Electronics – 96032

Amplifiers and Feedback Theory

Alessandro Spinelli
Phone: (02 2399) 4001
alessandro.spinelli@polimi.it home.deib.polimi.it/spinelli
 Disclaimer 2

Slides are supplementary

material and are NOT a
replacement for textbooks
and/or lecture notes

Alessandro Spinelli – Electronics 96032

 Outline 3

• Review: equivalent circuits

• Amplifiers
• Negative feedback
• Operational amplifiers

Alessandro Spinelli – Electronics 96032

 The origin 4

Voltage source equivalent circuit Current source equivalent circuit

Hermann von Lèon Charles Hans Ferdinand Edward Lawry

Helmholtz Thévenin Mayer Norton
(1821-1894) (1857-1926) (1895-1980) (1898-1983)

1853 1883 1926 1926

From [1]

Alessandro Spinelli – Electronics 96032

 Equivalent circuits 5

• Linear network: R, L, C with parameters not dependent on I or V

and V/I sources either constant or linearly dependent on other
voltages or currents
• Every linear network «seen» between any pair of 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
Alessandro Spinelli – Electronics 96032
 Equivalent circuits 6

Sources 𝐼𝐼
and linear 𝑉𝑉
elements Same 𝑍𝑍𝑒𝑒𝑒𝑒

𝐼𝐼 𝑍𝑍𝑒𝑒𝑒𝑒 𝐼𝐼
𝑍𝑍𝑒𝑒𝑒𝑒
𝑉𝑉 𝑉𝑉
𝑉𝑉𝑒𝑒𝑒𝑒 𝐼𝐼𝑒𝑒𝑒𝑒

Equivalence is only from the viewpoint of the external load.

Power dissipation, for example, is not equal
Alessandro Spinelli – Electronics 96032
 Element calculations 7

• 𝑉𝑉𝑒𝑒𝑒𝑒 is the open-circuit voltage at the terminals

• 𝐼𝐼𝑒𝑒𝑒𝑒 is the short-circuit current through the terminals
• 𝑍𝑍𝑒𝑒𝑒𝑒 = 𝑉𝑉𝑒𝑒𝑒𝑒 /𝐼𝐼𝑒𝑒𝑒𝑒 , or equivalently
• 𝑍𝑍𝑒𝑒𝑒𝑒 is the impedance between the terminals when
 Independent voltage sources are replaced by short-circuits
 Independent current sources are replaced by open circuits

Alessandro Spinelli – Electronics 96032

 Outline 8

• Review: equivalent circuits

• Amplifiers
• Negative feedback
• Operational amplifiers

Alessandro Spinelli – Electronics 96032

 Amplifiers 9

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

• Four kinds can be identified:

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

Alessandro Spinelli – Electronics 96032

 Voltage/current amplifiers 10

𝐼𝐼𝑖𝑖 𝐼𝐼𝑜𝑜
𝑅𝑅𝑜𝑜 𝑅𝑅𝑜𝑜
𝑉𝑉𝑖𝑖 𝑅𝑅𝑖𝑖 𝑉𝑉𝑜𝑜 𝑅𝑅𝑖𝑖
𝐴𝐴𝑉𝑉 𝑉𝑉𝑖𝑖 𝐴𝐴𝐼𝐼 𝐼𝐼𝑖𝑖

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

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

• Resistors will be considered for simplicity, though complex
impedances can be assumed
Alessandro Spinelli – Electronics 96032
 Source and load resistors (VA) 11

𝑅𝑅𝑠𝑠 𝑅𝑅𝑜𝑜
𝑉𝑉𝑠𝑠 𝑉𝑉𝑖𝑖 𝑅𝑅𝑖𝑖 𝑅𝑅𝐿𝐿 𝑉𝑉𝑜𝑜
𝐴𝐴𝑉𝑉 𝑉𝑉𝑖𝑖

𝑅𝑅𝑖𝑖 𝑅𝑅𝐿𝐿
𝑉𝑉𝑖𝑖 = 𝑉𝑉𝑆𝑆 𝑉𝑉𝑜𝑜 = 𝐴𝐴𝑉𝑉 𝑉𝑉𝑖𝑖
𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆 𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿

Alessandro Spinelli – Electronics 96032

 Voltage gain 12

𝑉𝑉𝑜𝑜 𝑅𝑅𝐿𝐿 𝑅𝑅𝑖𝑖

= 𝐴𝐴𝑉𝑉
𝑉𝑉𝑆𝑆 𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿 𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆
• Total gain is less than 𝐴𝐴𝑉𝑉
• Gain is dependent on 𝑅𝑅𝑆𝑆 and 𝑅𝑅𝐿𝐿
• To avoid these drawbacks, a voltage amplifier should have:
𝑅𝑅𝑖𝑖 = ∞ (very high input impedance)
𝑅𝑅𝑜𝑜 = 0 (very low output impedance)

Alessandro Spinelli – Electronics 96032

 Source and load resistors (CA) 13

𝑅𝑅𝑠𝑠 𝐼𝐼𝑖𝑖 𝑅𝑅𝑜𝑜 𝐼𝐼𝑜𝑜

𝐼𝐼𝑠𝑠 𝑅𝑅𝑖𝑖 𝑅𝑅𝐿𝐿
𝐴𝐴𝐼𝐼 𝐼𝐼𝑖𝑖

𝑅𝑅𝑆𝑆 𝑅𝑅𝑜𝑜
𝐼𝐼𝑖𝑖 = 𝐼𝐼𝑆𝑆 𝐼𝐼𝑜𝑜 = 𝐴𝐴𝐼𝐼 𝐼𝐼𝑖𝑖
𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆 𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿
Alessandro Spinelli – Electronics 96032
 Current gain 14

𝐼𝐼𝑜𝑜 𝑅𝑅𝑜𝑜 𝑅𝑅𝑆𝑆

= 𝐴𝐴𝐼𝐼
𝐼𝐼𝑆𝑆 𝑅𝑅𝑜𝑜 + 𝑅𝑅𝐿𝐿 𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑆𝑆
• Total gain is less than 𝐴𝐴𝐼𝐼
• Gain is dependent on 𝑅𝑅𝑆𝑆 and 𝑅𝑅𝐿𝐿
• To avoid these drawbacks a current amplifier should have:
𝑅𝑅𝑖𝑖 = 0 (very low input impedance)
𝑅𝑅𝑜𝑜 = ∞ (very high output impedance)

Alessandro Spinelli – Electronics 96032

 Summary 15

Type 𝑹𝑹𝒊𝒊 𝑹𝑹𝒐𝒐

Voltage amplifier ∞ 0
Current amplifier 0 ∞
Transconductance ampl. ∞ ∞
Transresistance ampl. 0 0

Alessandro Spinelli – Electronics 96032

 Outline 16

• Review: equivalent circuits

• Amplifiers
• Negative feedback
• Operational amplifiers

Alessandro Spinelli – Electronics 96032

 American telephone lines… 17

• 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…

Alessandro Spinelli – Electronics 96032

 The amplifier problem 18

• 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 intermodulation distorsion in multi-channel
systems

Alessandro Spinelli – Electronics 96032

 Negative-feedback concept 19

Harold S. Black (1898-1983)

From [2]

Alessandro Spinelli – Electronics 96032

 The theory 20

+ 𝐺𝐺𝑂𝑂𝑂𝑂
𝑆𝑆𝑖𝑖𝑖𝑖 - 𝜀𝜀 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝐹𝐹
𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝜀𝜀 = 𝑆𝑆𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 𝐺𝐺𝑂𝑂𝑂𝑂

= 𝐺𝐺 =
𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐺𝐺𝑂𝑂𝑂𝑂 𝜀𝜀 𝑆𝑆𝑖𝑖𝑖𝑖 1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹

Alessandro Spinelli – Electronics 96032

 Closed-loop gain 21

𝐺𝐺𝑂𝑂𝑂𝑂 Open-loop gain, no

𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 ≪ 1 → 𝐺𝐺 = ~𝐺𝐺𝑂𝑂𝑂𝑂
1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 feedback

𝐺𝐺𝑂𝑂𝑂𝑂 1 Ideal gain,

𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 ≫ 1 → 𝐺𝐺 = ~ = 𝐺𝐺𝑖𝑖𝑖𝑖
1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 𝐹𝐹 independent of 𝐺𝐺𝑂𝑂𝑂𝑂

Alessandro Spinelli – Electronics 96032

 Calculation of 𝐺𝐺𝑖𝑖𝑖𝑖 22

𝜀𝜀 = 0
+ 𝐺𝐺𝑂𝑂𝑂𝑂 = ∞
𝑆𝑆𝑖𝑖𝑖𝑖 - 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝐹𝐹
𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 1
𝜀𝜀 = = 0 ⇒ 𝑆𝑆𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 = 0 ⇒ = = 𝐺𝐺𝑖𝑖𝑖𝑖
𝐺𝐺𝑂𝑂𝑂𝑂 𝑆𝑆𝑖𝑖𝑖𝑖 𝐹𝐹

Alessandro Spinelli – Electronics 96032

 Loop gain – calculation 23

𝐺𝐺𝑂𝑂𝑂𝑂 −𝐺𝐺𝑂𝑂𝑂𝑂 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

-
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐹𝐹

• 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = −𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 measures the strength of the feedback

• The result is independent of the breaking point
• A good feedback system has 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 < 0 and |𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 | ≫ 1
Alessandro Spinelli – Electronics 96032
 Loop gain – interpretation 24

𝐺𝐺𝑂𝑂𝑂𝑂 1/𝐹𝐹 𝐺𝐺𝑖𝑖𝑖𝑖

𝐺𝐺 = = =
1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 1 + 1/𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 1 − 1/𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
• 1/|𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 | is the relative error between 𝐺𝐺 and 𝐺𝐺𝑖𝑖𝑖𝑖 :
Example: 𝐺𝐺𝑂𝑂𝑂𝑂 = 105 , 𝐹𝐹 = 10−2 ⇒ 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = −1000, 𝐺𝐺𝑖𝑖𝑖𝑖 = 100,
𝐺𝐺 = 99.9 ⇒ 𝜀𝜀 = 0.001 = 1/|𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 |
• In fact, the error signal 𝜀𝜀 = 𝑆𝑆𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 can be written as
𝜀𝜀 𝐺𝐺𝑖𝑖𝑖𝑖 − 𝐺𝐺 𝜀𝜀 𝐺𝐺 1 1
= 1 − 𝐹𝐹𝐺𝐺 = = = ~
𝑆𝑆𝑖𝑖𝑖𝑖 𝐺𝐺𝑖𝑖𝑖𝑖 𝑆𝑆𝑖𝑖𝑖𝑖 𝐺𝐺𝑂𝑂𝑂𝑂 1 − 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 |𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 |
Alessandro Spinelli – Electronics 96032
 Sensitivity to 𝐺𝐺𝑂𝑂𝑂𝑂 25

𝑑𝑑𝑑𝑑 1 𝐺𝐺 1
= 2
=
𝑑𝑑𝐺𝐺𝑂𝑂𝑂𝑂 1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 𝐺𝐺𝑂𝑂𝑂𝑂 1 − 𝐺𝐺𝑙𝑙𝑙𝑙𝑜𝑜𝑜𝑜
𝑑𝑑𝑑𝑑 𝑑𝑑𝐺𝐺𝑂𝑂𝑂𝑂 1 ≪1
=
𝐺𝐺 𝐺𝐺𝑂𝑂𝑂𝑂 1 − 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

𝐺𝐺𝑂𝑂𝑂𝑂 = 105 , 𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.9

𝐺𝐺𝑂𝑂𝑂𝑂 = 2 × 105 , 𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.95

Alessandro Spinelli – Electronics 96032

 Sensitivity to 𝐹𝐹 26

2
𝑑𝑑𝑑𝑑 𝐺𝐺𝑂𝑂𝑂𝑂 2
=− = −𝐺𝐺
𝑑𝑑𝐹𝐹 1 + 𝐺𝐺𝑂𝑂𝑂𝑂 𝐹𝐹 2
𝑑𝑑𝑑𝑑 𝑑𝑑𝐹𝐹 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ≈ −1
=
𝐺𝐺 𝐹𝐹 1 − 𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

𝐺𝐺𝑂𝑂𝑂𝑂 = 105 , 𝐹𝐹 = 0.01 ⇒ 𝐺𝐺 = 99.9

𝐺𝐺𝑂𝑂𝑂𝑂 = 105 , 𝐹𝐹 = 2 × 0.01 ⇒ 𝐺𝐺 = 49.98

Alessandro Spinelli – Electronics 96032

 Interpretation 27

+ 𝐺𝐺𝑂𝑂𝑂𝑂 • Changes in 𝐺𝐺𝑂𝑂𝑂𝑂 are

𝑆𝑆𝑖𝑖𝑖𝑖 - 𝜀𝜀 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 nulled by the feedback
𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 loop
𝐹𝐹

+ 𝐺𝐺𝑂𝑂𝑂𝑂
𝑆𝑆𝑖𝑖𝑖𝑖 - 𝜀𝜀 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 • Changes in 𝐹𝐹 cannot be
𝐹𝐹𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 compensated
𝐹𝐹
Alessandro Spinelli – Electronics 96032
 Outline 28

• Review: equivalent circuits

• Amplifiers
• Negative feedback
• Operational amplifiers

Alessandro Spinelli – Electronics 96032

 Feedback amplifier design 29

• Forward (open-loop) block 𝐺𝐺𝑂𝑂𝑂𝑂 must have high gain, to ensure

that |𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 | ≫ 1.
All active elements are placed here even if gain is not stable –
their fluctuations are reduced by 1/|𝐺𝐺𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 |
• Feedback block 𝐹𝐹 must be stable, to ensure a stable closed-loop
gain ⇒ usually made with passives

Alessandro Spinelli – Electronics 96032

 Operational amplifiers (OAs) 30

• Integrated voltage amplifiers 𝑉𝑉 +

used as forward gain blocks in + 𝜀𝜀
- 𝐴𝐴
feedback circuits
𝑉𝑉 −

• The ideal OA has

 𝐴𝐴 = ∞ (105 − 106 ) +
𝑉𝑉𝑜𝑜 = 𝐴𝐴(𝑉𝑉 + − 𝑉𝑉 − )
 𝑅𝑅𝑖𝑖 = ∞ (106 − 109 Ω) _
 𝑅𝑅𝑜𝑜 = 0 (≈ 100 Ω)

Alessandro Spinelli – Electronics 96032

 Typical circuit arrangements 31

+ 0 +
𝜀𝜀 𝜀𝜀
_ _
Feedback
loop is on the
negative pin 𝐹𝐹 𝐹𝐹

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

⇒ 𝑉𝑉 + = 𝑉𝑉 −

Alessandro Spinelli – Electronics 96032

 References 32

1. http://tcts.fpms.ac.be/cours/1005-01/equiv.pdf
2. https://www.wpi.edu/News/Transformations/2005Summer/tim
ecapsule.html

Alessandro Spinelli – Electronics 96032