#### Transcript Experiment 8 - Rensselaer Polytechnic Institute

Electronic Instrumentation Experiment 8 * Op Amp Circuits Review * Voltage Followers and Adders * Differentiators and Integrators * Analog Computers Op Amp Circuits Review Inverting Amplifier Non-inverting Amplifier Differential Amplifier Op Amp Analysis Inverting Amplifier Rf Vout Vin Rin Non Inverting Amplifier Rf Vout 1 Vin R1 Differential Amplifier Rf Vout V1 V2 Rin Op Amp Analysis Golden Rules of Op Amp Analysis • 1) The current at the inputs is 0 • 2) The voltage at the two inputs is the same These are theoretical assumptions which allow us to analyze the op-amp circuit to determine what it does. These rules essentially allow us to remove the op amp from the circuit. General Analysis Example(1) Assume we have the circuit above, where Zf and Zin represent any combination of resistors, capacitors and inductors. General Analysis Example(2) We remove the op amp from the circuit and write an equation for each input voltage. Note that the current through Zin and Zf is the same, because equation 1 is a series circuit. General Analysis Example(3) Since I=V/Z, we can write the following: Vin VA VA Vout I1] Z in Zf But VA = VB = 0, therefore: Vout Zin Zf Vin Zf Vout Vin Zin General Analysis Conclusion For any op amp circuit where the positive input is grounded, as pictured above, the equation for the behavior is given by: Zf Vout Vin Z in Voltage Followers and Adders What is a voltage follower? Why is it useful? Voltage follower limitations Adders What is a voltage follower? Vout 1 Vin analysis: 1] VA Vout VA VB 2] VB Vin therefore, Vout Vin Why is it useful? In this voltage divider, we get a different output depending upon the load we put on the circuit. Why? We can use a voltage follower to convert this real voltage source into an ideal voltage source. The power now comes from the +/- 15 volts to the op amp and the load will not affect the output. Voltage follower limitations Voltage followers will not work if their voltage or current limits are exceeded. Voltage followers are also called buffers and voltage regulators. Adders V1 V2 Vout R f R1 R2 Rf Vout if R1 R2 then V1 V2 R1 Weighted Adders Unlike differential amplifiers, adders are also useful when R1<>R2. This is called a “Weighted Adder” A weighted adder allows you to combine several different signals with a different gain on each input. You can use weighted adders to build audio mixers and digital-to-analog converters. Analysis of weighted adder I1 If I2 I f I1 I 2 V1 VA I1 R1 V2 VA I2 R2 VA Vout V1 VA V2 VA Rf R1 R2 Vout V1 V2 Rf R1 R2 Vout VA Vout If Rf VA VB 0 V1 V2 R f R1 R2 Differentiators and Integrators Ideal Differentiator Ideal Integrator Miller (non-ideal) Integrator Comparison of Integration and Differentiation Ideal Differentiator analysis: Zf Rf Vout j R f Cin 1 Vin Z in j Cin Analysis in time domain dVCin I Cin Cin VRf I Rf R f dt d (Vin VA ) VA Vout I Cin dt Rf therefore, Vout dVin R f Cin dt I Cin I Rf I VA VB 0 Problem with ideal differentiator Ideal Real Circuits will always have some kind of input resistance, even if it is just the 50 ohms from the function generator. Analysis of real differentiator Z in Rin 1 j Cin Zf Rf j R f Cin Vout 1 Vin Z in j RinCin 1 Rin j Cin Low Frequencies Vout j R f Cin Vin ideal differentiator High Frequencies Rf Vout Vin Rin inverting amplifier Comparison of ideal and non-ideal Both differentiate in sloped region. Both curves are idealized, real output is less well behaved. A real differentiator works at frequencies below c=1/RinCin Ideal Integrator analysis: Zf Vout Vin Z in 1 j C f Rin 1 j j RinC f RinC f Analysis in time domain VRin I Rin Rin I Cf C f dVCf I Cf I Rin I dt Vin VA d (VA Vout ) I Cf VA VB 0 Rin dt dVout 1 1 Vin Vout Vin dt ( VDC ) dt RinC f RinC f Problem with ideal integrator (1) No DC offset. Works ok. Problem with ideal integrator (2) With DC offset. Saturates immediately. What is the integration of a constant? Miller (non-ideal) Integrator If we add a resistor to the feedback path, we get a device that behaves better, but does not integrate at all frequencies. Behavior of Miller integrator Low Frequencies High Frequencies Zf Rf Vout Vin Z in Rin Zf Vout 0 0 Vin Z in Rin inverting amplifier signal disappears The influence of the capacitor dominates at higher frequencies. Therefore, it acts as an integrator at higher frequencies, where it also tends to attenuate (make less) the signal. Analysis of Miller integrator 1 j C f Rf Zf 1 j R f C f 1 Rf j C f Rf Rf Zf j R f C f 1 Rf Vout Vin Z in Rin j Rin R f C f Rin Low Frequencies Rf Vout Vin Rin inverting amplifier High Frequencies Vout 1 Vin j RinC f ideal integrator Comparison of ideal and non-ideal Both integrate in sloped region. Both curves are idealized, real output is less well behaved. A real integrator works at frequencies above c=1/RfCf Problem solved with Miller integrator With DC offset. Still integrates fine. Why use a Miller integrator? Would the ideal integrator work on a signal with no DC offset? Is there such a thing as a perfect signal in real life? • noise will always be present • ideal integrator will integrate the noise Therefore, we use the Miller integrator for real circuits. Miller integrators work as integrators at > c where c=1/RfCf Comparison original signal Differentiaion v(t)=Asin(t) Integration v(t)=Asin(t) mathematically dv(t)/dt = Acos(t) v(t)dt = -(A/cos(t) mathematical phase shift mathematical amplitude change H(j electronic phase shift electronic amplitude change +90 (sine to cosine) -90 (sine to –cosine) 1/ H(j jRC -90 (-j) H(j jRC = j/RC +90 (+j) RC RC The op amp circuit will invert the signal and modify the mathematical amplitude by RC (differentiator) or 1/RC (integrator) Analog Computers (circa. 1970) Analog computers use op-amp circuits to do real-time mathematical operations. Using an Analog Computer Users would hard wire adders, differentiators, etc. using the internal circuits in the computer to perform whatever task they wanted in real time. Analog vs. Digital Computers In the 60’s and 70’s analog and digital computers competed. Analog • Advantage: real time • Disadvantage: hard wired Digital • Advantage: more flexible, could program jobs • Disadvantage: slower Digital wins • they got faster • they became multi-user • they got even more flexible and could do more than just math Now analog computers live in museums with old digital computers: Mind Machine Web Museum: http://userwww.sfsu.edu/%7Ehl/mmm.html Analog Computer Museum: http://dcoward.best.vwh.net/analog/index.html