# This representation is compared to a binary representation of a number with N bits.

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1 Chapter 11 Analog-Digital Conversion One of the common functions that are performed on signals is to convert the voltage into a digital representation. The converse function, digital-analog is also common. We will start with the digital-analog, D/A conversion. A simple D/A conversion circuit is shown in Figure 1. This circuit is essentially an operational amplifier inverting, summing circuit followed by a unity gain inverter. Each switch connects to zero volts, or +V R. The output voltage is V out = R f (V 0 /R 0 + V 1 /R 1 + V 2 /R 2 + V 3 /R V N-1 /R N-1 ) where V 0 is the voltage at the switch 0, which is either +V R or 0, etc.. Figure 1. Simple digital to analog converter This representation is compared to a binary representation of a number with N bits. Binary Value = 2 N-1 a N a a a a 0 where a 0 can take on either 0 or 1, etc. If N = 8, then the binary number can take on any value between 0 and 255. In the D/A circuit, there would be 8 switches and input resistors. We want 256 discrete steps, The smallest or least significant step is controlled by switch 0. The output voltage from A-D Conversion 1

2 switch 0 is V R (R f /R 0 ), while the maximum output voltage is 255 times as high, when all switches are connected to V R. Thus, R 0 = 2R 1, R 1 = 2R 2, etc., so that each switch contributes exactly twice as much to the output voltage as the previous switch, and R 0 = 128R 7. The total range is 0-255V R (R f /R 0 ). We know that operational amplifier outputs cannot exceed the rails, or power supply voltages, so 255V R (R f /R 0 ) must be less than the power supply voltage. This limitation determines the relationship between the resistors and V R. For example, if we wish to limit the range to 10 volts with V R = 5 volts, then R f /R 0 = 2/255, or R 0 = 127.5R f. To make numbers convenient in the binary system, we usually make R 0 = 128R f. The resistor ratios in the input string are related by factors of two so that most significant resistor, R 7 = 1R f. Resistor values used in operational amplifier circuits are typically in the range of KOhms. Lower values require large currents while large resistor values may cause errors due to bias currents and offset voltages. With high quality operational amplifiers, it is possible to use this range of without serious problems, but extending the number of digits may cause significant errors which may exceed the least significant values. One way to overcome the large range in resistor values is to use a ladder network or the R/2R network shown in Figure 2. In the ladder, only two values are used, R and 2R. The overall gain is controlled by R a and R b. Figure 2. Ladder type D/A converter A-D Conversion 2

3 We will assume the op-amp has infinite input impedance and all switches are connected to ground so the resistance seen looking any direction from any node is 2R as seen in Figure 3. Here we see that the two resistors looking left and right from node V 0, are 2R. The parallel combination is R, so that the resistance looking up from node V 1 is also 2R. This process can be carried out down the entire string. Similarly, the same process is used to see that the resistance looking down from each node is also 2R. Figure 3. Ladder network showing resistances seen looking away from each node If switch S N-1 is connected to +V R, the voltage at node V N-1 is 1/3V R. This voltage is amplified by the op-amp as seen in Figure 2. Similarly, the voltage V 3 = 1/3V R when switch V 3 is up. However, the voltage V N-1 = 1/2V 3. Thus, the switch S 3 contributes only half the voltage to the op-amp. Similarly, each switch above, contributes successively less voltage by factors of two at each step. Up to now, we have simply shown switches in the A/D converters. However, solid state switches are universally used. These switches are most commonly CMOS switches as shown in Figure 4. The control voltage ussed is the same as the switched voltages to guarantee low resistance connections with the MOSFETS in the triode region when on. A-D Conversion 3

4 Exercises Figure 4. CMOS switch used in A/D converters 1. For a 4-bit D/A converter as shown in Figure 1, assume V R is 5 volts, and we want the range to be approximately 0-5 volts. the op-amp feedback resistor is R f = 2.2 kohm. Determine the necessary values of the input resistors. What is the resolution, or the smallest step? What is the highest output voltage? 2. A four-bit ladder type D/A converter is to be designed. Assume V R is 5 volts, and we want the range to be approximately 0-5 volts. What is the necessary gain of the op-amp? Analog - Digital Conversion Analog - digital conversion is the opposite to the process described above. In this case we want to convert an analog voltage to a digital representation. We see this function performed in digital voltmeters where a voltage is converted to digital and then shown in LED or LCD numerical displays. If we assume the input voltage can take on any value in the allowed range, such as 0-5 volts, the A/D converter will interpret that range in discrete steps. For example, if we are using an 8-bit converter, there are 256 steps within the 0-5 volt range. Each step represents a voltage range of 5/256 = volts, or about 20 mv. This discreteness means that any voltage between 0 and 19.5 mv will be represented as , while between 19.5 and 39 mv will be represented as , etc. The most straightforward method of doing an A/D conversion is to use a series of voltage comparators as shown in Figure 5. Here a string of resistors generates a string of voltage references for the comparators. The output of each comparator will switch to a high when the input analog voltage is above the reference voltage. There is a comparator for each step in the digital output above the zero step. Thus, in the case of an 8-bit conversion, there would be 255 comparators. These 255 signals would normally be encoded into the 8-bit digital representation. This method is very fast and is called a flash converter, but is impractical for all but the most critical applications where more than just a few steps are needed. A-D Conversion 4

5 Figure 5. Flash A/D converter A more practical, and more common, A/D converter using a counter and a D/A converter is shown in Figure 6. In this circuit, the counter starts at zero. The output of the counter goes to the D/A converter which presents its output to the voltage comparator. If the input voltage is above the output of the D/A, the counter counts up. As the conter counts up, the output of the D/A goes higher. When the D/A output goes above the input signal, the counter stops. At that point, the output of the counter is the digital representation of the input signal. The output of the voltage comparator can be used to tell the outside world when the conversion is completed. To take the next sample, the counter must be reset and the process starts over. The resolution is determined by the counter-d/a resolution and the range is determined by the range of output from the D/A converter. Of course, there must be a few more controls added such as stopping the counter when it reaches its maximum and an overflow signal enabled. Figure 6. Counting A/D converter Figure 7. Output of the D/A module The output signal of the D/A converter module would look like Figure 7. The counter starts at zero which gives zero output from the D/A. The counter counts until the D/A output rises above the input voltage. The counting A/D converter does not resolve the conversion until the counter has counted far enough. Conceivably, this time could be 2 N -1 counts. The maximum clock speed is A-D Conversion 5

6 limited to the time delays of the various modules; counter, D/A converter, comparator, and the AND gate. An 8-bit A/D with a 1 MHz clock could take 255 µs to resolve the output. This delay is very long compared to the flash A/D converter which would only take a few microseconds. We will look at two methods to try to speed this up; tracking A/D, and a successive approximation A/D Tracking A/D A tracking A/D converter uses an up/down counter and runs continuously. This type A/D converter is shown in Figure 8. The direction of the count is determined by the voltage comparator. Presumably, a change in the input analog voltage will be close to the previous voltage so the count does not take as long as a count from zero every time. This system has the added advantage of tracking the input voltage continuously. It also has the disadvantage that it continuously toggles around the input voltage. Figure 8. Tracking A/D converter Successive Approximation A/D The successive approximation converter uses the same diagram as the tracking A/D converter shown in Figure 8. However, the binary counter use additional logic to control the count sequence differently. The counter switches the most significant bit to a 1 first. If the output of the D/A is lower than V in, the bit stays a 1, otherwise, it is switched back to a 0 on the next clock pulse. Then the counter goes through the same sequence for the second most significant bit, etc., until all bits have been switched and compared. This type of A/D takes a maximum of 2N clock pulses to resolve the output, much faster than the maximum time of the simple counting A/D. Exercises 3. A simple counting 16-bit A/D converter uses a clock frequency of 2 MHz and has a range of 0-2 volts. What is the binary output when the input voltage is 1.5 volts? What is the settling time? 4. If the A/D converter in the previous problem is a successive approximation converter, what is the settling time? A-D Conversion 6

7 BCD Converters Typical digital voltmeters display the voltage in decimal representation. It would be appropriate for these voltmeters to maintain the digital representation in BCD to minimize encoding and decoding logic. The most logical way to handle BCD is to used BCD counters and BCD D/A converters in the feedback loop. BCD counters are well known, but we need to look at the coding of the D/A converters. In particular, we are interested in the resistor ladder networks that produce the BCD weighting. A simple BCD D/A converter is shown in Figure 9. In this case, the least significant decimal digit has resistance values a factor of ten time the corresponding resistor values for the most significant digit. An alternative implementation is given in Figure 10. In this case, the ratio between the digits is accomplished by the second summing inverting amplifier. Figure 9. BCD D/A converter A-D Conversion 7

8 Figure 10. Alternative BCD D/A converter A-D Conversion 8

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