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Lecture 4: Operational Amplifier Basics Having put in place the broad concepts of signal amplification and the circuit modelling of amplifiers, we now introduce what is probably the most important building block for analogue systems, the operational amplifier, or op amp as it is more popularly known. Despite its name, the op amp cannot really be used on its own for amplification purposes but in conjunction with additional components. In this lecture, we address the basic properties of the op amp and how it can be used with additional system components to function as a useful amplifier. Learning Outcomes: On completing this lecture, you will be able to: 4.1 List key properties of the near-ideal operational amplifier; Derive expressions for the closed loop gain of non-inverting and inverting op amp configurations; Demonstrate how the performance of these systems approaches the ideal; Explain the terms open loop gain, closed loop gain, and loop gain. The Near-Ideal Operational Amplifier The operational amplifier is a circuit of approximately 30 semiconductor transistors and some additional passive components (resistors) configured as a very high gain amplifier. The actual circuit diagram of one of the more popular commercially available devices, the so-called 741 op amp, is shown below: 4-1 The complete circuit is fabricated on a single crystal of silicon and thus constitutes an integrated circuit. We simply note the relatively large number of active semiconductor devices — transistors — and the relatively small number of resistors and one capacitor. Such a distribution of components would be typical for an analogue integrated circuit. How the internal circuitry of the op amp functions is not our concern; what does concern us are the overall electrical properties from the signal input nodes (designated noninverting and inverting) to the signal output node. The op amp is represented by the following circuit symbol: +VCC + - e2 vO e1 -VCC +VCC and –VCC designate the necessary positive and negative power supplies. Most frequently these lines are omitted from circuit diagrams but are always understood to be present. There are two signal inputs to the op amp: a so-called non-inverting input labelled e2 and denoted by the + sign within the triangle, and an inverting input labelled e1 and denoted by the – sign. What the internal circuitry actually does is to amplify the signal across the input terminals eD e2 e1 resulting in an output signal given by vO Av eD Av e2 e1 The distinguishing feature of the op amp is that the signal gain Av, or open loop gain as it is technically referred to, is typically very high, between 10 5 and 106 — ie between 100 and 120 dB. It is also worth emphasising that the op amp responds to the voltage difference eD e2 e1 rather than to e2 or e1 individually. The amplifier is accordingly classified as a differential amplifier. + eD - e2 vO e1 4-2 In the previous lecture, we covered the circuit model for a general amplifier. We now consider a corresponding model for the op amp and list some of the properties of what we call a near-ideal op amp. The relevant circuit model is RO iIN e2 eD RIN AveD vO e1 As with our previous model, the input voltage to which the amplifier is sensitive (e D in this instance) sees an input resistance RIN. The basic functioning of the op amp is captured by means of the signal source AveD with which we also associate a source resistance RO. A near ideal op amp has the following properties: The input resistance is regarded as indefinitely large implying that the input current to the op amp tends to be insignificantly small. Stated formally RIN i IN 0 The output resistance is regarded as insignificantly small and this implies that the output voltage is Av times eD: RO 0 vO Av e2 e1 The open loop gain Av is large, but not infinite. Av is a constant, independent of frequency. The amplifier produces no distortion. The amplifier produces no electrical “noise,” ie unwanted electrical fluctuations. The amplifier has no “offsets,” ie when e2 = e1 = 0, the output is indeed equal to zero. Basically what we are saying is that all the properties of the op amp are considered to be ideal except for the open loop gain; this parameter is large rather than infinitely large. It is worth noting that real op amps such as the 741 do approach the ideal in performance. 4.2 Regions of Operation Based on the above considerations and acknowledging the practicality of having supply voltages, we can represent the voltage transfer characteristic of a near-ideal op amp as follows: 4-3 vO +VCC eMAX ed -VCC k j k represents where the amplifier is operating linearly, ie where vO Av e2 e1 . Regions represent non-linear operation characterised as follows: Region if eD eMAX if eD eMAX then vO VCC then vO VCC Noting that VCC is typically 105, we can estimate eMAX as eMAX VCC 15 5 0.15mV Av 10 Thus we simply note that when the op amp is operating in the linear region the voltage difference across the input terminals is very small, at most a small fraction of a millivolt. Corresponding to the two regions of operation, we have two classes of op amp applications: Linear Circuits — these circuits, as we shall see, employ negative feedback which has the effect of forcing the op amp into the linear region of operation. Non-Linear Circuits — the op amp is used either open loop (ie without feedback) or with positive feedback. For the moment our main interest lies in linear circuit applications. 4.3 The Non-Inverting Amplifier When it comes to practical signal amplifiers, the real need is not so much towards achieving very high signal amplification as towards achieving a system whose gain can be precisely and readily controlled. Thus the op amp on its own is not all that useful; for any 4-4 particular 741 for example, all we know at the outset is that the gain is somewhere between 105 and 106. Now consider the circuit shown below: e2 + - e1 i2 iIN R2 vI vO R1 vF i1 We will refer, somewhat loosely, to the overall circuit as the “system,” since it is important to distinguish between the op amp and the overall circuit/system, of which the op amp is but one component. The input signal to the overall system is denoted vI, and this is the signal for amplification. vI is applied directly to the non-inverting input of the op amp; ie e2 vI The output from the op amp is also the system output vO and this signal is applied to a resistor divider network, R1 and R2, to produce a feedback voltage vF. This feedback voltage is applied to the inverting input of the op amp e 1 and it is this connection which produces the negative feedback. Carrying out a circuit analysis on the system, we begin by noting i2 i1 i IN However, because the op amp is near-ideal and Therefore RIN , we have i IN 0 . i2 i1 and as a consequence e1 v F R1 vO R2 R1 Negative feedback dictates that the op amp functions in its linear region so that vO Av e2 e1 R1 Av v I vO R2 R1 4-5 Re-arranging, we get R1 vO 1 Av Av v I R2 R1 Hence the overall gain of the system can be expressed vO vI Av R1 1 Av R2 R1 and we prefer to arrange this as R1 Av vO R2 R1 R2 R1 R1 vI R1 1 Av R2 R1 We tidy this up by noting R2 R1 R 1 2 R1 R1 and letting R1 R2 R1 vO R2 Av 1 v I R1 1 Av vO is the overall system gain or, as it is more usually termed, the closed vI loop gain Acl while the quantity Av is called the loop gain. While β is a fraction, the The quantity open loop gain Av is usually sufficiently large to ensure Av 1 permitting the approximation Av 1 1 Av Hence the closed loop gain is expressed R Acl 1 2 Acl id R1 That is, the closed loop tends towards the value closed loop gain Acl id. 4-6 1 R2 and this we define to be the ideal R1 The fundamental idea behind the circuit is that, ideally, the output is a scaled replica of the input and the scale factor 1 R2 is set by the resistive values of R2 and R1. Thus the R1 gain is indeed readily under the control of the user. For example, if an amplifier of gain 100 is required, we might choose R2 99k and R1 1k . The resulting system will approach this ideal closed loop gain of 100 to the extent that Av 1 . This is illustrated by the following example. Example 4.1 Consider a non-inverting amplifier having R2 99k and R1 1k . Calculate the actual closed loop gain for the following three near-ideal op amps: (i) Av 10 4 (ii) Av 10 5 (iii) Av 10 6 For all three solutions we use the expression Acl Acl id Av 1 Av where and R2 99 1 100 R1 1 R1 1 10 2 R2 R1 1 99 Acl id 1 10 10 100100 99.01 101 1 10 10 10 10 1001000 99.9 100 1001 1 10 10 10 10 10010,000 99.99 100 10,001 1 10 10 (i) Acl 100 (ii) Acl (iii) Acl 2 4 2 2 4 5 2 2 5 6 2 6 Note that the ideal closed loop gain is within 1% of each of the actual values calculated for the closed loop gain with the approximation getting better as the open loop gain increases. 4.4 The Inverting Amplifier R2 i2 R1 i1 iIN e1 vI e2 + vO 4-7 A second basic amplifying application of the op amp is shown above. Again, our task is to derive an expression for the gain vO/vI of the overall system on the basis that the op amp is near-ideal. Note again that the system output is fed back through resistor R 2 to the inverting input of the op amp ensuring that we have a negative feedback system and, hence, linear operation. In this instance, the non-inverting input of the op amp is connected directly to signal ground so that e2 0 Similar to the previous section, because that RIN , we have i IN 0 . From this it follows i2 i1 and thus the series combination of R1 and R2 constitutes a voltage divider driven by vI at one end and by vO at the other end. We can therefore write e1 R2 R1 vI vO R1 R2 R1 R2 Because the op amp is near-ideal, we have vO Av e2 e1 and substituting for e2 and e1 R2 R1 vO Av 0 vI vO R1 R2 R2 R1 RA R A vO 1 1 v 2 v v I R1 R2 R1 R2 Thus, the closed loop gain is given by R2 Av vO R1 R2 Acl R1 vI 1 Av R1 R2 R1 Av R2 R1 R2 R1 R1 1 Av R1 R2 Again letting R1 , we can re-write the closed loop gain as R2 R1 4-8 R Av Acl 2 R1 1 Av Av 1 Assuming, as before, that we get R Acl 2 Acl id R1 Again the ideal closed loop gain (in magnitude) is set by the resistor values R 2 and R1 and note the exact same expression and considerations for the loop gain. The – sign attached to R2 R1 simply means that there is a 180˚ degree phase shift between input and output; if the input is of the form form sin t , then the output is of the sin t . It is this phase reversing property which gives the system its generic title of inverting amplifier. 4.5 Concluding Remarks We have introduced and defined the near-ideal op amp and investigated two of its more important linear applications. In both cases, the closed loop gain incorporated a term with the loop gain divided by unity plus the loop gain. Arising from the loop gain typically being large, this term tended to unity and the closed loop gain tends to a quantity we called the ideal closed loop gain. This ideal closed loop gain was found to depend solely on the values of passive components connected as part of the system. Essentially, in designing op amp circuits we work with the ideal closed loop gain. 4-9