Analog to Digital (A-D) and Digital to Analog (D-A) Conversion

Transcription

1 UNIVERSITY of PENNSYLVANIA DEPARTMENT of ELECTRICAL and SYSTEMS ENGINEERING Electrical Circuits and Systems II Laboratory ESE206 Analog to Digital (A-D) and Digital to Analog (D-A) Conversion Goals To become familiar with digitally represented data. To build a D-A converter and an A-D converter. To become confident with the construction and trouble-shooting of large circuits. Introduction Our perceptual world is largely based on analog inputs. We hear music as a cascade of sounds which blend into each other. We see color in continuous hues and shades. We comment on the day being warm, but not as warm as yesterday. Remarkably, physics appears to be mostly an analog system also. Providing we stay well above the quantum regime, we find that light and sound consists of smooth waves. When we find an objects length, it could be any value and not limited to integer values of meters. We note that time doesn t actually tick by, but rather it flows evenly. For most of its history, electronics were analog. This makes sense, since the goal of any device is to make our world easier and our world is an analog one. However, at this point in history, technology is digital. That seems as odd as trying to discuss Russian literature in Portugese or trying to describe a painting with formulas. It s just the wrong language. But analog circuits have their own troubles. In the first lab we went to great pains to understand error. We came to understand that we will never own a exactly 1k resistor although we can easily own a 1.0k resistor. And while a power supply might output 5.00 V it will never output 5V. What s more, we experienced how difficult it is to say what a circuit will do when we involve this uncertainty. The root of that difficulty is because error accumulates. However, let s suppose that we could say that an input was either 5.V or 0.V and that our circuit would output either 5.V or 0.V. While these numbers still contain uncertainty, it would be possible using error analysis to conclude the result always to be within 0.5V. What s more, many such circuits could be applied in series and at the end, the result would still be certain to 0.5V. If we don t care about this small deviation, then in a sense, our circuit contains no error. However there s a price to pay. This new digital language is very limited. With one wire we can only say two things (0.V and 5.V), where before in the analog domain we could say V or any other such number if we chose. However, what if our ESE206 1

2 digital signal contained two outputs? Then we could say up to four statements through all the possible permutations of 0.V and 5.V. If we keep going and our circuit has eight outputs, we find that we now have a vocabulary of 256 words. The average person uses 2000 different words a week which could be accomplished using only 11 such outputs and an elaborate look-up table. At the end of the day, we find that we just don t have that much to say and the price we pay in limiting our vocabulary is worth being able to make concrete, errorless statements. A prime example is the comparison between cassette tapes and CD s. Why was music piracy not much of an issue in the day s of cassette tapes? They were actually easier to copy than CD s are. The reason was that the data on a cassette was encoded as an analog signal. Each copy lost some quality and so there was always a market for new cassettes since the new cassettes were closest to the master recording. However, a CD is encoded as digital data. It is possible to reproduce them exactly even from copies. While the CD suffers from the fact that each data value is made from a limited number of bits, each taking on one of two possible values, the vocabulary of values this outlines is large enough that your ear can t tell the difference. This lab will focus on the translation between the analog and digital domains. Such circuits are found in almost all devices. There are thousands that are commercially available. While you will never need to build one in real world, doing so will expose you to several key engineering concepts. The Relationship between Analog and Digital Data The relationship between analog and digital data is best illustrated pictorially. Suppose that we have a 4 bit system, meaning that we have 4 separate digital values by 4 through which we can specify a number. There are 2 = 16 different on/off combinations that could be made through these values. The range of these values is up to us. In this lab we will take the lowest possible analog value that -5V while the largest possible value will be +5V. These will correspond to the bottom of and the top 11112, respectively. In other words, VFS in Fig. 1 is 5V. FS stands for Full Scale. Notice that this outlines 16 different voltage bands marked off in red. Each band corresponds to a different combination of on/off values. These are shown in blue to the left where blue indicated on while white indicates off. The value of a waveform can therefore be indicated with these digital outputs. As the analog voltage varies, these V FS D0 D1 D2 D3 D0 D1 D2 D3 Figure 1: Pictorial representation of Analog-to-digital conversion. The bars on the left in blue give the digital equivalent of the analog wavevorm (4-bit). ESE206 2

3 V FS V FS V FS V FS Figure 2: Determination of consecutive bits. The top graph shows how the most significant bit it determined; the 2 nd graph shows the remainder generated by the first bit centered and rescaled to V FS to +V FS, and how that is used to determine the second bit. etc. digital values will change. This is shown below in white and black stripes. Notice that there is no pattern to when the waveform crosses the boundaries. This is an asynchronous A-D converter. Alternatively you could use a device called a ESE206 3

4 sample-and-hold to take snapshots of the analog input at fixed intervals. This is what is commonly done so that recorded data can be read back on at a fixed rate and be meaningful. However, we re not recording the data so that won t be necessary. Analog to Digital Converter (ADC) There are many ways of thinking about analog to digital conversion, but within this lab, it will be most convenient to consider it as a form of division. Determining the most significant bit (MSB) is trivial. We simply examine the sign of the input. This is pictured in Figure 2 a. The second bit is more difficult. We must examine the remainder of the first bit. This remainder must exist on the same scale ( V FS to +VFS ) as the original signal. Once this is clear, then the rest follows as the third bit is simply the remainder of the second bit and the fourth bit is simply the remainder of the third bit. Consider the following circuit. In op-amp A, the signal AIn is compared to ground. If it is higher the output of op-amp A goes to approximately -10.5V while if Ain is lower, the output of the op-amp goes to +10.5V. This signal is led into a voltage limiter. If the LED has a turn-on voltage of V LED (about 1.9 V) and the diode has a turnon voltage of (about 0.6 V), then the resulting voltage will be either V diode + ( V LED +V diode ) ( LED diode ) or V +V, depending on the sign of the input. What s more, one of the LED s will be lit, so you can tell what that sign is. In this case, this is the most significant bit, telling us which half of the space the value lies in. Remember the goal is to provide an analog output which represents the remainder of the first comparator. This remainder must be centered and amplified so that it goes from V to + V (Fig. 2b). FB FB Figure 4: AIn (blue), DOut (violet), and AOut (green) for the circuit in Fig. 3. Output of the voltage limiter is then combined with the original signal through a voltage divider. Remember that our comparator is providing a negative signal if the voltage is positive and a positive signal if it is negative. If we perform a weighted average of the output of the comparator with the original input, it will lower the positive voltages and raise the negative ones. At this point we don t care what the amplitude of this signal is, only that it is symmetric around ground. That is, ESE206 4

5 V = V V = a V in FS int FS Figure 3: Circuit of a 1-bit of the A.D converter. Vin = 12VFS Vint = 0 + Vin = 0 Vint = a VFS, Vin = 0 Vint = + a VFS Vin = 12VFS Vint = 0 Vin = -VFS Vint = a VFS where a is some number less than one. Many of these conditions are redundant. The signal must then be boosted to full scale again. This is performed by opamp B. A sample plot is shown above in which the blue curve is the original input signal Ain and the red one Vout. The signal from Aout could be the input to another stage. While the first stage determines the most significant bit, the second stage would determine the next bit. A four-bit ADC would look like the one shown in Fig. 5. Notice that we buffer and invert the digital outputs so that they are positive when the analog input is positive and so we can use them to drive a load. Figure 5: 4 bit analog-to-digital converter in a pipeline fashion consisting of 4 identical stages. ESE206 5

6 Digital to Analog Converter (DAC) After a CD player reads the values in binary, they must be converted to an analog signal so that it can be amplified and sent to the speakers. A very simple DAC can be constructed using only resisters. All the curvy resistors have the same variables If all the inputs are V dig, then the output Figure 6: 4 bit digital-to-analog converter. All resistors pictured as angled lines have the same at Vout will be Vdig. If all the inputs are value. + V dig, then the output will be +Vdig. It is left as part of the prelab to show that. 1 Aout = ( 8 D0 + 4 D1 + 2 D2 + D3 ) 15 In practice, this is not the most common style of DAC. Another called an R-2R resister network is far more prevalent, but the one pictured is far easier to analyze and will handle bipolar inputs better. Pre-Lab 1. Determine appropriate values R9, R10, R11, R12, and R13 such that this ADC has a range from -5V to +5V and the LEDs don t burn out. 2. Consider only the first stage. Given ± 5% on all resistor values, quantify the error in the remainder analog output of stage one. Given that uncertainty, how many bits are theoretically worthwhile. 3. Analytically show that the resistor network in the DAC is indeed a base two DAC. In otherwords, Aout = a 2 D0 + 2 D1 + 2 D2 + 2 D3)+ b, where a and b are arbitrary. Lab Experiment Equipment: ( HP function generator/waveform generator (HP 33120A) 2. Power supply 3. HP digital oscilloscope HP LM353 Op-Amps 5. LEDs 6. Diodes 7. Resistors In-Lab Experiment: ESE206 6

7 Figure 7: Overall structure of ADC inline with a DAC. By end of first week: Build the circuits pictured in Figs 3, 5, and 7. Do this in steps and verify the proper functioning of each stage sequentially. Capture the digital output of each stage relative to AIn. Capture the analog output of the first three stages relative to AIn. Capture a plot similar to the one shown in Fig 8 below. DO NOT TAKE APART YOUR CIRCUIT!!!! By end of second week: While the chunky vs smooth plot shows that you were qualitatively able to develop both the ADC and the DAC, we will now try to quantitatively characterize both devices separately. First, disconnect the ADC from the DAC. Instead of using the function generator as the analog input, use the +6 power supply so that you can vary AIN precisely. Determine at which analog voltages the LED s flip. (note that you will most likely not be able to determine the transition between and ) Plot these values with the expected results. You will have to be a little creative to get the +6 power supply to generate negative voltages. Second, use the old power supply to create two rails at and Use these as the digital drive your DAC. Plot the analog output as a function of digital input. Determine the differential non-linearity (DNL) of your DAC for each digital output. VFS ( V Expi, 1 VExpi, ) + n DNL 2 i = VFS n 2 The DNL is essentially the relative accuracy of the step sizes are from one digital output to another in term of the LSB. Also determine the integral non-linearity (INL) INL = Max Vexp, i V ideal, i The INL is essentially the maximum absolute error that the DAC will achieve. Third, while we rarely even consider both the waveform generator and the oscilloscope are actually digital devices. The waveform generator is a high-speed DAC while the oscilloscope is a high-speed ADC. As such, both suffer from discretization error. Experimentally determine the number of bits used in each device. ESE206 7

8 Figure 8: Analog signal (smooth) which has been rendered into digital data and then reformed back into an analog signal. Below is a plot of the LSB (D3). Brian Edwards February 5, 2007 ESE206 8