2. 0 Chap 2. Operational amplifiers (op-amps) Circuit symbol of an op-amp Widely used Often requires 2 power supplies + V Responds to difference between two signals

3. 1 2.2 Ideal op-amp Characteristics of an ideal op-amp R in = infinity R out = 0 A vo = infinity (A vo is the open-loop gain, sometimes A or A v of the op-amp) Bandwidth = infinity (amplifies all frequencies equally)

4. 2 Model of an ideal op-amp V+V+ V-V- V out = A(V + - V - ) + - + - Usually used with feedback Open-loop configuration not used much I-I- I+I+

5. 3 Summary of op-amp behavior V out = A(V + - V - ) V out /A = V + - V - Let A infinity then, V + -V - 0

6. 4 Summary of op-amp behavior V + = V - I + = I - = 0 Seems strange, but the input terminals to an op-amp act as a short and open at the same time

7. 5 To analyze an op-amp circuit Write node equations at + and - terminals (I + = I - = 0) Set V + = V - Solve for V out

8. 6 2.3 Inverting configuration Very popular circuit

9. 7 Analysis of inverting configuration I 1 = (V i - V - )/R 1 I 2 = (V - - V o )/R 2 set I 1 = I 2, (V i - V - )/R 1 = (V - - V o )/R 2 but V - = V + = 0 V i / R 1 = -V o / R 2 Solve for Vo V o / V i = -R 2 / R 1 Gain of circuit determined by external components I1I1 I2I2

10. 8 2.4 Applications of the inverting configuration V1 V2 V3 R1 R2 R3 Rf Current in R 1, R 2, and R 3 add to current in R f (V 1 - V - )/R 1 + (V 2 - V - )/R 2 + (V 3 - V - )/R 3 = (V - - Vo)/R f Set V - = V + = 0, V 1 /R 1 + V 2 /R 2 + V 3 /R 3 = - Vo/R f solve for V o, V o = -R f (V 1 /R 1 + V 2 /R 2 + V 3 /R 3 ) This circuit is called a weighted summer

11. 9 2.5 Noninverting configuration (0 - V - )/R 1 = (V - - V o )/R2 But, V i = V + = V-, ( - V i )/R 1 = (V i - V o )/R 2 Solve for Vo, V o = V i (1+R 2 /R 1 ) Vi I I

12. 10 Input resistance of noninverting amplifier Vout = A(V + -V - ) V- V+ R in = V in / I, from definition R in = V in / 0 R in = infinity I

13. 11 Input resistance of inverting amplifier V out = A(V + - V - ) V-V- V+V+ R in = V in / I, from definition I = (V in - V out )/R I = [V in - A (V + - V - )] / R But V + = 0 I = [V in - A( -V in )] / R R in = V in R / [V in (1+A)] As A approaches infinity, R in = 0 I

14. 12 Summary of op-amp behavior ViInverting configurationNoninverting configuration V o /V i = 1+R 2 /R 1 R in = infinity V o /V i = - R 2 /R 1 R in = R 1 Rin = 0 at this point Vi