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Operational Amplifiers Luke Gibbons CSUS Fall 2006 ME 114

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Intro An Operational Amplifier, or Op-Amp, is a component often used in circuits because of its wide range of abilities An Operational Amplifier, or Op-Amp, is a component often used in circuits because of its wide range of abilities Op-Amps can be used to amplify, invert, add, subtract, integrate, differentiate, filter and compare different input signals Op-Amps can be used to amplify, invert, add, subtract, integrate, differentiate, filter and compare different input signals Op-Amps were created to perform specific mathematic functions, such as a function to invert, integrate, and amplify different input signals Op-Amps were created to perform specific mathematic functions, such as a function to invert, integrate, and amplify different input signals

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Op-Amps

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Ideal Vs Practical We will concentrate our efforts to understanding ideal op-amps then analyze both ideal and practical op-amps using Simulink and Camp-G We will concentrate our efforts to understanding ideal op-amps then analyze both ideal and practical op-amps using Simulink and Camp-G Characteristics of ideal and practical op-amps are very high voltage gain and input impedance, very low output impedance, and wide bandwidth Characteristics of ideal and practical op-amps are very high voltage gain and input impedance, very low output impedance, and wide bandwidth

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Operational Amplifier Diagrams The Theop-ampblockdiagram basicop-amp

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Types of Op-Amps The type of op-amps we will analyze include: The type of op-amps we will analyze include: Integrating Integrating Differentiating Differentiating Inverting Inverting Non-Inverting Non-Inverting Summing Summing Subtracting Subtracting Filtering Filtering Comparing Comparing

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Integrating and Differentiating Op- Amps

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Bond Graphs of Integrating and Differentiating Op-Amps SF R SE C Integrator:

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Bond Graphs of Integrating and Differentiating Op-Amps SF R SE C Differentiator:

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Integrating and Differentiating Op- Amp Transfer Functions Integrating op-amp: Integrating op-amp: TF(s) = Vo(s)/Vi(s) TF(s) = -Z2(s)/Z1(s) TF = -(1/RC)/s Differentiating op-amp Differentiating op-amp TF(s) = Vo(s)/Vi(s) TF(s) = -Z2(s)/Z1(s) TF = -RCs

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*R)] [B] = [-1 1/R] [C] = [1/C] [D] = [0 0] TF (e8/f1) = e8/f1 = [ -R/[(C*R)*s + 1] 1/[(C*R)*s + 1] ] SF R SE C Integrator:

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*R)] [B] = [1 1/R] [C] = [-1/C] [D] = [0 1] TF (e8/f1) = e8/f1 = [ - R/[(C*R)*s + 1] (C*R)*s /[(C*R)*s + 1] ] SF R SE C Differentiator:

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Op-Amp Transfer Functions Notice how the transfer functions can either relate the input current or voltage to the output current or voltage Notice how the transfer functions can either relate the input current or voltage to the output current or voltage The common op-amp transfer function relates the input voltage to the output voltage, as shown a few slides back The common op-amp transfer function relates the input voltage to the output voltage, as shown a few slides back Because of the “geometry” of the bond graphs, we will relate the input current to the output voltage, as shown on the two previous slides Because of the “geometry” of the bond graphs, we will relate the input current to the output voltage, as shown on the two previous slides

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Integrating and Differentiating Op-Amps Say we have an input voltage of V input = R input *sin (t) Say we have an input voltage of V input = R input *sin (t) where R input is some input resistance and the current is represented as i = sin (t) Say C = 2, R = 2, R input = 1, V input = 1*sin (t) Say C = 2, R = 2, R input = 1, V input = 1*sin (t) The output voltages of the integrating and differentiating op-amps in comparison with the input voltage is shown on the next slide The output voltages of the integrating and differentiating op-amps in comparison with the input voltage is shown on the next slide

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Simulink Model of Integrating Op- Amps

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Simulink Model of Differentiating Op- Amps

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Simulink Comparison of Integrating and Differentiating Op-Amps Integrator: Differentiator: Sine Wave:

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Integrating and Differentiating Op-Amps Notice the expected behavior of the integrating and differentiating op-amps Notice the expected behavior of the integrating and differentiating op-amps Notice how we used the gain function to reduce the output voltage for the reference case Notice how we used the gain function to reduce the output voltage for the reference case Notice how SIMULINK forces the initial current/voltage to be zero Notice how SIMULINK forces the initial current/voltage to be zero

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Inverting & Non-Inverting Op-Amps

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Bond Graph of Inverting & Non- Inverting Op-Amps SF Ri SE Rf Inverting: C Need to add compliance element to make the system operate properly 109

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Bond Graph of Inverting & Non- Inverting Op-Amps Need to add compliance element to make the system operate properly SF Rf SE Ri Non-Inverting: C 8 7 9

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Inverting & Non-Inverting Op-Amp Transfer Functions Inverting op-amp: Inverting op-amp: TF(s) = Vo(s)/Vi(s) TF(s) = -Z2(s)/Z1(s) TF = -Rf/Ri Non-Inverting op-amp: TF(s) = Vo(s)/Vi(s) TF(s) = -Z2(s)/Z1(s) TF = (Ri+Rf)/Ri

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Non-Inverting Op-Amps Notice how the inverting and non-inverting op-amps have different inputs into each terminal Notice how the inverting and non-inverting op-amps have different inputs into each terminal In order to represent a non-inverting op-amp (and all op-amps which have the input voltage going into the positive terminal) we have to use either the a bond graph type similar to the bond graph from the inverting op- amp or the previous non-inverting op-amp bond graph, that does not fully work properly, but is suitable for our purpose, and manually invert the output signal In order to represent a non-inverting op-amp (and all op-amps which have the input voltage going into the positive terminal) we have to use either the a bond graph type similar to the bond graph from the inverting op- amp or the previous non-inverting op-amp bond graph, that does not fully work properly, but is suitable for our purpose, and manually invert the output signal

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*Ri) -1/(C*Rf)] [B] = [-1 1/Ri] [C] = [1/C] [D] = [0 0] TF (e8/f1) = e8/f1 = [ -Ri*Rf/[(C*Ri*Rf)*s+Ri+Rf] Rf/[(C*Ri*Rf)*s+Ri+Rf] ] SF Ri SE Rf Inverting: C 109

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*Ri) -1/(C*Rf)] [B] = [-1 0] [C] = [1/C] [D] = [0 0] TF (e8/f1) = e8/f1 = [ -Ri*Rf/[(C*Ri*Rf)*s+Ri+Rf] 0 ] SF Rf SE Ri Non-Inverting: C 8 7 9

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Inverting and Non-Inverting Op- Amps Say we have an input voltage of V input = R input *sin (t) Say we have an input voltage of V input = R input *sin (t) where R input is some input resistance and the current is represented as i = sin (t) Say C = 1, Ri = 2, Rf = 2, R input = 1, V input = 1*sin (t) Say C = 1, Ri = 2, Rf = 2, R input = 1, V input = 1*sin (t) The output voltages of the integrating and differentiating op-amps in comparison with the input voltage is shown on the next slide The output voltages of the integrating and differentiating op-amps in comparison with the input voltage is shown on the next slide

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Simulink Model of Inverting Op- Amps

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Simulink Model of Non-Inverting Op - Amps

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Simulink Comparison of Inverting and Non-Inverting Op-Amps Inverting: Non-Inverting: Sine Wave:

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Inverting and Non-Inverting Op- Amps Notice the expected behavior of the inverting and non-inverting op-amps Notice the expected behavior of the inverting and non-inverting op-amps Notice how we used the gain function to reduce the output voltage for the reference case Notice how we used the gain function to reduce the output voltage for the reference case Notice the lag involved with the inverting and non-inverting op-amps when compared with the reference case Notice the lag involved with the inverting and non-inverting op-amps when compared with the reference case

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Investigation: Input Voltage We will investigate a series of issues encountered while using CAMPG and MATLAB We will investigate a series of issues encountered while using CAMPG and MATLAB First, we will look into determining the transfer function from the output voltage to the input voltage First, we will look into determining the transfer function from the output voltage to the input voltage For most situations, it is more valuable to know the relationship between the input and output voltages than the input current to output voltage relationship determined on the previous slides For most situations, it is more valuable to know the relationship between the input and output voltages than the input current to output voltage relationship determined on the previous slides

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*R)] [B] = [-1/R 1/R] [C] = [1/C] [D] = [0 0] TF (e8/e1) = e8/e1 = [ -1/[(C*R)*s + 1] 1/[(C*R)*s + 1] ] SE R C Integrator:

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*R)] [B] = [1/R -1/R] [C] = [1/C] [D] = [-1 1] TF (e8/f1) = e8/e1 = [ -CR*s/[CR*s + 1] CR*s /[CR*s + 1] ] SE R C Differentiator:

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Transfer Function and Matrices from Camp-G [u] = [SF SE] [A] = [-1/(C*Ri) -1/(C*Rf)] [B] = [-1/Ri 1/Ri] [C] = [1/C] [D] = [0 0] TF (e8/e1) = e8/e1 = [ -Rf/[(C*Ri*Rf)*s+Ri+Rf] Rf/[(C*Ri*Rf)*s+Ri+Rf] ] SE Ri SE Rf Inverting: C 109

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Investigation: Derivative Causality Now we will look the CAMPG’s derivative causality error for the bond graphs on the following slides Now we will look the CAMPG’s derivative causality error for the bond graphs on the following slides CAMPG cannot interface to another program when there are any derivative causality errors CAMPG cannot interface to another program when there are any derivative causality errors CAMPG errors show up in red and can be diagnosed by using the peek and analyze functions CAMPG errors show up in red and can be diagnosed by using the peek and analyze functions

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Derivative Causality The derivative causality error is shown in red because CAMPG needs the integral form of the compliance element SF Ci SE R Cf 910 0

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Derivative Causality The derivative causality error is shown in red because CAMPG and MATLAB need the integral form of the compliance element SFSE Cf Ci 34 1

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Derivative Causality The derivative causality error is shown in red because CAMPG and MATLAB need the integral form of the compliance element SFSE Ci 34 1 R Cf 1112

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Investigation: Output Effort Location Now we will look into which terminal should be used as the output terminal Now we will look into which terminal should be used as the output terminal We will look at the relationship between the input voltage across the 1 st terminal and the output voltage across 2 different terminals We will look at the relationship between the input voltage across the 1 st terminal and the output voltage across 2 different terminals

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Output Effort Location As before, look at the effort output across the 8 th terminal, e8 SE R C Differentiator: [u] = [SF SE] [A] = [-1/(C*R)] [B] = [1/R -1/R] [C] = [1/C] [D] = [-1 1] TF (e8/f1) = e8/e1 = [ -CR*s/[CR*s + 1] CR*s /[CR*s + 1] ]

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Output Effort Location Now, look at the effort output across the 7 th Terminal, e7 SE R C Differentiator: [u] = [SF SE] [A] = [-1/(C*R)] [B] = [1/R -1/R] [C] = [0] [D] = [0 1] TF (e8/f1) = e8/e1 = ERROR USING sym.maple at offset 28, ‘)’ expected’ ERROR IN sym.collect at 36 r=reshape(maple(‘map’,’collect’,S(:(X),size(s)); ERROR IN campgsym at 135 H = collect H

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Investigation: Non-Inverting Op- Amps Now we will try to produce a non-inverting op amp in CAMPG Now we will try to produce a non-inverting op amp in CAMPG SE 1 represents the voltage going into the positive terminal of the op-amp SE 1 represents the voltage going into the positive terminal of the op-amp SE 2 represents the voltage going into the negative terminal of the op-amp SE 2 represents the voltage going into the negative terminal of the op-amp

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Non-Integrating Amplifier SE 1 Rb SE Ra Non-Integrator: Rc C SE The following configuration is the only one found that can be created without any derivative causality errors However, this bond graph does not get past the DOS Interface

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Filtering Op-Amps Single pole active low pass filter Single pole active low pass filter Single pole active high pass filter Single pole active high pass filter

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Summing & Subtracting Op-Amps

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Comparing Op-Amps

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