One turns ratio for a band

An amplifier with 4 $\text{nV}/\sqrt{\text{Hz}}$ of voltage noise and 0.6 $\text{pA}/\sqrt{\text{Hz}}$ of current noise is quietest from a source of 6.67 kΩ, and a microphone, a strain gauge, a thermocouple or a moving coil is tens of ohms. Connected directly, a 50 Ω source sees the amplifier’s voltage noise as a large fraction of its own, and the noise figure is many decibels. The classical repair is a step-up transformer: a turns ratio n makes the source look $n^2$ times larger to the amplifier, so $n = \sqrt{6670/50} = 11.5$ puts it at the optimum and the noise figure at the amplifier’s floor.

That calculation uses the optimum at one frequency, and the source resistance the flicker moves found the optimum does not stay at one value: with a voltage-noise corner at 10 Hz and a current-noise corner at 1 kHz it falls from 6.67 kΩ at high frequency to 699 Ω at a hertz. A transformer has one ratio for every frequency it carries. The drift essay’s last open question was which ratio suits a band, how much of the band optimum it recovers, and what a real transformer’s winding resistance adds. Two generators with one cause added a correlation to the same part; this essay sets the correlation aside and takes the drift and the transformer together.

The band’s own ratio #

Over a band from $f_1$ to $f_2$ each generator’s power integrates to its white part times the width plus its corner times $\ln(f_2/f_1)$ — the drift essay’s closed form, checked there and here by quadrature. With a source $R_s$ seen through a ratio n, the amplifier sees $n^2R_s$, and the noise factor over the band, still referred to the source’s own noise, is least at

$$n^2 = \frac{\sqrt{E/I}}{R_s},$$

with $E$ and $I$ the integrated voltage and current noise powers. That is the drift essay’s band optimum resistance divided by the source, nothing more; the transformer is the way of reaching it.

Over the audio band the best ratio is 10.7, where the amplifier sees 5.76 kΩ, and the noise figure is 1.298 dB. The ratio that matches the white optimum, 11.5, gives 1.310 — twelve thousandths of a decibel worse — and no transformer at all gives 13.2 dB. For audio the classical calculation is fine, and the reason is that the audio band’s integrated noise is dominated by its upper decades, where the densities are white: the band is three decades wide, and the two decades above 200 Hz carry most of its width.

Move the slider to 1 Hz to 100 Hz and the picture changes completely. The best ratio is 4.8, giving 5.44 dB, and the white optimum’s 11.5 gives 9.21 dB — nearly four decibels lost by a ratio computed from the data sheet’s headline densities. Over 0.1 Hz to 10 Hz the best ratio is 3.8 and gives 12.1 dB, the white ratio 18.6 and no transformer 20.6: the white-optimum transformer is barely better than none. For a low-frequency band, the band’s integrated noise is dominated by its bottom decade, where the flicker has moved the optimum to a fraction of its white value, and the ratio has to follow it.

Why the audio band escapes and the low band does not #

The difference between the bands is a matter of shares, and the integrals make it countable. Over 20 Hz to 20 kHz the voltage noise’s flicker term, its 10 Hz corner times the logarithm of a thousand, is 0.3 per cent of its integrated power; the current noise’s, its 1 kHz corner times the same logarithm, is 26 per cent of its. The band optimum is the white optimum times the square root of the ratio of the two integrals’ growth factors, $\sqrt{1.003/1.346}$ = 0.863, which is the 5.76 kΩ the figure found. The current noise’s flicker is a quarter of its band power, and it moves the optimum by fourteen per cent.

Over 1 Hz to 100 Hz the shares are 32 and 98 per cent. Almost all of the current noise in that band is flicker and a third of the voltage noise is too, and the band optimum falls to 1.17 kΩ, a sixth of the white value — which is the ratio of 11.5 against 4.8, squared. Over 0.1 Hz to 10 Hz the voltage noise is 82 per cent flicker and the current noise all but entirely, and the band is in the regime where the optimum has settled at its low-frequency limit.

The shares are what a designer should look at first, because they say in two numbers whether the headline densities are good enough. If the current noise’s corner times the logarithm of the band’s ratio is small against the band’s width, the white optimum is the band’s; if it is comparable or larger, the band needs its own.

The ratio each frequency would want #

At each frequency the ideal ratio is $\sqrt{R_{\text{opt}}(f)/R_s}$: 11.52 at 100 kHz, 9.73 at 1 kHz, 6.49 at 100 Hz, 4.33 at 10 Hz, 3.74 at 1 Hz, and tending at low frequency to 3.65 — the white ratio times the fourth root of the ratio of the two corners, since the optimum resistance went as the square root of that ratio. A transformer for the audio band gets 10.73, close to the spot ratio at a kilohertz and above; one for 1 to 100 Hz gets 4.84, close to the spot ratio at a few hertz; one for 0.1 to 10 Hz gets 3.83, near the low-frequency limit.

The pattern is the drift essay’s band result in a different coordinate. A band’s ratio is set where the band’s integrated noise is, and for any band reaching below the current noise’s corner that is near the bottom, because the flicker’s $1/f$ power is concentrated there and the band’s width is not. The band’s top decade has the most hertz in it and the least noise per hertz; the bottom has the fewest hertz and the most noise.

What one ratio costs #

A transformer cannot follow the optimum, but the cost of not following it can be bounded. A hypothetical ratio that tracked the spot optimum at every frequency would put the band’s noise figure at the spot floors’ excess integrated across the band, and no fixed ratio can do better.

For two-decade bands placed along the spectrum, the best fixed ratio’s excess over the tracking bound is never more than 0.24 dB, reached for the band from 2.6 Hz to 261 Hz, between the two flicker corners where the optimum moves most within two decades. Far above the corners the cost is a thousandth of a decibel — the optimum hardly moves — and far below them it is small again, because there the optimum has settled at its low-frequency limit. For 1 to 100 Hz the cost is 0.19 dB: 5.44 against 5.25.

So the large losses in the hero figure are not the price of having one ratio. They are the price of having the wrong one: the white optimum’s ratio loses nearly four decibels over 1 to 100 Hz, and the band’s own ratio loses a fifth of a decibel against perfection. The lesson is to compute the ratio from the band’s integrated densities, which takes two numbers off the data sheet’s noise plots and one logarithm, rather than from the headline densities, which takes none.

There is a floor under all of this that no ratio can move. Even the tracking ratio leaves the band’s noise figure at the spot floors’ excess integrated, and those floors rise below both corners as the product of the two densities does. For 1 to 100 Hz that bound is 5.25 dB and for 0.1 to 10 Hz 12.1, against 1.32 for the band from 100 Hz to 10 kHz. A transformer recovers what the wrong ratio threw away; it cannot recover what the amplifier’s flicker costs at the optimum, and for a sub-hertz band that cost is most of the figure. The only lever left there is the amplifier itself — a part with lower corners, or the correlation the two generators essay found able to bring a sub-hertz floor down by ten decibels.

The figure in volts #

A noise figure is a ratio, and for a source of known resistance it converts directly into the volts a listener or a measurement would notice. The 50 Ω source’s own thermal noise is 0.895 $\text{nV}/\sqrt{\text{Hz}}$. Through the audio band’s best transformer the whole chain, referred back to the source, is 1.039 $\text{nV}/\sqrt{\text{Hz}}$ — 147 nanovolts across 20 kHz — and through the white optimum’s ratio 1.041, the same to a part in a thousand. With the source connected straight to the amplifier it is 4.10 $\text{nV}/\sqrt{\text{Hz}}$, 580 nanovolts across the band: the transformer has bought a factor of four in noise voltage, twelve decibels, and the choice between its two candidate ratios has bought nothing that could be heard.

The same conversion for the low band shows where the ratio matters. Over 1 to 100 Hz the four decibels between the white ratio and the band’s own are a factor of 1.58 in noise voltage at the source — the difference between a seismometer resolving a tremor and losing it, for a sensor that is otherwise unchanged. The transformer is the same part in both cases; only its ratio, chosen from the wrong frequency, has thrown the factor away.

The winding resistance, counted twice #

A real transformer has copper in it, and the copper has resistance. Referred to the primary, the windings add a resistance $r_w$ in series with the source, so the source seen through the transformer is $n^2(R_s + r_w)$ while the noise figure is still referred to $R_s$ alone. The resistance contributes its own thermal noise, which is the obvious cost. It also sits in the path of the amplifier’s current noise, which now flows through $R_s + r_w$ rather than $R_s$ — and that is the cost the loss in front, counted twice warned about in a different circuit.

Re-optimising the ratio for each winding resistance, the best noise factor is exactly the lossless one times $1 + r_w/R_s$ — to a part in a billion across three decades of resistance. The winding multiplies the whole factor, the amplifier’s share included, because the optimum ratio is now set against $R_s + r_w$ and the amplifier’s best contribution is proportional to that. A winding resistance of a tenth of the source costs 0.41 dB, 1.712 against 1.298; adding only the winding’s own noise would have predicted 1.609, a cost of 0.31. At a hundredth of the source the cost is 0.04 dB, at three-tenths 1.14, and a winding as resistive as the source itself costs three decibels on top of the amplifier’s floor.

That sets a specification for the transformer that has nothing to do with its turns ratio. For a 50 Ω source a winding resistance of 5 Ω referred to the primary costs 0.41 dB at any band and any ratio, because the scaling is independent of both, and a designer who has fought for every tenth of a decibel in the amplifier can lose four of them in the copper.

Paralleled amplifiers: the same transformer without copper #

There is a second way to bring a low source up to an amplifier’s optimum, and it is not a component but a count. N identical amplifiers in parallel across the source, their outputs summed, have uncorrelated generators that add in power: the voltage noise of the combination is $e/\sqrt N$ and its current noise $i\sqrt N$, so its optimum is $R_{\text{opt}}/N$.

N amplifiers in parallel are an ideal transformer of ratio $\sqrt{N}$, noise for noise, to a part in $10^{12}$ at every N — the integrated densities scale by 1/N and N exactly as a transformer’s $n^2$ scales the source, and both flicker corners stay where they were. The best count for the audio band is 115, reaching the same 1.298 dB as the ideal 10.7:1 transformer. A real transformer with winding resistance of 5 per cent of the source reaches 1.510 dB, with 20 per cent 2.090.

The equivalence makes the choice explicit. The transformer pays in copper and, at low frequency, in size and susceptibility to magnetic pickup; the parallel amplifiers pay in parts, power and the summing network. For a low band the comparison shifts: the band ratio there is smaller, 4.8 for 1 to 100 Hz, which is 23 amplifiers — and a transformer for a band that low needs a large core to keep its magnetising inductance from shunting the source, which is exactly where its copper gets long.

What the drifting optimum asks of a designer #

The figures above reduce to a short procedure. Read both noise densities at the bottom and the top of the band from the data sheet’s plots, integrate each as white plus corner times logarithm, take the square root of their ratio as the band’s optimum, and divide by the source. That is the turns ratio, or the number of parallel parts. Then budget the copper separately, as a factor of $1 + r_w/R_s$ on the whole noise factor.

The headline densities, quoted at a kilohertz, are the right inputs only for a band whose noise is white — one that begins well above the current noise’s corner. For any band reaching below it they give a ratio too large, and the error grows the lower the band goes: twelve thousandths of a decibel for audio, nearly four decibels for 1 to 100 Hz, six and a half for 0.1 to 10 Hz.

What it leaves out #

The transformer’s own bandwidth. The transformer here is ideal: its ratio holds at every frequency. A real one’s magnetising inductance shunts the source at the bottom of the band and its leakage inductance and winding capacitance limit the top, and for a band reaching a tenth of a hertz the magnetising inductance needed is enormous. The noise argument says which ratio; whether a transformer of that ratio can carry the band is a separate question, and for the lowest band it is usually no. The magnetising inductance’s droop, and the ringing a series capacitor adds to it, are what the ring a blocking capacitor buys measures in a pulse transformer; the same inductance here is the low corner of the noise match.

Correlated generators. With a correlation between the generators the band’s optimum source acquires a reactance, which a transformer presents only through its own inductances. The two generators essay found the real part of a correlation moving the floor and the imaginary part moving the optimum off the resistance axis; a transformer can supply the first kind of optimum and only accidentally the second.

Still open: the transformer’s low end, a source that is reactive, and the summing network’s own noise #

The magnetising inductance as a noise source. At the bottom of the band the transformer’s magnetising inductance appears in parallel with the source and so changes the impedance the amplifier sees, frequency by frequency, as a drifting optimum would like. Whether a transformer can be designed so that its own low-frequency roll-off moves the presented impedance towards the drifting optimum — turning a defect into a correction — is a question of choosing the inductance as deliberately as the ratio.

A capacitive source. A piezoelectric or capacitive sensor is a reactance first, and its impedance falls with frequency as the optimum’s real part rises. Whether a single ratio can match such a source across a band at all, and what the band’s integrated optimum looks like for a source whose own impedance moves, is the drift essay’s second open question with a transformer added.

The summing network for parallel amplifiers. N amplifiers summed through resistors add those resistors’ noise, scaled by the gain of the sum. For a hundred parts the summing resistors’ contribution could undo much of what the parallel connection gained, and how large a summing network may be before it costs as much as a transformer’s copper is the parallel route’s equivalent of the winding-resistance figure above.