Corollary Theorems: Operational amplifiers



Operational Amplifiers

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Operational Amplifier (Op Amp or OA for short) is an analog integrated circuit developed in 1965 by Mr. Robert Widlar at Fairchild Semiconductors Laboratories. Now, the beauty with this OA is, we work with it mathematically--well, almost mathematically.

In terms of construction, OAs are based on transistors' analog amplification [mostly FET transistors, but there are a few old OAs built with BJTs], and each OA contains tens or hundreds of transistors. However, one OA workd as just one, functional, complex unit block, and it has many specific characteristics.

Working with analog signals is far from being easy, therefore OAs were designed to help us do the job. As hardware designers, the best thing is to avoid analog signals as much as it is possible. If you have an analog input, the first thing to do is, send it to an Analog-to-Decimal channel, and then use it as a digital value inside controller's memory. Unfortunately, that is not always possible. Although we live in a digital world today, there are two aspects which will forever require some analog processing: the inputs, and the outputs.

This Operational Amplifiers page is going to be a tough one to explain, because the topic is quite complex--though it is also extremely important, therefore it requires increased attention. The classic μA741 OA has been employed for this exemplification, and also for the simulation models used.

The following structure presents, summarily, a few general topics about working with Operational Amplifiers:
1. Operational Amplifier Specifications
2. Summing Amplifier
3. Noninverting Amplifier
4. Voltage Follower
5. Difference Amplifier
6. Integrator Amplifier
7. Differentiator Amplifier
8. Logarithmic Amplifier
9. Comparator Amplifier
10. Active Filters
11. Other Operational Amplifier Circuits


The basic notions highlighted in this page are related to a few electronic design topics presented in the first part, Hardware Design, of

The best thing to do is to download the Data Sheet (a PDF file) of μA741. O course, you will discover nothing impressive in there: just lists of specifications and a few graphs.

An OA has the following main characteristics:
1. High Voltage Gain, noted with Av, and having 103..109 levels of amplification
2. High input impedance; the ideal case is (Zi --> ∞ ohms) [read this as: the input impedance tends to reach infinite values in ohms]
3. Wide Bandwidth 
4. Low output impedance; Zo --> 0 ohms

Let's take a closer look at μA741 using a few common schematic models.


Schematic Model
Fig 1: μA741 pin names assignment

Note that the OA has no direct connection to the board (circuit) ground. However, the circuits inside 741 are referenced to ground virtually, via external references, and you will see this aspect exemplified in the next pictures.

On pins N2 and N1 it is (usually) wired a 10K potentiometer having the cursor connected to (V-). That potentiometer allows for input voltage and current offset adjustments.
741 ground reference model Fig 2: External (virtual) ground referenced model

This model allows us to understand the following relations:

Ii = (V1-V2) / Zi
Vo = Av * (V1-V2) - Io* Zo
741 model with feedback
Fig 3: Inverted OA with "feedback"; model used to calculate OA's performances

Model approximations:

I2 = 0 (infinite input impedance)
Vo = -Av * V2 (zero output impedance)

Voltage Gain:  Av = -Vo / Vi = - (Zf / Zi) or
Vi / Zi = - (Vo / Zo)

A few terms and formulas used to calculate OA schematics need to be explained. 


Input Offset Voltage

It happens some OAs do output little voltage, although there are no inputs. Therefore, we need to add a minor input offset voltage in order to have a perfect 0 V output. According to the Data Sheet, the Input Offset Voltage for μA741 is 1 mV, on average.

Input Offset Current

Normally, the two input currents should be equal when the output is 0 V. However, there is a slight imbalance. For μA741 it takes about 20 nA to compensate--which is negligible in most cases.

Input Bias Current

This is the DC current needed at the amplifier input terminal to establish the operation in the linear region.
For the Inverted Feedback circuit presented above, the Input Bias Current Ib is:

Ib (appx.) = Vo / Rf
Slew rate Fig 4: Slew Rate

SR = dV / dT

The "Slew Rate" is caused by the compensating capacitors (internal or external). For μA741 SR is 0.5 V/us (dT should be 10 us in Fig 4).

[us] is read as "micro second"; "u" is a common notation in electronics instead of "μ" (micro)

For sine-waves, SR allows us to calculate the maximum frequency of the OA:

fmax = SR / 2*PI*Vpp 
Vpp being "the voltage peak to peak" 

Gain Bandwidth Product
We have seen that the Slew Rate limits the maximum frequency. In addition, we have the capacitive reactance that limits the maximum frequency. Please remember the formula used to calculate capacitive reactance:

Xc = 1 / 2*PI*f*C
with f being the frequency and C the capacitance

The higher is the frequency the smaller is Xc--as mentioned in previous Design Notes. In addition we have:

Av = (Rc || Xc) / Re   [Re comes from the input impedance Zi = 2*β*Re]
with Rc being the AC collector resistance of the OA input transistor
||  means: in parallel with
Re is the AC emitter resistance of the OA input transistor

Based on values particular to each OA type, the Av(f) graph is built, and it is named a "Baude diagram".

One decibel is defined as: 1 db = 20 log(Vo/Vi)--in this case log is in base 10

The decibel notation allows us to express Av in decibels when the frequency increases ten times,  fx = f * 10, meaning, Voltage Gain decreases ten times: Avx = Av / (-20 db) with x marking the new value. In other words, Voltage Gain (Av) is inversely proportional to the frequency by a factor of 10.

When Av = 1 (at 0 db) the frequency limit is 1 MHz (for μA741). That 1 MHz value is named "Small-Signal Unity Gain Frequency (Ft)". However, sometimes another parameter is given in Data Sheets named "Transient Response Rise Time (Tr)", defined as the time it takes a waveform to raise from 10% to 90% of the final amplitude.

Unity Gain Frequency Bandwidth (BW) is calculated with:   BW = 0.35 / Tr

The Gain Bandwidth Product is: GBP = BW * Av
with Av being the closed-loop (feedback) Voltage Gain of the OA circuit

BW = GBP / Av
BW = 1 MHz / 100 = 10 kHz for μA741

Power Supply and virtual ground
As mentioned, the ground is virtual (related to) for (some) OAs, and this is an important concept because we relate all voltage and current formulas to that virtual ground.

For μA741 we need two power supplies: one negative and one positive. The virtual ground is between the two supplied voltages. Newer/different models require only one power supply, plus the ground connection. This facilitates working with OA, though the formulas used remain the same

In all schematics/circuits presented here (V+) and (V-) pins of μA741 are not wired/connected, though they do need to be wired appropriately.

First Approximation

The difference in potential between the two input terminals of an OA is approximately zero.

Av = Vo / Vi

for μA741 we have:
Vi = Vo / Av = 10 V /200000 = 50 uV (a negligible value)

Second Approximation

No current flows in or out the OA input terminals.

Ii = Vi / Zi

for μA741 we have:
Ii = 50 uV / 2 Mohms = 25 pA (a negligible value)



The summing OA circuit performs the mathematical operation of addition.


Summing OA
Fig 5: Summing OA

The Inverting Input pin (-) is also the summing node. The voltage at the summing node is 0 V.
V1, V2, V3 drop all their voltages on R1, R2, R3

We calculate this circuit with the following formulas:

I1 + I2 + I3 = If
Vo = -Rf * (V1/R1 + V2/R2 + V3/R3)


The summing OA schematic is used for DC and AC voltages/currents as well. Because each input voltage is dropped on its resistors, there is no signal mixing/distortion.

The Averaging OA
schematic is a particular case: each input resistors is equal to Ri,
and Rf = Ri / N; N is the number of input resistors.

Considering three inputs we have:
Vo = -Ri/3 * 1/Ri * (V1 + V2 + V3) = -(V1+V2+V3) / 3



You may have noticed that we can work with OAs as inverting or noninverting. We have seen the inverting circuit; time has come to look at the noninverting one.


Noninverting OA
Fig 6: A noninverting OA

We calculate this circuit with:

Ii = Vi / Ri and Vf = Ii * Rf
Vf = Vi * (Rf/Ri)
Vo = Vi + Vi * (Rf/Ri) = Vi * (1+ Rf/Ri)
Av = Vo/Vi = 1 + Rf/Ri


There is no negative sign in the above Av formula. That means, there is no phase inversion and the Voltage Gain (Av) is never less than unity (1).

The Input Impedance of the noninverting schematic is the impedance of the OA itself, which is very high.





Voltage follower Fig 7: Voltage Follower

This is a very simple circuit which duplicates the input DC or AC signals (with unity amplification).

Due to the very high input impedance this circuit isolates the output. In addition, there is no phase shift, therefore the output follows exactly the input. This is a safe input buffer circuit.



The difference amplifier is a combination of inverting and noninverting amplifiers.



Difference OA Fig 8: A difference Amplifier

If Ri = Ra = Rf = Rb 
Vo = Vi1 + Vi2

If Ri = Ra and Rf = Rb
Vo = (Rf/Ri) * (Vi2-Vi1)

The Difference Amplifier circuit and the calculations presented above are used (mostly) in Instrumentation Amplifier schematics.



 This is another mathematical function which OAs perform. The schematics required are presented next.


Square-wave integrator OA Graph Square-wave integrator OA Fig 9: Square-wave Integrator

Red trace is Vi
Blue trace is Vo

The output is:

Vo = (Vipp*Tdc) / (Ri*C)

Tdc = duty cycle (50% in this case)
Vipp = peak-to-peak input voltage
Sine-wave Integrator OA Graph Sine-wave Integrator OA
Fig 10: Sine-wave Integrator

This circuit is perfectly similar to the one above. Only the input signal is changed to a sine-wave.

Blue trace is Vi
Red trace is Vo

In this case the output is:

Vo = Vipp / 2*PI*f*Ri*C

Due to the added capacitance, we have a lagging phase shift of up to 90 degrees (PI/2)



Mathematically, the differentiation is the reverse of integration. The following simulation graphs exemplify this operation.


Sine-wave differentiation Graph Sine-wave differentiation Fig 11: Sine-wave differentiation

Red trace is Vi
Blue trace is Vo

Square-wave differentiation Graph Square-wave differentiation Fig 12: Square-wave differentiation

Red trace is Vi
Blue trace is Vo

The derivation of the square-wave is obvious
Trangle-wave differentiation Graph Trangle-wave differentiation Fig 13: Triangle-wave differentiation

Red trace is Vi
Blue trace is Vo

Vopp= -Rf*C*(dVi/dt)



For the analog logarithmic schematic, the output is proportional to the logarithm of the input signal. The antilogarithm performs the reverse of the logarithm. No simulation graphs are provided, because tuning these circuits (finding the right values for their components) is very difficult and time consuming.

The good news is, there are a few ICs specially designed to perform the log and antilog functions having all needed/biasing components already built-in.



Log OA
Fig 14: Logarithmic amplifier

This is a simplified schematic of the practical-implementation circuit.

Vo = K * log (Id/Iri)

K = proportional constant
Id = current through D
Iri = current through Ri

Antilog OA Fig 15: Antilogarithmic amplifier

This simplified schematic of the antilogarithm amplifier performs the inverse function of the above circuit



Comparator amplifier circuits are used a lot in hardware design. Following is a schematic with feedback and memory (hysterensis). For practical implementations of comparator OA in electronic circuits please consult

Comparator OA Graph Comparator OA Fig 16: Comparator with Feedback and  Hysterensis (Memory)

Red trace is Vi
Blue trace is Vo (transposed)

This is a "zero crossing" type of comparator



This topic is presented in Design Notes 9 - Filters.


There are very many types of OA schematics. A few of them are briefly listed next--this topic is way too vast to deal with in one Internet page.
1. Sine-wave oscillator
2. Square-wave oscillator
3. Voltage regulators
4. Voltage-to-current and current-to-voltage converters
5. Sample and hold circuits
6. Current differencing amplifier (CDA)
7. Programmable Operational Amplifiers

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November 01, 2014
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