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Lecture 4: Operational Amplifier Basics
Having put in place the broad concepts of signal amplification and the circuit modelling of
amplifiers, we now introduce what is probably the most important building block for
analogue systems, the operational amplifier, or op amp as it is more popularly known.
Despite its name, the op amp cannot really be used on its own for amplification purposes
but in conjunction with additional components. In this lecture, we address the basic
properties of the op amp and how it can be used with additional system components to
function as a useful amplifier.
Learning Outcomes:
On completing this lecture, you will be able to:




4.1
List key properties of the near-ideal operational amplifier;
Derive expressions for the closed loop gain of non-inverting and inverting op amp
configurations;
Demonstrate how the performance of these systems approaches the ideal;
Explain the terms open loop gain, closed loop gain, and loop gain.
The Near-Ideal Operational Amplifier
The operational amplifier is a circuit of approximately 30 semiconductor transistors and
some additional passive components (resistors) configured as a very high gain amplifier.
The actual circuit diagram of one of the more popular commercially available devices, the
so-called 741 op amp, is shown below:
4-1
The complete circuit is fabricated on a single crystal of silicon and thus constitutes an
integrated circuit. We simply note the relatively large number of active semiconductor
devices — transistors — and the relatively small number of resistors and one capacitor.
Such a distribution of components would be typical for an analogue integrated circuit.
How the internal circuitry of the op amp functions is not our concern; what does concern
us are the overall electrical properties from the signal input nodes (designated noninverting and inverting) to the signal output node. The op amp is represented by the
following circuit symbol:
+VCC
+
-
e2
vO
e1
-VCC
+VCC and –VCC designate the necessary positive and negative power supplies. Most
frequently these lines are omitted from circuit diagrams but are always understood to be
present.
There are two signal inputs to the op amp: a so-called non-inverting input labelled e2 and
denoted by the + sign within the triangle, and an inverting input labelled e1 and denoted
by the – sign. What the internal circuitry actually does is to amplify the signal across the
input terminals
eD  e2  e1
resulting in an output signal given by
vO  Av eD  Av e2  e1 
The distinguishing feature of the op amp is that the signal gain Av, or open loop gain as
it is technically referred to, is typically very high, between 10 5 and 106 — ie between 100
and 120 dB. It is also worth emphasising that the op amp responds to the voltage
difference eD  e2  e1 rather than to e2 or e1 individually. The amplifier is accordingly
classified as a differential amplifier.
+
eD
-
e2
vO
e1
4-2
In the previous lecture, we covered the circuit model for a general amplifier. We now
consider a corresponding model for the op amp and list some of the properties of what we
call a near-ideal op amp. The relevant circuit model is
RO
iIN
e2
eD
RIN
AveD
vO
e1
As with our previous model, the input voltage to which the amplifier is sensitive (e D in this
instance) sees an input resistance RIN. The basic functioning of the op amp is captured by
means of the signal source AveD with which we also associate a source resistance RO. A
near ideal op amp has the following properties:

The input resistance is regarded as indefinitely large implying that the input
current to the op amp tends to be insignificantly small. Stated formally
RIN    i IN  0

The output resistance is regarded as insignificantly small and this implies that the
output voltage is Av times eD:
RO  0  vO  Av e2  e1 

The open loop gain Av is large, but not infinite.

Av is a constant, independent of frequency.

The amplifier produces no distortion.

The amplifier produces no electrical “noise,” ie unwanted electrical fluctuations.

The amplifier has no “offsets,” ie when e2 = e1 = 0, the output is indeed equal to
zero.
Basically what we are saying is that all the properties of the op amp are considered to be
ideal except for the open loop gain; this parameter is large rather than infinitely large. It
is worth noting that real op amps such as the 741 do approach the ideal in performance.
4.2
Regions of Operation
Based on the above considerations and acknowledging the practicality of having supply
voltages, we can represent the voltage transfer characteristic of a near-ideal op amp as
follows:
4-3
vO
+VCC
eMAX
ed
-VCC
k
j
k
 represents where the amplifier is operating linearly, ie where vO  Av e2  e1  .
Regions  represent non-linear operation characterised as follows:
Region
if
eD  eMAX
if
eD  eMAX
then vO  VCC
then vO  VCC
Noting that VCC is typically 105, we can estimate eMAX as
eMAX 
VCC
15
 5  0.15mV
Av 10
Thus we simply note that when the op amp is operating in the linear region the voltage
difference across the input terminals is very small, at most a small fraction of a millivolt.
Corresponding to the two regions of operation, we have two classes of op amp
applications:
Linear Circuits — these circuits, as we shall see, employ negative feedback which has
the effect of forcing the op amp into the linear region of operation.
Non-Linear Circuits — the op amp is used either open loop (ie without feedback) or
with positive feedback.
For the moment our main interest lies in linear circuit applications.
4.3
The Non-Inverting Amplifier
When it comes to practical signal amplifiers, the real need is not so much towards
achieving very high signal amplification as towards achieving a system whose gain can be
precisely and readily controlled. Thus the op amp on its own is not all that useful; for any
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particular 741 for example, all we know at the outset is that the gain is somewhere
between 105 and 106. Now consider the circuit shown below:
e2
+
-
e1
i2
iIN
R2
vI
vO
R1
vF
i1
We will refer, somewhat loosely, to the overall circuit as the “system,” since it is
important to distinguish between the op amp and the overall circuit/system, of which the
op amp is but one component. The input signal to the overall system is denoted vI, and
this is the signal for amplification. vI is applied directly to the non-inverting input of the
op amp; ie
e2  vI
The output from the op amp is also the system output vO and this signal is applied to a
resistor divider network, R1 and R2, to produce a feedback voltage vF. This feedback
voltage is applied to the inverting input of the op amp e 1 and it is this connection which
produces the negative feedback.
Carrying out a circuit analysis on the system, we begin by noting
i2  i1  i IN
However, because the op amp is near-ideal and
Therefore
RIN   , we have i IN  0 .
i2  i1 and as a consequence
e1  v F 
R1
vO
R2  R1
Negative feedback dictates that the op amp functions in its linear region so that
vO  Av e2  e1 


R1
 Av  v I 
vO 
R2  R1 

4-5
Re-arranging, we get


R1
vO 1 
Av   Av v I
 R2  R1 
Hence the overall gain of the system can be expressed
vO

vI
Av
R1
1
Av
R2  R1
and we prefer to arrange this as
R1
Av
vO R2  R1 R2  R1

R1
vI
R1
1
Av
R2  R1
We tidy this up by noting
R2  R1
R
 1 2
R1
R1
and letting

R1
R2  R1
vO  R2  Av

 1 
v I 
R1  1  Av
vO
is the overall system gain or, as it is more usually termed, the closed
vI
loop gain Acl while the quantity Av is called the loop gain. While β is a fraction, the
The quantity
open loop gain Av is usually sufficiently large to ensure
Av  1
permitting the approximation
Av
1
1  Av
Hence the closed loop gain is expressed
 R 
Acl  1  2   Acl id
R1 

That is, the closed loop tends towards the value
closed loop gain Acl id.
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1
R2
and this we define to be the ideal
R1
The fundamental idea behind the circuit is that, ideally, the output is a scaled replica of
the input and the scale factor
1
R2
is set by the resistive values of R2 and R1. Thus the
R1
gain is indeed readily under the control of the user. For example, if an amplifier of gain
100 is required, we might choose R2  99k and R1  1k . The resulting system will
approach this ideal closed loop gain of 100 to the extent that
Av  1 . This is illustrated
by the following example.
Example 4.1 Consider a non-inverting amplifier having
R2  99k and R1  1k .
Calculate the actual closed loop gain for the following three near-ideal op amps:
(i) Av  10
4
(ii) Av  10
5
(iii) Av  10
6
For all three solutions we use the expression
Acl  Acl id
Av
1  Av
where
and
R2
99
 1
 100
R1
1
R1
1


 10 2
R2  R1 1  99
Acl id  1 
10 10   100100  99.01
101
1  10 10 
10 10   1001000  99.9
 100
1001
1  10 10 
10 10   10010,000  99.99
 100
10,001
1  10 10 
(i)
Acl  100
(ii)
Acl
(iii)
Acl
2
4
2
2
4
5
2
2
5
6
2
6
Note that the ideal closed loop gain is within 1% of each of the actual values calculated
for the closed loop gain with the approximation getting better as the open loop gain
increases.
4.4
The Inverting Amplifier
R2
i2
R1
i1
iIN
e1
vI
e2
+
vO
4-7
A second basic amplifying application of the op amp is shown above. Again, our task is to
derive an expression for the gain vO/vI of the overall system on the basis that the op amp
is near-ideal. Note again that the system output is fed back through resistor R 2 to the
inverting input of the op amp ensuring that we have a negative feedback system and,
hence, linear operation.
In this instance, the non-inverting input of the op amp is connected directly to signal
ground so that
e2  0
Similar to the previous section, because
that
RIN   , we have i IN  0 . From this it follows
i2  i1 and thus the series combination of R1 and R2 constitutes a voltage divider
driven by vI at one end and by vO at the other end. We can therefore write
e1 
R2
R1
vI 
vO
R1  R2
R1  R2
Because the op amp is near-ideal, we have
vO  Av e2  e1 
and substituting for e2 and e1


R2
R1
vO  Av 0 
vI 
vO 
R1  R2
R2  R1 


RA 
R A
vO 1  1 v    2 v v I
R1  R2
 R1  R2 
Thus, the closed loop gain is given by
R2
Av
vO
R1  R2
Acl 

R1
vI
1
Av
R1  R2
R1
Av
R2 R1  R2

R1
R1
1
Av
R1  R2
Again letting

R1
, we can re-write the closed loop gain as
R2  R1
4-8
 R   Av 
Acl    2  

 R1  1  Av 
Av  1
Assuming, as before, that
we get
 R 
Acl    2   Acl id
 R1 
Again the ideal closed loop gain (in magnitude) is set by the resistor values R 2 and R1 and
note the exact same expression and considerations for the loop gain.
The – sign attached to
R2
R1
simply means that there is a 180˚ degree phase shift
between input and output; if the input is of the form
form
sin t  , then the output is of the
 sin t  . It is this phase reversing property which gives the system its generic title
of inverting amplifier.
4.5
Concluding Remarks
We have introduced and defined the near-ideal op amp and investigated two of its more
important linear applications. In both cases, the closed loop gain incorporated a term with
the loop gain divided by unity plus the loop gain. Arising from the loop gain typically
being large, this term tended to unity and the closed loop gain tends to a quantity we
called the ideal closed loop gain. This ideal closed loop gain was found to depend solely
on the values of passive components connected as part of the system. Essentially, in
designing op amp circuits we work with the ideal closed loop gain.
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