General Control Loop block diagram:
Given schematic below, how to know which is which? Knowing which
is which is critical, as it allows the use of theory to achieve the best
result.
Equations that allowed us to identify corresponding blocks:
voltage at inverting pin of amp is just vin summed up with
voltage drop across Ri


vn = vin + Ri * i

eq 1


since amp input is of high input impedance, current through Ri,
Rf is just


i = (vout  vin) / (Ri + Rf)

eq 2


we see that vout is just opamp gain (Aamp) multiply by its input
voltage


vout = (0  vn) * Aamp


vout =  vn* Aamp

eq 3


eq 2 > eq 1


vn = vin + Ri * (vout  vin) / (Ri + Rf)


let α = Ri/(Ri + Rf)


vn = vin + α * (vout  vin)


vn = (1α)*vin + α * vout

eq 4


eq 4 > eq 3


vout =  {(1α)*vin + α * vout} * Aamp


vout =  {(1α)*vin* Aamp + α * vout* Aamp}


vout * (1 + α*Aamp)
=  {(1α)*vin* Aamp}


vout/vin =  {(1α)* Aamp} /(1 + α*Aamp)

eq 5


from close loop equation of control theory:


Vout/Vin = TF = Aforward/(1 + Aforward*β)

eq 6


comparing eq 6 with eq 5 , we see that


Aforward =  {Rf/(Ri+Rf)} *Aamp

eq 7


Aforward*β = α*Aamp

eq 8


and


β = α * Aamp/Aforward

rearrange eq 8


β = { Ri/(Ri+Rf) } { (Ri+Rf)/Rf }

substitue Aforward from eq 8


β =  Ri/Rf


when loop gain (Aforward*β) is large enough, we see that


1 + Aforward*β ~= Aforward*β


TF = Aforward/ Aforward*β


TF = 1/β = Rf/Ri


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