objective of lecture discuss analog computing and the application of 1 st order operational...
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Objective of LectureDiscuss analog computing and the
application of 1st order operational amplifier circuits.
Derive the equations that relate the output voltage to the input voltage for a differentiator and integrator.
Explain the source of the phase shift between the output and input voltages.
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Mechanical Analog Computers
Designed by Vannevar Bush in 1930 and used to control position of artillery through WWII. Replaced by electrical analog computers towards the end of WWII, which performed the needed calculations much faster. http://www.science.uva.nl/museum/AnalogComputers.html
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Why Use an Analog Computer? Calculations performed in real time without the use
of a ‘real’ computer.Can be integrated into the instrumentation circuitry.
Commonly used in control circuits to rapidly monitor and change conditions without the need to communicate back and forth with a digital computer.
Power consumption is not high.Input can be any value between V+ and V-.
Can be designed to handle large (or small) voltages.No digitizing errors.
Analog computers are more robustLess susceptible to electromagnetic radiation damage
(cosmic rays), electrostatic discharge, electromagnetic interference (pick-up of electric noise from the environment), etc.
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DisadvantageSlow
Maximum frequency is less than 10 MHz Compare this to the clock speed of your digital
computer.Voltage transfer function is nonlinear over
entire range of input voltages.Timing of inputs needs to be carefully
considered. Any time delays can cause errors in the calculations
performed.
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SubsystemsMultipliers
Inverting and non-inverting amplifiers Typically fixed number, which means fixed resistor
values in amplifiers
Adders and SubtractorsSumming and difference amplifiers
DifferentiatorsIntegrators
1st order op amp circuits
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Capacitors
)()(1
)(
)(
1
oC
t
t
CC
CC
tvdttiC
tv
dt
dvCti
o
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Differentiator
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Ideal Op Amp ModelVirtual ground
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Op Amp ModelVirtual ground
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AnalysisSince current is not
allowed to enter the input terminals of an ideal op amp.
dt
tdvRCtv
dt
dvC
R
vR
vti
dt
dvC
dt
dvCti
tvtv
titi
So
So
oR
SCC
SC
RC
)()(
)(
)(
)()(
)()(
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Example #1Suppose vS(t) = 3V u(t-5s)
The voltage source changes from 0V to 3V at t = 5s. Initial condition of VC = 0V when t <5s.
Final condition of VC = 3V when t > 5RC.
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Example #1 (con’t)
seVtv
eVVVtv
eVVVtv
Vtv
sstC
sstC
ttCCCC
C
o
initialfinalinitial
5 when t 3)(
t when t 030)(
t when t )(
t when t0)(
8.05
o8.05
o
o
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Example #1 (con’t)
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Co
eVtv
eVFxxs
tv
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tdvRCtv
8.05
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o
3)(
)3)(1040)(1020(8.0
1)(
9F))(405(20k5s when t0)(
RC where,5 t when t0)(
5s when t0)(
)()(
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Example #2Let R = 2 k, C = 0.1F, and vS(t) = 2V
sin(t) at t = 0s
0s when t0)(
0s when t )500cos(2.0)(
)500cos(50022.0)(
)500sin(2)10)(2000()(
)(
)()( Since
7
Vtv
tVtv
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dt
dvRCtv
tvtv
o
o
o
o
So
SC
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Cosine to Sine Conversion
, the output voltage lags the input voltage by 90 degrees.
)500sin(2)( As
)90500sin(2.0)(
)18090500sin(2.0)(
)90500sin(2.0)(
)500cos(2.0)(
tVtv
tVtv
tVtv
tVtv
tVtv
S
oo
ooo
oo
o
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PSpice Simulation
Shows the 90 degree phase shift as well as the deamplification.
vS(t) vo(t)
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IntegratorVirtual ground
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Op Amp Model
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Integrator
)()(1
)(
0)()(
0)()(
0
)()()(
)()(
12
1
1
2
1
tvdttvRC
tv
RC
tv
dt
tdvdt
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R
tv
mAii
tvtvvtvdt
dvCi
R
tv
R
vtvi
o
t
t
So
So
oS
CR
ooC
CC
SSR
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Example #3Let R = 25 k, C = 5nF, at
t=0s
since vo(t) = -vC(t) and the voltage across a capacitor can’t be discontinuous.
stVts
radkVtV
tVts
radkVtV
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ino
0 when 85.39024.6sin85.3)(
)(24.6cos85.3)(
24.6sin3)5(25
1)(
)()(1
)(
122
12
2
12
2
1
2
1
2
1
t
s
radkVtvS 24.6sin3)(
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PSpice Simulation
Shows that the output voltage leads the input voltage by +90 degree and the voltage offset due to the Vo(t1) term.
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Example #4Initial Charge on Capacitor
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Example #4 (con’t) If there is an initial charge that produces a
voltage on the capacitor at some time, to:
The voltage on the negative input of the op amp is:v1 = VC + VR1
v1= v2 = 0V
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Example #4 (con’t) The current flowing through R1 is the same
current flowing through C.
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R
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R
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R
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R
Vti
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0)(,0)( , tas
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)()](0[)]([)(
)()(
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CR
tt
oCC
C
CR
tt
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CC
CC
RC
o
o
etvRdt
tdvCi
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dt
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tv
dt
tdvC
titi
1
1
1
1
1
1
)(1)(
)()(
0)()(
0)()(
0)()(
R1C is the time constant, .
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CR
tt
oCo
ooR
CR
o
etvR
Rtv
R
tv
R
tvVi
ii
1
1
2
222
2
)()(
)()(0
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SummaryDifferentiator and integrator circuits are 1st
order op amp circuits.When the C is connected to the input of the op
amp, the circuit is a differentiator. If the input voltage is a sinusoid, the output voltage
lags the input voltage by 90 degrees. The output voltage may be discontinuous.
When the C is connected between the input and output of the op amp, the circuit is an integrator. If the input voltage is a sinusoid, the output voltage
leads the input voltage by 90 degrees. The output voltage must be continuous.