op amps - lesson 7 - mathematical operations - calculus

7/31/2019 Op Amps - Lesson 7 - Mathematical Operations - Calculus

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2012 Mladen Hruska This work is licensed under a CreativeCommons Attribution-ShareAlike 2.5 Canada License

Background image courtesy of [MrLightman]/FreeDigitalPhotos.net

Operational Amplifiers

Lesson 7 Mathematical Operations -Calculus


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Outline

Integrator

Integrator Practical

Differentiator

Differentiator Practical


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2012 Mladen Hruska This work is licensed under a Creative

Commons Attribution-ShareAlike 2.5 Canada LicenseBackground image courtesy of [MrLightman]/FreeDigitalPhotos.net

Integrator

Very useful for wave shaping and real timecalculations

Place a capacitor in the feedback network

Use the charging / discharging to integrate

Output voltage proportional to the inputvoltage integrated over time


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Integrator

Vsat

t(s)

Vi

Vo

t(s)


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Integrating Square Wave Results in Triangle Wave


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Integrating Sine Wave Resultsin -Cosine Wave


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Integrating Triangle Wave Results in Parabolic Wave


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Basic Integrator


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Basic Integrator

Where T (period) < (Time constant R1C

F)

Circuit very operation frequency and voltagedependent

When input DC, XC1

open, AV

open loop

Unstable, impractical operation

Add RF to set low frequency gain


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Practical Integrator


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Response without RF

-RF/R1

fa

fb

-3dB

AMPLIFIER

INTEG

RAT

OR

AV

f(HZ)


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fa is the frequency where XCF = RF

Circuit integrates where XCF

> RF

Circuit gain unity at fb

where XCF

= R1

Circuit operation between fa

and fb

After fa

gain decreases by 20dB/dec

fa =1

2RFCFfb =

1

2R1

CFwhere RFCF T R1CF


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Differentiator

Find rate of change of waveform

Used in triggering and modulation circuits

Replace R1 with a capacitor

Use capacitor charging / discharging

Differentiation within set frequency range

Output voltage proportional to theinstantaneous rate of change of inputvoltage


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Differentiator

t(s)Vi

Vo t(s)


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Differentiating Sine Wave Results in Cosine Wave


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Differentiating Triangle Wave Results in Square Wave


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Differentiating Square Wave Results in Impulses


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Basic Differentiator


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Basic Differentiator

Where T (period) > (Time constant RFC

1)

Circuit very operation frequency and voltagedependent

When input DC, XC1

open, AV

= 0

Unstable high frequency operation

Add R1 and CF to set high frequency gain


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Practical Differentiator


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Responsewithout R1

fa fb

-20dB/dec

PracticalDifferentiator

AV

f(HZ)

20dB/dec

Op-Amp OpenLoop Gain

0dB


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Circuit gain unity at fa where XC1 = RF

fb

is the frequency where XCF

= RF

& XC1

= R1

Circuit gain increases by 20dB/dec DC to fb

Circuit gain decreases by 20dB/dec after fb

Gain limited by R1

and CF

fa =1

2RFC1fb =

1

2R1C

1

=1

2RFCFwhere T RFC1


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Credits and Attributions

Reference Text: Gayakwad, R.A., (1988). Op-Amps and Linear Integrated

Circuits.

Software:

Schematic diagram(s) drawn using National InstrumentsMultisim 12


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2012 Mladen Hruska This work is licensed under a CreativeCommons Attribution-ShareAlike 2.5 Canada LicenseBackground image courtesy of [MrLightman]/FreeDigitalPhotos.net

Copyright (c) 2012 by Mladen Hruska. This work is made availableunder the terms of the Creative Commons Attribution-ShareAlike 2.5Canada license

All images, diagrams, charts, etc. are the copyright work of MladenHruska if not immediately attributed otherwise.

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op amps - lesson 7 - mathematical operations - calculus

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