29 September 2011

Tutorial 15a: Analog Signal Conversion

Looking back on the past tutorials, we really only have two more major peripherals to learn. Today, we'll start taking a look at the Analog to Digital Converter (ADC), then learn about serial communication methods in preparation for an actual scientific experiment.

A corrolary to the Analog to Digital Converter is the Digital to Analog Converter. Unfortunately, it's very difficult to fully understand either of these without understanding the other, but we can learn a great deal about each to start. To understand the actual inner-workings of an ADC or DAC system, we can start with the basic ideas and then learn how they are implemented. So first, let's examine what it is that's different between an analog and a digital signal.

An example of an analog signal
An analog signal is what we're most likely to experience in the physical world. An incandescent light bulb can be put on a circuit where by rotating a knob the brightness can be adjusted. The oven in your kitchen can adjust its internal temperature to any value between about 170 and 500 deg F. The music you listen to pulls the membrane of a speaker in and out to create pressure waves that our ears interpret as sound. For the most part, the universe around us is analog: any measurement can take any value within a continuous range. As a visual example, consider the sine wave. No matter how closely you look at this function, it's always smooth--each of the infinite number of values between -1 and 1 is found in the curve.

While there are digital equivalents in the universe, for the most part our common encounters with digital signals reside in the realm of computers and electronics. The key difference is that measurements can only take discrete values; it's like saying the value can be 1, 2, or 3, but not 1.3 or 2.14. Consider the discrete version of the sine wave shown here.
A digital representation of the analog signal

The red lines represent the digital values that approximate the sine curve (shown in cyan for reference). While this example may not look so great, picture what would happen if we can have more than the 9 possible values available in this example---you can probably imagine that more values in the discretization would give us a better approximation. The simplest ADC peripheral in the MSP430 is a 10 bit system, which gives us 2**10 (or 1024) values. The sine approximation looks quite good with this set:

10-bit digital approximation of the analog signal

Obviously more bits gives a better representation, but it does come at a cost, in both complexity and speed. The reasons for that are more apparent when we understand how an ADC comes up with its values, so let's take a look at the ADC itself.

In reality, we've already looked at elementary ADC by using the comparator. When we convert an analog value (in this case a voltage) to a digital value (a number), the result tells us something about the magnitude of the analog signal. For the comparator, a particular voltage threshold is set for the analog value, and the number is either 0 or 1, depending on if the signal is greater than or less than the threshold. This makes a 1-bit ADC; the number is expressed in a single bit.

While very useful for more applications than you might expect from a single bit, the comparator is limited in how  much it can say about the analog signal coming in. If the signal changes about the threshold, we see the bit change between 0 and 1. But if the signal is changing above or below the threshold without crossing it, we have no way of seeing that occur.

The simple-minded solution would be to add another comparator with a different threshold. If we set the threshold on the first comparator to 1 V, and the second to 2 V, with both returning 1 when Vin is greater than the threshold, we would have the following possible results:

  • 0,0:  both comparators are below threshold, so the voltage must be less than 1 V.
  • 0,1:  the first comparator is above threshold, but the second is below, so the voltage must be greater than 1 V, but less than 2 V.
  • 1,1:  both comparators are above threshold, so the voltage must be greater than 2 V.
This setup would work, but notice that it's not completely efficient. Particularly, it is impossible to get a result of 1,0, because the voltage cannot be greater than 2 V but less than 1 V. So adding comparators can give you a broader view of the analog signal, but it's not the best way to do it.

The MSP430 ADC10 peripheral uses a Successive-Approximation-Register ADC, or SAR ADC. This fancy name is simply a description of how the ADC makes its measurement. Have you ever played the number guessing game? Say I ask you to guess the number I'm thinking of between 1 and 100, and I'll tell you if you're low or high. What's the most efficient way of getting there? If you guess 90, and I say you're low, then you've narrowed it down to 10 possible values in just one guess! On the other hand, if I say you're high, then you've only eliminated 10 possible values. When you consider that compromise, it's clear the best starting guess would be 50-- you're guaranteed to eliminate half of the possibilities. What next? Well, it would make sense to cut the remaining possibilities in half-- if 50 was low, guess 75; if high, guess 25. If you continue on with this algorithm, it won't take you too long to come up with the number I'm thinking of. (It will take you at most 7 guesses.)

SAR works in the same way; note that in any binary representation, a 1 followed by zeros is half the total possible range in the same number of bits. (Eg. 0b1000 is 8, while the upper limit 0b1111 is 15.) If we take a single comparator and use half of our reference voltage, we get our first bit-- if it's above 1/2 Vref, set it to 1; if below, set it to 0. Then we set the comparator reference to either 1/4 or 3/4 Vref (depending on the value we just got), and compare again to get the next bit. Using this method, you can come up with the digital value in as many measurements as you have bits-- in the case of ADC10, it takes 10 measurements.

Note that instead of adding another fixed comparator, we use only one as opposed to the 1,024 we would have needed to get the same resolution. The compromise is that now we have to change the comparison voltage and make multiple measurements. These two factors lead to a limit on how quickly we can make a measurement; to get very fast measurements, we need fast settling times on both the reference divider and the recording to the data register (which depends on the number and frequency of clock cycles in the CPU, of course). The ADC10 is rated to measurements up to about 200,000 samples per second (or 200 ksps, in the nomenclature used for the datasheets).

The actual mechanism used to make the measurement is pretty simple; you use a DAC of some sort to set the reference according to the bits we've already set. The details are fascinating, but beyond the scope of this particular tutorial. Feel free to read in the Family User's Guide or in a copy of MSP430 Microntroller Basics to get more information. Search online as well. That concludes our introduction to ADC. This is only a basic introduction, of course, and the ADC10 has a wealth of powerful operating modes. Next time we'll look at how to configure the ADC10 peripheral.

2 comments:

Anonymous said...

Why in you blog not visible picture?

David said...

The pictures appear fine in my browser window; I don't think the problem is in the blog itself. Hope you can find the problem. Fortunately, the images aren't really essential for the tutorial.