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Operational Amplifiers: The
answer to many analogue
questions.
A/Prof Cesar Ortega-Sanchez
© Curtin University 2020
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Contents
• Background
• Inverting Amplifier
• Non-inverting Amplifier
• Voltage Follower
• Summing Amplifier
• Differential Amplifier
• Other applications
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The world is continuous
Physical phenomena in the universe present
continuous behaviours (except at quantum level).
What are continuous variables?
“Continuous variables are numeric
variables that have an infinite number
of values between any two values.”
minitab.com
For example: temperature, pressure, time,
luminosity, sound level, chemical concentration,...
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Analogue signals
What is an analogue signal?
“An analogue signal is any continuous signal for which the time-varying
feature (variable) of the signal is a representation of some other time
varying quantity, i.e., analogous to another time varying signal.”
Wikipedia
Vibration and Noise in
a Washing Machine
To visualise, we need to
convert vibration to
voltage using a transducer.
Voltage is analogous to
vibration.
https://www.com sol.com /blogs/sim ulating-vibration-and-noise-in-a-washing-m achine/
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What is an Op-Amp?
• Operational amplifier (op-amp) is a very useful device in
analogue electronics.
• It consists of several circuit elements, however, in this unit it
is considered as a building block.
• Op-amps can be used to perform a variety of mathematical
operations on analogue signals such as addition, subtraction,
multiplication, division, differentiation, and integration with
the use of a minimal number of external elements.
• Op-amps are commercially available in integrated circuit
packages.
https://www.westfloridacomponents.com/mm5/graphics/Q04/DG409DJ.jpg
• Op-amps are used to process both DC and AC signals.
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Op-Amp symbol
• Two inputs: an inverting input marked with minus (-)
and a non-inverting input marked with plus (+).
• One output. The amplified difference of the inputs.
• Typically two power supplies, one positive and one
negative. However, can also have a single power supply.
• The output voltage is limited by the voltages of the
power supplies,
−𝑉!! ≤ 𝑜𝑢𝑡𝑝𝑢𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 ≤ +𝑉!!
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Op-Amp equivalent circuit
𝑣" = 𝐴#$ 𝑣% − 𝑣&
𝐴#$ is the open loop gain.
For an ideal op-amp
• Infinite input resistance (Ri à ∞), making the
current going into the op-amp equal to zero.
• Zero output resistance (Ro = 0).
• Infinite gain Aod,
• v1 = v2 for a finite output voltage v0
• Practical op-amps are not ideal.
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Procedure to analyse Op-Amp circuits
1. Calculate the voltage at the non-inverting
input (v2)
2. Apply v1 = v2
3. Write the KCL for the Op-Amp input nodes
to find the output voltage
Next stop: Inverting amplifier
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Op-Amp as inverting amplifier
In this case, 𝑣!" is applied to the inverting (negative) input.
Considering an ideal op-amp,
𝒗𝑶 = 𝒗𝒊𝒏 −
𝑣# − 𝑣$ = 0
As 𝑣# = 0, 𝑣$ = 0 makes node 1 a virtual ground.
Also, the currents going in to the op-amp are zero,
𝐼%! = 𝐼%"
𝑹𝟐
𝑹𝟏
Applying KCL to Node 1 gives,
𝑣!" − 𝑣$
𝑣$ − 𝑣&
=
𝑅$
𝑅#
As 𝑣$ = 0, we can say, the closed loop gain 𝐴' ,
𝒗𝟎
𝑹𝟐
𝑨𝒗 =
=−
𝒗𝒊𝒏
𝑹𝟏
vo
vin
The negative sign indicates that there is 180o phase difference between
the input and the output.
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Inverting amplifier example
If 𝑅! = 10 kΩ, and 𝑅" = 25 k Ω, and 𝑣#$ = 0.5 V,
calculate the output voltage 𝑣% and the current
through the 10 kΩ resistor.
As,
&.
& /0
'
= − 1,
'2
'
"(
𝑣% = − '" 𝑣#$ = − !% × 0.5 𝑉
!
𝒗𝟎 = −𝟏. 𝟐𝟓𝑽
Current through 𝑅! is given by,
𝐼'! =
Next stop: Non-inverting amplifier
&() *&!
'!
=
%.(*%
!%
𝑰𝑹𝟏 = 50 mA
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Op-Amp as non-inverting amplifier
R2
IR2
R1
In this case, 𝑣!" is applied to the non-inverting (positive) input.
Considering an ideal op-amp, 𝑣# − 𝑣$ = 0
IR1
-
v
v
v0
1
𝑣# = 𝑣!" , 𝑣$ = 𝑣!"
Also, as the currents going in to the op-amp is zero, 𝐼%! = 𝐼%"
+
2
As
vin +-
Applying KCL to Node 1 gives,
𝒗𝑶 = 𝒗𝒊𝒏 𝟏 +
0 − 𝑣$
𝑣$ − 𝑣&
=
𝑅$
𝑅#
As 𝑣$ = 𝑣!" , we can say, the closed loop gain 𝐴' ,
𝑹𝟐
𝑹𝟏
Note that the output is
always greater than the input
𝐴' =
𝑣&
𝑅#
=1+
𝑣!"
𝑅$
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Non-inverting amplifier example
If 𝑅$ = 10 k Ω, and 𝑅# = 25 k Ω, and 𝑣!" = 0.5 V, Calculate the
output voltage 𝑣& and the current through the 10 kΩ resistor.
R2
IR2
R1
IR1
v
v
-
1
2
+
vin +-
As,
v0
'(
')*
%!
%
#(
%!
$&
𝑣& = (1 + ")𝑣!" = (1 +
)× 0.5 𝑉
𝑣& = 1.75𝑉
Current through 𝑅$ is given by,
𝐼%! =
Next stop:
Voltage follower
%
=1+ "
&)'!
%!
=
&)&.(
$&
𝐼%! = -50 mA
This means the actual direction of the current 𝐼%! is in the opposite
of the direction marked in the figure.
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Op-Amp as voltage follower
In this case, the feedback resistor of the non-inverting amplifier
𝑅! is made zero (i.e., short circuited) and 𝑅" is open circuited.
v1
v
2
-
v0
Considering an ideal op-amp, 𝑣! − 𝑣" = 0
+
vin +-
As
𝑣! = 𝑣#$ , 𝑣" = 𝑣#$
And
𝑣" = 𝑣% = 𝑣#$
Hence, for a voltage follower, the voltage gain
𝒗𝑶 = 𝒗𝒊𝒏
𝑨𝒗 =
𝒗𝟎
=𝟏
𝒗𝒊𝒏
The voltage follower is very useful as an intermediate stage
buffer (amplifier) in isolating one circuit from another.
It minimises interaction between the two stages and eliminates
inter-stage loading.
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Use of Op-Amp as follower
Observe the circuit in the figure.
• The Op-Amp is configured as a follower.
• Input is the output of a voltage divider.
Thinking time:
• If we did not use the Op-Amp, what would
happen to the output of the voltage divider as
soon as we connected any load to it?
The Op-Amp configured as follower
behaves like a virtual short circuit.
But it isn’t.
• What happens when we connect the Op-Amp?
• Do you see the advantage of having infinite input
resistance and zero output resistance?
Next stop: Adder amplifier
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Op-Amp as adder amplifier
Rf
R1
v
i1
i=0
i2
i=0
v
2
R3
𝑖 = 𝑖1 + 𝑖2 + 𝑖3
-
1
R2
The output of an adder or summing amplifier is is the
weighted sum of the inputs. Applying KCL at the
inverting node gives
i
As
+
i3
𝑣 −0
𝑖1 = 1
v
3
𝑅1
and
𝑣 −0
, 𝑖2 = 2
𝑅2
𝑖=
0−𝑣0
𝑅𝑓
𝑣 −0
, 𝑖3 = 3
𝑅3
,
0 − 𝑣0 𝑣1 − 0 𝑣2 − 0 𝑣3 − 0
=
+
+
𝑅𝑓
𝑅1
𝑅2
𝑅3
𝒗𝟎 = −
𝑹𝒇
𝑹𝒇
𝑹𝒇
𝒗𝟏 +
𝒗𝟐 +
𝒗
𝑹𝟏
𝑹𝟐
𝑹𝟑 𝟑
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Adder amplifier example
Calculate the output voltage and
current in the following circuit
Rf= 8kW
R1= 20kW
i1
v1= 1.5V
R2= 10kW
i2
R3= 6kW
i3
i=0
i=0
v2= 2V
-
Applying
i
𝑣0 = −
8
i0
v0
8
8
𝑣0 = − 20 (1.5) + 10 (2) + 6 (1.2)
𝑣0 = −3.8𝑉
+
v3= 1.2V
𝑅𝑓
𝑅𝑓
𝑅𝑓
𝑣1 + 𝑣2 + 𝑣3
𝑅1
𝑅2
𝑅3
4kW
Applying KCL at the output node,
Next stop: Differential amplifier
𝑖0 =
𝑣0 − 0
𝑣0 − 0
+
= −1.425 𝑚𝐴
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Op-Amp as differential amplifier
Applying KCL at the two nodes give
!! "!"#
##
+
!! "!$
#%
=0
(1)
!& "!"%
#'
+
!& "$
#(
=0
(2)
and
From equation (1):
The difference amplifier is an op-amp
circuit that amplifies the difference
between two inputs, and rejects any
signals common to the two inputs
(common noise).
#
#
%
#
𝑣% = 𝑣& 1 + ## − 𝑣'( #%
#
(3)
(
𝑣& = 𝑣) = 𝑣'* # +#
and
'
(
By substituting 𝑣& in (3), it can be shown that
𝑣$ =
𝑅* 1 + 𝑅(⁄𝑅*
𝑅*
𝑣'* − 𝑣'(
𝑅( 1 + 𝑅,⁄𝑅𝑅(
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Op-Amp as differential amplifier
𝑣$ =
𝑅* 1 + 𝑅(⁄𝑅*
𝑅*
𝑣'* − 𝑣'(
𝑅( 1 + 𝑅,⁄𝑅𝑅(
Since the difference amplifier rejects a signal
common to the two inputs, the amplifier should
have the property that 𝑣. = 0 when 𝑣'(= 𝑣'*. This
exists when
The difference amplifier is an op-amp
circuit that amplifies the difference
between two inputs, and rejects any
signals common to the two inputs
(common noise).
##
#'
=
and then the output voltage becomes
#%
#(
𝒗𝟎 =
𝑹𝟐
𝒗 − 𝒗𝑰𝟏
𝑹𝟏 𝑰𝟐
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Differential amplifier example
https://projectiot123.com /2019/02/12/operational-am plifier-as-differential-am plifier/
• Most microphones capture sound by converting it to a voltage between two terminals.
But this voltage is not referenced to ground! This is called a differential voltage.
• A differential amplifier amplifies the differential voltage while eliminating common-mode
noise, and delivers an output that is referenced to the ground of the system.
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Op-Amp as differentiator
The nodal equation at the inverting input terminal's node is
Current going
into the node
https://www.tutorialspoint.com
If RC = 1 sec, then the output voltage VO will be
𝑽𝑶 = −
𝒅𝑽𝒊
𝒅𝒕
In this case the output will be the derivative of Vi.
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Op-Amp as integrator
The nodal equation at the inverting input terminal's
node is
Current going
out of the node
https://www.tutorialspoint.com
Integrating both sides of the equation we obtain
1
⇒ 𝑉# = −
( 𝑉$ 𝑑𝑡
𝑅𝐶
If RC = 1 sec, then the output voltage VO
will be
𝑽𝑶 = − & 𝑽𝒊 𝒅𝒕
In this case the output will be the
integral of Vi.
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Op-Amp as low-pass filter
This circuit allows (passes) only Vi’s low frequency
components and rejects (blocks) all other high
frequency components. This means that as the
frequency of Vi increases, the amplitude of the output
decreases.
https://www.tutorialspoint.com
We can choose the values of Rf an
R1 to obtain the desired gain at the
output.
From Lab 2 we know that the electric network
connected to the non-inverting terminal of the opamp is a passive low pass filter. So, the input of a
non-inverting terminal of an op-amp is the output of
a passive low pass filter.
The circuit resembles a non-inverting amplifier.
+
Hence, it produces an output, which is 1 + % times
+&
the input present at the non-inverting terminal.
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Op-Amp as high-pass filter
This circuit allows (passes) only Vi’s high frequency
components and rejects (blocks) all other frequency
components. This means that as the frequency of Vi
increases, the amplitude of the output increases.
The electric network connected to the non-inverting
terminal of the op-amp is a passive high pass filter.
So, the input of a non-inverting terminal of an op-amp
is the output of a passive high pass filter.
https://www.tutorialspoint.com
We can choose the values of Rf an
R1 to obtain the desired gain at the
output.
The circuit resembles a non-inverting amplifier.
+%
Hence, it produces an output, which is 1 + +
&
times
the input present at the non-inverting terminal.
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Complex systems
Each of the circuits with Op-Amps covered can be considered a functional block in more
complex systems.
To analyse such systems, it is possible to analyse individual blocks independently and
then combine the results to find the output of the complete system.
https://fccid.io/2ACRG-39859F49/Block-Diagram /Block-Diagram -2370143
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Op-Amp as band-pass filter
By connecting the output of an active high pass filter,
to the input of an active low pass filter it is possible to
obtain the output in such a way that it contains only a
particular band of frequencies.
When the output of one circuit is connected to the
input of another, the circuits are cascaded.
The values of RA, CA, RB and CB must be carefully
calculated so that the cut-off frequency of the highpass filter is greater than the cut-off frequency of the
low-pass filter.
https://www.tutorialspoint.com
(See diagrams in a couple of slides)
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Op-Amp as band-stop filter
https://www.tutorialspoint.com
The block diagram of an active band-stop filter consists of two blocks in its first stage: an
active low-pass filter and an active high-pass filter. The outputs of these two blocks are
applied as inputs to a summing amplifier that produces the amplified version of the sum of
the outputs of the active low pass filter and the active high pass filter.
Therefore, the output of the above block diagram will be the output of an active band-stop
filter when we choose the cut-off frequency of the low-pass filter to be smaller than the
cut-off frequency of the high-pass filter.
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Op-Amp as band-stop filter
The circuit diagrams of an active
low-pass filter, an active high-pass
filter and a adder amplifier have
been covered.
Observe that the circuit diagram of
the active band-stop filter was
obtained by replacing the blocks in
the block diagram for an active
band-stop filter with the
respective circuit diagrams.
https://www.tutorialspoint.com
Circuit diagram of an active band-stop filter
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Frequency response of different filters
The function of a filter is to
block specific frequencies
while allowing others to
propagate.
Study these graphs to
understand how filters work.
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Where to from here?
The Swiss army knife is getting quite fat!
Superposition
Theorem
Phasors
Current
dividers
Inductors
AC and DC
sources
Norton’s
Theorem
Complex
numbers
KCL
Complex
impedance
Ohm’s Law
Semiconductors
Diodes
Block diagrams
Phase
shift
Op-Amps
KVL
Filters
Thevenin’s
Theorem
RMS value
Source
transformation
Integrator
Differentiator
MPTT
Voltage dividers
Capacitors
Transistors
Adders
https://www.traveluniverse.com.au/victorinox-huntsman-swiss-army-knife-desert-wood/
Frequency
response
Followers
Differential
amplifiers
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