Operational Amplifier Circuits

Transcription

1 Operational Amplifier Circuits eview: deal Op-amp an open loop configuration p p + i _ + i + Ai o o n n _ An ideal op-amp is characterized with fite open loop ga A The other relevant conditions for an ideal op-amp are: p n 0 i o0 deal op-amp a negative feedback configuration When an op-amp is arranged with a negative feedback the ideal rules are:. p n 0 : put current constrat 2. n p : put voltage constrat These rules are related to the requirement/assumption for large open-loop ga A, and they form the basis for op-amp circuit analysis. The voltage n tracks the voltage p and the control of n is accomplished via the feedback network. Chaniotakis and Cory Sprg 2006 Page

2 Operational Amplifier Circuits as Computational Devices So far we have explored the use of op amps to multiply a signal by a constant. For the 2 vertg amplifier the multiplication constant is the ga and for the non vertg 2 amplifier the multiplication constant is the ga. Op amps may also perform other mathematical operations rangg from addition and subtraction to tegration, differentiation and exponentiation. We will next explore these fundamental operational circuits. Summg Amplifier A basic summg amplifier circuit with three put signals is shown on Figure N F F Current conservation at node N gives Figure. Summg amplifier 2 3 F (.) By relatg the currents, 2 and 3 to their correspondg voltage and resistance by Ohm s law and notg that the voltage at node N is zero (ideal op-amp rule) Equation (.) becomes 2 3 (.2) 2 3 F The term operational amplifier was first used by John agazzi et. al a paper published 947. The relevant historical quotation from the paper is: As an amplifier so connected can perform the mathematical operations of arithmetic and calculus on the voltages applied to its puts, it is hereafter termed an operational amplifier. John agazzi, obert andall and Frederick ussell, Analysis of Problems Dynamics by Electronics Circuits, Proceedgs of E, ol. 35, May 947 Chaniotakis and Cory Sprg 2006 Page 2

3 And so is F F F (.3) The put voltage is a sum of the put voltages with weightg factors given by the values of the resistors. f the put resistors are equal =2=3=, Equation (.3) becomes F 2 3 (.4) The put voltage is thus the sum of the put voltages with a multiplication constant given by F. The value of the multiplication constant may be varied over a wide range and for the special case when F = the put voltage is the sum of the puts 2 3 (.5) The put resistance seen by each source connected to the summg amplifier is the correspondg series resistance connected to the source. Therefore, the sources do not teract with each other. Chaniotakis and Cory Sprg 2006 Page 3

4 Difference Amplifier This fundamental op amp circuit, shown on Figure 2, amplifies the difference between the put signals. The subtractg feature is evident from the circuit configuration which shows that one put signal is applied to the vertg termal and the other to the nonvertg termal N F 2 4 Figure 2. Difference Amplifier Before we proceed with the analysis of the difference amplifier let s thk ab the overall behavior of the circuit. Our goal is to obta the difference of the two put signals 2 -. Our system is lear and so we may apply superposition order to fd the resultg put. We are almost there once we notice that the contribution of the signal 2 to the put is 4 2 (.6) and the contribution of signal is 2 - (.7) And the put voltage is (.8) Chaniotakis and Cory Sprg 2006 Page 4

5 Note that order to have a subtractg circuit which gives =0 for equal puts, the weight of each signal must be the same. Therefore (.9) which holds only if 4 2 (.0) 3 The put voltage is now (.) which is a difference amplifier with a differential ga of 2/ and with zero ga for the common mode signal. t is often practical to select resistors such as 4=2 and 3=. The fundamental problem of this circuit is that the put resistance seen by the two sources is not balanced. The put resistance between the put termals A and B, the differential put resistance, id (see Figure 3) is id A B Figure 3. Differential amplifier Sce + = -, 3 and thus id 2. The desire to have large put resistance for the differential amplifier is the ma drawback for this circuit. This problem is addressed by the strumentation amplifier discussed next. nstrumentation Amplifier Figure 4 shows our modified differential amplifier called the strumentation amplifier (A). Op amps U and U2 act as voltage followers for the signals and 2 which see the fite put resistance of op amps U and U2. Assumg ideal op amps, the voltage Chaniotakis and Cory Sprg 2006 Page 5

6 at the vertg termals of op amps U and U2 are equal to their correspondg put voltages. The resultg current flowg through resistor is 2 (.2) Sce no current flows to the termals of the op amp, the current flowg through resistor 2 is also given by Equation (.2). U U U2 02 Figure 4. nstrumentation Amplifier circuit Sce our system is lear the voltage at the put of op-amp U and op-amp U2 is given by superposition as (.3) (.4) Next we see that op amp U3 is arranged the difference amplifier configuration examed the previous section (see Equation (.)). The put of the difference amplifier is (.5) The differential ga, 4 22, may be varied by changg only one resistor:. 3 Chaniotakis and Cory Sprg 2006 Page 6

7 Current to voltage converters A variety of transducers produce electrical current response to an environmental condition. Photodiodes and photomultipliers are such transducers which respond to electromagnetic radiation at various frequencies rangg from the frared to visible to rays. A current to voltage converter is an op amp circuit which accepts an put current and gives an put voltage that is proportional to the put current. The basic current to voltage converter is shown on Figure 5. This circuit arrangement is also called the transresistance amplifier. N Figure 5. Current to voltage converter represents the current generated by a certa transducer. f we assume that the op amp is ideal, KCL at node N gives 0 0 (.6) The ga of this amplifier is given by. This ga is also called the sensitivity of the converter. Note that if high sensitivity is required for example /µ then the resistance should be M. For higher sensitivities unrealistically large resistances are required. A current to voltage converter with high sensitivity may be constructed by employg the T feedback network topology shown on Figure 6. n this case the relationship between and is 2 2 (.7) Chaniotakis and Cory Sprg 2006 Page 7

8 N 2 Figure 6. Current to voltage converter with T network Chaniotakis and Cory Sprg 2006 Page 8

9 oltage to Current converter A voltage to current (-) converter accepts as an put a voltage and gives an put current of a certa value. n general the relationship between the put voltage and the put current is S (.8) Where S is the sensitivity or ga of the - converter. Figure 7 shows a voltage to current converter usg an op-amp and a transistor. The opamp forces its positive and negative puts to be equal; hence, the voltage at the negative put of the op-amp is equal to. The current through the load resistor, L, the transistor and is consequently equal to /. We put a transistor at the put of the op-amp sce the transistor is a high current ga stage (often a typical op-amp has a fairly small put current limit). cc L Figure 7. oltage to current converter Chaniotakis and Cory Sprg 2006 Page 9

10 Amplifiers with reactive elements We have seen that op amps can be used with negative feedback to make simple lear signal processors. Examples clude amplifiers, buffers, adders, subtractors, and for each of these the DC behavior described the apparent behavior over all frequencies. This of course is a simplification to treat the op amp ideally, as through it does not conta any reactive elements. Providg we keep the operatg conditions of the slew rate limit then this is a reasonable model. Here we wish to extend this picture of op amp operation to clude circuits that are designed to be frequency dependent. This will enable the construction of active filters, tegrators, differentiators and oscillators. The feedback network of an op-amp circuit may conta, besides the resistors considered so far, other passive elements. Capacitors and ductors as well as solid state devices such as diodes, BJTs and MOSFETs may be part of the feedback network. n the general case the resistors that make up the feedback path may be replaced by generalized elements with impedance Z and Z2 as shown on Figure 8 for an vertg amplifier. Z 2 Z Figure 8. nvertg amplifier with general impedance blocks the feedback path. For an ideal op-amp, the transfer function relatg to is given by Z 2 (.9) Z Now, the ga of the amplifier is a function of signal frequency () and so the analysis is to be performed the frequency doma. This frequency dependent feedback results some very powerful and useful buildg blocks. Chaniotakis and Cory Sprg 2006 Page 0

11 The ntegrator: Active Low Pass Filter The fundamental tegrator circuit (Figure 9) is constructed by placg a capacitor C, the feedback loop of an vertg amplifier. C C Figure 9. The tegrator circuit Assumg an ideal op-amp, current conservation at the dicated node gives C d C dt (.20) earrangg Equation (.20) and tegratg from 0 to t, we obta ( ) d d ( t) ( ) d (0) (.2) t C C 0 The put voltage is thus the tegral of the put. The voltage (0) tegration and corresponds to the capacitor voltage at time t = 0. is the constant of The frequency doma analysis is obtaed by expressg the impedance of the feedback components the complex plane. The transfer function may thus be written as ZC jc j Z C (.22) Chaniotakis and Cory Sprg 2006 Page

12 The above expression dicates that there is a 90 o phase difference between the put and the put signals. This 90 o phase shift occurs at all frequencies. The ga of the amplifier given by the modulus is also a function of frequency. For dc signals with C =0 the ga is fite and it falls at a rate of 20dB per decade of frequency change. The fite ga for dc signals represents a practical problem for the circuit configuration of Figure 27. Sce the equivalent circuit of a capacitor for =0 is an open circuit, the feedback path is open. This lack of feedback results a drift (cumulative summg) of the put voltage due to the presence of small dc offset voltages at the put. This problem may be overcome by connectg a resistor, F, parallel with the feedback capacitor C as shown on Figure 0. C C F Figure 0. Active Low Pass filter The feedback path consists of the capacitor C parallel with the resistor F. The equivalent impedance of the feedback path is Z F F jc F jc F C F j C F C F (.23) The transfer function Z F becomes Z Z F F F Z j j FC (.24) Chaniotakis and Cory Sprg 2006 Page 2

13 Where H FC (.25) Figure shows the logarithmic plot of versus frequency. At frequencies much less than H ( << H ) the voltage ga becomes equal to, while at frequencies higher than H ( >> H ) the ga decreases at a rate of 20dB per decade. F

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