Wednesday, 23 May 2012

Feedback loop element identification: Which is which for Inverting Opamp circuit?

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 = (1-α)*vin + α * vout eq 4 eq 4 -> eq 3 vout = - {(1-α)*vin + α * vout} * Aamp vout = - {(1-α)*vin* Aamp + α * vout* Aamp} vout * (1 + α*Aamp) = - {(1-α)*vin* Aamp} vout/vin = - {(1-α)* 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 re-arrange 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