amplifiers and feedback theory - intranet deibhome.deib.polimi.it/spinelli/corsi/ele/l01.pdf · •...

Amplifiers and Feedback Theory

Alessandro SpinelliPhone: (02 2399) 4001

[email protected]/spinelli

Electronics 96032 Alessandro Spinelli

Slides are supplementary material and are NOT a

replacement for textbooks and/or lecture notes

Outline

• Review: equivalent circuits• Amplifiers• Negative feedback• Operational amplifiers


The origin


Hermann von Helmholtz

(1821-1894)

Lèon Charles Thévenin

(1857-1926)

Hans Ferdinand Mayer

(1895-1980)

Edward LawryNorton

(1898-1983)

1853 1883 1926 1926

From [1]

Equivalent circuits

• Every linear network «seen» between any pairof terminals behaves as if composed by a source and an impedance only

• Thévenin equivalent circuit: voltage source with impedance in series

• Norton equivalent circuit: current source with impedance in parallel


Equivalent circuits

Equivalence is only from the viewpoint of the external load. Power dissipation, for example, is notequal

Same 𝑍𝑍𝑒𝑒𝑒𝑒


𝑉𝑉𝑒𝑒𝑒𝑒

𝑍𝑍𝑒𝑒𝑒𝑒𝑉𝑉

𝐼𝐼

𝐼𝐼𝑒𝑒𝑒𝑒

𝑍𝑍𝑒𝑒𝑒𝑒𝑉𝑉

𝐼𝐼

𝑉𝑉𝐼𝐼Sources

and linear elements

Element calculations

• 𝑉𝑉𝑒𝑒𝑒𝑒 is the open-circuit voltage at the terminals• 𝐼𝐼𝑒𝑒𝑒𝑒 is the short-circuit current through the

terminals• 𝑍𝑍𝑒𝑒𝑒𝑒 = 𝑉𝑉𝑒𝑒𝑒𝑒/𝐼𝐼𝑒𝑒𝑒𝑒, or equivalently• 𝑍𝑍𝑒𝑒𝑒𝑒is the impedance between the terminals

when– voltage sources are replaced by short-circuits– current sources are replaced by open circuits


Outline



Amplifiers

• We consider a «black box» approach with equivalent circuits

• Four kinds can be identified:

In Out TypeV V Voltage ampl.I I Current ampl.V I Transconductance ampl.I V Transresistance ampl.


Voltage/current amplifiers

• One-directional amplifiers (no reverse transfer from output to input)

• Resistors will be considered for simplicity, though complex impedances can be assumed

Voltage-controlled voltage source (VCVS) Current-controlled current source (CCCS)

𝑅𝑅𝑖𝑖𝐴𝐴𝑉𝑉𝑉𝑉𝑖𝑖

𝑅𝑅𝑜𝑜𝑉𝑉𝑜𝑜𝑉𝑉𝑖𝑖 𝑅𝑅𝑖𝑖

𝐴𝐴𝐼𝐼𝐼𝐼𝑖𝑖

𝑅𝑅𝑜𝑜𝐼𝐼𝑜𝑜𝐼𝐼𝑖𝑖


Source and load resistors (VA)

𝑉𝑉𝑖𝑖 = 𝑉𝑉𝑆𝑆𝑅𝑅𝑖𝑖

𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆𝑉𝑉𝑜𝑜 = 𝐴𝐴𝑉𝑉𝑉𝑉𝑖𝑖

𝑅𝑅𝐿𝐿𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿

𝑅𝑅𝑖𝑖

𝑅𝑅𝑠𝑠𝑉𝑉𝑠𝑠

𝐴𝐴𝑉𝑉𝑉𝑉𝑖𝑖

𝑅𝑅𝑜𝑜𝑅𝑅𝐿𝐿 𝑉𝑉𝑜𝑜𝑉𝑉𝑖𝑖


Voltage gain

• Total gain is less than 𝐴𝐴𝑉𝑉• Gain is dependent on 𝑅𝑅𝑆𝑆 and 𝑅𝑅𝐿𝐿• To avoid these drawbacks, a voltage amplifier

should have𝑅𝑅𝑖𝑖 = ∞ (very high input impedance)𝑅𝑅𝑜𝑜 = 0 (very low output impedance)

𝑉𝑉𝑜𝑜𝑉𝑉𝑆𝑆

= 𝐴𝐴𝑉𝑉𝑅𝑅𝐿𝐿

𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿𝑅𝑅𝑖𝑖

𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆


Source and load resistors (CA)

𝐼𝐼𝑖𝑖 = 𝐼𝐼𝑆𝑆𝑅𝑅𝑆𝑆

𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆𝐼𝐼𝑜𝑜 = 𝐴𝐴𝐼𝐼𝐼𝐼𝑖𝑖

𝑅𝑅𝑜𝑜𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿

𝑅𝑅𝑖𝑖𝑅𝑅𝑠𝑠

𝐼𝐼𝑠𝑠𝐴𝐴𝐼𝐼𝐼𝐼𝑖𝑖

𝑅𝑅𝑜𝑜𝑅𝑅𝐿𝐿

𝐼𝐼𝑜𝑜𝐼𝐼𝑖𝑖


Current gain

• Total gain is less than 𝐴𝐴𝐼𝐼• Gain is dependent on 𝑅𝑅𝑆𝑆 and 𝑅𝑅𝐿𝐿• To avoid these drawbacks a current amplifier

should have𝑅𝑅𝑖𝑖 = 0 (very low input impedance)𝑅𝑅𝑜𝑜 = ∞ (very high output impedance)

𝐼𝐼𝑜𝑜𝐼𝐼𝑆𝑆

= 𝐴𝐴𝐼𝐼𝑅𝑅𝑜𝑜

𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿𝑅𝑅𝑆𝑆

𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆


Summary

Type 𝑹𝑹𝒊𝒊 𝑹𝑹𝒐𝒐Voltage amplifier ∞ 0Current amplifier 0 ∞Transconductance ampl. ∞ ∞Transresistance ampl. 0 0


Outline



American telephone lines…

• First transcontinental telephone line built in 1914 (announced 1915), upgraded in 1921 to three channels and using twelve amplifiers

• Second line built in 1923 with four channels and twenty amplifiers

• A further increase in the number of channels was very, very challenging…


The amplifier problem

• Signal is attenuated as it propagates along the wires and must be regenerated

• Gain of vacuum-tube amplifiers changes with plate voltage, temperature, aging,…

• Non-linearity creates intermodulationdistorsion in multi-channel systems


Negative-feedback concept

Harold S. Black (1898-1983)


From [2]

The theory

+-

𝜀𝜀 = 𝑆𝑆𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐺𝐺𝑂𝑂𝐿𝐿 𝜀𝜀

𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜𝑆𝑆𝑖𝑖𝑖𝑖

= 𝐺𝐺 =𝐺𝐺𝑂𝑂𝐿𝐿

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹


𝐺𝐺𝑂𝑂𝐿𝐿

𝐹𝐹

𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜𝑆𝑆𝑖𝑖𝑖𝑖

𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝜀𝜀

Closed-loop gain

𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹 ≪ 1 → 𝐺𝐺 =𝐺𝐺𝑂𝑂𝐿𝐿

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹~𝐺𝐺𝑂𝑂𝐿𝐿

Open-loop gain, no feedback

Ideal gain, independent of 𝐺𝐺𝑂𝑂𝐿𝐿

𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹 ≫ 1 → 𝐺𝐺 =𝐺𝐺𝑂𝑂𝐿𝐿

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹~

1𝐹𝐹

= 𝐺𝐺𝑖𝑖𝑖𝑖


Loop gain – calculation

• 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙 = −𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹 measures the strength of the feedback

• A good feedback system has 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙 < 0 and |𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙| ≫ 1

-



𝐹𝐹𝑆𝑆𝑜𝑜𝑒𝑒𝑠𝑠𝑜𝑜

−𝐺𝐺𝑂𝑂𝐿𝐿𝑆𝑆𝑜𝑜𝑒𝑒𝑠𝑠𝑜𝑜

−𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹𝑆𝑆𝑜𝑜𝑒𝑒𝑠𝑠𝑜𝑜

Loop gain – interpretation

• 1/|𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙| is the error between 𝐺𝐺 and 𝐺𝐺𝑖𝑖𝑖𝑖:Ex: 𝐺𝐺𝑂𝑂𝐿𝐿 = 105, 𝐹𝐹 = 10−2 ⇒ 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙 = −1000,

𝐺𝐺𝑖𝑖𝑖𝑖 = 100,𝐺𝐺 = 99.9 ⇒ 𝜀𝜀 = 0.001 = 1/|𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙|

• In fact, the error signal 𝜀𝜀 = 𝑆𝑆𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 is


𝐺𝐺 =𝐺𝐺𝑂𝑂𝐿𝐿

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹=

1/𝐹𝐹1 + 1/𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹

=𝐺𝐺𝑖𝑖𝑖𝑖

1 − 1/𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙

𝜀𝜀𝑆𝑆𝑖𝑖𝑖𝑖

= 1 − 𝐹𝐹𝐺𝐺 =𝐺𝐺𝑖𝑖𝑖𝑖 − 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖

=1

1 − 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙~

1|𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙|

Sensitivity to 𝐺𝐺𝑂𝑂𝐿𝐿𝑑𝑑𝐺𝐺𝑑𝑑𝐺𝐺𝑂𝑂𝐿𝐿

=1

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹 2 =𝐺𝐺𝐺𝐺𝑂𝑂𝐿𝐿

11 − 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙

𝑑𝑑𝐺𝐺𝐺𝐺

=𝑑𝑑𝐺𝐺𝑂𝑂𝐿𝐿𝐺𝐺𝑂𝑂𝐿𝐿

11 − 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙

𝐺𝐺𝑂𝑂𝐿𝐿 = 105,𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.9𝐺𝐺𝑂𝑂𝐿𝐿 = 2 × 105,𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.95


<<1

Sensitivity to 𝐹𝐹

𝑑𝑑𝐺𝐺𝑑𝑑𝐹𝐹

= −𝐺𝐺𝑂𝑂𝐿𝐿2

1 + 𝐺𝐺𝑂𝑂𝐿𝐿𝐹𝐹 2 = −𝐺𝐺2

𝑑𝑑𝐺𝐺𝐺𝐺

=𝑑𝑑𝐹𝐹𝐹𝐹

𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙1 − 𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙

𝐺𝐺𝑂𝑂𝐿𝐿 = 105,𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.9𝐺𝐺𝑂𝑂𝐿𝐿 = 105,𝐹𝐹 = 2 × 0.01 ⇒ 𝐺𝐺 = 49.98


∼-1

Interpretation

• Changes in 𝐺𝐺𝑂𝑂𝐿𝐿are nulled by the feedback loop

• Changes in 𝐹𝐹cannot be compensated


+-


𝐹𝐹

𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜𝑆𝑆𝑖𝑖𝑖𝑖𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝜀𝜀

+-


𝐹𝐹

𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜𝑆𝑆𝑖𝑖𝑖𝑖𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝜀𝜀

Outline



Feedback amplifier design

• Forward (open-loop) block 𝐺𝐺𝑂𝑂𝐿𝐿 must havehigh gain, to ensure that |𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙| ≫ 1.All active elements are placed here even ifgain is not stable – their fluctuations are reduced by 1/𝐺𝐺𝑙𝑙𝑜𝑜𝑜𝑜𝑙𝑙

• Feedback block 𝐹𝐹 must be stable, to ensure a stable closed-loop gain.Usually made with passives


Operational amplifiers (OAs)

• Integrated voltageamplifiers used asforward gain blocks in feedback circuits

• The ideal OA has–𝐴𝐴 = ∞ (105 − 106)–𝑅𝑅𝑖𝑖 = ∞ (106 − 109Ω)–𝑅𝑅𝑜𝑜 = 0 (≈ 100 Ω)

Electronics 96032

-+

+_𝑉𝑉𝑜𝑜 = 𝐴𝐴(𝑉𝑉+ − 𝑉𝑉−)

Alessandro Spinelli

𝑉𝑉+

𝑉𝑉−

𝜀𝜀𝐴𝐴

Typical circuit arrangements

In ideal feedback loops 𝜀𝜀 = 0 ⇒ ideal OAs keep𝑉𝑉+ − 𝑉𝑉− = 0 ⇒ 𝑉𝑉+ = 𝑉𝑉−


𝐹𝐹

+_

𝐹𝐹

+_

0

References

1. http://tcts.fpms.ac.be/cours/1005-01/equiv.pdf

2. https://www.wpi.edu/News/Transformations/2005Summer/timecapsule.html


amplifiers and feedback theory - intranet deibhome.deib.polimi.it/spinelli/corsi/ele/l01.pdf · •...

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