Amplifiers and Feedback Theory Phone: (02 2399) 4001 alessandro.spinelli@polimi.it home.deib.polimi.it/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 Lawry Norton (1898-1983) 1853 1883 1926 1926 From [1]
Equivalent circuits 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
Equivalent circuits Sources and linear elements II VV Same ZZ eeee ZZ eeee II VV ZZ eeee II VV VV eeee II eeee Equivalence is only from the viewpoint of the external load. Power dissipation, for example, is not equal
Element calculations VV eeee is the open-circuit voltage at the terminals II eeee is the short-circuit current through the terminals ZZ eeee = VV eeee /II eeee, or equivalently ZZ eeee is the impedance between the terminals when voltage sources are replaced by short-circuits current sources are replaced by open circuits
Outline Review: equivalent circuits Amplifiers Negative feedback Operational amplifiers
Amplifiers 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.
Voltage/current amplifiers RR II ii II oo oo RR oo VV ii RR ii VV oo RR ii AA VV VV ii AA II II ii 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
Source and load resistors (VA) RR ss RR ii VV ss AA VV VV ii RR oo RR LL VV oo VV ii VV ii = VV SS RR ii RR ii + RR SS VV oo = AA VV VV ii RR LL RR oo + RR LL
Voltage gain VV oo VV SS = AA VV RR LL RR oo + RR LL RR ii RR ii + RR SS Total gain is less than AA VV Gain is dependent on RR SS and RR LL To avoid these drawbacks, a voltage amplifier should have RR ii = (very high input impedance) RR oo = 0 (very low output impedance)
Source and load resistors (CA) RR ss II ii RR oo II oo II ss RR ii RRLL AA II II ii II ii = II SS RR SS RR ii + RR SS II oo = AA II II ii RR oo RR oo + RR LL
Current gain II oo II SS = AA II RR oo RR oo + RR LL RR SS RR ii + RR SS Total gain is less than AA II Gain is dependent on RR SS and RR LL To avoid these drawbacks a current amplifier should have RR ii = 0 (very low input impedance) RR oo = (very high output impedance)
Summary Type RR ii RR oo Voltage amplifier 0 Current amplifier 0 Transconductance ampl. Transresistance ampl. 0 0
Outline Review: equivalent circuits Amplifiers Negative feedback Operational amplifiers
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 intermodulation distorsion in multi-channel systems
Negative-feedback concept Harold S. Black (1898-1983) From [2]
The theory SS iiii + - εε GG OOOO SS oooooo FFSS oooooo FF εε = SS iiii FFSS oooooo SS oooooo = GG OOOO εε SS oooooo SS iiii = GG = GG OOOO 1 + GG OOOO FF
Closed-loop gain GG OOOO FF 1 GG = GG OOOO 1 + GG OOOO FF ~GG OOOO Open-loop gain, no feedback GG OOOO FF 1 GG = GG OOOO 1 + GG OOOO FF ~ 1 FF = GG iiii Ideal gain, independent of GG OOOO
Loop gain calculation SS tttttttt - GG OOOO FFSS tttttttt GG OOOO FF GG OOOO SS tttttttt GG llllllll = GG OOOO FF measures the strength of the feedback A good feedback system has GG llllllll < 0 and GG llllllll 1
GG = Loop gain interpretation GG OOOO 1 + GG OOOO FF = 1/FF 1 + 1/GG OOOO FF = GG iiii 1 1/GG llllllll 1/ GG llllllll is the error between GG and GG iiii : Ex: GG OOOO = 10 5, FF = 10 2 GG llllllll = 1000, GG iiii = 100, GG = 99.9 εε = 0.001 = 1/ GG llllllll In fact, the error signal εε = SS iiii FFSS oooooo is εε = 1 FFGG = GG iiii GG 1 = ~ 1 SS iiii GG iiii 1 GG llllllll GG llllllll
Sensitivity to GG OOOO dddd ddgg OOOO = 1 1 + GG OOOO FF 2 = GG 1 GG OOOO 1 GG lllloooo dddd GG = ddgg OOOO 1 <<1 GG OOOO 1 GG llllllll GG OOOO = 10 5, FF = 0.01 GG = 99.9 GG OOOO = 2 10 5, FF = 0.01 GG = 99.95
Sensitivity to FF dddd ddff = dddd GG = ddff FF 2 GG OOOO 1 + GG OOOO FF 2 = GG2 GG llllllll 1 GG llllllll -1 GG OOOO = 10 5, FF = 0.01 GG = 99.9 GG OOOO = 10 5, FF = 2 0.01 GG = 49.98
Interpretation SS iiii + - εε FFSSoooooo GG OOOO FF SS oooooo Changes in GG OOOO are nulled by the feedback loop SS iiii + - εε FFSSoooooo GG OOOO FF SS oooooo Changes in FF cannot be compensated
Outline Review: equivalent circuits Amplifiers Negative feedback Operational amplifiers
Feedback amplifier design Forward (open-loop) block GG OOOO must have high gain, to ensure that GG llllllll 1. All active elements are placed here even if gain is not stable their fluctuations are reduced by 1/GG llllllll Feedback block FF must be stable, to ensure a stable closed-loop gain. Usually made with passives
Operational amplifiers (OAs) Integrated voltage amplifiers used as forward gain blocks in feedback circuits The ideal OA has AA = (10 5 10 6 ) RR ii = (10 6 10 9 Ω) RR oo = 0 ( 100 Ω) VV + VV + - εε AA + _ VV oo = AA(VV + VV )
Typical circuit arrangements + _ 0 + _ FF FF In ideal feedback loops εε = 0 ideal OAs keep VV + VV = 0 VV + = VV
References 1. http://tcts.fpms.ac.be/cours/1005-01/equiv.pdf 2. https://www.wpi.edu/news/transformations /2005Summer/timecapsule.html