### Experiment 8 - Rensselaer Polytechnic Institute

```Electronic Instrumentation
Experiment 8
* Op Amp Circuits Review
* Differentiators and Integrators
* Analog Computers
Op Amp Circuits Review

Inverting Amplifier
 Non-inverting Amplifier
 Differential Amplifier
 Op Amp Analysis
Inverting Amplifier
Rf
Vout

Vin
Rin
Non Inverting Amplifier
Rf
Vout
 1
Vin
R1
Differential Amplifier
Rf
Vout

V1  V2 Rin
Op Amp Analysis

Golden Rules of Op Amp Analysis
• 1) The current at the inputs is 0
• 2) The voltage at the two inputs is the same


These are theoretical assumptions which
allow us to analyze the op-amp circuit to
determine what it does.
These rules essentially allow us to
remove the op amp from the circuit.
General Analysis Example(1)

Assume we have the circuit above,
where Zf and Zin represent any
combination of resistors, capacitors and
inductors.
General Analysis Example(2)

We remove the op amp from the circuit
and write an equation for each input
voltage.

Note that the current through Zin and Zf is
the same, because equation 1 is a series
circuit.
General Analysis Example(3)

Since I=V/Z, we can write the following:
Vin  VA VA  Vout
I1] 

Z in
Zf

But VA = VB = 0, therefore:
 Vout

Zin
Zf
Vin
Zf
Vout

Vin
Zin
General Analysis Conclusion

For any op amp circuit where the positive input
is grounded, as pictured above, the equation for
the behavior is given by:
Zf
Vout

Vin
Z in

What is a voltage follower?
 Why is it useful?
 Voltage follower limitations
What is a voltage follower?
Vout
1
Vin
analysis:
1] VA  Vout
VA  VB
2] VB  Vin
therefore, Vout  Vin
Why is it useful?

In this voltage divider, we get a
different output depending upon
the load we put on the circuit.
 Why?

We can use a voltage follower to convert this real
voltage source into an ideal voltage source.
 The power now comes from the +/- 15 volts to the
op amp and the load will not affect the output.
Voltage follower limitations

Voltage followers will not work if
their voltage or current limits are
exceeded.
 Voltage followers are also called
buffers and voltage regulators.
 V1 V2 
Vout   R f   
 R1 R2 
Rf
Vout
if R1  R2 then

V1  V2
R1




also useful when R1<>R2.
This is called a “Weighted Adder”
A weighted adder allows you to combine
several different signals with a different
gain on each input.
You can use weighted adders to build audio
mixers and digital-to-analog converters.
I1
If
I2
I f  I1  I 2
V1  VA
I1 
R1
V2  VA
I2 
R2
VA  Vout V1  VA V2  VA


Rf
R1
R2
 Vout V1 V2
 
Rf
R1 R2
Vout
VA  Vout
If 
Rf
VA  VB  0
 V1 V2 
  R f   
 R1 R2 
Differentiators and Integrators

Ideal Differentiator
 Ideal Integrator
 Miller (non-ideal) Integrator
 Comparison of Integration and
Differentiation
Ideal Differentiator
analysis:
Zf
Rf
Vout


  j R f Cin
1
Vin
Z in
j Cin
Analysis in time domain
dVCin
I Cin  Cin
VRf  I Rf R f
dt
d (Vin  VA ) VA  Vout
I  Cin

dt
Rf
therefore, Vout
dVin
  R f Cin
dt
I Cin  I Rf  I
VA  VB  0
Problem with ideal differentiator
Ideal
Real
Circuits will always have some kind of input resistance,
even if it is just the 50 ohms from the function generator.
Analysis of real differentiator
Z in  Rin 
1
j Cin
Zf
Rf
j R f Cin
Vout



1
Vin
Z in
j RinCin  1
Rin 
j Cin
Low Frequencies
Vout
  j R f Cin
Vin
ideal differentiator
High Frequencies
Rf
Vout

Vin
Rin
inverting amplifier
Comparison of ideal and non-ideal
Both differentiate in sloped region.
Both curves are idealized, real output is less well behaved.
A real differentiator works at frequencies below c=1/RinCin
Ideal Integrator
analysis:
Zf
Vout


Vin
Z in
1
j C f
Rin
1
j


j RinC f  RinC f
Analysis in time domain
VRin  I Rin Rin
I Cf  C f
dVCf
I Cf  I Rin  I
dt
Vin  VA
d (VA  Vout )
I
 Cf
VA  VB  0
Rin
dt
dVout
1
1

Vin Vout  
Vin dt ( VDC )

dt
RinC f
RinC f
Problem with ideal integrator (1)
No DC offset.
Works ok.
Problem with ideal integrator (2)
With DC offset.
Saturates immediately.
What is the integration of a constant?
Miller (non-ideal) Integrator

If we add a resistor to the feedback path, we
get a device that behaves better, but does
not integrate at all frequencies.
Behavior of Miller integrator
Low Frequencies
High Frequencies
Zf
Rf
Vout


Vin
Z in
Rin
Zf
Vout
0


0
Vin
Z in
Rin
inverting amplifier

signal disappears
The influence of the capacitor dominates at higher
frequencies. Therefore, it acts as an integrator at higher
frequencies, where it also tends to attenuate (make less) the
signal.
Analysis of Miller integrator
1
j C f
Rf
Zf 

1
j R f C f  1
Rf 
j C f
Rf 
Rf
Zf
j R f C f  1
Rf
Vout



Vin
Z in
Rin
j Rin R f C f  Rin
Low Frequencies
Rf
Vout

Vin
Rin
inverting amplifier
High Frequencies
Vout
1

Vin
j RinC f
ideal integrator
Comparison of ideal and non-ideal
Both integrate in sloped region.
Both curves are idealized, real output is less well behaved.
A real integrator works at frequencies above c=1/RfCf
Problem solved with Miller integrator
With DC offset.
Still integrates fine.
Why use a Miller integrator?

Would the ideal integrator work on a signal with
no DC offset?
 Is there such a thing as a perfect signal in real
life?
• noise will always be present
• ideal integrator will integrate the noise

Therefore, we use the Miller integrator for real
circuits.
 Miller integrators work as integrators at  > c
where c=1/RfCf
Comparison
original signal
Differentiaion
v(t)=Asin(t)
Integration
v(t)=Asin(t)
mathematically
dv(t)/dt = Acos(t)
v(t)dt = -(A/cos(t)
mathematical
phase shift
mathematical
amplitude change
H(j
electronic phase
shift
electronic
amplitude change
+90 (sine to cosine)
-90 (sine to –cosine)

1/
H(j  jRC
-90 (-j)
H(j jRC = j/RC
+90 (+j)
RC
RC

The op amp circuit will invert the signal and modify the
mathematical amplitude by RC (differentiator) or 1/RC
(integrator)
Analog Computers (circa. 1970)
Analog computers use op-amp circuits to do real-time
mathematical operations.
Using an Analog Computer
Users would hard wire adders, differentiators, etc. using the
internal circuits in the computer to perform whatever task they
wanted in real time.
Analog vs. Digital Computers


In the 60’s and 70’s analog and digital computers
competed.
Analog

Digital
• Advantage: more flexible, could program jobs