Operational Amplifiers

David E. Thompson, Professor

University of Idaho



In signal conditioning, perhaps the most useful and flexible active device available is the Operational amplifier, or op-amp.  The op-amp was named because of its ability to perform several mathematical operations (add, subtract, multiply by constant, differentiate, integrate, and more).  For many applications, it has been superseded by the digital computer, but op-amps are an essential element of many instrumentation solutions.

Text Box:  
Figure 1.  Tube-based & modern IC op-amps.
Figure 1 portrays a K2-W tube-based general purpose computing Op-Amp from George A. Philbrick Research first introduced in 1952. The op-amp is shown with and without its bakelite shell. The first solid-state monolithic op-amp (not shown) was manufactured by Fairchild Semiconductors and sold for $300 in 1963 as the µA702.  It had a tendency to burn out when loaded, contained just nine transistors, and sales were primarily Military and Aerospace consumers.  This was followed up by the µA709 with higher gain, a larger bandwidth, lower input current, more user-friendly supply voltage requirements, and a $70 price. Today op-amps sell for pennies in large quantities, and are infinitely more powerful and useable than their predecessors.


The op-amp is constructed from several transistor stages and commonly employs a differential-input stage, an intermediate-gain stage and a push-pull output stage. The differential amplifier consists of a matched pair of bipolar transistors or FETs. The push-pull amplifier has the potential to provide large currents to the op-amp’s load and hence we state that the op-amp has a small output impedance.  It typically also has an extremely high input impedance and thus limits the current drain from sensors and other instrumentation elements to very small levels (nano- to pico-amps).  This ability to serve as a buffer to shield a sensor from the load of a measurement system is important, but the ability to manipulate the signal to improve its readability and reduce noise is much more important.


The op-amp is a highly linear device with its output voltage proportional to the input voltage, or Vout=A*Vin.   An important property of the op-amp is that the open-loop gain, A, is a very large number (typically 106 to 1015).  This gain is so large that feedback must be used to obtain a more useable gain, frequency response (transfer function), and stability.


Like most highly successful and widely used integrated circuits, costs can be quite small.  The cheap models operate (with external components limiting the overall gain to 1) from DC to about 10MHz, while the high-performance models may operate up to 1 GHz.  A popular device ($0.18 each) is the LM411 dual op-amp whose open loop gain is shown in Error! Reference source not found..  Note that this is a log-log plot of gain (output/input) versus frequency, and shows that the response curve is flat down to very low frequencies and even to a zero frequency input voltage (DC).  The gain drops 1 decibel/decade from around 2 Hz.  Electrically, this model amplifier has an input bias current of about 50pA and total harmonic distortion of 0.02%.  Op-amps are integrated circuits (IC’s) and usually available in an 8-pin dual, in-line package (DIP) as well as other forms.  Some ICs have more than one op-amp on the same chip to improve the density on printed circuit boards.



Before proceeding we define a few terms:

linear amplifier - the output is directly proportional to the amplitude of input signal.

open-loop gain, A - the voltage gain without feedback (about 106).

closed-loop gain, G - the voltage gain with negative feedback.

negative feedback - the output is connected to the inverting input forming a feedback loop (usually through a feedback resistor, Rf).

Text Box:  
Figure 2. Open loop gain characteristics of an operational amplifier.

Open-Loop Amplifiers

Figure 3.a shows a complete diagram of an operational amplifier. A more common shorthand version of the diagram is shown in Error! Reference source not found.b, where missing parts are assumed to exist that result in different gains for different inputs. The inverting input means that the output signal will be of opposite polarity with the signal applied to this input.  For harmonic signals, that means a 180 degrees phase shift. On the diagram, typical power supply voltages are V+ = +15Vdc, and, V- = -15Vdc.

For dual supply amplifiers, Vcc is typically, but not necessarily, ±15V. The positive and negative voltages are necessary to allow the amplification of both positive and negative signals without introducing a bias to the voltages.  Since many systems must interface to computers, many op-amps have been designed to operate from a single supply voltage at +5Vdc and the reference voltage for alternating signals raised to approximately 2.5vdc.

For a linear amplifier (and a differential amplifier) the open-loop gain is A and the output is given as

                                                                                                          Equation 1


Figure 3. Operational amplifier a) Schematic diagram, b) Shorthand.


As implied in Figure 3 and Equation 1, two conditions must be satisfied for linear operation:

  1. The actual input voltage at the amplifier must operate within a very narrow voltage range since A is so large, or

  1. For no clipping or saturation, the output voltage swing must be restricted to

Figure 4.  Input-output relationship for an open-loop op-amp.


Ideal Amplifier Approximation

The following are properties of an ideal amplifier, which to a good approximation are obeyed by an operational amplifier:

  1. Large forward transfer function (high open loop gains, low output impedance).
  2. Virtually nonexistent reverse transfer function.
  3. Large input impedance, (any signal can be supplied to the op-amp without loading problems).
  4. Small output impedance, (the power supplied by the op-amp is not limited).
  5. Wide bandwidth.
  6. Seemingly infinite gain, A>>>1.

If these approximations are followed, two rules can be used to analyze op-amp circuits:

Rule 1:  The input currents are negligible.

Rule 2:  The input voltages are essentially equal.

To apply these rules requires negative feedback.

Feedback is used to control and stabilize the amplifier gain. The open-loop gain is too large to be useful since even the slightest input noise will causes the circuit to clip and/or saturate. Stabilization is obtained by feeding the output back into the input (closed negative feedback loop). In this way the closed-loop gain does not depend on the amplifier characteristics.


Derivation of Gain Equation

Consider the circuit shown in Figure 5.  One can sum voltages and currents using Thevanin’s Theorem results in the following equations. 

Figure 5.  Inverting amplifier with currents and voltages shown.

See if you can identify the loops used to generate these equations.

                                                                                                                         Equation 2

                                                                                                                       Equation 3

As shown in Figure 4, we can also write following equation for the linear region of the Op-Amp:

                                                                                                                                        Equation 4

Summing the currents at the location of eb yields (since ib is negligible)

                                                                                                                       Equation 5

Solving these equations simultaneously yields the gain equation for the amplifier as shown in Equation 4.

                                                                                                Equation 6

For large open loop gain, A>>1, this reduces to G = -Rf/Ri.  This is the classic gain for an inverting operational amplifier.  One can use this general derivation approach for any configuration of components that modify the gain of an operational amplifier.



Non-inverting Amplifiers


Figure 6 shows a non-inverting amplifier, sometimes referred to as a voltage follower.

Figure 6. Circuit and shorthand diagram for a non-inverting, unity-gain amplifier.


Applying our rules to this circuit we have Ein=Eout and iin=0.  The amplifier has a unit closed-loop gain, G=1, and does not change the sign of the input signal (no phase change for a harmonic signal).  This configuration is often used to buffer the input to an amplifier since the input resistance is high, there is unit gain and no inversion. The buffer amplifier is also used to isolate a signal source from a load.


The use of inverting amplifiers is often of the greatest utility and one can achieve a non-inverting amplifier by cascading two inverting amplifiers.  The general schematic for an inverting amplifier is shown in Figure 7.  Here, the gain, G = - Rf/Ri allowing us to multiply a signal by a constant.

Figure 7.  A simple inverting op-amp circuit.

Similarly, one may use a modification of this circuit to add a number of voltages at different gains.

Figure 8.  Summation of voltages.

What other operations can we perform with op-amps?  How about integration and differentiation as shown in Figure 9a and Figure 9b, respectively.

Figure 9.  Integrator and differentiator circuits.


Frequency Response

In attempting to manipulate signals with frequency content, it is important to note that Gain is not a simple number, but a complex number with both amplitude and phase that are frequency dependent.  We will cover this in detail later.  If we consider an op-amp configured as a simple amplifier, what is the frequency range of operation for a gain of 1000?  The answer is labeled as the DC-100X curve in Figure 10.

Figure 10.  DC and AC coupled amplifier harmonic response curves.

Similarly, if a small capacitor is put in front of such an amplifier and the gain adjusted to G=1, the response curve will be the curve AC-1X as shown.  As an engineer, when you calibrate an instrumentation system, it is important to vary the input frequency and determine the shape of the gain curve from Gain, G = Eout/Ein as well as any phase changes.  We will discuss determining the complete harmonic response of an instrument later.


Practical Circuit Considerations


In any practical circuit, there may be deviations from the ideal op-amp.  In general, these are often negligible, but for precision circuits we may need to add circuitry to improve the result.



Non-Inverting Single Supply Circuits

Even though modern op-amps are great, they may still have a bothersome offset or bias voltage inherent to the integrated circuit.  In a voltage follower circuit, often a feedback resistor is used as shown in

Figure 11 along with a resistor to ground to introduce a variable gain.  Note that we still have negative feedback for stability.



Figure 11. Non-inverting amplifier with variable gain.

For this circuit, the gain, G = 1+(Rf/Ri).  This is especially useful for circuits which can only have a single supply voltage (typically +5v in digital applications).  The input impedance is simply Ri.


Reducing Offset Due to Bias Currents

An improvement in the circuit shown in Figure 6 allows us to compensate for the small voltage offsets, Eoff, resulting from the bias currents, ib, at the input of a real amplifier as shown in Figure 12.

Figure 12. Inverting amplifier with bias compensation.


Differential Amplifier

One is frequently faced with the necessity to perform a differential operation.  The schematic for this operation is shown in Figure 13.  If you study this configuration, you will note that both the output signal and the ground reference are feedback in this application.

Figure 13. A balanced differential amplifier.


Current-to-Voltage Amplifier

There are a growing number of modern sensors that produce an output current that is proportional to the input variable being measured (measurand). A simple current-to-voltage converter is shown in Figure 14.  Applying our ideal amplifier rules gives and output voltage Eout = iinRf.

Figure 14. Current-to-voltage converter.

The use of current-producing sensors is helpful in lengthy cable runs where the voltage losses due to the resistance of the cable will affect the measurements.


Integration Circuit

The output of an integration circuit as shown in Figure 15 is stable because it effectively integrates out any input noise.  On the other hand, a small DC offset error on the input will cause the output to grow continually with time until the op-amp saturates.  When the output reaches either the positive or negative power supply rail, the op-amp cannot drive the output higher, and the output is said to “saturate”.  It is therefore important to avoid even the smallest offset error on the input to an integrator circuit.

Figure 15.  An integrator circuit with a gain of -1/RiCf.


Differentiation Circuit

The output of a differentiation circuit is often unusable because of input noise.  Because the derivative of a signal includes high frequency noise and the output of a differentiator is proportional to its frequency, we have to limit the bandwidth of the circuit. 

Figure 16. A practical differentiator circuit.

The frequency response of this system is shown in as depicted in Figure 16.  This looks like a combination of a differentiator and an integrator circuit, which it is. You will note that the maximum gain is limited by Gmax = -Rf/Ri.   The slope of an ideal differentiator on the log-log plot is +1 since d/dt(ejwt) = jwejwt.

Figure 17. Frequency response of ideal integrator (red), differentiator (blue), and amplifier(gold) and their combination (green).



We can combine the above inverting, summing, offset, differentiation and integration circuits to build an analog computer that can solve differential equations. However, in modern applications, differentiator and integrator circuits are mainly used to condition signals.  Another important application is supporting digital data acquisition. 


Sample-Hold Circuit

There are times when we wish to grab a voltage and hold it for some period of time.  This is supported by the Sample-Hold configuration shown in Figure 18.  The capacitor, Cs, is the sampling capacitor, and it’s leakage is a key factor in how long the sample is retained.

Figure 18. Sample-Hold amplifier configuration.


Instrumentation Amplifiers

The circuit in Figure 19 is an instrumentation amplifier with an overall gain of 90 and uses matched 0.1% resistors.  This design was provided by Analog Devices who sells a single IC Instrumentation Amp to replace such a circuit.  Their product (AD623) uses only a single external resistor to set the gain and can be studied in more detail at the URL below:


This example has excellent common-mode voltage and noise rejection, and an excellent application is to amplify the output from a Wheatstone bridge. 

Text Box:  
Figure 19.  Instrumentation amplifier example using single-supply op-amps, with an overall gain of 90 and an offset of +2.5vdc.


The following are some internet website references to common op-amps.


National Semiconductor

Company website


Low Power Dual PDIP (8) OpAmps   ($0.18)


Low Drift & Offset Dual JFET MDIP (8) OpAmps   ($0.45)


Low Drift & Offset Dual JFET MDIP (8) OpAmps   ($0.55)



Analog Devices

            Company website


AD623, single or dual supply, rail-to-rail instrumentation amplifier ($1.25)


OP284, dual amp, single or dual supply, rail-to-rail operational amplifier ($2.70)