1. Negative vs. positive feedback What is the difference? Consider what happens when there is a perturbation Positive feedback drives op amp into saturation: v OUT  ± V S

2. Static Analysis of Positive Feedback Circuit

3. Representing dynamics of op amp: time dependence Let us consider this circuit and analyze its dynamics to build insight

4. Time behavior of v O

5. Consider a Small Disturbance to v 0 (Noise) v - > v + v + > v - v + = v -

6. One use of instability: Build Comparator

7. Now, Use Positive Feedback

8. Without Hysteresis

9. Oscillator: Can Create a Clock Assume v 0 =V S at t=0, v C =0

10. Clock in Digital Systems We build an oscillator using an op amp Why do we use a clock in a digital system? a)1,1,0? b)When is the signal valid? Common timebase – when to “look” at a signal (e.g. whenever the clock is high) Discretization of time: one bit of information associated with an interval of time (cycle)

11. Physical Limitations of Op Amp Voltage limit: V S - < v out <V S +

12. Frequency Response Limits The product of gain and band width in any given op- amp is constant, i.e. A  = K Increasing the gain-bandwidth product by means of amplifiers in cascade. Offset voltage limit Slew rate limit