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It is relatively simple to construct a DAC by taking advantage of the summing ampli er illustrated in 12 Consider the circuit shown in Figure 1524, where each bit in the word to be converted is represented by means of a 5-V source and a switch When the switch is closed, the bit takes a value of 1 (5 V); when the switch is open, the bit has value 0 Thus, the output of the DAC is proportional to the word bn 1 bn 2 b1 b0 You will recall that a property of the summing ampli er is that the sum of the currents at the inverting node is zero, yielding the relationship va = RF bi 5 Ri i = 0, 1, , n 1 (1524)
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RF Vin Rn 1 bn 1 Rn 2 bn 2 + va
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where Ri is the resistor associated with each bit and bi is the decimal value of the ith bit (ie, b0 = 20 , b1 = 21 , and so on) It is easy to verify that if we select Ri = R0 2i (1525)
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R0 b0
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Figure 1524 n-bit digital-to-analog converter (DAC)
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we can obtain weighted gains for each bit so that va = RF n 1 (2 bn 1 + + 21 b1 + 20 b0 ) Vin R0 (1526)
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and so that the analog output voltage is proportional to the decimal representation of the binary word As an illustration, consider the case of a four-bit word; a reasonable choice for R0 might be R0 = 10 k , yielding a resistor network consisting of 10-, 5-, 25-, and 125-k resistors, as shown in Figure 1525 The largest decimal value of a four-bit word is 24 1 = 15, and so it is reasonable to divide this range into steps of 1 volt (ie, v = 1 V) Thus, the full-scale value of va is 15 V: 0 va 15 V and we select RF according to the following expression: RF = v R0 1 104 = =2k Vin 5
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2 k
125 k b3 25 k 5 k b1 10 k b0 va + b2
4-bit DAC with 0-15 V range
The corresponding four-bit DAC is shown in Figure 1525 The DAC transfer characteristic is such that the analog output voltage, va , has a steplike appearance, because of the discrete nature of the binary signal The coarseness of the staircase can be adjusted by selecting the number of bits in the binary representation The practical design of a DAC is generally not carried out in terms of discrete components, because of problems such as the accuracy required of the resistor value Many of the problems associated with this approach can be solved by designing the complete DAC circuit in integrated circuit (IC) form The speci cations stated by the IC manufacturer include the resolution, that is, the minimum nonzero voltage; the full-scale accuracy; the output range; the output settling time; the power supply requirements; and the power dissipation The following examples illustrate the use of integrated circuit DACs
Figure 1525 Four-bit DAC
15
Electronic Instrumentation and Measurements
EXAMPLE 156 DAC Resolution
Problem
Determine the smallest step size, or resolution, of an 8-bit DAC
Solution
Known Quantities: Maximum analog voltage Find: Resolution v Schematics, Diagrams, Circuits, and Given Data: va, max = 12 V Analysis: Using equation 1523, we compute:
v =
va, max va, min 12 0 = 8 = 471 mV 28 1 2 1
Comments: Note that the resolution is dependent not only on the number of bits, but also on the analog voltage range (12 V in this case)
EXAMPLE 157 Determining the Required Number of Bits in a DAC
Problem
Find an expression for the required number of bits in a DAC using the de nitions of range and resolution
Solution
Known Quantities: Range and resolution of DAC Voltage level corresponding to logic 1 Find: Number of DAC bits required Schematics, Diagrams, Circuits, and Given Data:
Range: the analog voltage range of the DAC = va, max va, min Resolution: the minimum step size v Vin = voltage level corresponding to logic 1 0 V = voltage level corresponding to logic 0
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