Fullscreen HTML5Med Res (??MB)HTML4

Precision Electronics Part 5: Noise So far, this series has mostly been concerned with errors arising from component matching and unwanted currents and fixed voltages. There is another type of unwanted signal that can cause all sorts of problems that we refer to as noise. So, how can we quantify its effects on our circuits and reduce the resulting errors? By Andrew Levido I n electronics, the term “noise” can refer to any form of unwanted signal that masks a signal of interest. This includes noise that is imposed upon a circuit from external sources such as radio-frequency interference or mains hum, which can be reduced or eliminated by filtering, shielding or other design techniques. However, there are sources of noise that are intrinsic to the components themselves. They come from within the circuit, not outside, so they cannot be reduced by shielding. In this article, we will look at this type of truly random noise that is caused by various physical phenomena within the components we use. As we have discussed in earlier articles, precision electronics design is all about understanding and quantifying the sources of uncertainty, and noise is another error source that we cannot always ignore. To understand noise, we will have to start with a bit of theory, but I will try to keep it to a minimum. We will then get into the practical side with a full noise analysis of a simple audio amplifier. Let’s begin by looking at the main types of noise of concern to electronics designers. Johnson noise Johnson noise, sometimes also called Nyquist or thermal noise, is essentially the electrical signal produced in lossy components by the random movement of charge carriers (usually electrons) due to temperature. Remember that temperature relates to the motion of atoms and other elements of a material; they are only still when the material is at absolute zero. For example, a 10kW resistor at room temperature will develop a noise voltage across its terminals of 76 Silicon Chip around 1.3µV RMS if measured with a high-impedance AC voltmeter with a 10kHz bandwidth. If we were to short-circuit the resistor, we would measure a noise current of about 130pA (1.3µV ÷ 10kW) over the same bandwidth. This phenomenon was first described by John Bertrand Johnson in 1927 and characterised by Harry Nyquist in 1928. Nyquist showed that the noise voltage density in a resistor is given by the equation Vrms = √4kTRfb, where k is Boltzmann’s constant (1.38 × 10-23), T is the absolute temperature in Kelvin, R is the resistance and fb is the bandwidth in hertz (Hz). This highlights the first important thing to keep in mind when discussing noise: we can only quantify a noise voltage or current if we also specify a bandwidth over which to measure it. If we don’t know the bandwidth of concern, we can only describe noise in terms of a voltage or current per unit frequency, called the voltage or current noise density. The units of voltage noise density are V/√Hz and those for current noise density are A/√Hz. You have to be careful to distinguish between an absolute value of noise (an RMS [root-mean-squared] voltage or current) and its density. The relationship between them is analogous to the relationship between the mass and density of a material. Density is a property intrinsic to the material, but the mass of an object made of the material depends on the amount of it we are dealing with in a specific case. The noise voltage density of a resistor, for example, is a property of the resistor at a given temperature, but the noise voltage developed across it depends on the bandwidth we use to measure it (or over which it has an effect). In the case of Johnson noise Australia's electronics magazine (and any white noise, as we will see below), the relationship between noise density and voltage is the square root of the bandwidth. Johnson noise is an inescapable result of the thermal agitation of electrons that occurs anywhere that charges are free to move. Fortunately, you can normally ignore the Johnson noise generated in conductors like wires, since their resistance is so low that any noise that they may contribute is negligible. In fact, mostly lossless devices like capacitors and inductors (up to a point) do not contribute to the overall noise in a circuit. They can, however, impact bandwidth and therefore can influence the noise voltage or current. Shot noise Johnson noise occurs even when no current is flowing. On the other hand, shot noise occurs because a flowing current is made up of discrete ‘chunks’ of charge (electrons or holes). If the moving charges act independently of each other, the resulting randomness of the current flow causes noise. This phenomenon does not occur in metallic conductors, where the moving electrons influence each other and therefore don’t act randomly. It does occur in semiconductors, though; for example, when charge carriers are diffusing across a semiconductor junction. The shot noise density is given by the formula in = √2qIdc and is expressed in units of A/√Hz. Here, q is the charge on an electron (1.6 × 10-19C) and Idc is the average current. The RMS value of shot noise is therefore irms = √2qIdcfb . This means that a steady 1A current passing through a semiconductor junction will see a random variation of 57nA RMS when measured over a siliconchip.com.au 10kHz bandwidth. That corresponds to about 0.057ppm (parts per million) – small enough to be immaterial in most situations. Shot noise gets relatively larger as the current reduces because there are fewer moving charge carriers. For example, a 1µA current will have a superimposed shot noise of 57pA RMS, which corresponds with 57ppm; it is becoming more significant. 1 ∕f noise Shot and Johnson noise are both types of ‘white noise’, ie, they carry equal power per unit of frequency (hertz) across the spectrum. This is why we can simply multiply the noise density by the square root of the bandwidth to calculate the noise voltage or current. 1∕ noise (sometimes called flicker f noise) differs from Johnson or shot noise, which are the result of atomic-­ level physical phenomena. 1∕f noise is created by a variety of mechanisms (not all well understood) relating to materials and construction techniques. 1∕ noise has the defining characf teristic of having an equal energy per decade of spectrum. In other words, it has the same power in the 1-10Hz range as it does in the 10-100Hz and 1-10kHz ranges. This means the power per hertz is inversely proportional to frequency – hence the 1∕f name. Noise with this power spectrum is known as ‘pink noise’ and it occurs in a wide variety of places, including the flow of traffic, ocean currents and the loudness profile of classical music! From an electronics perspective, 1∕ noise does occur in some resistors f depending on their construction (carbon composition resistors were notoriously bad for this) but it can usually be safely ignored in modern resistors. It can become significant in op amp circuits, which is why we need to know about it. Burst and avalanche noise There are two other sources of noise in electronics that are worth mentioning: burst noise and avalanche noise. Burst noise, also known as popcorn noise, is a low frequency (<100Hz) ‘popping’ phenomenon caused by manufacturing imperfections in semiconductor materials. It used to be a problem in the early days of integrated circuits, but improved manufacturing processes have all but eliminated it. Avalanche noise is similar to shot noise but occurs during the reverse breakdown of semiconductor junctions. Its amplitude can be very high, so it is often used when we want to deliberately to create an analog white noise source. We don’t usually operate our circuits with semiconductor junctions in reverse breakdown, so we can ignore avalanche noise most of the time. There is one major exception: zener diodes with voltage ratings above about 5.5V operate in a controlled avalanche breakdown mode. Those rated below 5.5V use the Zener effect, which is a different phenomenon altogether (interestingly, a 5.6V zener diode uses a mixture of both!). If you have these in your circuits, you may have to take avalanche noise into account. Now we have covered the sources of noise, we have to consider one more important point that will allow us to analyse noise in practical circuits. Gaussian distributions Johnson, shot and 1∕f noise are all considered Gaussian, which means the amplitude of instantaneous voltage is distributed according to a Gaussian, sometimes called ‘normal’, curve as shown on the left of Fig.1. The average amplitude of noise is zero, but the peak value at any given instant will be a matter of probability. Higher-amplitude excursions are less likely than those close to the mean due to the ‘bell’ shape of the Gaussian distribution. The probability that the instantaneous voltage will be between any two values is given by the area under the curve between those values. Statisticians love these curves, but I don’t find them a particularly intuitive way to look at amplitude. I prefer the relative occurrence chart on the righthand side of Fig.1. The vertical scale is the fraction of time the instantaneous voltage will exceed some multiple of the RMS voltage. For example, the instantaneous amplitude will exceed twice the RMS voltage approximately 4.6% of the time, but will exceed four times the RMS voltage only 0.006% of the time. You can also see from the occurrence chart that the instantaneous noise voltage will be below the RMS value approximately 2/3 of the time. This, plus the fact that the different sources of noise in a given circuit will be uncorrelated (they behave completely Fig.1: the probability of a given instantaneous voltage occurring in white noise is distributed according to a Gaussian curve (left). The relative occurrence curve on the right instead shows the fraction of time that the peak voltage will exceed a specific amplitude, which I find easier to understand. siliconchip.com.au Australia's electronics magazine March 2025  77 Fig.3: the noise equivalent circuit for a parallel RC circuit. The resistor and capacitor form a low-pass filter for the resistor’s Johnson noise source. The resulting noise voltage depends only on the temperature and the capacitor value. Fig.2: the noise equivalent circuit of a resistor is a noiseless resistor in series with a voltage source with a value given by the Johnson noise equation. independently), means that adding the RMS values of different noise sources will over-estimate the resulting noise. So, instead of adding noise in the normal arithmetic way, we add noise as the root sum of squares. This means we square the values, add them together, then take the square root to arrive at a value that is statistically equivalent to the sum. If we are adding more than two noise sources, we just extend the sum by adding more squared values before taking the square root. The root sum of squares method leads to a shortcut that can help simplify noise calculations significantly. If one of the quantities being added is smaller than another by a factor of 10 or more, you can ignore it altogether with very little resulting error. For example, a 10µV and 1µV source sum to 10.05µV using root-sum-of-squares, so we could simply ignore the 1µV source in that case. A simple example Noise calculations in circuits can get quite complex since almost every component contributes to or shapes noise. For this reason, it is important to take a step-by-step approach, breaking the circuit down into manageable chunks. For each ‘chunk’ of circuit, we need to analyse the ‘noise equivalent circuit’ to calculate the total noise. We must understand the noise equivalent circuits of the components to accomplish this. The noise equivalent circuit of a resistor (Fig.2) is a good place to start. We have already seen that a resistor will exhibit Johnson noise with a voltage of √4kTRfb so it can be modelled by a noiseless resistor in series with a voltage source of that value. Resistors in parallel or series can be reduced to a single equivalent resistor, then converted to the noise equivalent circuit. We mentioned above the capacitors 78 Silicon Chip don’t themselves contribute noise, but Fig.3 shows the noise equivalent circuit of a capacitor in parallel with a resistor. If we replace the resistor with its noise equivalent circuit from Fig.2, we have a noise source feeding a simple single-pole RC low-pass filter. This filter will reduce the noise bandwidth and thus the noise voltage seen at the terminals of the RC pair. The -3dB bandwidth of the RC filter will be 1 ÷ (2πRC), but we can’t just use this in noise calculations since such a filter lets through frequencies higher than the corner frequency, albeit in an attenuated form. We therefore need to convert the -3dB bandwidth to an ‘equivalent noise bandwidth’ (ENBW), which is the bandwidth of a perfect ‘brick wall’ filter that lets through the same amount of noise. The scale factor for a single-pole filter turns out to be π ÷ 2, so the ENBW for a single-pole RC filter is 1 ÷ (4RC). Substituting this into the equation for Johnson noise gives an expression for the resulting noise voltage: Vrms = √kT ÷ C. A parallel RC circuit therefore has the noise equivalent circuit shown on the right in Fig.3. Notice that the noise voltage is dependent only on the temperature and the capacitor value – the resistor value is irrelevant! Noise can be weird sometimes. Op amp noise equivalent circuit A typical op amp exhibits Johnson and shot noise with a flat power spectrum, as well as 1∕f noise, which has a power spectrum biased towards lower frequencies as shown in Fig.4. At low frequencies, the 1∕f noise dominates, while at high frequencies, the Johnson and shot noise dominate. At some frequency, fc, the amplitudes of these two noise components will cross over. When specifying op amp noise, manufacturers condense everything down to three things: a figure or graph for fc, an input noise voltage density (en) referred to the non-­ inverting input, and an input noise current density (in) at each input. Fig.5 shows the noise equivalent circuit of an op amp. It consists of a noiseless op amp with noise sources at its inputs. These current sources Fig.4: op amps exhibit both pink (1∕f ) noise, which dominates at low frequencies, and white (Johnson and shot) noise, which dominates at high frequencies. The frequency at which they cross over is the noise corner frequency, fc. This figure is usually specified in the data sheet. Australia's electronics magazine siliconchip.com.au Fig.5: the noise equivalent circuit of an op amp includes noise current sources associated with each input of a noiseless op amp, plus a noise voltage source in series with its non-inverting input. Fig.6: a simple audio amplifier stage circuit we will analyse for noise performance. produce noise voltages across the source resistances at the op amp’s inputs. The noise crossover frequency is used together with these sources to calculate the overall op amp noise. shown in row 1 of the noise budget table (Table 1). Next, we have the op amp’s noise voltage, shown in Fig.7(c), which is given by the voltage noise density (11nV/√Hz) and the bandwidth. We have seen that for white noise, we simply multiply the noise density by the square root of the circuit bandwidth to get the voltage. However, we know that the op amp produces pink 1∕f noise up to 150Hz – well within our 1Hz to 25kHz bandwidth. We take this into account by applying a modification factor to the bandwidth, a bit like we did to determine the ENBW from the -3dB bandwidth. When the bandwidth of interest straddles fc, we have to use the formula fb + fcloge(fh ÷ fl) to calculate an equivalent bandwidth. In this equation, fb is the nominal bandwidth, fc is the noise corner frequency, fh is the upper bandwidth limit and fl is the lower limit of bandwidth. So the op amp noise bandwidth evaluates to 26.5kHz instead of 25kHz. Plugging this into the op amp’s input noise density gives us a noise voltage at 1.79µV. We can now consider the noise voltage generated by the op amp’s current noise. This is a bit trickier. Fig.7(d) shows the superposition circuit for this source. We have a current source in parallel with a resistor A practical example Let’s apply this to a real-world example. Consider a simple op-amp based audio amplifier circuit, as per Fig.6. The input signal is AC-­coupled to the op amp via C1. This forms a high-pass filter with R1, which has a -3dB cutoff frequency of about 1.6Hz. The op amp is configured as a non-inverting amplifier with a signal gain of 11. C2 rolls off the amplifier’s frequency response at around 16kHz. The example circuit uses a TLV2460 general-­ purpose rail-to-rail input/output (RRIO) op amp with en = 11nV/√Hz, in = 0.13pA/√Hz and fc = 150Hz. We will analyse the noise in this amplifier in a step-by-step manner, using the principle of superposition. This analysis technique allows us to analyse linear circuits with multiple sources by calculating the effect of each one in isolation and adding the results. We replace any voltage sources we are ignoring with short circuits, and any current sources we are ignoring with open circuits. We will build up a noise budget table (Table 1) as we go. Before we start, we need to define the nominal bandwidth over which we will calculate the noise. Since our circuit has a single-pole 1.6Hz highpass filter at the input and a single-pole 16kHz low-pass filter around the op amp, our overall bandwidth is well defined. However, just like the parallel RC case above, we have to calculate the equivalent noise bandwidth, since the roll-offs are far from abrupt. Recalling that the ENBW scale factor for a single pole filter is π ÷ 2, the lower ENBW limit becomes 1Hz and the upper one becomes 25kHz. So, the overall noise bandwidth of the circuit is 25kHz – 1Hz ≈ 25kHz. We start our analysis at the non-­ inverting input of the op amp. Since the source voltage is short-circuited, R1 and C1 are in parallel as far as noise is concerned. Fig.7(a) shows the noise equivalent circuit of this part of the circuit. There are three noise sources: the R1/ C1 parallel noise voltage, the op amp’s voltage noise source and the current noise source, both at the non-­inverting input of the op amp. We will look at each in turn and use the superposition principle. Fig.7(b) shows the R1/C1 source. We have already seen that the noise voltage in this case is √kT ÷ C. Plugging in a temperature of 300K (26.85°C) and the 1µF value of C1 gives a noise voltage of 64.3nV. This calculation is Raspberry Pi Pico W BackPack The new Raspberry Pi Pico W provides WiFi functionality, adding to the long list of features. This easy-to-build device includes a 3.5-inch touchscreen LCD and is programmable in BASIC, C or MicroPython, making it a good general-purpose controller. This kit comes with everything needed to build a Pico W BackPack module, including components for the optional microSD card, IR receiver and stereo audio output. $85 + Postage ∎ Complete Kit (SC6625) siliconchip.com.au/Shop/20/6625 siliconchip.com.au Australia's electronics magazine March 2025  79 Fig.7: the process of analysing the input stage noise equivalent circuit for Fig.6(a) shows all the noise sources together and the subsequent circuits show how the noise contribution of the individual sources are evaluated. 80 Silicon Chip Fig.8: the process for analysing the noise sources at the op amp’s output is similar to that for the input. This time, there are four noise sources, including the input noise multiplied by the op amp’s noise gain. Australia's electronics magazine (and a capacitor, but let’s park that for a moment), so we can replace these with the Thévenin equivalent voltage source in series with the resistor, as shown in Fig.7(e). In case you are not aware of Thévenin’s theorem, it states, “Any linear electrical network containing only voltage sources, current sources and resistances can be replaced by a voltage source in series with a resistance.” It is a powerful tool for simplifying circuits and well worth reading about if you don’t understand it (see https://w.wiki/9XaJ). The current noise density is 0.13pA/√Hz and the resistance is 100kW, so the Thévenin noise voltage density is 13nV/√Hz. Now let’s bring the capacitor back into the picture. This forms a low-pass filter with R1, which will impact the bandwidth we should use to calculate the RMS voltage at the op amp’s input. The filter’s ENBW is given by 1 ÷ (4RC), as we saw above, so we use this figure (2.5Hz) in the calculation. The resulting voltage will be 20.6nV (line 3 of the Table 1). To complete the superposition process, we just have to add all three voltages together using root-sum-ofsquares to get a single equivalent noise voltage at the non-inverting input of the op amp. Since the op amp’s input noise voltage is more than an order of magnitude higher than either of the other two, we can safely ignore the others. This leaves us with a total voltage noise voltage at the op amp input of 1.79µV RMS. Now we turn to the op amp output. Fig.8(a) shows the equivalent circuit. This time, there are four sources of noise voltage: the output noise of the op amp due to the amplified input noise we just calculated, two resistor noise voltages (R2 and R3) and the current noise at the op amp’s inverting input. We use superposition again to calculate each contributing part of the noise voltage individually. Fig.8(b) shows that the first of these is easy; it is just the 1.79µV input noise voltage multiplied by a gain of 11, giving 19.7µV. For a non-inverting amplifier, the noise gain is the same as the signal gain, since the noise source is on the same input as the signal. This isn’t necessarily true for all op amp configurations, so you will need to scrutinise each circuit for noise sources. For example, for a standard inverting siliconchip.com.au amplifier configuration, the noise gain equals the signal gain plus one. Remember that the op amp’s voltage noise equivalent is always referred to the non-inverting input. The noise due to R2/C2, as shown in Fig.8(c), is also easy to calculate since this is just another RC parallel circuit that we are already familiar with. The noise voltage here is 2.03µV. The noise due to resistor R3, shown in Fig.8(d), is also easy using the formula for Johnson noise on page 76. This works out to be about 643nV. The noise voltage due to the op amp’s current noise can be calculated using the Thévenin equivalent circuit shown in Figs.8(e) & 8(f). In this case, the noise voltage works out to be 20.6nV. The only trick here is to use the 26.5kHz augmented bandwidth to account for the 1∕f noise. The total noise at the output of the circuit is calculated by root-sum-ofsquares of the voltages. It turns out to be 19.8µV RMS. Since the circuit was designed to deliver signals at 1V RMS, the signal-to-noise ratio (SNR) is 20log10(1V ÷ 19.8µV) ≈ 94dB. Power supply noise This analysis so far ignores power supply noise. Power supply noise will be coupled into the op amp’s output to some degree, although op amp designers try hard to maximise the rejection of such noise. The op amp’s power supply rejection ratio (PSRR) defines the degree to which a disturbance on the power supply rail reaches the output. The TLV2460 has a power supply rejection ratio (at 25°C) of >80dB up to 20kHz. This means any power supply noise will be attenuated by a factor of 10,000 over the frequencies we care about. To calculate the power supply noise contribution, you reduce the RMS noise present on the power supply by the PSRR and add the result as you would for any other noise voltage source on the op amp’s output. Put another way, if we can keep the power supply noise in our example circuit under 20mV RMS, the op amp’s PSRR will reduce this to 2µV at the op amp output. This is one tenth of the ~20µV of noise we just calculated, so will make no meaningful contribution to the overall circuit noise. Minimising noise We have seen that noise is an inescapable phenomenon, with causes linked to the fundamentals of physics. There is nothing we can do to eliminate it, short of cooling our circuit down to absolute zero. However, you can minimise the noise in any circuit using a few techniques. They may not all be possible or relevant in your application, but the following ideas are worth considering. 1. Reduce bandwidth: we have seen that noise voltage depends highly on bandwidth and that reducing bandwidth reduces noise amplitude. You may not be able to reduce the overall bandwidth of the circuit, but even limiting bandwidth in parts of the circuit may help. 2. Use oversampling: this is a kind of bandwidth reduction. If you are measuring a noisy quantity with an ADC, you can take multiple samples and average the results. Since noise is Gaussian with zero mean, the average of several samples tends toward zero as the number of samples increases. 3. Minimise gain: the gain elements in your circuit amplify input-side noise voltages. Use the minimum gain necessary to amplify your signal. 4. Use lower value resistors where possible. A 1kW resistor has a noise voltage density of ~4nV/√Hz, compared to ~40nV/√Hz for a 100kW resistor. A 100W resistor is even better at ~1.3nV/√Hz. 5. Choose the right op amps. There is a huge range of ‘low noise’ op amps and their headline specifications don’t always tell the full story. Low voltage noise density does not always mean low 1∕f noise and vice versa. Depending on your bandwidth, you might need to balance one parameter with another. 6. Pay attention to power supplies. Power supply noise can be coupled into your signal path in all sorts of ways. Linear regulators are generally quieter than switch-mode ones (or you could use a switch-mode regulator with a linear post-regulator). 7. Pay attention to power supply and ground routing, especially where high current circuits are present on the same board. Use decoupling capacitors thoughtfully and consider using LC filters or capacitance multipliers to create a low noise supply to particularly sensitive portions of the circuit. Keep high-current ground networks separate from signal grounds. As a concrete example, refer to our circuit shown in Fig.6. We calculated its SNR as being close to 94dB. If we intended to use it to process an audio signal, we probably want to reduce the noise a bit, for an SNR of at least 100dB. That could most effectively be achieved by using a lower noise op amp. An op amp like the NE5534, for example, with a voltage noise density of 3.5nV√Hz, a current noise density of 1.5pA√Hz and an fc of about 1kHz might be a better choice. A quick estimate using these figures gives an op amp voltage noise (line 2 of the table) of 0.66µV compared to 1.79µV, and the overall circuit noise figure reduces to about 7.5µV giving SC an SNR of just over 102dB. Table 1 – a noise budget for our example audio amplifier circuit Line Noise Source Figure Notes Result (RMS) 1 R1/C1 (100kW, 1μF) 7(b) Parallel RC: Vn = √kT ÷ C 64.3nV 2 Op amp voltage noise (11nV/√Hz, 150Hz) 7(c) 3 Op amp current noise (0.13pA/√Hz) 7(d)(e)(f) Thévenin equivalent and LPF: en = inR1, fb = 1 ÷ (4RC) 20.6nV 4 Total input noise (Lines 1-3) − Root sum of squares, Line 2 dominates 1.79µV 5 Output noise due to input noise 8(b) Line 4 times noise gain of 11 19.7µV 6 R2/C2 (10kW, 1nF) 8(c) Parallel RC: Vn = √kT ÷ C 2.03µV 7 R3 (1kW) 8(d) Resistor: Vn = √4kTRfb 643nV 8 Op amp current noise (0.13pA/√Hz) 8(e)(f) Thévenin equivalent: Vn = inR3√fb + fc loge(fh ÷ fl) 21.2nV 9 Total output noise (Lines 5-8) − Root sum of squares, Line 5 & 6 dominate 19.8µV siliconchip.com.au Bandwidth straddles fc so use Vn = en√fb + fc loge(fh ÷ fl) 1.79µV Australia's electronics magazine March 2025  81