1. Summing Amplifier -+-+ RFRF R4R4 + IFIF I4I4 VoVo R3R3 + I3I3 V3V3 V4V4 R2R2 + I2I2 V2V2 R1R1 + I1I1 V1V1 RLRL V id

2. Summing Amplifier I F = I 1 + I 2 + I 3 + I 4 I 1 = V 1 /R 1, I 2 = V 2 /R 2, I 3 = V 3 /R 3, I 4 = V 4 /R 4 I F = -V 0 /R F -V 0 /R F = V 1 /R 1 + V 2 /R 2 + V 3 /R 3 + V 4 /R 4 V 0 = -R F (V 1 /R 1 + V 2 /R 2 + V 3 /R 3 + V 4 /R 4 )

3. Practical Summing Amplifier R B included for bias current compensation R B = R F II R 1 II R 2 II ….. II R n Bandwidth of the summing amplifier is decreased in proportion to the effective resistance between inverting terminal and ground -+-+ RFRF RBRB VoVo R2R2 + I2I2 V2V2 R1R1 + I1I1 V1V1 RLRL

4. Practical Summing Amplifier Actual bandwidth of summer is dependent on noise gain of circuit Noise gain A N = 1 + (R F /R in-eq ) R in-eq = R 1 II R 2 II….. II R n Actual bandwidth BW = GBW / A N

5. Averagers Special case of summing amplifier V avg = (V 1 + V 2 + V 3 + …+ V n )/n To design an averager, the input resistor value is chosen such that R = n R F V o = - V avg

6. Additional Summing Amplifier Circuit By superposition theorem V o = (V o1 +V o2 +V 03 ) – V o1 output due to source V 1 – V o2 output due to source V 2 – V 03 output due to source V 3 -+-+ RFRF RBRB VoVo R2R2 I2I2 V2V2 R1R1 I1I1 V1V1 RLRL V3V3 RARA IFIF V RB

7. Additional Summing Amplifier Circuit V o1 and V o2 are determined by assuming source V o3 to be zero – R A and R B will be in parallel Circuit reduces to a simple summing amplifier V o1 = -V 1 (R F /R 1 ) V o2 = -V 2 (R F /R 2 ) V o1 + V o2 = - (V 1 R F /R 1 + V 2 R F /R 2 ) V o3 is determined by assuming sources V o1 and V o2 to be zero – R 1 and R 2 will be in parallel (R eq )

8. Additional Summing Amplifier Circuit V RB = V 3 [R B /(R A +R B )] V o3 = V 3 [R B /(R A +R B )] [(R eq +R F )/R eq ] V o = V o1 +V o2 +V 03 V o = - R F (V 1 /R 1 + V 2 /R 2 ) + V 3 [R B /(R A +R B )] [(R eq +R F )/R eq ]

9. Differential Amplifier VoVo -+-+ RFRF R1R1 V1V1 V2V2 R’ 1 R’ F RLRL Special case of amplifier similar to additional summing amplifier

10. Differential Amplifier Op-amp should amplify V 1 and V 2 equally but opposite amounts (gain for inverting and noninverting must be equal) Above is possible if R 1 = R’ 1 and R F = R’ F V 0 = V 2 {R’ F /(R’ 1 + R’ F )} {(R 1 +R F )/R 1 } – (V 1 R F /R 1 ) =V 2 (R F /R 1 ) – V 1 (R F /R 1 ) V 0 = (V 2 – V 1 ) (R F /R 1 )

11. Differential Amplifier In ideal case, V o = 0 when V 1 = V 2, producing common-mode input voltage (V iCM ) Real differential amplifier will produce an O/P in response to the common-mode input Response of amplifier to common-mode voltage V oCM = A CM V iCM Rejection of common-mode signals is quantified by common-mode rejection ratio (dB) CMRR = 20 log I A d /A CM l CMRR should be as large as possible Limiting factors is matching between R 1 -R’ 1 and R F - R’ F (laser-trimmed precision resistors used)

12. Instrumentation Amplifier VoVo -+-+ RFRF R1R1 V1V1 V2V2 R’ 1 R’ F RLRL -+-+ -+-+ V1V1 V2V2

13. Instrumentation Amplifier Differential amplifier optimized in terms of dc parameters CMRR, offset current, offset voltage, bias current and temperature drift specifications are tightly controlled Buffered inputs are used to minimize loading of input source(s) Application: processing of the output of a bridge circuit

14. Instrumentation Amplifier VoVo -+-+ RFRF R5R5 V1V1 V2V2 R’ 5 R’ F RLRL -+-+ -+-+ V1V1 V2V2 R 4 =R R 3 =R R 2 =R R 1 =R + ΔR V ref T° Instrumentation amplifier used to sense temperature changes