Chapter 1

The Basics

Signals

An audio signal can be transmitted either as a voltage or as a current. The construction of the universe is such that almost always the voltage mode is more convenient; consider for a moment an output driving more than one input. Connecting a series of high-impedance inputs to a low-impedance output is simply a matter of connecting them in parallel, and if the ratio of the output and input impedances is high there will be negligible variations in level. To drive multiple inputs with a current output it is necessary to have a series of floating current-sensor circuits that can be connected in series. This can be done [1], as pretty much anything in electronics can be done, but it requires a lot of hardware, and probably introduces performance compromises. The voltage-mode connection is just a matter of wiring.

Obviously, if there’s a current, there’s a voltage, and vice versa. You can’t have one without the other. The distinction is in the output impedance of the transmitting end (low for voltage mode, high for current mode) and in what the receiving end responds to. Typically, but not necessarily, a voltage input has a high impedance; if its input impedance was only 600 Ω, as used to be the case in very old audio distribution systems, it is still responding to voltage, with the current it draws doing so a side issue, so it is still a voltage amplifier. In the same way, a current input typically, but not necessarily, has a very low input impedance. Current outputs can also present problems when they are not connected to anything. With no terminating impedance, the voltage at the output will be very high, and probably clipping heavily; the distortion is likely to crosstalk into adjacent circuitry. An open-circuit voltage output has no analogous problem.

Current-mode connections are not common. One example is the Krell Current Audio Signal Transmission (CAST) technology, which uses current mode to interconnect units in the Krell product range. While it is not exactly audio, the 4–20 mA current loop format is widely used in instrumentation. The current-mode operation means that voltage drops over long cable runs are ignored, and the zero offset of the current (i.e. 4 mA = zero) makes cable failure easy to detect: if the current suddenly drops to zero, you have a broken cable.

The old DIN interconnection standard was a form of current-mode connection in that it had voltage output via a high output impedance, of 100 kΩ or more. The idea was presumably that you could scale the output to a convenient voltage by selecting a suitable input impedance. The drawback was that the high output impedance made the amount of power transferred very small, leading to a poor signal-to-noise ratio. The concept is now wholly obsolete.

Amplifiers

At the most basic level, there are four kinds of amplifier, because there are two kinds of signal (voltage and current) and two types of port (input and output). The handy word ‘port’ glosses over whether the input or output is differential or single-ended. Amplifiers with differential input are very common – such as all op-amps and most power amps – but differential outputs are rare and normally confined to specialized telecoms chips.

The four kinds of amplifier are summarized in Table 1.1.

Voltage Amplifiers

These are the vast majority of amplifiers. They take a voltage input at a high impedance and yield a voltage output at a low impedance. All conventional op-amps are voltage amplifiers in themselves, but they can be made to perform as any of the four kinds of amplifier by suitable feedback connections. Figure 1.1(a) shows a high-gain voltage amplifier with series voltage feedback. The closed-loop gain is (R1 + R2)/R2.

Transconductance Amplifiers

The name simply means that a voltage input (usually differential) is converted to a current output. It has a transfer ratio A = I_out/V_in, which has dimensions of I/V or conductance, so it is referred to as a transconductance or, less commonly, a transadmittance amplifier. It is possible to make a very simple, though not very linear, voltage-controlled amplifier with transconductance technology: differential-input operational transconductance amplifier (OTA) Integrated circuits (ICs) have an extra pin that gives voltage control of the transconductance, which when used with no negative feedback gives gain control (see Chapter 19 for details). Performance falls well short of that required for quality hi-fi or professional audio. Figure 1.1(b) shows an OTA used without feedback; note the current-source symbol at the output.

Current Amplifiers

These accept a current in, and give a current out. Since, as we have already noted, current-mode operation is rare, there is not often a use for a true current amplifier in the audio business. They should not be confused with current feedback amplifiers (CFAs), which have a voltage output, the ‘current’ bit referring to the way the feedback is applied in current mode [2]. The bipolar transistor is sometimes described as a current amplifier, but it is nothing of the kind. Current may flow in the base circuit but this is just an unwanted side-effect. It is the voltage on the base that actually controls the transistor.

Transimpedance Amplifiers

A transimpedance amplifier accepts a current in (usually single-ended) and gives a voltage out. It is sometimes called an I–V converter. It has a transfer ratio A = V_out/I_in, which has dimensions of V/I or resistance. That is why it is referred to as a transimpedance or transresistance amplifier. Transimpedance amplifiers are usually made by applying shunt voltage feedback to a high-gain voltage amplifier. An important use is as virtual-earth summing amplifiers in mixing consoles (see Chapter 17). The voltage-amplifier stage (VAS) in most power amplifiers is a transimpedance amplifier. They are used for I–V conversion when interfacing to digital-to-analog converters (DACs) with current outputs (see Chapter 21). Transimpedance amplifiers are sometimes incorrectly described as ‘current amplifiers’.

Figure 1.1(c) shows a high-gain voltage amplifier transformed into a transimpedance amplifier by adding the shunt voltage feedback resistor R1. The transimpedance gain is simply the value of R1, though it is normally expressed in V/mA rather than ohms.

Negative Feedback

Negative feedback is one of the most useful and omnipresent concepts in electronics. It can be used to control gain, to reduce distortion and improve frequency response, and to set input and output impedances, and one feedback connection can do all these things at the same time. Negative feedback comes in four basic modes, as in the four basic kinds of amplifier. It can be taken from the output in two different ways (voltage or current feedback) and applied to the amplifier input in two different ways (series or shunt). Hence there are four combinations.

However, unless you’re making something exotic like an audio constant-current source, the feedback is always taken as a voltage from the output, leaving us with just two feedback types, series and shunt, both of which are extensively used in audio. When series feedback is applied to a high-gain voltage amplifier, as in Figure 1.1(a), the following statements are true:

• Negative feedback reduces voltage gain.

• Negative feedback increases gain stability.

• Negative feedback increases bandwidth.

• Negative feedback increases amplifier input impedance.

• Negative feedback reduces amplifier output impedance.

• Negative feedback reduces distortion.

• Negative feedback does not directly alter the signal-to-noise ratio.

If shunt feedback is applied to a voltage amplifier to make a transimpedance amplifier, as in Figure 1.1(c), all the above statements are still true, except since we have applied shunt rather than series negative feedback, the input impedance is reduced.

The basic feedback relationship is Equation 1.1, which is dealt with at length in any number of textbooks, but it is of such fundamental importance that I feel obliged to include it here. The open-loop gain of the amplifier is A, and β is the feedback fraction, such that if in Figure 1.1(a) R1 is 2 kΩ and R2 is 1 kΩ, β is ⅓. If A is very high, you don’t even need to know it; the 1 on the bottom becomes negligible, and the As on the top and bottom cancel out, leaving us with a gain of almost exactly 3.

Negative feedback can, however, do much more than stabilizing gain. Anything unwanted occurring in the amplifier, be it distortion or DC drift, or any of the other ills that electronics is prone to, is also reduced by the negative feedback factor (NFB factor for short). This is equal to:

What negative feedback cannot do is improve the noise performance. When we apply feedback the gain drops, and the noise drops by the same factor, leaving the signal-to-noise ratio the same. Negative feedback and the way it reduces distortion is explained in much more detail in one of my other books [3].

Nominal Signal Levels

The absolute level of noise in a circuit is not of great significance in itself – what counts is how much greater the signal is than the noise – in other words the signal-to-noise ratio. An important step in any design is the determination of the optimal signal level at each point in the circuit. Obviously a real audio signal, as opposed to a test sine wave, continuously varies in amplitude, and the signal level chosen is purely a nominal level. One must steer a course between two evils:

• If the signal level is too low, it will be contaminated unduly by noise.

• If the signal level is too high, there is a risk it will clip and introduce severe distortion.

You will note that the first evil is a certainty, while the second is more of a statistical risk. The consequences of either must be considered when choosing a level, and the wider the gap between them the greater the dynamic range. If the best possible signal-to-noise is required in a studio recording, then the internal level must be high, and if there is an unexpected overload you can always do another take. In live situations it will often be preferable to sacrifice some noise performance to give less risk of clipping. The internal signal levels of mixing consoles are examined in detail in Chapter 12.

If you seek to increase the dynamic range, you can either increase the maximum signal level or lower the noise floor. The maximum signal levels in op-amp-based equipment are set by the voltage capabilities of the op-amps used, and this usually means a maximum signal level of about 10 Vrms or +22 dBu. Discrete transistor technology removes the absolute limit on supply voltage, and allows the voltage swing to be at least doubled before the supply rail voltages get inconveniently high. For example, ±40 V rails are quite practical for small-signal transistors and permit a theoretical voltage swing of 28 Vrms or +31 dBu. However, in view of the complications of designing your own discrete circuitry, and the greater space and power it requires, those nine extra decibels of headroom are dearly bought.

Gain Structures

There are some very basic rules for putting together an effective gain structure in a piece of equipment. Like many rules, they are subject to modification or compromise when you get into a tight corner. Breaking them reduces the dynamic range of the circuitry, either by worsening the noise or restricting the headroom; whether this is significant depends on the overall structure of the system and what level of performance you are aiming at. Three simple rules are:

1. Don’t amplify then attenuate.

2. Don’t attenuate then amplify.

3. The signal should be raised to the nominal internal level as soon as possible to minimize contamination with circuit noise.

Amplification then Attenuation

Put baldly it sounds too silly to contemplate, but it is easy to thoughtlessly add a bit of gain to make up for a loss somewhere else, and immediately a few decibels of precious and irretrievable headroom are gone for good. This assumes that each stage has the same power rails and hence the same clipping point, which is usually the case in op-amp circuitry.

Figure 1.2(a) shows a system with a gain control designed to keep 10 dB of gain in hand. In other words, the expectation is that the control will spend most of its working life set somewhere around its ‘0 dB’ position where it introduces 10 dB of attenuation, as is typically the case for a fader on a mixer. To maintain the nominal signal level at 0 dBu we need 10 dB of gain, and a +10 dB amplifier (Stage 2) has been inserted just before the gain control. This is not a good decision. This amplifier will clip 10 dB before any other stage in the system, and introduces what one might call a headroom bottleneck.

There are exceptions. The moving-coil phono head-amp described in Chapter 8 appears to flagrantly break this rule, as it always works at maximum gain even when this is not required. But when considered in conjunction with the following RIAA stage, which also has considerable gain, it makes perfect sense, for the stage gains are configured so that the second stage always clips first, and there is actually no loss of headroom.

Attenuation then Amplification

In Figure 1.2(b) the amplifier is now after the gain control, and noise performance rather than headroom suffers. If the signal is attenuated, any active device will inescapably add noise in restoring the level. Any conventional gain-control block has to address this issue. If we once more require a gain variable from +10 dB to off, i.e. −∞ dB, as would be typical for a fader or volume control, then usually the potentiometer is placed before the gain stage as in Figure 1.2(b) because as a rule some loss in noise performance is more acceptable than a permanent 10 dB reduction in system headroom. If there are options for the amplifier stages in terms of a noise/cost trade-off (such as using the 5532 versus a TL072) and you can only afford one low-noise stage, then it should be Stage 2.

If all stages have the same noise performance this configuration is 10 dB noisier than the previous version when gain is set to 0 dB.

Raising the Input Signal to the Nominal Level

Getting the incoming signal up to the nominal internal level in one jump is always preferable as it gives the best noise performance. Sometimes it has to be done in two amplifier stages; typical examples are microphone preamps with wide gain ranges and phono preamps that insist on performing the RIAA equalization in several goes. (These are explored in their respective chapters.) In these cases the noise contribution of the second stage may no longer be negligible.

Consider a signal path which has an input of −10 dBu and a nominal level of 0 dBu. The first version has an input amplifier with 10 dB of gain followed by two unity-gain circuit blocks, A and B. All circuit blocks are assumed to introduce noise at −100 dBu. The noise output for the first version is −89.2 dBu. Now take a second version of the signal path that has an input amplifier with 5 dB of gain, followed by block A, another amplifier with 5 dB of gain, then block B. The noise output is now −87.5 dB, 1.7 dB worse, due to the extra amplification of the noise from block A. There is also more hardware, and the second version is clearly an inferior design.

Active Gain Controls

The previous section should not be taken to imply that noise performance must always be sacrificed when a gain control is included in the signal path. This is not so. If we move beyond the idea of a fixed-gain block, and recognize that the amount of gain present can be varied, then less gain when the maximum is not required will reduce the noise generated. For volume-control purposes it is essential that the gain can be reduced to near-zero, though it is not necessary for it to be as firmly ‘off’ as the faders or sends of a mixer.

An active volume-control stage gives lower noise at lower volume settings because there is less gain. The Baxandall active configuration also gives excellent channel balance as it depends solely on the mechanical alignment of a dual linear pot – all mismatches of its electrical characteristics are canceled out, and there are no quasi-log dual slopes to induce anxiety.

Active gain controls are looked at in depth in Chapter 9.

Noise

Noise here refers only to the random noise generated by resistances and active devices. The term is sometimes used to include mains hum, spurious signals from demodulated RF and other non-random sources, but this threatens confusion and I prefer to call the other unwanted signals ‘interference’. In one case we strive to minimize the random variations arising in the circuit itself, in the other we are trying to keep extraneous signals out, and the techniques are wholly different.

When noise is referred to in electronics it means white noise unless it is specifically labeled as something else, because that is the form of noise that most electronic processes generate. There are two elemental noise mechanisms that make themselves felt in all circuits and active devices. These are Johnson noise and shot noise, which are both are forms of white noise. Both have Gaussian probability density functions. These two basic mechanisms generate the noise in both bipolar junction transistors (BJTs) and field-effect transistors (FETs), though in rather different ways.

There are other forms of noise that originate from less fundamental mechanisms such as device processing imperfections that do not have a white spectrum; examples are 1/f (flicker) noise and popcorn noise. These noise mechanisms are described later in this chapter.

Non-white noise is given a color that corresponds to the visible spectrum; thus, red noise has a larger low-frequency content than white noise, while pink is midway between the two.

White noise has equal power in equal absolute bandwidth, i.e. with the bandwidth measured in Hz. Thus, there is the same power between 100 and 200 Hz as there is between 1100 and 1200 Hz. It is the type produced by most electronic noise mechanisms.

Pink noise has equal power in equal ratios of bandwidth, so there is the same power between 100 and 200 Hz as there is between 200 and 400 Hz. The energy per Hz falls at 3 dB per octave as frequency increases. Pink noise is widely used for acoustic applications like room equalization and loudspeaker measurement as it gives a flat response when viewed on a third-octave or other constant-percentage-bandwidth spectrum analyzer.

Red noise has energy per Hz falling at 6 dB per octave rather than 3. It is important in the study of stochastic processes and climate models, but has little application in audio. The only place you are likely to encounter it is in the oscillator section of analog synthesizers. It is sometimes called Brownian noise as it can be produced by Brownian motion; hence its alternative name of random-walk noise. Brown here is a person and not a color.

Blue noise has energy per Hz rising at 3 dB per octave. Blue noise is used for dithering in image anti-aliasing, but has, as far as I am aware, no application to audio. The spectral density of blue noise (i.e. the power per Hz) is proportional to the frequency. It appears that the light-sensitive cells in the retina of the mammalian eye are arranged in a pattern that resembles blue noise [4]. Great stuff, this evolution.

Violet noise has energy per Hz rising at 6 dB per octave (I imagine you saw that one coming). It is also known as ‘differentiated white noise’ as a differentiator circuit has a frequency response rising at 6 dB per octave. It is sometimes called purple noise.

Gray noise is pink noise modified by a psychoacoustic equal loudness curve, such as the inverse of the A-weighting curve, to give the perception of equal loudness at all frequencies.

Green noise really does exist, though not in the audio domain. It is used for stochastic half-toning of images, and consists of binary dither patterns composed of homogeneously distributed minority pixel clusters. I think we had better leave it there.

Johnson Noise

Johnson noise is produced by all resistances, including those real resistances hiding inside transistors (such as r_bb, the base spreading resistance). It is not generated by the so-called intrinsic resistances, such as r_e, which are an expression of the V_be/I_c slope and not a physical resistance at all. Given that Johnson noise is present in every circuit, and often puts a limit on noise performance, it is a bit surprising that it was not discovered until 1928 by John B. Johnson at Bell Labs [5].

The rms amplitude of Johnson noise is easily calculated with the classic formula:

where v_n is the rms noise voltage, T is absolute temperature (°K), B is the bandwidth in Hz, k is Boltzmann’s constant, and R is the resistance in ohms.

The only thing to be careful with here (apart from the usual problem of keeping the powers of 10 straight) is to make sure you use Boltzmann’s constant (1.380662 × 10⁻²³), and not the Stefan–Boltzmann constant (5.67 × 10⁻⁸), which relates to black-body radiation and will give some spectacularly wrong answers. Often the voltage noise is left in its squared form for ease of summing with other noise sources. Table 1.2 gives a feel for how resistance affects the magnitude of Johnson noise. The temperature is 25°C and the bandwidth is 22 kHz.

Johnson noise theoretically goes all the way to daylight, but in the real world is ultimately band-limited by the shunt capacitance of the resistor. Johnson noise is not produced by circuit reactances – i.e. pure capacitance and inductance. In the real world, however, reactive components are not pure, and the winding resistances of transformers can produce significant Johnson noise; this is an important factor in the design of moving-coil cartridge step-up transformers. Capacitors with their very high leakage resistances approach perfection much more closely, and the capacitance has a filtering effect. They usually have no detectable effect on noise performance, and in some circuitry it is possible to reduce noise by using a capacitative potential divider instead of a resistive one [6].

The noise voltage is of course inseparable from the resistance, so the equivalent circuit is of a voltage source in series with the resistance present. While Johnson noise is usually represented as a voltage, it can also be treated as a Johnson noise current, by means of the Thevenin–Norton transformation, which gives the alternative equivalent circuit of a current source in shunt with the resistance. The equation for the noise current is simply the Johnson voltage divided by the value of the resistor it comes from:

When it is first encountered, this ability of resistors to generate electricity from out of nowhere seems deeply mysterious. You wouldn’t be the first person to think of connecting a small electric motor across the resistance and getting some useful work out – and you wouldn’t be the first person to discover it doesn’t work. If it did, then by the First Law of Thermodynamics (the law of conservation of energy), the resistor would have to get colder, and such a process is flatly forbidden by … the Second Law of Thermodynamics. The Second Law is no more negotiable than the First Law, and it says that energy cannot be extracted by simply cooling down one body. If you could it would be what thermodynamicists call a perpetual motion machine of the second kind, and they are no more buildable than the common sort of perpetual motion machine.

It is interesting to speculate what happens as the resistor is made larger. Does the Johnson voltage keep increasing, until there is a hazardous voltage across the resistor terminals? Obviously not, or picking up any piece of plastic would be a lethal experience. Johnson noise comes from a source impedance equal to the resistor generating it, and this alone would prevent any problems. Table 1.2 ends with a couple of silly values to see just how this works; the square root in the equation means that you need a peta-ohm resistor (1 × 10¹⁵ Ω) to reach even 600 mVrms of Johnson noise. Resistors are made up to at least 100 GΩ but peta-ohm resistors (PΩ?) would really be a minority interest.

Shot Noise

It is easy to forget that an electric current is not some sort of magic fluid, but is actually composed of a finite though usually very large number of electrons, so current is in effect quantized. Shot noise is so called because it allegedly sounds like a shower of lead shot being poured on to a drum, and the name emphasizes the discrete nature of the charge carriers. Despite the picturesque description the spectrum is still that of white noise, and the noise current amplitude for a given steady current is described by a surprisingly simple equation (as Einstein said, the most incomprehensible thing about the universe is that it is comprehensible) that runs thus:

where q is the charge on an electron (1.602 × 10⁻¹⁹ C), I_dc is the mean value of the current, and B is the bandwidth examined.

As with Johnson noise, often the shot noise is left in its squared form for ease of summing with other noise sources. Table 1.3 helps to give a feel for the reality of shot noise. As the current increases, the shot noise increases too, but more slowly as it depends on the square root of the DC current; therefore, the percentage fluctuation in the current becomes less. It is the small currents that are the noisiest.

The actual level of shot noise voltage generated if the current noise is assumed to flow through a 100 Ω resistor is rather low, as the last column shows. There are very few systems that will be embarrassed by an extra noise source of even −99 dBu unless it occurs right at the very input. To generate this level of shot noise requires 1 A to flow through 100 Ω, which naturally means a voltage drop of 100 V and 100 W of power dissipated. These are not often the sort of circuit conditions that exist in preamplifier circuitry. This does not mean that shot noise can be ignored completely, but it can usually be ignored unless it is happening in an active device where the noise is amplified.

1/f Noise (Flicker Noise)

This is so called because it rises in amplitude proportionally as the frequency examined falls. Unlike Johnson noise and shot noise, it is not a fundamental consequence of the way the universe is put together, but the result of imperfections in device construction. 1/f noise appears in all kinds of active semiconductors, and also in resistors. For a discussion of flicker noise in resistors, see Chapter 2.

Popcorn Noise

This form of noise is named after the sound of popcorn being cooked, rather than eaten. It is also called burst noise or bistable noise, and is a type of low-frequency noise that appears primarily in integrated circuits, appearing as low-level step changes in the output voltage, occurring at random intervals. Viewed on an oscilloscope this type of noise shows bursts of changes between two or more discrete levels. The amplitude stays level up to a corner frequency, at which point it falls at a rate of 1/f². Different burst-noise mechanisms within the same device can exhibit different corner frequencies. The exact mechanism is poorly understood, but is known to be related to the presence of heavy-metal ion contamination, such as gold. This is very much a quality-control issue at the device fabrication stage, and the only measure that can be taken against it is to examine the reputation of each potential component supplier. Like 1/f noise, popcorn noise does not have a Gaussian amplitude distribution.

Summing Noise Sources

When random noise from different sources is summed, the components do not add in a 2 + 2 = 4 manner. Since the noise components come from different sources, with different versions of the same physical processes going on, they are uncorrelated and will partially reinforce and partially cancel, so root-mean-square (rms) addition holds, as shown in Equation 1.5. If there are two noise sources with the same level, the increase is 3 dB rather than 6 dB. When we are dealing with two sources in one device, such as a bipolar transistor, the assumption of no correlation is slightly dubious, because some correlation is known to exist, but it does not seem to be enough to cause serious calculation errors.

Any number of noise sources may be summed in the same way, by simply adding more squared terms inside the square root, as shown by the dotted lines. When dealing with noise in the design process, it is important to keep in mind the way that noise sources add together when they are not of equal amplitude. Table 1.4 shows how this works in decibels. Two equal voltage noise sources give a sum of +3 dB, as expected. What is notable is that when the two sources are of rather unequal amplitude, the smaller one makes very little contribution to the result.

If we have a circuit in which one source is twice the rms amplitude of the other, i.e. with a 6 dB difference, then it only increases the sum by 0.97 dB, a change that would be barely detectable on critical listening. If one source is 10 dB below the other, it only raises the result by 0.4 dB, which in most cases could be ignored. At 20 dB down, the extra noise contribution is lost in measurement error.

This mathematical property of uncorrelated noise sources is exceedingly convenient, because it means that in practical calculations we can neglect all except the most important noise sources with minimal error. Since all semiconductors have some variability in their noise performance, it is rarely worthwhile to make the calculations to great accuracy.

Noise in Amplifiers

There are basic principles of noise design that apply to all amplifiers, be they discrete or integrated, single-ended or differential. Practical circuits, even those consisting of an op-amp and two resistors, have multiple sources of noise. Typically one source of noise will dominate, but this cannot be taken for granted and it is essential to evaluate all the sources and the ways that they add together if a noise calculation is going to be reliable. Here I add the complications one stage at a time.

Figure 1.3 shows that most useful of circuit elements, the perfect noiseless amplifier. (These seem to be unaccountably hard to find in catalogs.) It is assumed to have a definite gain without bothering about whether it is achieved by feedback or not, and an infinite input impedance. To emulate a real amplifier noise sources are concentrated at the input, combined into one voltage noise source and one current noise source. These can represent any number of actual noise sources inside the real amplifier. Figure 1.3 shows two ways of drawing the same situation.

It does not matter on which side of the voltage source the current source is placed; the ‘perfect’ amplifier has an infinite input impedance, and the voltage source a zero impedance, so either way all of the current noise flows through whatever is attached to the input.

Figure 1.4 shows the first step to a realistic situation, with a signal source now connected to the amplifier input. The signal source is modeled as a perfect zero-impedance voltage source with added series resistance R_s. Many signal sources are modeled accurately enough for noise calculations in this way. Examples are low-impedance dynamic microphones, moving-coil phono cartridges, and most electronic outputs. In others cases, such as moving-magnet phono cartridges and capacitor microphone capsules, there is a big reactive component that has a major effect on the noise behavior and cannot be ignored or treated as a resistor. The magnitude of the reactances tends to vary from one make to another, but fortunately the variations are not great enough for the circuit approach for optimal noise to vary greatly. It is pretty clear that a capacitor microphone will have a very high source impedance at audio frequencies, and will need a special high-impedance preamplifier to avoid low-frequency roll-off (see Chapter 13 for details). It is perhaps less obvious that the series inductance of a moving-magnet phono cartridge becomes the dominating factor at the higher end of the audio band, and designing for the lowest noise with the 600 Ω or so series resistance alone will give far from optimal results. This is dealt with in Chapter 7.

There are two sources of voltage noise in the circuit of Figure 1.4:

1. the amplifier voltage noise source v_n at the input;

2. the Johnson noise from the source resistance R_s.

These two voltage sources are in series and sum by rms addition as they are uncorrelated.

There is only one current noise component: the amplifier noise current source i_n across the input. This generates a noise voltage when its noise current flows through R_s. (It cannot flow into the amplifier input because we are assuming an infinite input impedance.) This third source of voltage noise is also added in by rms addition, and the total is amplified by the voltage gain A and appears at the output. The noise voltage at the input is the equivalent input noise (EIN). This is not easy to measure, so the noise at the amplifier output is divided by A to get the EIN. Having got this, we can compare it with the Johnson noise from the source resistance R_s; with a noiseless amplifier there would be no difference, but in real life the EIN will be higher by a number of decibels, which is called the noise figure (NF). This gives a concise way of assessing how noisy our amplifier is and if it is worth trying to improve it. Noise figures rarely appear in hi-fi literature, probably because most of them wouldn’t look very good. For the application of noise figures to phono cartridge amplifiers, see Chapters 7 and 8.

Noise in Bipolar Transistors

An analysis of the noise behavior of discrete bipolar transistors can be found in many textbooks, so this is something of a quick summary of the vital points. Two important transistor parameters for understanding noise are r_bb, the base spreading resistance, and r_e, the intrinsic emitter resistance. r_bb is a real physical resistance – what is called an extrinsic resistance. The second parameter r_e is an expression of the V_be/I_c slope and not a physical resistance at all, so it is called an intrinsic resistance.

Noise in bipolar transistors, as in amplifiers in general, is best dealt with by assuming a noiseless transistor with a theoretical noise voltage source in series with the base and a theoretical noise current source connected from base to emitter. These sources are usually described simply as the ‘voltage noise’ and the ‘current noise’ of the transistor.

Bipolar Transistor Voltage Noise

The voltage noise v_n is made up of two components:

1. the Johnson noise generated in the base spreading resistance r_bb;

2. the collector current (I_c) shot noise creating a noise voltage across r_e, the intrinsic emitter resistance.

These two components can be calculated from the equations given earlier, and rms-summed thus:

where k is Boltzmann’s constant (1.380662 × 10⁻²³), q is the charge on an electron (1.602 × 10⁻¹⁹ C), T is absolute temperature (°K), I_c is the collector current, and r_bb is the base resistance in ohms.

The first part of this equation is the usual expression for Johnson noise, and is fixed for a given transistor type by the physical value of r_bb, so the lower this is the better. The absolute temperature is a factor; running your transistor at 25°C rather than 125°C reduces the Johnson noise from r_bb by 1.2 dB. Input devices usually run cool but this may not be the case with moving-coil preamplifiers where a large I_c is required, so it is not impossible that adding a heat-sink would give a measurable improvement in noise.

The second (shot noise) part of the equation decreases as collector current I_c increases; this is because as I_c increases, r_e decreases proportionally, following r_e = 25/I_c, where I_c is in mA. The shot noise, however, is only increasing as the square root of I_c, and the overall result is that the total v_n falls – though relatively slowly – as collector current increases, approaching asymptotically the level of noise set by the first part of the equation. The only way you can reduce this is by changing to another transistor type with a lower r_bb.

There is an extra voltage noise source resulting from flicker noise produced by the base current flowing through r_bb; this is only significant at high collector currents and low frequencies due to its 1/f nature, and is not usually included in design calculations unless low-frequency quietness is a special requirement.

Bipolar Transistor Current Noise

The current noise i_n is mainly produced by the shot noise of the steady current I_b flowing through the transistor base. This means it increases as the square root of I_b increases. Naturally I_b increases with I_c. Current noise is given by:

where q is the charge on an electron and I_b is the base current.

So, for a fixed collector current, you get less current noise with high-beta transistors because there is less base current.

The existence of current noise as well as voltage noise means it is not possible to minimize transistor noise just by increasing the collector current to the maximum value the device can take. Increasing I_c reduces voltage noise, but it increases current noise, as in Figure 1.5. There is an optimum collector current for each value of source resistance, where the contributions are equal. Because both voltage and current noise are proportional to the square root of I_c, they change slowly as it alters, and the combined noise curve is rather flat at the bottom. There is no need to control collector current with great accuracy to obtain optimum noise performance.

I must emphasize that this is a simplified noise model. In practice both voltage and current noise densities vary with frequency. I have also ignored 1/f noise. However, it gives the essential insight into what is happening and leads to the right design decisions so we will put our heads down and press on.

A quick example shows how this works. In a voltage amplifier we want the source impedances seen by the input transistors to be as low as possible, to minimize Johnson noise, and to minimize the effects of current noise. If we are lucky it may be as low as 100 Ω. How do we minimize the noise from a single transistor faced with a 100 Ω source resistance?

We assume the temperature is 25°C, the bandwidth is 22 kHz, and the r_bb of our transistor is 40 Ω. (Why don’t they put this on spec sheets anymore?) The h_fe (beta) is 150. Set I_c to 1 mA, which is plausible for an amplifier input stage, step the source resistance from 1 to 100,000 Ω in decades, and we get Table 1.5.

Column 1 shows the source resistance and column 2 the Johnson noise density it generates by itself. Factor in the bandwidth actual noise voltage, and you get columns 3 and 4, which show the voltage in nV and dBu respectively. Column 5 is the noise density from the transistor, the rms sum of the voltage noise and the voltage generated by the current noise flowing in the source resistance. Column 6 gives total noise density when we sum the source resistance noise density with the transistor noise density. Factor in the bandwidth again, and the resultant noise voltage is given in columns 7 and 8. The final column 9 gives the noise figure (NF), which is the amount by which the combination of transistor and source resistance is noisier than the source resistance alone. In other words, it tells how close we have got to perfection, which would be a noise figure of 0 dB. The results for the 100 Ω source show that the transistor noise is less than the source resistance Johnson noise; there is little scope for improving things by changing transistor type or operating conditions.

The results for the other source resistances are worth looking at. The lowest noise output (−134.9 dBu) is achieved by the lowest source resistance of 1 Ω, as you would expect, but the NF is very poor at 17.3 dB, because the r_bb at 40 Ω is generating a lot more noise than the 1 Ω source. This gives you some idea why it is hard to design quiet moving-coil head amplifiers. The best noise figure, and the closest approach to theoretical perfection, is with a 1000 Ω source, attained with a greater noise output than 100 Ω. As source resistance increases further, NF worsens again; a transistor with I_c = 1 mA has relatively high current noise and performs poorly with high source resistances.

Since I_c is about the only thing we have control over here, let’s try altering it. If we increase I_c to 3 mA we find that for 100 Ω source resistance, our amplifier is only a marginal 0.2 dB quieter (see Table 1.6, which skips the intermediate calculations and just gives the output noise and NF).

At 3 mA the noise with a 1 Ω source is 0.7 dB better, due to slightly lower voltage noise, but with 100 kΩ noise is higher by no less than 9.8 dB as the current noise is much increased.

If we increase I_c to 10 mA, this makes the 100 Ω noise worse again, and we have lost that slender 0.2 dB improvement.

At 1 Ω the noise is 0.3 dB better, which is not exactly a breakthrough, and for the higher source resistances things are worse again, the 100 kΩ noise increasing by another 5.2 dB. It therefore appears that a collector current of 3 mA is actually pretty much optimal for noise with our 100 Ω source resistance.

If we now pluck out our ‘ordinary’ transistor and replace it with a specialized low-r_bb part like the much-lamented 2SB737, with its superbly low r_bb of 2 Ω, the noise output at 1 Ω plummets by 10 dB, showing just how important low r_bb is for moving-coil head amplifiers. The improvement for the 100 Ω source resistance is much less at 1.0 dB.

If we go back to the ordinary transistor and reduce I_c to 100 μA, we get the last two columns in Table 1.6. Compared with I_c = 3 mA, noise with the 1 Ω source worsens by 5.7 dB, and with the 100 Ω source by 2.6 dB, but with the 100 kΩ source there is a hefty 12.4 dB improvement, due to reduced current noise. Quiet BJT inputs for high impedances can be made by using low collector currents, but junction field-effect transistors (JFETs) usually give better noise performance under these conditions.

The transistor will probably be the major source of noise in the circuit, but other sources may need to be considered. The transistor may have a collector resistor of high value, to optimize the stage gain, and this naturally introduces its own Johnson noise. Most discrete transistor amplifiers have multiple stages, to get enough open-loop gain for linearization by negative feedback, and an important consideration in discrete noise design is that the gain of the first stage should be high enough to make the noise contribution of the second stage negligible. This can complicate matters considerably. Precisely the same situation prevails in an op-amp, but here someone else has done the worrying about second-stage noise for you; if you’re not happy with it, all you can do is pick another type of op-amp.

Noise in JFETs

JFETs operate completely differently from bipolar transistors, and noise arises in different ways. The voltage noise in JFETs arises from the Johnson noise produced by the channel resistance, the effective value of which is the inverse of the transconductance (g_m) of the JFET at the operating point we are looking at. An approximate but widely accepted equation for this noise is:

where k is Boltzmann’s constant (1.380662 × 10⁻²³) and T is absolute temperature (°K).

FET transconductance goes up proportionally as the square root of drain current I_d. When the transconductance is inserted into the equation above, it is again square-rooted, so the voltage noise is proportional to the fourth root of drain current, and varies with it very slowly. There is thus little point in using high drain currents.

The only current noise source in a JFET is the shot noise associated with the gate leakage current. Because the leakage current is normally extremely low, the current noise is very low, which is why JFETs give a good noise performance with high source resistances. However, don’t let the JFET get hot, because gate leakage doubles with each 10°C rise in temperature; this is why JFETs can actually show increased noise if the drain current is increased to the point where they heat up.

The g_m of JFETs is rather variable, but at I_d = 1 mA ranges over about 0.5–3 mA/V (or mMho) so the voltage noise density varies from 4.7 to . Comparing this with column 5 in Table 1.5, we can see that the BJTs are much quieter except at high source impedances, where their current noise makes them noisier than JFETs.

Noise in Op-Amps

The noise behavior of an op-amp is very similar to that of a single input amplifier, the difference being that there are now two inputs to consider, and usually more associated resistors.

An op-amp is driven by the voltage difference between its two inputs, and so the voltage noise can be treated as one voltage v_n connected between them (see Figure 1.6, which shows a differential amplifier).

Op-amp current noise is represented by two separate current generators i_n+ and i_n−, one in parallel with each input. These are assumed to be equal in amplitude and not correlated with each other. It is also assumed that the voltage and current noise sources are likewise uncorrelated, so that rms addition of their noise components is valid. In reality things are not quite so simple and there is some correlation, and the noise produced can be slightly higher than calculated. In practice the difference is small compared with natural variations in noise performance.

Calculating the noise is somewhat more complex than for the simple amplifier of Figure 1.4. You must do the following:

1. Calculate the voltage noise from the voltage noise density.

2. Calculate the two extra noise voltages resulting from the noise currents flowing through their associated components.

3. Calculate the Johnson noise produced by each resistor.

4. Allow for the noise gain of the circuit when assessing how much each noise source contributes to the output.

5. Add the lot together by rms addition.

There is no space to go through a complete calculation, but here is a quick example. Suppose you have an inverting amplifier like that in Figure 1.9(a) below. This is simpler because the non-inverting input is grounded, so the effect of i_n+ disappears, as it has no resistance to flow through and cannot give rise to a noise voltage. This shunt-feedback stage has a ‘noise gain’ that is greater than the signal gain. The input signal is amplified by −1, but the voltage noise source in the op-amp is amplified by 2 times, because the voltage noise generator is amplified as if the circuit was a series-feedback gain stage.

Low-Noise Op-Amp Circuitry

The rest of this chapter deals with designing low-noise op-amp circuitry, dealing with op-amp selection and the minimization of circuit impedances. It also shows how adding more stages can actually make the circuitry quieter. This sounds somewhat counter-intuitive, but as you will see, it is so.

When you are designing for low noise, it is obviously important to select the right op-amp, the great divide being between bipolar and JFET inputs. This chapter concentrates mainly on using the 5532, as it is not only a low-noise op-amp with superbly low distortion, but also a low-cost op-amp, due to its large production quantities. There are op-amps with lower noise, such as the AD797 and the LT1028, but these are specialized items and the cost penalties are high. The LT1028 has a bias-cancellation system that increases noise unless the impedances seen at each input are equal, and since audio does not need the resulting DC precision, it is not useful. The new LM4562 is a dual op-amp with somewhat lower noise than the 5532, but at present it also is much more expensive.

The AD797 runs its bipolar input transistors at high collector currents (about 1 mA), which reduces voltage noise but increases current noise. The AD797 will therefore only give lower noise for rather low source resistances; these need to be below 1 kΩ to yield benefit for the money spent. There is much more on op-amp selection in Chapters 4 and 14.

Noise Measurements

There are difficulties in measuring the low noise levels we are dealing with here. The Audio Precision System 1 test system has a noise floor of −116.4 dBu when its input is terminated with a 47 Ω resistor. When it is terminated in a short-circuit, the noise reading only drops to −117.0 dBu, demonstrating that almost all the noise is internal to the AP and the Johnson noise of the 47 Ω resistor is much lower. The significance of 47 Ω is that it is the lowest value of output resistor that will guarantee stability when driving the capacitance of a reasonable length of screened cable. This resistor value will keep cropping up in the noise situations we are about to examine.

To delve below this noise floor, we can subtract this figure from the noise we measure (on the usual rms basis) and estimate the noise actually coming from the circuit under test. This process is not very accurate when circuit noise is much below that of the test system, because of the subtraction involved, and any figure below −120 dBu should be regarded with caution. Cross-checking against the theoretical calculations and SPICE results is always wise; in this case it is essential.

We will now look at a number of common circuit scenarios and see how low-noise design can be applied to them.

How to Attenuate Quietly

Attenuating a signal by 6 dB sounds like the easiest electronic task in the world. Two equal-value resistors to make up a potential divider, and voila! This knotty problem is solved. Or is it?

To begin with, let us consider the signal going into our divider. Wherever it comes from, the source impedance is not likely to be less than 50 Ω. This is also the lowest output impedance setting for most high-quality signal generators (though it’s 40 Ω on my AP SYS-2702). The Johnson noise from 50 Ω is −135.2 dBu, which immediately puts a limit – albeit a very low one – on the performance we can achieve. The maximum signal handling capability of op-amps is about +22 dBu, so we know at once our dynamic range cannot exceed 135 + 22 = 157 dB. This comfortably exceeds the dynamic range of human hearing, which is about 130 dB if you are happy to accept ‘instantaneous ear damage’ as the upper limit.

In the scenario we are examining, there is only one variable – the ohmic value of the two equal resistors. This cannot be too low, or the divider will load the previous stage excessively, increasing distortion and possibly reducing headroom. On the other hand, the higher the value, the greater the Johnson noise voltage generated by the divider resistances that will be added to the signal, and the greater the susceptibility of the circuit to capacitative crosstalk and general interference pickup. In Table 1.7 the trade-off is examined.

What happens when our signal with its −135.2 dBu noise level encounters our 6 dB attenuator? If it is made up of two 1 kΩ resistors, the noise level at once jumps up to −125.2 dBu, as the effective source resistance from two 1 kΩ resistors effectively in parallel is 500 Ω. Ten decibels of signal-to-noise ratio are irretrievably gone already, and we have only deployed two passive components. There will no doubt be more active and passive circuitry downstream, so things can only get worse.

However, a potential divider made from two 1 kΩ resistors in series presents an input impedance of only 2 kΩ, which is too low for most applications; 10 kΩ is normally considered the minimum input impedance for a piece of audio equipment in general use, which means we must use two 5 kΩ resistors, and so we get an effective source resistance of 2.5 kΩ. This produces Johnson noise at −118.2 dBu, so the signal-to-noise ratio has been degraded by another 7 dB simply by making the input impedance reasonably high.

In some cases 10 kΩ is not high enough, and a 100 kΩ input impedance is sought. Now the two resistors have to be 50 kΩ, and the noise is 10 dB higher again, at −108.2 dBu. That is a worrying 27 dB worse than our signal when it arrived.

If we insist on an input impedance of 100 kΩ, how can we improve on our noise level of −108.2 dBu? The answer is by buffering the divider from the outside world. The output noise of a 5532 voltage-follower is about −119 dBu with a 50 Ω input termination. If this is used to drive our attenuator, the two resistors in it can be as low as the op-amp can drive. The 5532 has a most convenient combination of low noise and good load-driving ability, and the divider resistors can be reduced to 500 Ω each, giving a load of 1 kΩ and a generous safety margin of drive capability (pushing the 5532 to its specified limit of a 500 Ω load tends to degrade its superb linearity by a small but measurable amount) (see Figure 1.7).

The noise from the resistive divider itself has now been lowered to −128.2 dBu, but there is of course the extra −119 dBu of noise from the voltage-follower that drives it. This, however, is halved by the divider just as the signal is, so the noise at the output will be the rms sum of −125 and −128.2 dBu, which is −123.3 dBu. A 6 dB attenuator is actually the worst case, as it has the highest possible source impedance for a given total divider resistance. Either more or less attenuation will mean less noise from the divider itself.

So, despite adding active circuitry that intrudes its own noise, the final noise level has been reduced from −108.2 to −123.3 dBu, an improvement of 15.1 dB.

How to Amplify Quietly

OK, we need a low-noise amplifier. Let’s assume we have a reasonably low source impedance of 300 Ω, and we need a gain of four times (+12 dB). Figure 1.8(a) shows a very ordinary circuit using half a 5532 with typical values of 3 and 1 kΩ in the feedback network, and the noise output measures as −105.0 dBu. The Johnson noise generated by the 300 Ω source resistance is −127.4 dBu, and amplifying that by a gain of 4 gives −115.4 dBu. Compare this with the actual −105.0 dBu we get, and the noise figure is 10.4 dB – in other words, the noise from the amplifier is three times the inescapable noise from the source resistance, making the latter essentially negligible. This amplifier stage is clearly somewhat short of noise-free perfection, despite using one of the quieter op-amps around.

We need to make things quieter. The obvious thing to do is to reduce the value of the feedback resistances; this will reduce their Johnson noise and also reduce the noise produced in them by the op-amp current noise generators. Figure 1.8(b) shows the feedback network altered to 360 and 120 Ω, adding up to a load of 480 Ω, pushing the limits of the lowest resistance the op-amp can drive satisfactorily. This assumes of course that the next stage presents a relatively light load so that almost all of the driving capability can be used to drive the negative-feedback network; keeping tiny signals free from noise can involve throwing some serious current about. The noise output is reduced to −106.1 dBu, which is only an improvement of 1.1 dB, and only brings the noise figure down to 9.3 dB, leaving us still a long way from what is theoretically attainable. However, at least it cost us nothing in extra components.

If we need to make things quieter yet, what can be done? The feedback resistances cannot be reduced further, unless the op-amp drive capability is increased in some way. An output stage made of discrete transistors could be added, but it would almost certainly compromise the low distortion we get from a 5532 alone. For one answer see the next section on ultra-low-noise design.

How to Invert Quietly

Inverting a signal always requires the use of active electronics (OK, you could use a transformer). Assume that an input impedance of 47 kΩ is required, along with a unity-gain inversion. A straightforward inverting stage as shown in Figure 1.9(a) will give this input impedance only if both resistors are 47 kΩ. These relatively high-value resistors contribute to a noise output of −101.4 dBu, which is also due to the fact that the op-amp is working at a noise gain of 2 times.

The only way to improve this noise level is to add another active stage. It sounds paradoxical – adding more non-silent circuitry to reduce noise – but that’s the way the universe works. If a voltage-follower is added to the circuit given in Figure 1.9(b), then the resistors around the inverting op-amp can be greatly reduced in value without reducing the input impedance, which can now be pretty much as high as we like. The ‘Noise buffered’ column in Table 1.8 shows that if R is reduced to 2.2 kΩ the total noise output is lowered by 8.2 dB, which is a very useful improvement. If R is further reduced to 1 kΩ, which is perfectly practical with a 5532’s drive capability, the total noise is reduced by 9.0 dB compared with the 47 kΩ case. The ‘Noise unbuffered’ column gives the noise output with specified R value but without the buffer, demonstrating that adding the buffer does degrade the noise slightly, but the overall result is still far quieter than the unbuffered version with 47 kΩ resistors. In each case the circuit input is terminated to ground via 50 Ω.

How to Balance Quietly

The design of low- and ultra-low-noise balanced amplifiers is thoroughly examined in Chapter 14.

Ultra-Low-Noise Design with Multipath Amplifiers

Are the above circuit structures the ultimate? Is this as low as noise gets? No. In the search for low noise, a powerful technique is the use of parallel amplifiers with their outputs summed. This is especially useful where source impedances are low and therefore generate little noise compared with the associated electronics.

If there are two amplifiers connected, the signal gain increases by 6 dB due to the summation. The noise from the two amplifiers is also summed, but since the two noise sources are completely uncorrelated (coming from physically different components) they partially cancel and the noise level only increases by 3 dB. Thus there is an improvement in signal-to-noise ratio of 3 dB. This strategy can be repeated by using four amplifiers, in which case the signal-to-noise improvement is 6 dB. Table 1.9 shows how this works for increasing numbers of amplifiers.

In practice the increased signal gain is not useful, and an active summing amplifier would compromise the noise improvement, so the output signals are averaged rather than summed, as shown in Figure 1.10. The amplifier outputs are simply connected together with low-value resistors; the gain is unchanged but the noise output falls. The amplifier outputs are nominally identical, so very little current should flow from one op-amp to another. The combining resistor values are so low that their Johnson noise can be ignored.

Obviously there are economic limits on how far you can take this sort of thing. Unless you’re measuring gravity waves or something equally important, 256 parallel amplifiers is probably not a viable choice.

Ultra-Low-Noise Voltage Buffers

The multiple-path philosophy works well even with a minimally simple circuit such as a unity-gain voltage buffer. Table 1.10 gives calculated results for 5532 sections (the noise output is too low to measure reliably even with the best testgear) and shows how the noise output falls as more op-amps are added. The distortion performance is not affected.

The 10 Ω output resistors combine the op-amp outputs, and limit the currents that would flow from output to output as a result of DC offset errors. AC gain errors here will be very small indeed as the op-amps have 100% feedback. If the output resistors were raised to 47 Ω they would as usual give HF stability when driving screened cables or other capacitances, but the total output impedance is usefully halved to 23.5 Ω. Another interesting bonus of this technique is that we have doubled the output drive capability; this stage can easily drive 300 Ω.

Ultra-Low-Noise Amplifiers

We now return to the problem studied earlier; how to make a really quiet amplifier with a gain of 4 times. We saw that the minimum noise output using a single 5532 section and a 300 Ω source resistance was −106.1 dBu, with a not particularly impressive noise figure of 9.3 dB. Since almost all the noise is being generated in the amplifier rather than the source resistance, the multiple-path technique should work well here. And it does.

There is, however, a potential snag that needs to be considered. In the previous section, we were combining the outputs of voltage-followers, which have gains very close indeed to unity because they have 100% negative feedback and no resistors are involved in setting the gain. We could be confident that the output signals would be near-identical and unwanted currents flowing from one op-amp to the other would be small despite the low value of the combining resistors.

The situation here is different; the amplifiers have a gain of 4 times, so there is a smaller negative feedback factor to stabilize the gain, and there are two resistors with tolerances that set the closed-loop gain for each stage. We need to keep the combining resistors low to minimize their Johnson noise, so things might get awkward. It seems reasonable to assume that the feedback resistors will be 1% components. Considering the two-amplifier configuration in Figure 1.11, the worst case would be to have R1A 1% high and R2A 1% low in one amplifier, while the other had the opposite condition of R1B 1% low and R2B 1% high. This highly unlikely state of affairs gives a gain of 4.06 times in the first amplifier and 3.94 times in the second. Making the further assumption of a 10 Vrms maximum voltage swing, we get 10.15 Vrms at the first output and 9.85 Vrms at the second, both applied to the combining resistors, which here are set at 47 Ω. The maximum possible current flowing from one amplifier output into the other is therefore 0.3 V/(47 Ω + 47 Ω), which is 3.2 mA; in practice it will be much smaller. There are no problems with linearity or headroom, and distortion performance is indistinguishable from that of a single op-amp.

Having reassured ourselves on this point, we can examine the circuit of Figure 1.11, with two amplifiers combining their outputs. This reduces the noise at the output by 2.2 dB. This falls short of the 3 dB improvement we might hope for because of a significant Johnson noise contribution from source resistance, and doubling the number of amplifier stages again only achieves another 1.3 dB improvement. The improvement is greater with lower source resistances; the measured results with one, two, three, and four op-amps for three different source resistances are summarized in Table 1.11.

The results for 200 and 100 Ω show that the improvement with multiple amplifiers is greater for lower source resistances, as these resistances generate less Johnson noise of their own.

References

[1]  J. Smith, Modern Operational Circuit Design, Wiley-Interscience, 1971, p. 129.

[2]  Intersil Application NoteAN9420.1, Current Feedback Amplifier Theory and Applications (April 1995).

[3]  D. Self, Audio Power Amplifier Design Handbook, fifth ed. Focal Press, 2009 (Chapter 2).

[4]  J.I. Yellott Jr., Spectral consequences of photoreceptor sampling in the rhesus retina, Science 221 (1983), pp. 382–385.

[5]  J. Johnson, Thermal agitation of electricity in conductors, Phys. Rev. 32 (1928) 97.

[6]  Sansen Chang, Low-Noise Wideband Amplifiers in Bipolar and CMOS Technologies, Kluwer, 1991, p. 106.