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EXAMPLE 1211 Gain-Bandwidth Product Limit in an Op-Amp
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Determine the maximum allowable closed-loop voltage gain of an op-amp, if the ampli er is required to have an audio-range bandwidth of 20 kHz
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Known Quantities: Gain-bandwidth product Find: AV
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Schematics, Diagrams, Circuits, and Given Data: A0 = 106 ; 0 = 2 5 rad/s Assumptions: Assume gain-bandwidth product limited op-amp Analysis: The gain-bandwidth product of the op-amp is:
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A0 0 = K = 106 2 5 = 107 rad/s The desired bandwidth is max = 2 20,000 rad/s, and the maximum allowable gain will therefore be: Amax = K 107 = = 250 max 4 104 V V
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For any closed-loop voltage gain greater than 250, the ampli er would have reduced bandwidth
Comments: If we desired to achieve gains greater than 250 and maintain the same
bandwidth, two options would be available: (1) use a different op-amp with greater gain-bandwidth product or (2) connect two ampli ers in cascade, each with lower gain and greater bandwidth, such that the product of the gains would be greater than 250 To further explore the rst option, you may wish to look at the device data sheets for different op-amps (in the accompanying CD-ROM, or in the device templates in the Electronics WorkbenchTM libraries), and verify that op-amps can be designed (at a cost!) to have substantially greater gain-bandwidth product than the ampli er used in this example The second option is examined in the next example
EXAMPLE 1212 Increasing the Gain-Bandwidth Product by Means of Ampli ers in Cascade
Problem
Determine the overall 3-dB bandwidth of the cascade ampli er of Figure 1247
Solution
Known Quantities: Gain-bandwidth product and gain of each ampli er Find: 3 dB of cascade ampli er
ampli er RF /RS = 100 for each ampli er
Schematics, Diagrams, Circuits, and Given Data: A0 0 = K = 4 106 for each
12
Operational Ampli ers
RF RF RS VS
RS Vout
+ + Vout
Figure 1247 Cascade ampli er
Assumptions: Assume gain-bandwidth product limited (otherwise ideal) op-amps Analysis: Let A1 and 1 denote the gain and the 3-dB bandwidth of the rst ampli er, respectively, and A2 and 2 those of the second ampli er The 3-dB bandwidth of the rst ampli er is:
1 =
K 4 106 = = 4 104 A1 102 K 4 106 = = 4 104 A2 102
rad s
The second ampli er will also have: 2 = rad s
Thus, the approximate bandwidth of the cascade ampli er is 4 104 and the gain of the cascade ampli er is A1 A2 = 100 100 = 104 Had we attempted to achieve the same gain with a single-stage ampli er having the same K, we would have achieved a bandwidth of only: 3 = K 4 106 = = 4 102 A3 104 rad s
Comments: In practice, the actual 3-dB bandwidth of the cascade ampli er is not quite as large as that of each of the two stages, because the gain of each ampli er starts decreasing at frequencies somewhat lower than the nominal cutoff frequency The calculation of the actual 3-dB bandwidth of the cascade ampli er is illustrated in Check Your Understanding Exercise 1217
Input Offset Voltage Another limitation of practical op-amps results because even in the absence of any external inputs, it is possible that an offset voltage will be present at the input of an op-amp This voltage is usually denoted by Vos and it is caused by mismatches in the internal circuitry of the op-amp The offset voltage appears as a differential input voltage between the inverting and noninverting input terminals The presence of an additional input voltage will cause a DC bias error in the ampli er output, as illustrated in Example 1213 Typical and maximum values of Vos are quoted in manufacturers data sheets The worst-case effects due to the presence of offset voltages can therefore be predicted for any given application
Part II
Electronics
EXAMPLE 1213 Effect of Input Offset Voltage on an Ampli er
Problem
Determine the effect of the input offset voltage Vos on the output of the ampli er of Figure 1248
R1 +
+ V os _
Solution
Known Quantities: Nominal closed-loop voltage gain; input offset voltage Find: The offset voltage component in the output voltage, vout,os Schematics, Diagrams, Circuits, and Given Data: Anom = 100; Vos = 15 mV Assumptions: Assume input-offset-voltage-limited (otherwise ideal) op-amp Analysis: The ampli er is connected in a noninverting con guration; thus its gain is:
+ vout
Figure 1248 Op-amp input offset voltage
= 100 = 1 +
RF RS
The DC offset voltage, represented by an ideal voltage source, is represented as being directly applied to the noninverting input; thus: Vout,os = AV
nom Vos
= 100Vos = 150
Thus, we should expect the output of the ampli er to be shifted upward by 150 mV
Comments: The input offset voltage is not, of course, an external source, but it
represents a voltage offset between the inputs of the op-amp Figure 1251 depicts how such an offset can be nulled The worst-case offset voltage is usually listed in the device data sheets (see the data sheets in the accompanying CD-ROM, or in the device templates in the Electronics WorkbenchTM libraries, for an illustration) Typical values are 2 mV for the 741c general-purpose op-amp and 5 mV for the FET-input TLO81
Input Bias Currents Another nonideal characteristic of op-amps results from the presence of small input bias currents at the inverting and noninverting terminals Once again, these are due to the internal construction of the input stage of an operational ampli er Figure 1249 illustrates the presence of nonzero input bias currents (IB ) going into an op-amp Typical values of IB depend on the semiconductor technology employed in the construction of the op-amp Op-amps with bipolar transistor input stages may see input bias currents as large as 1 A, while for FET input devices, the input bias currents are less than 1 nA Since these currents depend on the internal design of the op-amp, they are not necessarily equal One often designates the input offset current Ios as Ios = IB+ IB (1284)
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