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Operational Amplifiers

ME 4447: Intro to MechatronicsFebruary 18, 2004

Jonathon LeekeBenjamin Macdonald

Anthony Storc

Outline

IntroductionTheoryHistoryReal vs Ideal

Types of Op-AmpsApplicationsConclusion

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Theory

Measurements of physical parameters usually begin with an electrical voltageThe amplitude of the voltages produced are usually lowNecessary to amplify to a level that can be processed by instrumentation

Op-Amp History

First developed in the early 1940sEarly Op-Amps were made from vacuum tubesFirst modular solid state Op-Amps in 1962Originally developed for operations of integration and summation for differential equations with analog computersBuilding blocks of analog electronic circuits

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741 Op-Amp

Developed by Fairchild in 1968Most prolific Op-AmpPerforms close to an ideal Op-Amp with little limitationsInexpensive

Op-Amps Replace

Low reliability and convenienceHigh costs

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741 Pin ConnectionsPin 1: Nulls the DC offsetPin 2: Inverting inputPin 3: Non-inverting inputPin 4: Negative DC power

supplyPin 5: Nulls the DC offsetPin 6: Output voltagePin 7: Positive DC power

supplyPin 8: No connection

741 Op-Amp Diagram

Complex circuitry with 20 transistors and 11 resistors.

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Op-Amp Diagram

Reliable and ruggedCost about 10¢

Transfer Relation

+ Saturation,- Saturation,Linear, ( )−+ −= vvAvout

CCout vv +=

CCout vv −=

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Ideal Op-Amp

Inputs(+) Non-Inverting (-) Inverting

OutputsFunction of Inputs

AOL is the open loop gain

( )−+ −= vvAv OL0

Ideal Op-Amp, Assumptions

Infinite input impedanceNo current flows back into the inputs

Infinite gainOpen loop gain approaches infinity

Infinite band widthOpen loop gain is not a function of frequency

Zero DC bias currentsNo DC current flows into inputs, a problem with real op-amps

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Ideal Op-Amp, Assumptions

Zero DC offset voltageNo DC voltage flows into inputs, a problem with real op-amps

An infinite slew rateOutput voltage responds instantaneously to changes in either input

Fortunately, real op-amps operate very closely to that of ideal op-amps

Ideal Op-Amp, Assumptions

Ideal Op-Amp Mathematical Summary:

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Ideal Op-Amp, Saturation Curve

Large voltage gain, typically 105 or 106

Amplifies in linear regionOutput voltage lies between the power-supply voltagesOnly used while saturated

Output voltage approaches supply voltage

Op-Amps, Feedback Loop

Positive and negative feedback Negative feedback

Typically usedMakes the system stable

Positive feedbackDrives the system to extremes

Feedback loop is integral in gain control

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Types of Op-Amps

Basic circuits of op-ampsInvertingNon-invertingSummingDifferencingIntegratingDifferential

Inverting Op-Amp

Voltage gain:

Output and input voltages are 180°out of phaseAn increased gain is the result of R2>R1

1

2

RR

VV

in

out −=

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Inverting Op-Amp, Derived

Begin with the basic inverting op-ampGround, the reference voltage, is assumed to equal zeroCircuit can be reduced

Inverting Op-Amp, Derived

From Vx, make nodal equations so:

Vx

1

2

21

21

0 assume

0

0

RR

VV

RV

RV

VRVV

RVVi

in

out

outin

x

outxinx

out

−=

=−

=

=−

+−

=∑

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Non-Inverting Op-Amp

1

21RR

VVin

out +=

Voltage gain:

Input and output voltage signals are in phaseIf Rf=0, you have a unity gain buffer where Vin=Vout

R

R1

R2

Non-Inverting Op-Amp, Derived

Begin with the basic inverting op-ampNote the virtual short that makes Vx=Vin

Circuit can be reduced

R2

R1

R

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Non-Inverting Op-Amp, Derived

1

2

1

2

21

21

1

1

)(

00

RR

VV

VV

VVV

RR

RVV

RV

RVV

RV

in

out

in

out

in

outin

outinin

outinin

+=

−=−

=−

−−=

=−

+−

Nodal analysis at Vin

R1

R2

Summing Op-Amp

Allows you to add several signals together.Applications:

Measuring temperature, offset to zeroAudio mixer, vocal & instrumental together

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Summing Op-Amp cont.

Sum of voltage,

where,2

2

1

1

Rv

Rviin +=

×+×−=−=

22

11 R

RvRRvRiv FF

Finout

Summing then Inverting

Outputs the same phase as the input

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Differencing Op-Amp

Output Voltage,

If, R1=R2

( )211

vvRRv F

out −=

Integrating Op Amp

Capacitor integrates current

Assume:In this case:Current through resistor and capacitor

dtdVC

dtdqi

CVq

==

=

)()( +− =VV

0)( =+V

Rv

Rvv

i ininR =

−= −)(

dtdVC

dtVVd

CdtdvCi outout

C −=−

== − )( )(

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Integrating Op Amp, cont.

Infinite impedance (i(-) = 0), so:

Integrate and solvedtdVC

Rv

ii

outin

CR

−=

=

∫−= dtVRC

VtV inoutout1)0()(

Differentiating Op Amp

Similar to Integrating, but resistor and capacitor are switched, giving:

dtdVRCV

dtdVC

RV

ii

inout

inout

CR

−=

=−

=

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Applications

Analog ComputationFilteringControl SystemsSignal Amplification

As in:EKG Heart Monitors – FilteringMicrophone Signals - FilteringInstrumentation – Analog Computation

Conclusion

Many assumptions made when using Op-AmpsA reliable, low cost solutionOperate in linear region of transfer functionDifferent functions and computationsWide range of applications

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References

http://batronix.com/pdf/uA741.pdfThomas E Brewer, Experiments in Analog and Digital ElectronicsJ.R. Cogdell, Foundations of Electrical Engineeringhttp://ece.wpi.edu/~sjbitar/ee2011/handouts/opamphandout.pdfhttp://users.ece.gatech.edu/~gte395r/ece3710/docs/hw6_solutions.pdf

Example

Find the gain of the following circuit:

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Example, solution

1.Find the nodal voltage

2.Realize the nodal voltage is the input voltage to the inverting op-amp

3.Use inverting op-amp gain equation

inx VRRR

RRV321

32

||||

+=

xout VRRV

3

4−=

+

−=321

32

3

4

||||

RRRRR

RR

VV

in

out

Questions?