controlled by the valve made the weights spin around:  if the pressure rose, the weights would spin more quickly.  In turn, the weights would open the valve more, thereby lowering the pressure.  This represents a case of negative feedback because the output (the pressure in the steam engine) was made to decrease automatically if it became larger than the desired value.

 The concept of feedback has extremely wide applications to other fields of study, including mathematical biology and economics.  For example, predator-prey relationships rely upon several feedback loops that determine the stable size of populations.  If the number of predators (the system's output) increases, then the number of prey animals (the input) will decrease.  The negative feedback in this system occurs because an increase in the output (predator population) results in a decrease in the input (prey population).  There will in general be an equilibrium ratio of populations as a result of the stabilizing influence of the negative feedback, with, of course, many other factors entering in to establish their exact sizes.

        

In another example, computerized trading of stocks on the stock market represents a prime example of the pernicious effects of positive feedback.  Extremely large investors, such as pension funds, can buy and sell stocks using computer programs set to make trading decisions based on the behavior of a market index, such as the Dow Jones Industrial Average.    These trading decisions are the system's inputs.  When the stock market index (the system's output) begins to drop, these programs are designed to quickly sell off stocks in order to minimize investors' losses.  However, a large investor can further depress the stock market index by selling off its stocks (i.e. the system has positive feedback).  This system can lead to wild oscillations, or even a crash, should many large investors use such programs during a period of sharply falling prices.  The Securities and Exchange Commission decided to regulate computerized trading after this practice was implicated in the major crash of 1987.

Golden Rules

        

One can show mathematically that the results of feedback in op-amps circuits are summarized by two "golden rules", which we will take as our starting points for figuring out how op-amp circuits will function:

1)       The output of the op-amp attempts to do whatever is necessary to make the voltage difference between the inputs almost equal to zero.  (That's the function of high gain of the amplifier and the negative feedback; if the output voltage rises too much, it drives the input voltage difference down.)  The voltage difference between the inputs is so close to zero that we can assume that it really is zero in analyzing circuits.  It turns out that if A=10⁶, then the error in this assumption is only about one part in a million.

2)       The inputs draw almost no current, because the input impedance of the transistors attached to the inputs are very high.

       

Figure 3:  Inverting amplifier circuit (the triangle at the bottom denotes power supply ground).

        Using the golden rules, the negative feedback "inverting amplifier" circuit shown in figure 3 can be analyzed.  From golden rule number 1, the voltage at the inverting input must be at ground because V₊  is at ground.  (The inverting input isn't actually connected to ground, rather the internal circuitry of the op-amp labors to keep it very near ground.  We call such a voltage a virtual ground.)   From golden rule number 2, all of the current through R₁ must flow through R_F, because no current flows into the op-amp inputs.   We arbitrarily take the direction of the conventional (positive) current to be to the right.  Then applying Kirchoff's laws gives

             

                                                                                                                                      

This means that you can choose the gain by selecting the values of the resistors!  Also, the gain is not dependent on the details of the particular op-amp (e.g. the exact value of the open loop gain A₀), but rests only on the open loop gain being large, and on the values of the resistors.  (The largeness of A₀ is the underlying justification for Golden Rule 1.)   Since the output is negative for positive inputs, the amplifier called an inverting amplifier. 

Experimental Procedure

Experiment 1:  Op-amp in an open loop configuration

        You will be using the LF411 op-amp in today's lab.  (The F refers to the original manufacturer, Fairchild, and the other numbers specify one of many different integrated circuits made by this company.)   It comes as a eight-legged DIP ("dual in-line package") IC (integrated circuit) shown in figure 4.  The LF 411 has been inserted into your breadboard for you.

                                       

Figure 4:  Connections for the LF 411 op-amp (top view)

        The "offset null" connections will not be used here; their purpose is to allow the user to make small adjustments if desired so that the output is precisely zero when the input voltage difference is precisely zero. 

Figure 5:  Open loop circuit (the triangle at the bottom is power supply ground).

1)       Assemble the circuit shown in figure 5.  Here a voltage divider is used to provide the op-amp with an input voltage (relative to power supply ground) varying from +15V to -15V, with a potentiometer controlling the exact value. 

2)       Use a DMM to adjust your input voltage until it is very close to zero.  Notice that the output voltage is far from zero because of the very high open loop gain of the op-amp.  Try adjusting the input and sketch the output voltage as a function of the input voltage.  Explain briefly why your graph looks the way it does.

Experiment 2: Inverting Amplifier with negative feedback

        Figure 3 (presented earlier) employs negative feedback (from V_out to V_-) to give a stable gain that is controlled by the ratio of two resistors.  In this portion of the lab you will construct an op-amp circuit with negative feedback.

1)       Set up the circuit shown in figure 3 using a 411 op-amp.  Remember to connect the +/-15V DC supply, and also the ground (common) from the DC supply!  Use your function generator's sine wave setting and the LO output set to a very low voltage to supply V_in.  The feedback is set up by connecting pin 6 to pin 2 with R_F.  Choose R_F=100 KW, and use a resistor for R₁ that gives a closed-loop gain of 100.

2)       Plot both the input and output voltages as a function of time on your oscilloscope, using a roughly 1 kHz signal.  You should start with an extremely low amplitude input signal to see a clear sine wave output.  Sketch this display, carefully noting the amplitudes and phase relationships of the two signals.   Determine the gain of your circuit from this data. 

3)       Now, try increasing the amplitude of your input signal until you start to see saturation (i.e. flattening as the output reaches a maximum positive or negative value)  at some times.  How large an input is required to do this?  What determines whether the output is saturated?

Experiment 3: Weighted summing circuit

        Op-amps can be used to do mathematical computations.  For example, they can be used to compute the sum of many inputs, with each one multiplied by a selectable gain.  For example, the output of the circuit shown in figure 6 is given by the following expression:

                    .                   

                                                             

This can easily be generalized to an arbitrary number of inputs by adding terms to the equation.

Figure 6:  Weighted summing circuit.

       

        Another useful function of an op-amp is to compute the integral or derivative of an input signal.   It may be surprising to realize that all the integrals we compute mathematically can be calculated electronically by the simple op-amp circuit shown above.  Figure 7 shows an op-amp circuit for which the negative feedback is now through an RC circuit rather than just a resistor.  From the golden rules, the current flowing through R and C must be equal just as for the resistors in figure 3 above.  (Note that R_S carries insignificant current because you will choose it to be very large (>>10´) compared to R.  It is there to discharge the capacitor for very long times when the input voltage is turned off, or if the function-generator has a DC offset.  Otherwise, charge will build up gradually and saturate the op-amp.)  So

                                            

                                                                                         

Therefore, the output voltage at a given time is the integral of the input voltage from some initial time (when the feedback was connected) to the present time (t). 

        However, in this case, a small deviation from ideal behavior in the op-amp can cause problems.  As you will have noted earlier, there can be a small offset voltage between the inputs of an op-amp even when the output is zero.  In an amplifier circuit (e.g. experiment 2), this can often be ignored.  In an integrator, however, it is a real problem, because the small constant offset voltage will be integrated to produce a gradually growing output that eventually dominates over the desired integration of the real input signal.  This linearly increasing spurious output is manifested by a gradual accumulation of charge on the capacitor.  The function of R_S is to allow this charge to leak off slowly.  This prevents the "drift", but it also means that the circuit will not act as an integrator for signals whose frequencies are too low (smaller than 1/R_SC).  A differentiator can be constructed by small modifications of this circuit, but we shall not do so here.

1)       Assemble the circuit shown in figure 7 and test its performance at a frequency w that is large compared to 1/R_SC by using an input signal from the LO output of your function generator.  Try 1/R_sC  100 s^-1 with C = 0.1 µF, and adjust the frequency so that the output signal has a convenient amplitude.  To get a substantial gain you will need w << 1/RC.  (Apply Eq. 6 to a sinusoidal input signal.)  Compare the amplitudes of the input and output against the theoretical prediction.

2)       Use a variety of input signals from your function generator, starting with a square wave.  Be sure to try out sine waves and triangle waves too.  Sketch sample input and output signals in your brief report.  Explain your results qualitatively.

3)       Also check one case (say, the square wave) as a function of frequency.  Find out what happens if the frequency is made either too high or too low.  

        In this experiment you have explored only a few of the hundreds of useful tasks that op-amps can perform.  Other circuits that can be easily built with op-amps include current-to-voltage converters; oscillators of various kinds; precision current sources; precision AC-to-DC converters; filters of various kinds; timers; log generators (output is the log of the input); precision power supplies.  Most of these circuits are far superior to circuits that can be constructed from individual semiconductor devices.  The fundamental reason is the high gain and high input impedance of the op-amp.