# Op-Amps Experiment Theory

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1 EE 4/00 Operational mplifiers Op-mps Experiment Theory. Objective The purpose of these experiments is to introduce the most important of all analog building blocks, the operational amplifier ( op-amp for short). This handout gives an introduction to these amplifiers and a smattering of the various configurations that they can be used in. part from their most common use as amplifiers (both inverting and non-inverting), they also find applications as buffers (load isolators), adders, subtractors, integrators, logarithmic amplifiers, impedance converters, filters (low-pass, high-pass, band-pass, band-reject or notch), and differential amplifiers. So let s get set for a fun-filled adventure with op-amps!. Introduction: mplifier Circuit efore jumping into op-amps, let s first go over some amplifier fundamentals. n amplifier has an input port and an output port. ( port consists of two terminals, one of which is usually connected to the ground node.) In a linear amplifier, the output signal input signal, where is the amplification factor or gain. Depending on the nature of the input and output signals, we can have four types of amplifier gain: voltage gain (voltage out / voltage in), current gain (current out / current in), transresistance (voltage out / current in) and transconductance (current out / voltage in). Since most op-amps are voltage/voltage amplifiers, we will limit the discussion here to this type of amplifier. The circuit model of an amplifier is shown in Figure (center dashed box, with an input port and an output port). The input port plays a passive role, producing no voltage of its own, and is modelled by a resistive element i called the input resistance. The output port is modeled by a dependent voltage source i in series with the output resistance o, where i is the potential difference between the input port terminals. Figure shows a complete amplifier circuit, which consists of an input voltage source s in series with the source resistance s, and an output load resistance L. From this figure, it can be seen that we have voltage-divider circuits at both the input port and the output port of the amplifier. This requires us to re-calculate i and o whenever a different source and/or load is used: i i s s () i

2 EE 4/00 Operational mplifiers o L i o () L S S i INPUT POT i i o OUTPUT POT o L SOUCE MPLIFIE LOD Figure : Circuit model of an amplifier circuit.. The Operational mplifier: Ideal Op-mp Model The amplifier model shown in Figure is redrawn in Figure showing the standard op-amp notation. n op-amp is a differential to single-ended amplifier, i.e. it amplifies the voltage difference p n i at the input port and produces a voltage o at the output port that is referenced to the ground node of the circuit in which the op-amp is used. i p p i i n i i o o p i i o n n Figure : Standard op-amp Figure : Ideal op-amp The ideal op-amp model was derived to simplify circuit analysis and is commonly used by engineers for first-order approximation calculations. The ideal model makes three simplifying assumptions: Gain is infinite: () Input resistance is infinite: i (4)

3 EE 4/00 Operational mplifiers Output resistance is zero: o 0 (5) pplying these assumptions to the standard op-amp model results in the ideal op-amp model shown in Figure. ecause i and the voltage difference p n i at the input port is finite, the input currents are zero for an ideal op-amp: i n i p 0 (6) Hence there is no loading effect at the input port of an ideal op-amp: i s (7) In addition, because o 0, there is no loading effect at the output port of an ideal op-amp: Finally, because and o must be finite, i p n 0, or o i (8) p n (9) Note: lthough Equations -5 constitute the ideal op-amp assumptions, Equations 6 and 9 are used most often in solving op-amp circuits. p I n p n out n in p in out out I in Figure 4a: Non-inverting amplifier Figure 5a: oltage follower Figure 6a: Inverting amplifier out out out > in in <0 in Figure 4b: oltage transfer curve Figure 5b: oltage transfer curve Figure 6b: oltage transfer curve of non-inverting amplifier of voltage follower of inverting amplifier

4 EE 4/00 Operational mplifiers out out out power power power > in in <0 in -power -power -power Figure 4c: ealistic transfer curve Figure 5c: ealistic transfer curve Figure 6c: ealistic transfer curve of non-inverting amplifier of voltage follower of inverting amplifier 4. Non-Inverting mplifier n ideal op-amp by itself is not a very useful device, since any finite non-zero input signal would result in infinite output. (For a real op-amp, the range of the output signal is limited by the positive and negative power-supply voltages.) However, by connecting external components to the ideal opamp, we can construct useful amplifier circuits. Figure 4a shows a basic op-amp circuit, the non-inverting amplifier. The triangular block symbol is used to represent an ideal op-amp. The input terminal marked with a (corresponding to p ) is called the non-inverting input; the input terminal marked with a (corresponding to n ) is called the inverting input. To understand how the non-inverting amplifier circuit works, we need to derive a relationship between the input voltage in and the output voltage out. For an ideal op-amp, there is no loading effect at the input, so p i (0) Since the current flowing into the inverting input of an ideal op-amp is zero, the current flowing through is equal to the current flowing through (by Kirchhoff s Current Law -- which states that the algebraic sum of currents flowing into a node is zero -- to the inverting input node). We can therefore apply the voltage-divider formula find n : n out () 4

5 EE 4/00 Operational mplifiers From Equation 9, we know that in p n, so () out in The voltage transfer curve ( out vs. in ) for a non-inverting amplifier is shown in Figure 4b. Notice that the gain ( out / in ) is always greater than or equal to one. The special op-amp circuit configuration shown in Figure 5a has a gain of unity, and is called a voltage follower. This can be derived from the non-inverting amplifier by letting and 0 in Equation. The voltage transfer curve is shown in Figure 5b. frequently asked question is why the voltage follower is useful, since it just copies input signal to the output. The reason is that it isolates the signal source and the load. We know that a signal source usually has an internal series resistance ( s in Figure, for example). When it is directly connected to a load, especially a heavy (high conductance) load, the output voltage across the load will degrade (according to the voltagedivider formula). With a voltage-follower circuit placed between the source and the load, the signal source sees a light (low conductance) load -- the input resistance of the op-amp. t the same time, the load is driven by a powerful driving source -- the output of the op-amp. 5. Inverting mplifier Figure 6a shows another useful basic op-amp circuit, the inverting amplifier. It is similar to the noninverting circuit shown in Figure 4a except that the input signal is applied to the inverting terminal via and the non-inverting terminal is grounded. Let s derive a relationship between the input voltage in and the output voltage out. First, since n p and p is grounded, n 0. Since the current flowing into the inverting input of an ideal op-amp is zero, the current flowing through must be equal in magnitude and opposite in direction to the current flowing through (by Kirchhoff s Current Law): Since n 0, we have: in n out n () out in The gain of inverting amplifier is always negative, as shown in Figure 6b. (4) 5

6 EE 4/00 Operational mplifiers 6 6. Operation Circuit Figure 7 shows an operation circuit, which combines the non-inverting and inverting amplifier. Let s derive the relationship between the input voltages and the output voltage out. We can start with the non-inverting input node. pplying Kirchhoff s Current Law, we obtain: p p p p (5) pplying Kirchhoff s Current Law to the inverting input node, we obtain: out n n n n (6) Since n p (from Equation 9), we can combine Equations 5 and 6 to get out (7) where and Thus this circuit adds, and and subtracts, and. Different coefficients can be applied to the input signals by adjusting the resistors. If all the resistors have the same value, then ( ) ( ) out (8) out n p Figure 7: Operation circuit

7 EE 4/00 Operational mplifiers 7. Integrator y adding a capacitor in parallel with the feedback resistor in an inverting amplifier as shown in Figure 8, the op-amp can be used to perform integration. n ideal or lossless integrator ( ) performs the computation out indt. Thus a square wave input would cause a triangle wave C output. However, in a real circuit ( < ) there is some decay in the system state at a rate proportional to the state itself. This leads to exponential decay with a time constant of τ C. C n in p out Figure 8: Integrator 8. Differentiator y adding a capacitor in series with the input resistor in an inverting amplifier, the op-amp can be used to perform differentiation. n ideal differentiator ( 0) has no memory and performs the computation din C. Thus a triangle wave input would cause a square wave output. dt out However, a real circuit ( > 0) will have some memory of the system state (like an lossy integrator) with exponential decay of time constant τ C. 9. Differential mplifier Figure 9 shows the differential amplifier circuit. s the name suggests, this op-amp configuration amplifies the difference of two input signals. out ( ) (0) If the two input signals are the same, the output should be zero, ideally. To quantify the quality of the amplifier, the term Common Mode ejection atio (CM) is defined. It is the ratio of the 7

8 EE 4/00 Operational mplifiers output voltage corresponding to the difference of the two input signals to the output voltage corresponding to common part of the two signals. good op-amp has a high CM out Figure 9: Differential amplifier 0. Frequency esponse of Op-mp The frequency response of any circuit is the magnitude of the gain in decibels (d) as a function of the frequency of the input signal. The decibel is a common unit of measurement for the relative loudness of a sound or, in electronics, for the relative magnitude of two power levels. ( decibel is one-tenth of a "el", a seldom-used unit named for lexander Graham ell, inventor of the telephone.) The gain expressed in d is 0 log 0 G. The frequency response of an op-amp is a low pass characteristic (passing low-frequency signals, attenuating high-frequency signals), Figure 0. Gain (log scale) G - d point Freq (Hz ) Figure 0: Frequency response of op-amp. The bandwidth is the frequency at which the power of the output signal is reduced to half that of the maximum output power. This occurs when the power gain G drops by d. In Figure 0, the 8

9 EE 4/00 Operational mplifiers bandwidth is Hz. For all op-amps, the Gain*andwidth product is a constant. Hence, if the gain of an op-amp is decreased, its operational bandwidth increases proportionally. This is an important trade-off consideration in op-amp circuit design. In Sections through 8 above, we assumed that the op-amp has infinite bandwidth. 0. More on Op-mps ll of the above op-amp configurations have one thing in common. There exists a path from the output of the op-amp back to its inverting input. When the output is not railed to a supply voltage, negative feedback ensures that the op-amp operates in the linear region (as opposed to the saturation region, where the output voltage is saturated at one of the supply voltages). mplification, addition/subtraction, and integration/differentiation are all linear operations. Note that both C signals and DC offsets are included in these operations, unless we add a capacitor in series with the input signal(s) to block the DC component.. The LMC 648 Operational mplifier Guidelines for usage: Two power-supply voltages (one positive, one negative) must be connected to the LM74. The input voltage should fall within the range between the two power-supply voltages. The power-supply voltages limit the output voltage swing. The voltage-transfer characteristics of the non-inverting amplifier, the voltage follower and the inverting amplifier are illustrated in Figures 4c, 5b and 5c, respectively. The output swing also depends on the load resistor L. In order for the LMC 648 to work like an ideal op-amp, don t connect too heavy a load (resistor of low resistance) to it. void continuous operation under maximum current output; otherwise it can burn your finger! The input resistance of the LMC 648 is not tremendously large, so resistors like and shown in Figure 4 and 6 should not be too large. Of course they should not be too small, since neither the external signal source nor the LMC 648 itself may have sufficient current driving capacity. 9

10 EE 4/00 Operational mplifiers LMC 648 pins: The op-amp is packaged in an epoxy mini-dip ( dual inline package ) which has 4 pins emerging from each of the two sides of the package. The most important part of the identification is to note that Pin is marked by a small circular indentation in the lower left-hand corner of the package when viewed from the top side with the manufacturer s marking reading upright. The pin numbers on ICs always start with Pin being this labeled pin and increase counter-clockwise as read from the top side of the package. Thus on this particular package, Pin 8 is directly across from Pin, and Pin 5 is diagonally across from Pin. You will need to use only 5 of the 8 pins. These are: Inverting Input Pin, Non-Inverting Input Pin, Negative Supply Pin4, Output Pin6 and Positive Supply Pin7. Other pins are available for an optional offset null adjustment (Pins & 5), which is unnecessary for the circuits used in this experiment. Do not ground these pins, as this will result in failure of the op-amp. Pin 8 has no internal connection into the silicon chip inside. 0

11 EE 4/00 Operational mplifiers Op-mps: Experiment Guide In this lab, we are going to study operational amplifiers and circuits with op-amps. The op-amp chip that we are going to use is LMC 648 from National Instrument. The configuration of the chip is shown below. It has two amplifiers in one chip with 8 pins. The pin configuration is also shown in the same figure (There is a node on the chip indicating pin ). The power supply to the chip is -4 for - and 6 for in this lab (Maximum - - is 0). For more information, please refer to the device specification. Part : Noninverting mplifier (a) DC measurements: () uild up the noninverting amplifier as shown in Fig. Use 5 channel and -5 channel of the DC power supply for the DD and SS, the 5 should be set up to 6 and -5 channel should be set up to -4. Use 6 channel of the DC power supply for in, and measure both input and output using oscilloscope. is 5k and is 5k. Change in from - to to verify the proper amplification range of DC inputs. () Fix the DC input 0.5, measure the amplifier gain ( out / in ) for k, 5k, 0kΩ (turn ) and compare with the calculated gain. (You need to take out the pot from the circuit to measure its value.) (b) C measurement: () Now, set the input signal to a khz, 0.5 PP, 0 DC offset (on the function generator display) Sine wave from the function generator. Use a 0kΩ potentiometer as. djust to see the gain change. Can you get a gain less than unity by turning? Why?

12 EE 4/00 Operational mplifiers () Turn the potentiometer until the gain is and then adjust the pp and DC offset to the input signal. Observe the input and output waveforms as you vary the DC offset for large pp (say.5). (c) Clipping: () Turn the potentiometer until the gain is ~ and then add a DC offset to the input signal. Observe the input and output waveforms as you vary the DC offset from 0DC to.0dc (the FG should display 0DC to 0.5DC). () Draw the input and output for a case that gives clipping, and include measured voltages of the peak, trough, and saturation levels referenced to ground. Explain the conditions under which distortion occurred. Do not disassemble this circuit. emove i and change to a fixed 4.7k to be used in Part. Fig 0: Grounding Unused Op-amps 8 DD 4 SS o i Fig :Non-inverting mplifier

13 EE 4/00 Operational mplifiers Part : Inverting mplifier Using the unused op-amp of the IC, build the inverting amplifier as shown below (please use the unused op-amp now). While you are building a circuit, it is safer for the circuits if you turn the DC power supply OUTPUT OFF. Let the input signal be a khz,.0 PP sine wave, 0 DC offset, turn to max. What s happening to the output signal? djust the input offset to make the output more complete. Now adjust the potentiometer and observe the resulting change in the amplitude and offset of the output. djust these two parameters until the gain is at its maximum and there s no clipping. What range of gain do you have in this circuit? erify the correct amplification of both C and DC signals. What is the phase difference between out and in and where is it from? Is it a delay? DD i 5 4 o SS Part : Cascaded connection Now we will study a cascade connection of two amplifiers. Connect the output of the inverting amp to act as the input voltage for the non-inverting amp. Use 0k in the inverting circuit and 5k in the noninverting circuit. The input signal should be a khz, 50m PP (on the function generator display) sine wave and you have to pick the correct offset for the circuit to amplify linearly. djust the input signal to make sure there is no clipping in the circuit. Measure the gain of each stage separately and then the overall gain of this cascaded circuit. Part 4: Integrator Put a 0. uf capacitor instead of in a new inverting amplifier (Fig ) and measure the time constant. Use a 60 Hz, 500m PP square wave as input. fter getting the waveforms and triggering correct, measure time constant C (how will you measure it? Hint: your prelab question 4).

14 EE 4/00 Operational mplifiers Compare measured time constant with theory. Now change the function generator back to a sine wave input, sweep frequency from Hz to 00kHz and observe the change of the gain with frequency. Note on op-amp integrator The circuit in figure 4 violates one of the cardinal rules of op-amp circuit design - ``there must always be a DC feedback path to the inverting input or the op-amp output will go to the rail.'' The general problem with this integrator circuit is that a small error current, input offset current, will be integrated by the capacitor to be large output voltages, and eventually drive the op-amp output into saturation. The LMC 648 op-amp you are using has remarkably low input offset currents, so that you may not see this effect in a short time. If you want to see this effect, ask your T for another pin-compatible op-amp such as the LM64, substitute in the integrator circuit, and see if you observe any difference in the average DC level of the output. (Typically, a real integrator is made with a zero-reset, or a large resistor in parallel with the integrator capacitor). Part 5: Differentiator uild the inverting amplifier but put 0. uf capacitor in stead of as shown in Fig 4. Use 5k Input a 500 Hz 500 mpp triangle wave. Zoom into the waveform to measure time constant C (Hint: prelab question 5). Compare measured time constant with theory. dd DC offset to the input signal, is there any change on the output signal? Why? What happens when the input is a triangle wave? 4

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