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Part 1: Introduction Precision Electronics This is the first article in a series covering the basics of precision electronics design. The practical series will cover a range of topics, including precision op amps, instrumentation amplifiers, signal switching and noise. I will use real examples and real components to demonstrate the concepts. By Andrew Levido W hile I aim to cover this topic from a practical perspective rather than a theoretical one, some theory is unavoidable. Along with explaining the concepts, I hope to give a few tips and tricks along the way. Since most devices built today include a microcontroller, we will also look at analog-­to-digital and digital-to-analog conversion. What is precision? We should start by defining ‘precision’ in the context of precision circuits. We should also distinguish between precision and accuracy, two often confused terms. Both precision and accuracy are ways of looking at the error in the measurement of a physical or electrical quantity. Accuracy describes how closely a measurement or series of measurements matches the ‘true’ value. In practice, that more likely means how closely it matches an accepted proxy for the quantity, probably traceable to some international standard. Precision describes how closely a series of measurements match each other. It relates to the repeatability of a measurement – how confident we can be that another measurement in one minute, tomorrow, or next year will be the same as the one taken now. Alternatively, it could indicate how confident we are that the measurement taken by the second, 100th or 10,000th unit off the production line will perform identically to the first one. Fig.1 illustrates this nicely. It is a histogram of 16 different measurements of a nominal 10.0V source taken over time. The mean of the samples is Fig.1: 16 samples of a nominally 10V source. The measurement accuracy is the difference between the sample mean and the ‘true’ value of 10V, while the precision is the spread of samples about the mean. Here, the precision is ±0.2V (absolute) or ±2% (relative). 42 Silicon Chip Australia's electronics magazine 9.9V, with a spread of ±0.2V around this (from 9.7V to 10.1V). The mean differs from the ‘true’ 10V value by 0.1V. Therefore, we can say that our accuracy is within ±0.1V of 10.0V or ±1%. The precision of our measurement is ±0.2V around the 9.9V mean, so within ±2%. Precision and accuracy are related but independent quantities. We can have precision without accuracy and accuracy without precision (although the latter would be of limited value). Note that in the example above, an accuracy of ±1% does not mean that every measurement will be within ±1% of the actual value since the measurement precision is not good enough to allow that. Accuracy is all about traceability and calibration, whereas precision is all about understanding and controlling the sources of uncertainty or error in our circuits. It is not always about achieving the highest levels of precision – it is about getting ‘good enough’ results for the application, which requires us to know what the precision of our circuit is. From the example above, you will have seen that we talk about precision in both absolute terms, such as ±0.2V, or in relative terms using percentages (±2%). We also use parts per million (ppm) for relative precision when the numbers get very small; for example, 0.01% equals 100ppm. If we have extremely good precision, we might even talk about parts per billion (ppb)! We can always measure the precision of a circuit after it is built, but we have just seen that one sample isn’t enough. Also, we usually want to be sure our design will meet the precision targets before we commit to mass manufacture. Precision circuit design is the process of keeping careful track siliconchip.com.au of errors and uncertainties and how they accumulate to impact the overall precision of the circuit of interest. Sources of uncertainty Before we get into a practical example, it might help to understand where these errors and uncertainties come from. Many errors result from complex interactions of various causes, but it helps to think of them in some broad categories: Limitations of physics Real-world limitations introduce errors. For example, there is no such thing as a perfect insulator, so leakage currents occur. It is impossible to source or sink infinite current, so devices must have some finite output impedance, which means outputs will change with load. Noise Another inescapable result of physics is the electrical noise caused by the random movement of electrical charges in certain materials. This can significantly impact measurements involving small quantities (microvolts and microamps, or even nanovolts and nanoamps!). Noise is a whole topic in itself that we will cover later in this series. Temperature Sadly, almost everything in electronics changes with temperature, and usually not for the better. Resistor values change, noise increases and offsets drift. The wider the temperature range your device will be subject to, the more this will be a problem you must address. Frequency and time Like temperature, frequency changes almost everything. A parameter specified at DC may vary considerably as frequency increases. Some things get worse over time, too. MLCC capacitors lose capacitance with age, and even the frequency of crystals can drift over time. It’s not the biggest problem you will likely encounter, but it is worth being aware of. Manufacturing variation Even a well-designed component, using the best materials and a good manufacturing process, will have some degree of variation between parts. It is impossible to make them all siliconchip.com.au absolutely identical. Common examples include resistor tolerances and op amp input offset voltages. There will be a natural spread of these values around a mean (the nominal resistance for resistors or 0mV for op amp offset voltages). Understanding component limitations There are no perfect components, just as there are no perfect circuits. Optimising for one parameter may have a detrimental effect on another. One example that springs to mind is the common multi-layer ceramic capacitor (MLCC). Many of these use a dielectric material that allows the manufacturers to cram a huge amount of capacitance into a tiny volume for a ridiculously low price. The downside is that the capacitance is highly sensitive to temperature, applied voltage and ageing. The component variation with these conditions can easily be two or three times the nominal tolerance of the capacitance. That is the price you pay for 10¢ 10µF 0402-size capacitors. Sticking with the example of ceramic capacitors, do you know what it means when a capacitor is labelled X7R, X5R, Y5V, C0G, NP0 etc? It is related to the temperature range and how much the capacitance varies over it, but it is actually much more than that. For example, these codes also affect how capacitance changes with voltage. This shows why it pays to do your homework! Manufacturers are not always as forthcoming about a part’s limitations as they are about its features (especially on the front page of the data sheet). Be wary of typical values compared to worst-case values. You must read the data sheets carefully and thoroughly. Don’t just read the data tables – often, the graphs give useful information about how a device will perform that is quite different from the flattering conditions under which the nominal values are derived. A practical example Despite all this, it is, of course, possible to design high-precision circuits, and there are a few handy tricks that can help us get there. To get started, we will use a simple example that we can build upon in subsequent articles. Imagine we are designing a DC power Australia's electronics magazine Fig.2: our first attempt at a current-measuring circuit. The 0-1A current to be measured (Il) flows through Rs and the resulting voltage is amplified by IC1 to produce a 0-2.5V output. It uses regular 1% resistors and a lowcost rail-to-rail op amp. supply to power a microcontroller-­ based circuit. We want to measure the current consumed by our device over the range of 0A to 1A. We would ultimately like to measure currents down to the microamp level (or lower) if possible, since our device may go into sleep mode. This isn’t easy to achieve. We will develop the idea over the next few articles, but let’s start by working out what sort of performance is possible with some very basic components and a straightforward circuit. Fig.2 shows the circuit we will begin with. On the left is a 0.1W resistor used as a current shunt. For the time being, we will assume it is ground-referenced. This shunt will drop 100mV across it at the full 1A load. We need to amplify this signal to get it into the range of an analog-to-digital converter, say to around 2.5V, which means we need an amplifier gain of 25. I have used a low-cost general-­ purpose rail-to-rail input and output (RRIO) op amp, the LM7301, to start with since its inputs and outputs can swing to the rails. We’ll also use standard 1% tolerance resistors to set the gain. Initially, we will power this part of the circuit with a single 5V supply. To estimate the precision that we can expect from this circuit, we need to move through the circuit one element at a time, find its contribution to the overall error and sum them somehow. We will take this very slowly initially to illustrate the process. At node A, we will see a voltage proportional to the load current but with some uncertainty due to the resistor November 2024  43 Parameter Test Conditions TYP MAX Ta = 25°C 0.03mV 6mV Ta = Tj N/A 8mV 2μV/°C Measured Data Error Current Vout Abs. Rel. 0.0 25.0 25.0 1.0% N/A 99.7 251.9 2.7 0.1% Fig.3: this extract from the LM7301 data sheet shows the expected input offset voltage. At 25°C, it is specified to be ±30µV (typical) and ±6mV (maximum) – quite a range! I suggest using the latter in your designs. 199.8 515.2 15.7 0.6% 299.7 769.6 20.4 0.8% 399.9 1021.3 21.6 0.9% 499.9 1272.5 22.8 0.9% 599.9 1523.9 24.2 1.0% 699.9 1777.0 27.3 1.1% 800.0 2030.1 30.1 1.2% 900.0 2282.1 32.1 1.3% 1000.0 2533.3 33.3 1.3% Vos – input offset voltage TCVos – input offset voltage average drift Ta = Tj Adding two quantities with errors: (z + Δz) = (x + Δx) + (y + Δy) = (x + y) + (Δx + Δy) → z = x + y, Δz = Δx + Δy Multiplying two quantities with errors: (z + Δz) = (x + Δx)•(y + Δy) = x•y + x•Δy + y•Δx + Δx•Δy → z = x•y Δz Δx Δy and ≈ + Δz ≈ x•Δy + y•Δx z x y Fig.4: when adding or subtracting quantities with uncertainties, the uncertainty of the result is the sum of the absolute uncertainties, shown at the top. When multiplying or dividing, the uncertainty of the result is approximated by the sum of relative uncertainties, shown below. Table 1 – measured results from the Fig.2 circuit using a single supply (+5V). Units: Current (mA), Vout (mV), Absolute (mV), Relative (%). tolerance. The resistor tolerance is 1%, so it will have an absolute resistance value of 100±1mW. We will therefore see a voltage across it of 100±1mV at full load. We will also see the op amp’s input offset voltage appearing at node A. Fig.3 shows the relevant extract from the LM7301 data sheet. The input offset voltage at 25°C is specified to be ±30µV (typical) and ±6mV (maximum). The maximum offset is more than 100 times the typical figure! We will use the worst-case value for reasons I will discuss below. We now have two quantities (voltage across the resistor and the op amp offset voltage), each with its own uncertainty, that we need to sum. The error in the total value will simply be the sum of the absolute errors of each part. This probably seems obvious, gain-setting resistors will be 25±2%, or 25±0.5 in absolute terms. The total error at the circuit output (Node B) will therefore be the sum of the relative errors of the Node A voltage (±7%) and the gain (±2%), or ±9%. This corresponds to about ±225mV absolute error in the 2.5V full-scale signal. Clearly, that is not acceptable. The op amp offset voltage is the biggest contributor by far and is pretty easy to deal with. But how will this circuit perform in real life? but you can see the maths that proves it in Fig.4. That figure also shows the less obvious result: that the total error when two quantities are multiplied is approximated by the sum of the relative errors of each quantity. The approximation works because we can ignore the Δx•Δy term if the errors are small. This leads to an important rule for precision circuit design: If adding or subtracting quantities, sum the absolute errors; if multiplying or dividing, sum the relative errors. So, back to our circuit. Summing the absolute errors at node A gives a total error of ±7mV. You can probably already see this is a potential problem (no pun intended), but let’s keep going. At node B, we will see the voltage at node A multiplied by the gain of the op amp stage. The gain with two 1% Practical results I built this circuit and measured the results shown in Table 1. You won’t be surprised that they are much better than the worst-case estimate of ±9%. This is because the errors result from statistical variation, and there is a much higher probability that any given Fig.5: at left is a plot of the measured results from the Fig.2 circuit; note the subtle kink in the curve near zero. The closeup on the right clearly shows that the output is too high at 0A due to the op amp’s limited output swing. 44 Silicon Chip Australia's electronics magazine siliconchip.com.au Measured Data Error Measured Data Error Current Vout Abs. Rel. Current Vout Abs. Rel. 0.0 -41.5 -41.5 -1.7% 0.0 12.8 12.8 0.5% 97.9 203.7 -41.1 -1.6% 97.9 253.9 9.2 0.4% 198.2 454.6 -41.9 -1.6% 198.2 500.7 5.2 0.2% 298.3 693.3 -52.5 -2.1% 298.3 735.5 -10.3 -0.4% 398.3 944.1 -51.7 -2.1% 398.3 982.1 -13.6 -0.5% 498.3 1197.2 -48.6 -1.9% 498.3 1231.1 -14.7 -0.6% 598.3 1447.5 -48.3 -1.9% 598.3 1477.2 -18.5 -0.7% 698.0 1728.3 -16.7 -0.7% 698.0 1753.4 8.4 0.3% 798.0 1982.2 -212.8 -0.5% 798.0 2003.1 8.1 0.3% 898.0 2235.2 -9.8 -0.4% 898.0 2252.0 7.0 0.3% 998.0 2488.8 -6.2 -0.2% 998.0 2501.4 6.4 0.3% Table 2 – raw results from the Fig.6 circuit with a dual supply (±5V). Table 3 – the Table 2 data after applying a fixed offset and gain corrections. sample will be near the mean or nominal value than an outlier. The full-scale error was 33mV, or 1.3%, and the errors reduce at lower currents except at the bottom of the range, where there seems to be some kind of anomaly. You can see this also in the plot of the results in Fig.5, on the left. The full set of results looks OK except for the zero-current reading, which is slightly off. The first three readings, along with the ideal response, are shown on the ‘zoomed in’ plot on the right of Fig.5. There is clearly a problem at or near zero current. We know the op amp offset voltage is not causing this, because that would appear as a consistent vertical shift of the measurements above or below the ideal line. It is not caused by gain error, because that would appear as a variation in the slope compared to the ideal line. Something else is going on – there is a small but definite ‘bend’ in the measured results at the bottom end. The culprit is the op amp’s output swing. While the LM7301 claims to be a “rail-to-rail” output op amp, a close look at the data reveals that with a 5V supply and a 10kW load, the output typically won’t go below 70mV (and isn’t guaranteed to go below 120mV). We are measuring 25mV, which is better than claimed. This is a very good swing, better than most op amps, but it isn’t rail-to-rail as advertised! We would rather avoid non-linearities like this because they are harder to deal with than purely linear errors such as fixed offsets or gain errors, as we shall see. I refined my circuit by adding a negative supply rail (Fig.6). Running the tests again produced the data shown in Table 2 and plotted in Fig.7. In some ways, this looks worse than our first test! The most significant error is just over -52mV or 2.1% of full scale. This error occurred mid-scale, with the absolute error at zero being -42mV; at full scale, it is only -6mV (0.2%). The good news is that the points are fairly linear. The dotted line in Fig.7 is a line of best fit, using the equation shown on the graph. This line suggests there is a fixed offset error of -54.5mV and a gain error (the difference between the slope of the line and the ideal slope of 2.5) of about 1.7%. The fixed error comes mainly from the op amp’s offset voltage, which must be around -2.2mV (taking the gain of 25 into account). The gain error comes largely from the resistor tolerances. The good news is that there is no longer a bend in the plot. Note that the op amp offset is less than the quoted worst-case figure (±6mV), but by no means does it fall within the typical figure of ±30µV. This is just one sample, but it does illustrate the danger of assuming your results will match the ‘typical’ figures in the data sheet. We will improve this result next time by selecting a ‘better’ op amp and tighter tolerance resistors. But just for a moment, let’s look at another solution. We could compensate for both of these errors (offset & gain) by adding a fixed correction – either through analog trimming or, more likely these days, in software on the microcontroller. Just because we can, let’s look at how much we could improve these readings by applying gain and offset correction using the values from the Fig.6: powering the op amp from dual supply rails (±5V) fixes its output swing problem. Otherwise, this circuit is identical to Fig.2. Fig.7 (right): the measured result of the Fig.6 circuit, along with a calculated line of best fit (dotted). There is now a fixed offset and gain error that can be trimmed out in either the analog or digital domains. siliconchip.com.au Australia's electronics magazine November 2024  45 line of best fit. Table 3 shows the corrected results. Now the absolute error is never worse than about ±20mV, or 0.75% of full scale. Not bad, given the parts we have chosen. This is one of the big secrets of precision design. You can usually trim out fixed offset or gain errors to some significant degree. The emphasis should be on the word “fixed”. It’s way more difficult to trim out non-linearities or errors that change over time, such as temperature drift. Temperature effects To examine the effect of temperature, I want to introduce the idea of the error budget table. This is just a way of capturing the uncertainties we discussed above in a neat tabular form. Table 4 shows an example. You can use any format you like, but this is how I generally do it. Under the “At Nominal 25°C” section, you will see each step we went through in the above example, capturing the nominal value and relative and/or absolute uncertainty. For example, Line 1 is the shunt resistor and Line 3 is the op amp offset. Lines 2 and 4 are calculated values and are shown in bold text. I always show both the absolute and relative errors on calculated lines. At Line 8, we get to the ±225mV and ±9% error figures calculated above. The second part of the table brings the temperature-dependent errors into the picture. We obviously have to know the temperature range of interest to calculate these uncertainties. I have chosen a range of 0°C to 50°C (±25°C either side of the nominal 25°C) in this example. The data sheet for the shunt resistor I used (Stackpole CSR1225) tells me that its temperature coefficient (tempco) is 100ppm/°C. This means we will see a resistance change of up to ±2500ppm or ±0.25% over the range of interest on top of the 1% tolerance. Similarly, the op amp’s offset voltage has a drift of ±2µV/°C, corresponding to ±50µV. This is already more than the ±30µV ‘typical’ offset at 25°C claimed in the data – another reason to take ‘typical’ values with a grain of salt. If we continue with the rest of the analysis in the same way, we arrive at a variation of about ±0.8% over the proposed operating temperature range. Even if we could trim out all of the 25°C error in software, we are left with a temperature-dependent error approaching 1%. We will look at how we can reduce this in further instalments. Optimist or pessimist? One objection that frequently comes up when we are summing worst-case errors in this way is that we are being overly pessimistic in our design. We are assuming that errors will accumulate in the worst possible way. For example, we have assumed that our gain error is 2%, which would only be the case when both gain-setting resistors are at the extremes of their tolerances and in opposite directions. If they were both high or low by the same percentage, this would cancel out, and the gain would be unaffected. Is it reasonable to take this pessimistic view? What if our circuit had 10 gain-setting resistors instead of two? Would it be reasonable to assume they would all be at their tolerance extremes in the worst way? There is no correct answer to the question, but I can suggest some guidelines. Uncertainty is a statistical game – it’s all about probabilities and consequences. If the likelihood of the worst case occurring is low and its consequences are not severe, it is probably OK to make some concessions. But if the probability of an error Table 4: Error Budget Table for our Application occurring is high (eg, if you are making a lot of something), or the consequences of any errors are significant (dangerous, expensive or embarrassing), a cautious approach is better. One concession you might choose to make is to assume that the sources of error are uncorrelated. In such cases, it is possible to add errors (absolute or relative) as the root sum of squares. In our example of 10 gain-setting resistors, each with a 1% tolerance, we would come up with a gain error of ±3.1% instead of 10%. But I urge caution. The root sum of squares is just another statistical tool – it works best when there are a great many samples in a truly random and uncorrelated distribution. We do use this type of summation for noise, which fits these criteria, as we shall see in a later article. Remember that if some resistors have the same value, they will likely come from the same batch. In fact, they will probably have been manufactured sequentially. So they will very likely be off by roughly the same amount and in the same direction. In other words, the errors won’t be uncorrelated at all! In some cases, that can help you; eg, if you’re relying on matched resistor values. Still, you must examine the specific circuit to determine whether correlated errors will help or hurt your precision. Summary At this stage, it has become clear that our simple circuit is probably not up to the job of monitoring the current in our supply if we want anything better than a couple of percent resolution. We can trim out the worst of the ±9% error down to a little better than 1%, but we will have another 1% or so of error over the temperature range. This 2% error means a ±20mA uncertainty. We’ll have to do better next time! SC At Nominal 25°C Error Nominal Value Shunt Resistor: Stackpole CSR1225 (1% 100ppm/°C) 100mW Node A Voltage due to I × R shunt 100mV 1mV Op Amp: LM7301 (Vos ±6mV, 2μV/°C) 0mV 6mV Node A Voltage total (Line 2 + Line 3) 100mV 7mV Op Amp Gain Resistor R1: Yageo RC0805 (1% 100ppm/°C) 1kW 1.00% 0.25% Op Amp Gain Resistor R2: Yageo RC0805 (1% 100ppm/°C) 24kW 1.00% 0.25% Op Amp Gain (R1 + R2) ÷ R1 25 0.5 2.00% 0.125 0.50% Vout (Line 4 × Line 7) 2.5V 0.225V 9.00% 0.02V 0.80% 46 Silicon Chip Abs. Error Rel. Error 0-50°C (Nominal ±25°C) Abs. Error 1.00% Australia's electronics magazine 1.00% Rel. Error 0.25% 0.25mV 0.25% 0.05mV 7.00% 0.3mV 0.30% siliconchip.com.au