Designing for Predictable Amplifier Gain Gain is hard to control Varies with operating point Non-constant gain causes distortion Gain varies from one transistor to the next Sensitive to temperature 1EEE 3308

Amplifier Gain Varies a Lot 2EEE 3308 Gain varies with operating point Non-constant gain causes distortion

Input, Output, Source & Load Impedance Variations Affect Gain 3EEE 3308 Z in Z out A vo v i vivi vsvs ZsZs ZLZL v out

Input, Output, Source & Load Impedance Variations Affect Gain 4EEE 3308 Z in Z out A vo v i vivi vsvs ZsZs ZLZL v out Impedances vary with frequency, too.

So How Can We Possibly Design Amps That Just Work? 5EEE 3308 How to get gain that is stable, predictable, temperature-independent? How to get stable biasing? How to get desired input and output impedances?

So How Can We Possibly Design Amps That Just Work? 6EEE 3308 How to get gain that is stable, predictable, temperature-independent? How to get stable biasing? How to get desired input and output impedances? FEEDBACK!

Classic Feedback Example: The Non-Inverting Feedback Amplifier 7EEE 3308 R2R2 R1R1 vsvs vovo

Non-Inverting Feedback Amplifier 8EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 9EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 10EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 11EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 12EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 13EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 14EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 15EEE 3308 A is the “open-loop gain” is the “feedback factor” A CL is the “closed-loop gain” R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 16EEE 3308 T = Aβ is the “loop gain” R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 17EEE 3308 R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Non-Inverting Feedback Amplifier 18EEE 3308 If T is big enough, the closed-loop gain is independent of the amplifier gain A. R2R2 R1R1 Av i vivi vsvs vfvf R2R2 R1R1 vsvs vovo vovo

Feedback Analysis Using Loop Gain and A ∞ 19EEE 3308 The A-β approach works OK for the non-inverting amp example, but it doesn’t generalize well: - Many circuits don’t split cleanly into β and A parts - Results depend on arbitrary assumptions about amp - Some of the results are significantly wrong - Not all feedback circuits are amplifiers

Feedback Analysis Using Loop Gain and A ∞ 20EEE 3308 The A-β approach works OK for the non-inverting amp example, but it doesn’t generalize well: - Many circuits don’t split cleanly into β and A parts - Results depend on arbitrary assumptions about amp - Some of the results are significantly wrong - Not all feedback circuits are amplifiers Loop gain (T) is the key parameter for feedback analysis A ∞ generalizes the ideal op amp Combining separate analyses is design-oriented

Finding Loop Gain 21EEE 3308

Finding Loop Gain 22EEE 3308 Represent the amplifier by its linearized small-signal equivalent circuit.

Finding Loop Gain 23EEE 3308 Turn off independent voltage and current sources, replacing them by their internal resistances (short for voltage sources, open for current sources).

Finding Loop Gain 24EEE 3308 R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 25EEE 3308 Choose a branch through which the feedback signal flows... R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 26EEE 3308 Choose a branch through which the feedback signal flows... R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 27EEE 3308 Choose a branch through which the feedback signal flows... R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 28EEE 3308 Break the branch. R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 29EEE 3308 Call the input side the x port, and the output side the y port. x y R2R2 R1R1 RiRi RoRo A vo v i vivi Feedback signal flow

Finding Loop Gain 30EEE 3308 Find the resistance, call it R ix, looking into the x port with port y shorted. y vxvx y R2R2 R1R1 RiRi RoRo A vo v i vivi R ix

Finding Loop Gain 31EEE 3308 Find the resistance, call it R ix, looking into the x port with port y shorted. y vxvx y R2R2 R1R1 RiRi RoRo A vo v i vivi (R 1 ||R i )

Finding Loop Gain 32EEE 3308 R1R1 RiRi RoRo A vo v i vivi vxvx R2R2 Find the resistance, call it R ix, looking into the x port with port y shorted.

Finding Loop Gain 33EEE 3308 vyvy R1R1 RiRi RoRo A vo v i vivi vxvx R2R2 Place a copy of R ix across the y port.

Finding Loop Gain 34EEE 3308 Find the loop gain T = -v y /v x using standard amplifier analysis. vyvy R1R1 RiRi RoRo A vo v i vivi vxvx R2R2

Finding Loop Gain 35EEE 3308 In this case, vyvy vxvx R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 36EEE 3308 You get the same thing for T if you break the loop in other places. R2R2 R1R1 RiRi RoRo A vo v i vivi

Finding Loop Gain 37EEE 3308 R2R2 R1R1 RiRi RoRo A vo v i vivi You get the same thing for T if you break the loop in other places.

Finding Loop Gain 38EEE 3308 R2R2 R1R1 RiRi RoRo A vo v i vivi You get the same thing for T if you break the loop in other places.

Finding Loop Gain 39EEE 3308 R2R2 R1R1 RiRi RoRo A vo v i vivi T is a key property of any feedback circuit. T is independent of how you find it. It’s independent of where any inputs may be applied or any outputs are defined.

40EEE 3308 Finding Loop Gain: Summary

41EEE 3308 Finding Loop Gain: Summary Represent the amplifier by its linearized small-signal equivalent circuit.

42EEE 3308 Turn off independent voltage and current sources, replacing them by their internal resistances (short for voltage sources, open for current sources). Finding Loop Gain: Summary

43EEE 3308 Choose a branch through which the feedback signal flows. R2R2 R1R1 RiRi RoRo A vo v i vivi Finding Loop Gain: Summary

44EEE 3308 Break the branch. R2R2 R1R1 RiRi RoRo A vo v i vivi Finding Loop Gain: Summary x y

45EEE 3308 Find the resistance, call it R ix, looking into the x port with port y shorted. y vxvx y R2R2 R1R1 RiRi RoRo A vo v i vivi R ix Finding Loop Gain: Summary

46EEE 3308 vyvy R1R1 RiRi RoRo A vo v i vivi vxvx R2R2 Place a copy of R ix across the y port. Finding Loop Gain: Summary

47EEE 3308 Find the loop gain T = -v y /v x. vyvy R1R1 RiRi RoRo A vo v i vivi vxvx R2R2 Finding Loop Gain: Summary

48EEE 3308 Finding A ∞ R2R2 R1R1 Av i vsvs vfvf vovo RiRi vivi i i+ i i- A ∞ is the source-to-output gain when the controlled source gain A goes to infinity. R2R2 R1R1 vsvs vovo i i- vivi

49EEE 3308 Finding A ∞ R2R2 R1R1 Av i vsvs vfvf vovo RiRi vivi i i+ i i- A ∞ is the source-to-output gain when the controlled source gain A goes to infinity. R2R2 R1R1 vsvs vovo i i- vivi

50EEE 3308 Finding A ∞ R2R2 R1R1 Av i vsvs vfvf vovo RiRi vivi i i+ i i- A ∞ is the source-to-output gain when the controlled source gain A goes to infinity. R2R2 R1R1 vsvs vovo i i- vivi

51EEE 3308 Finding A ∞ R2R2 R1R1 Av i vsvs vfvf vovo RiRi vivi i i+ i i- A ∞ is the source-to-output gain when the controlled source gain A goes to infinity. These are equivalent to the ideal op assumptions: R2R2 R1R1 vsvs vovo i i- vivi

EEE Finding A ∞ : The Ideal Op Amp Assumptions R2R2 R1R1 Av i vsvs vfvf R2R2 R1R1 vsvs vovo vovo RiRi vivi i i+ i i- Ideal Op Amp Assumptions: i i+ i i- vivi

53EEE 3308 Finding A ∞ The A ∞ approach can be applied to any feedback circuit, even when there is no op amp as such. In general, A ∞ is the overall source-to-output gain when the signal controlling the controlled source is forced to be zero because of infinite controlled-source gain. As with the ideal op amp, assuming infinite gain leads to simpler circuit analysis.

54EEE 3308 Putting It All Together Once you know T and A ∞ you can find the overall gain using The loop gain is a measure of how close the circuit is to ideal.