Amplifiers and Feedback Theory - polimi.it
Transcript of Amplifiers and Feedback Theory - polimi.it
Electronics โ 96032
Alessandro SpinelliPhone: (02 2399) [email protected] home.deib.polimi.it/spinelli
Amplifiers and Feedback Theory
Alessandro Spinelli โ Electronics 96032
Slides are supplementary material and are NOT a
replacement for textbooks and/or lecture notes
Disclaimer 2
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Acquisition chain 3
Sensor Filter ADC
small signal
noise
amplifiedsignal
amplifiednoise
amplifiedsignal
reducednoise
Amp
next lessons
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โข We begin our study with the analysis and design of simpleamplifiers
โข Next lessons will deal with Basic amplifier principles and the feedback amplifier concept (this
lesson) Linear applications of OpAmps Feedback amplifier properties Stability of feedback amplifiers Instrumentation amplifiers and single-supply OpAmp circuits
Purpose of the lesson 4
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โข Review: equivalent circuitsโข Amplifiersโข Negative feedbackโข Operational amplifiers
Outline 5
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The origin 6
Hermann von Helmholtz
(1821-1894)
Lรจon Charles Thรฉvenin
(1857-1926)
Hans Ferdinand Mayer
(1895-1980)
Edward LawryNorton
(1898-1983)
1853 1883 1926 1926
From [1]
Voltage source equivalent circuit Current source equivalent circuit
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โข Linear network: R, L, C with parameters not dependent on I or V and V/I sources either constant or linearly dependent on othervoltages or currents
โข Every linear network ยซseenยป between any pair of terminalsbehaves as if composed by a source and an impedance only Thรฉvenin equivalent circuit: voltage source with impedance in series Norton equivalent circuit: current source with impedance in parallel
Equivalent circuits 7
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Equivalence is only from the viewpoint of the external load. Power dissipation, for example, is not equal
Equivalent circuits 8
Same ๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐๐๐
๐ผ๐ผ
๐ผ๐ผ๐๐๐๐
๐๐๐๐๐๐๐๐
๐ผ๐ผ
๐๐๐ผ๐ผSources
and linear elements
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โข ๐๐๐๐๐๐ is the open-circuit voltage at the terminals
โข ๐ผ๐ผ๐๐๐๐ is the short-circuit current through the terminals
โข ๐๐๐๐๐๐ = ๐๐๐๐๐๐/๐ผ๐ผ๐๐๐๐, or equivalently
โข ๐๐๐๐๐๐ is the impedance between the terminals when Independent voltage sources are replaced by short-circuits Independent current sources are replaced by open circuits
Element calculations 9
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โข Review: equivalent circuitsโข Amplifiersโข Negative feedbackโข Operational amplifiers
Outline 10
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โข We consider a ยซblack boxยป approach with equivalent circuitsโข Four kinds can be identified:
Amplifiers 11
In Out TypeV V Voltage ampl.I I Current ampl.V I Transconductance ampl.I V Transresistance ampl.
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โข One-directional amplifiers (no reverse transfer from output to input)โข Resistors will be considered for simplicity, though complex
impedances can be assumed
Voltage/current amplifiers 12
Voltage-controlled voltage source (VCVS) Current-controlled current source (CCCS)
๐ ๐ ๐๐๐ด๐ด๐๐๐๐๐๐
๐ ๐ ๐๐๐๐๐๐๐๐๐๐ ๐ ๐ ๐๐
๐ด๐ด๐ผ๐ผ๐ผ๐ผ๐๐
๐ ๐ ๐๐๐ผ๐ผ๐๐๐ผ๐ผ๐๐
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Source and load resistors (VA) 13
๐๐๐๐ = ๐๐๐๐๐ ๐ ๐๐
๐ ๐ ๐๐ + ๐ ๐ ๐๐๐๐๐๐ = ๐ด๐ด๐๐๐๐๐๐
๐ ๐ ๐ฟ๐ฟ๐ ๐ ๐๐ + ๐ ๐ ๐ฟ๐ฟ
๐ ๐ ๐๐
๐ ๐ ๐ ๐ ๐๐๐ ๐
๐ด๐ด๐๐๐๐๐๐
๐ ๐ ๐๐๐ ๐ ๐ฟ๐ฟ ๐๐๐๐๐๐๐๐
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โข Total gain is less than ๐ด๐ด๐๐โข Gain is dependent on ๐ ๐ ๐๐ and ๐ ๐ ๐ฟ๐ฟโข To avoid these drawbacks, a voltage amplifier should have:๐ ๐ ๐๐ = โ (very high input impedance)๐ ๐ ๐๐ = 0 (very low output impedance)
Voltage gain 14
๐๐๐๐๐๐๐๐
= ๐ด๐ด๐๐๐ ๐ ๐ฟ๐ฟ
๐ ๐ ๐๐ + ๐ ๐ ๐ฟ๐ฟ๐ ๐ ๐๐
๐ ๐ ๐๐ + ๐ ๐ ๐๐
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Source and load resistors (CA) 15
๐ผ๐ผ๐๐ = ๐ผ๐ผ๐๐๐ ๐ ๐๐
๐ ๐ ๐๐ + ๐ ๐ ๐๐๐ผ๐ผ๐๐ = ๐ด๐ด๐ผ๐ผ๐ผ๐ผ๐๐
๐ ๐ ๐๐๐ ๐ ๐๐ + ๐ ๐ ๐ฟ๐ฟ
๐ ๐ ๐๐๐ ๐ ๐ ๐ ๐ผ๐ผ๐ ๐
๐ด๐ด๐ผ๐ผ๐ผ๐ผ๐๐
๐ ๐ ๐๐๐ ๐ ๐ฟ๐ฟ
๐ผ๐ผ๐๐๐ผ๐ผ๐๐
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โข Total gain is less than ๐ด๐ด๐ผ๐ผโข Gain is dependent on ๐ ๐ ๐๐ and ๐ ๐ ๐ฟ๐ฟโข To avoid these drawbacks a current amplifier should have:
๐ ๐ ๐๐ = 0 (very low input impedance)๐ ๐ ๐๐ = โ (very high output impedance)
Current gain 16
๐ผ๐ผ๐๐๐ผ๐ผ๐๐
= ๐ด๐ด๐ผ๐ผ๐ ๐ ๐๐
๐ ๐ ๐๐ + ๐ ๐ ๐ฟ๐ฟ๐ ๐ ๐๐
๐ ๐ ๐๐ + ๐ ๐ ๐๐
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Type ๐น๐น๐๐ ๐น๐น๐๐Voltage amplifier โ 0Current amplifier 0 โTransconductance ampl. โ โTransresistance ampl. 0 0
Summary 17
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โข Review: equivalent circuitsโข Amplifiersโข Negative feedbackโข Operational amplifiers
Outline 18
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โข First transcontinental telephone line built in 1914 (announced 1915), upgraded in 1921 to three channels and using twelve amplifiers
โข Second line built in 1923 with four channels and twenty amplifiersโข A further increase in the number of channels was very, very
challengingโฆ
American telephone linesโฆ 19
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โข Signal is attenuated as it propagates along the wires and must be regenerated
โข Gain of vacuum-tube amplifiers changes with plate voltage, temperature, aging,โฆ
โข Non-linearity creates intermodulation distorsion in multi-channelsystems
The amplifier problem 20
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Negative-feedback concept 21
Harold S. Black (1898-1983)
From [2]
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The theory 22
+-
๐๐ = ๐๐๐๐๐๐ โ ๐น๐น๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = ๐บ๐บ๐๐๐ฟ๐ฟ ๐๐
๐๐๐๐๐๐๐๐๐๐๐๐๐๐
= ๐บ๐บ =๐บ๐บ๐๐๐ฟ๐ฟ
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น
๐บ๐บ๐๐๐ฟ๐ฟ
๐น๐น
๐๐๐๐๐๐๐๐๐๐๐๐๐๐
๐น๐น๐๐๐๐๐๐๐๐
๐๐
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Closed-loop gain 23
๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น โช 1 โ ๐บ๐บ =๐บ๐บ๐๐๐ฟ๐ฟ
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น~๐บ๐บ๐๐๐ฟ๐ฟ Open-loop gain, no
feedback
Ideal gain, independent of ๐บ๐บ๐๐๐ฟ๐ฟ
๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น โซ 1 โ ๐บ๐บ =๐บ๐บ๐๐๐ฟ๐ฟ
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น~
1๐น๐น
= ๐บ๐บ๐๐๐๐
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โข ๐บ๐บ๐๐๐๐๐๐๐๐ = โ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น measures the strength of the feedbackโข The result is independent of the breaking pointโข A good feedback system has ๐บ๐บ๐๐๐๐๐๐๐๐ < 0 and |๐บ๐บ๐๐๐๐๐๐๐๐| โซ 1
Loop gain โ calculation 24
-๐บ๐บ๐๐๐ฟ๐ฟ
๐น๐น๐๐๐๐๐๐๐ ๐ ๐๐
โ๐บ๐บ๐๐๐ฟ๐ฟ๐๐๐๐๐๐๐ ๐ ๐๐
โ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น๐๐๐๐๐๐๐ ๐ ๐๐
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๐บ๐บ =๐บ๐บ๐๐๐ฟ๐ฟ
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น=
1/๐น๐น1 + 1/๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น
=๐บ๐บ๐๐๐๐
1 โ 1/๐บ๐บ๐๐๐๐๐๐๐๐โข Example:
๐บ๐บ๐๐๐ฟ๐ฟ = 105, ๐น๐น = 10โ2
โ ๐บ๐บ๐๐๐๐๐๐๐๐ = โ1000, ๐บ๐บ๐๐๐๐ = 100, ๐บ๐บ = 99.9โข The relative error between ๐บ๐บ and ๐บ๐บ๐๐๐๐ is
๐๐ =100 โ 99.9
100= 0.001 =
1|๐บ๐บ๐๐๐๐๐๐๐๐|
Loop gain โ interpretation 25
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Error signal
+-
๐บ๐บ๐๐๐ฟ๐ฟ
๐น๐น
๐๐๐๐๐๐๐๐๐๐๐๐๐๐
๐น๐น๐๐๐๐๐๐๐๐
๐๐
๐๐๐๐๐๐๐๐
= 1 โ ๐น๐น๐บ๐บ =๐บ๐บ๐๐๐๐ โ ๐บ๐บ๐บ๐บ๐๐๐๐
๐๐๐๐๐๐๐๐
=๐บ๐บ๐บ๐บ๐๐๐ฟ๐ฟ
=1
1 โ ๐บ๐บ๐๐๐๐๐๐๐๐~
1|๐บ๐บ๐๐๐๐๐๐๐๐|
26
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Error signal and calculation of ๐ฎ๐ฎ๐๐๐๐ 27
+-
๐๐ =๐๐๐๐๐๐๐๐๐บ๐บ๐๐๐ฟ๐ฟ
= 0 โ ๐๐๐๐๐๐ โ ๐น๐น๐๐๐๐๐๐๐๐ = 0 โ๐๐๐๐๐๐๐๐๐๐๐๐๐๐
=1๐น๐น
= ๐บ๐บ๐๐๐๐
๐บ๐บ๐๐๐ฟ๐ฟ = โ
๐น๐น
๐๐๐๐๐๐๐๐๐๐๐๐๐๐
๐น๐น๐๐๐๐๐๐๐๐
๐๐ = 0
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๐๐๐บ๐บ๐๐๐บ๐บ๐๐๐ฟ๐ฟ
=1
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น 2 =๐บ๐บ๐บ๐บ๐๐๐ฟ๐ฟ
11 โ ๐บ๐บ๐๐๐๐๐๐๐๐
๐๐๐บ๐บ๐บ๐บ
=๐๐๐บ๐บ๐๐๐ฟ๐ฟ๐บ๐บ๐๐๐ฟ๐ฟ
11 โ ๐บ๐บ๐๐๐๐๐๐๐๐
๐บ๐บ๐๐๐ฟ๐ฟ = 105,๐น๐น = 0.01 โ ๐บ๐บ = 99.9๐บ๐บ๐๐๐ฟ๐ฟ = 2 ร 105,๐น๐น = 0.01 โ ๐บ๐บ = 99.95
Sensitivity to ๐ฎ๐ฎ๐ถ๐ถ๐ถ๐ถ 28
โช 1
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๐๐๐บ๐บ๐๐๐น๐น
= โ๐บ๐บ๐๐๐ฟ๐ฟ2
1 + ๐บ๐บ๐๐๐ฟ๐ฟ๐น๐น 2 = โ๐บ๐บ2
๐๐๐บ๐บ๐บ๐บ
=๐๐๐น๐น๐น๐น
๐บ๐บ๐๐๐๐๐๐๐๐1 โ ๐บ๐บ๐๐๐๐๐๐๐๐
๐บ๐บ๐๐๐ฟ๐ฟ = 105,๐น๐น = 0.01 โ ๐บ๐บ = 99.9๐บ๐บ๐๐๐ฟ๐ฟ = 105,๐น๐น = 2 ร 0.01 โ ๐บ๐บ = 49.98
Sensitivity to ๐ญ๐ญ 29
โ โ1
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โข Changes in ๐บ๐บ๐๐๐ฟ๐ฟ are nulled by the feedback loop
โข Changes in ๐น๐น cannot be compensated
Qualitative interpretation 30
+-
๐บ๐บ๐๐๐ฟ๐ฟ
๐น๐น
๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐น๐น๐๐๐๐๐๐๐๐
๐๐
+-
๐บ๐บ๐๐๐ฟ๐ฟ
๐น๐น
๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐น๐น๐๐๐๐๐๐๐๐
๐๐
Alessandro Spinelli โ Electronics 96032
โข Review: equivalent circuitsโข Amplifiersโข Negative feedbackโข Operational amplifiers
Outline 31
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โข Forward (open-loop) block ๐บ๐บ๐๐๐ฟ๐ฟ must have high gain, to ensurethat |๐บ๐บ๐๐๐๐๐๐๐๐| โซ 1.All active elements are placed here even if gain is not stable โtheir fluctuations are reduced by 1/|๐บ๐บ๐๐๐๐๐๐๐๐|
โข Feedback block ๐น๐น must be stable, to ensure a stable closed-loopgain โ usually made with passives
Feedback amplifier design 32
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Operational amplifiers (OAs)
โข Integrated voltage amplifiersused as forward gain blocks in feedback circuits
โข The ideal OA has ๐ด๐ด = โ (105 โ 106) ๐ ๐ ๐๐ = โ (106 โ 109ฮฉ) ๐ ๐ ๐๐ = 0 (โ 100 ฮฉ)
33
-+
+_
๐๐๐๐ = ๐ด๐ด(๐๐+ โ ๐๐โ)
๐๐+
๐๐โ
๐๐๐ด๐ด
Alessandro Spinelli โ Electronics 96032
In ideal feedback loops, ๐๐ = 0 โ ideal OAs keep ๐๐+ โ ๐๐โ = 0โ ๐๐+ = ๐๐โ
Typical circuit arrangements 34
๐น๐น
+_
๐น๐น
+_
0
Feedback loop is on the negative pin
๐๐ ๐๐
Alessandro Spinelli โ Electronics 96032
1. http://tcts.fpms.ac.be/cours/1005-01/equiv.pdf2. https://www.wpi.edu/News/Transformations/2005Summer/tim
ecapsule.html
References 35