ELECTRONIC DESIGN NOTES #8  OPERATIONAL AMPLIFIERS

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 mathematicallywell, 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 works 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 AnalogtoDecimal Conversion channel, and then use it as a decimal value inside controller's memory.
However, 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 complexthough
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
NOTE The basic
notions highlighted in this page are related to a few electronic
design topics presented in the first part, Hardware Design, of
LEARN HARDWARE FIRMWARE AND SOFTWARE
DESIGN.

1. OPERATIONAL AMPLIFIER SPECIFICATIONS
The best thing to do is to download the datasheet (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 10^{3}..10^{9}
levels of amplification. 2. High input impedance; the ideal case is (Z_{i
}> ∞ ohms)
[read this as: the input
impedance tends to reach infinite values in ohms]. 3. Wide Bandwidth. 4.
Low output impedance; Z_{o }> 0 ohms.
Let's take a closer look at μA741 using a few common schematic models.


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

GENERAL OPERATIONAL AMPLIFIER SPECIFICATION DATA AND FORMULAS 
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
datasheet, 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 compensatewhich 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 I_{b} is:
I_{b} (appx.) = V_{o} / R_{f} 

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).
ATTENTION
[us] is read as "micro second"; "u" is a common notation in electronics instead of
"μ" (micro)
For sinewaves, SR allows us to calculate the maximum frequency of the OA:
f_{max} = SR / 2*PI*V_{pp}
V_{pp} 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:
X_{c} = 1 / 2*PI*f*C
with f being the frequency and C the capacitance
The higher is the frequency the smaller is X_{c}as mentioned in previous Design Notes. In
addition we have:
Av = (Rc  Xc) / Re [Re comes from the input impedance Z_{i} = 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(V_{o}/V_{i})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 "SmallSignal Unity Gain Frequency (F_{t})". However, sometimes another
parameter is given in Data Sheets named "Transient Response Rise Time (T_{r})",
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 / T_{r}
The Gain Bandwidth Product is: GBP = BW * Av
with Av being the closedloop (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
in 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
ATTENTION
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.
Exemplification:
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.
Exemplification:
Ii = Vi / Zi
for μA741 we have:
Ii = 50 uV / 2 MΩ = 25 pA (a negligible value) 

2. SUMMING AMPLIFIER
The summing OA circuit performs the mathematical operation of addition.

SUMMING OPERATIONAL AMPLIFIER 

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:
I_{1} + I_{2} + I_{3} = I_{f}
V_{o} = R_{f} * (V_{1}/R_{1} + V_{2}/R_{2} + V_{3}/R_{3}) 
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 

3. NONINVERTING AMPLIFIER
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 OPERATIONAL AMPLIFIER 

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. 

4. VOLTAGE FOLLOWER
VOLTAGE FOLLOWER OPERATIONAL AMPLIFIERS 

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 replicates exactly the input. This is a safe input buffer circuit. 

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

DIFFERENCE OPERATIONAL AMPLIFIER 

Fig 8:
A difference Amplifier
If Ri = Ra = Rf = Rb
Vo = V_{i1} + V_{i2
}
If Ri = Ra and Rf = Rb
Vo = (Rf/Ri) * (V_{i2}V_{i1})

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

6. INTEGRATOR AMPLIFIER
This is another mathematical function OAs perform. The schematics
required are presented next.

INTEGRATOR OPERATIONAL AMPLIFIER 


Fig 9: Squarewave Integrator
Red trace is Vi
Blue trace is Vo
The output is:
Vo = (V_{ipp}*T_{dc}) / (R_{i}*C)
T_{dc} = duty cycle (50% in this case)
Vipp = peaktopeak input voltage



Fig 10: Sinewave Integrator
This circuit is perfectly similar to the one above. Only the input signal is changed to
a sinewave.
Blue trace is Vi
Red trace is Vo
In this case the output is:
Vo = V_{ipp} / 2*PI*f*Ri*C
Due to the added capacitance, we have a lagging phase shift of up to 90 degrees (PI/2)


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


8. LOGARITHMIC AMPLIFIER
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 a time consuming task (even as a simulation).
The good news is, there are a few ICs specifically designed to perform the log and antilog functions having all
the needed/biasing components already builtin.

LOGARITHMIC OPERATIONAL AMPLIFIER 

Fig 14:
Logarithmic amplifier
This is a simplified schematic of the practicalimplementation circuit.
Vo = K * log (Id/Iri)
K = proportional constant
Id = current through D
Iri = current through Ri 

Fig 15:
Antilogarithmic amplifier
This simplified schematic of the antilogarithm amplifier performs the inverse function of the above
circuit 

9. COMPARATOR AMPLIFIER
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
LEARN HARDWARE
FIRMWARE AND SOFTWARE DESIGN.

COMPARATOR WITH HYSTERENSIS AMPLIFIER 


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 

10. ACTIVE FILTERS
This topic is presented in Design Notes 9  Filters.
11. OTHER OPERATIONAL AMPLIFIER CIRCUITS
There are very many types of OA schematics. A few of them are briefly listed nextthis topic is way too vast to
deal with in one Internet page.
1. Sinewave oscillatorS 2. Squarewave oscillatorS 3. Voltage regulators 4. Voltagetocurrent and currenttovoltage converters 5. Sample and hold circuits 6. Current differencing amplifierSS (CDA) 7. Programmable Operational Amplifiers

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