barcode reader vb.net codeproject Voltage Supply Limits in an Op-Amp Integrator in Software

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EXAMPLE 1210 Voltage Supply Limits in an Op-Amp Integrator
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Compute and sketch the output voltage of the integrator of Figure 1230
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Known Quantities: Resistor, capacitor, and supply voltage values; input voltage Find: vout (t)
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15 V; VS = 15 V; vS (t) = 05 + 03 cos(10t) vout (0) = 0
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Schematics, Diagrams, Circuits, and Given Data: RS = 10 k ; CF = 20 F; VS+ = Assumptions: Assume supply voltage limited op-amp The initial condition is
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Analysis: For an ideal op-amp integrator the output would be:
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vout (t) =
1 R S CF
t
vS (t )dt =
1 02
t
05 + 03 cos(10t ) dt
= 25t + 15 sin(10t) However, the supply voltage is limited to 15 V, and the integrator output voltage will therefore saturate at the lower supply voltage value of 15 V as the term 25t increases with time Figure 1244 depicts the output voltage waveform
16 14 12 10 8 Ideal integrator output 6 Integrator output 4 affected by DC offset 2 0 2 0 1 2 3 4 5 6 7 8 9 10 Time (s)
Figure 1244 Effect of DC offset on integrator
Comments: Note that the DC offset in the waveform causes the integrator output voltage
to increase linearly with time The presence of even a very small DC offset will always cause integrator saturation One solution to this problem is to include a large feedback resistor in parallel with the capacitor; this solution is explored in the homework problems
Focus on Computer-Aided Solutions: An Electronics WorkbenchTM simulation of a
practical integrator (including the feedback resistance mentioned in the comments above) can be found in the accompanying CD-ROM Try removing the feedback resistor to verify that saturation at the supply voltages is a real problem
Frequency Response Limits Another property of all ampli ers that may pose severe limitations to the op-amp is their nite bandwidth We have so far assumed, in our ideal op-amp model, that the open-loop gain is a very large constant In reality, AV (OL) is a function of
Integrator voltage (V)
12
Operational Ampli ers
frequency and is characterized by a low-pass response For a typical op-amp, AV (OL) (j ) = A0 1 + j / 0 (1282)
The cutoff frequency of the op-amp open-loop gain, 0 , represents approximately the point where the ampli er response starts to drop off as a function of frequency, and is analogous to the cutoff frequencies of the RC and RL circuits of 6 Figure 1245 depicts AV (OL) (j ) in both linear and dB plots for the fairly typical values A0 = 106 and 0 = 10 It should be apparent from this gure that the assumption of a very large open-loop gain becomes less and less accurate for increasing frequency Recall the initial derivation of the closed-loop gain for the inverting ampli er: In obtaining the nal result, Vout /VS = RF /RS , it was assumed that AV (OL) This assumption is clearly inadequate at the higher frequencies
105 10 8
Gain
Open-loop gain of practical op-amp 120 100
Gain, dB
Open-loop gain of practical op-amp (dB plot)
6 4 2 0 10 1 100 101 102 103 104 Radian frequency (logarithmic scale) 105
80 60 40 10 1 100 102 103 104 101 Radian frequency (logarithmic scale) 105
Figure 1245 Open-loop gain of practical op-amp
AV dB 20 log10 A0 20 log10 A1
20 log10 A2 0 1 2 Log scale
The nite bandwidth of the practical op-amp results in a xed gainbandwidth product for any given ampli er The effect of a constant gainbandwidth product is that as the closed-loop gain of the ampli er is increased, its 3-dB bandwidth is proportionally reduced, until, in the limit, if the ampli er were used in the open-loop mode, its gain would be equal to A0 and its 3-dB bandwidth would be equal to 0 The constant gain-bandwidth product is therefore equal to the product of the open-loop gain and the open-loop bandwidth of the ampli er: A0 0 = K When the ampli er is connected in a closed-loop con guration (eg, as an inverting ampli er), its gain is typically much less than the open-loop gain and the 3-dB bandwidth of the ampli er is proportionally increased To explain this further, Figure 1246 depicts the case in which two different linear ampli ers (achieved through any two different negative feedback con gurations) have been designed for the same op-amp The rst has closed-loop gain A1 , and the second has closed-loop gain A2 The bold line in the gure indicates the open-loop frequency response, with gain A0 and cutoff frequency 0 As the gain decreases from the open-loop gain, A0 , to A1 , we see that the cutoff frequency increases from 0 to 1 If we further reduce the gain to A2 , we can expect the bandwidth to increase to 2 Thus, the product of gain and bandwidth in any given op-amp is constant That is, A0 0 = A1 1 = A2 2 = K (1283)
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