Two generators with one cause

The floor a circuit has gave an amplifier two noise generators — a voltage $e$ in series with its input and a current $i$ into it — and from them the noise figure against source resistance: a bowl whose bottom is at $e/i$ and whose floor is $1 + 2ei/4kT$. The bowl, and the bottom of it measured its width; the source resistance the flicker moves gave both generators a flicker spectrum and watched the bottom drift from 6.67 kΩ at high frequency to 699 Ω at a hertz, the floor rising to 15.1 dB.

Every one of those essays treated the two generators as independent. That is the textbook default and it is often not true. Inside a bipolar input stage the base current’s noise flows through the base resistance and so appears in both generators at once; inside any device, one physical fluctuation can move both the voltage an input needs and the current it draws. When the two share a cause they are correlated, and the drift essay closed by asking what that does: whether the optimum becomes a complex impedance and how much of the low-frequency floor a correlated part could recover.

A cross term the source can use #

The amplifier is the earlier essays’ part, 4 $\text{nV}/\sqrt{\text{Hz}}$ and 0.6 $\text{pA}/\sqrt{\text{Hz}}$. Its generators now have a correlation $\langle e\,i^*\rangle = \gamma\,e\,i$, with γ a complex number of magnitude at most one. A source of impedance $Z = R + jX$ adds its own thermal noise $4kTR$, and the noise at the input is $e + iZ$ plus that; its mean square is

$$e^2 + i^2|Z|^2 + 2\operatorname{Re}(\gamma\,e\,i\,Z^*) + 4kTR.$$

The middle term is new. It is linear in R through γ’s real part and linear in X through its imaginary part, and its sign is γ’s: a correlation can make the total smaller than the sum of the parts, because the current generator’s contribution through the source partly cancels the voltage generator’s.

Minimising the noise factor over X and then over R gives the optimum source in closed form,

$$X_{\text{opt}} = -\gamma_i\,\frac{e}{i},\qquad R_{\text{opt}} = \frac{e}{i}\sqrt{1-\gamma_i^2},\qquad F_{\min} = 1 + \frac{2ei}{4kT}\left(\gamma_r + \sqrt{1-\gamma_i^2}\right),$$

and every figure here holds that against a direct search of the same expression over the impedance plane; the two agree to a part in a hundred thousand or better.

A real correlation leaves the bottom where it was. At γ = −0.3 the bowl’s bottom is at 6.67 kΩ, the independent value to a part in a million, and the floor has fallen from 1.138 dB to 0.827. The cross term $2\gamma eiR$ grows in exact proportion to the source’s own thermal noise $4kTR$, so it adds a fixed fraction to the noise factor at every resistance and the bowl moves vertically without moving sideways. The floor’s excess over one scales as $1 + \gamma$: at +0.6 it is 1.70 dB, at −0.6 about half a decibel.

Off the resistance axis, and circles #

An imaginary correlation does something a real one cannot: it gives the source a reason to have a reactance.

At γ = 0.5j the quietest source is 5.77 kΩ in series with 3.33 kΩ of capacitive reactance, and the floor is 1.002 dB — lower than the independent part’s, because the reactance lets the cross term work. The contours of constant noise figure are circles. That is not a coincidence of the numbers: setting the noise factor equal to a constant gives an equation in R and X whose quadratic terms are $i^2(R^2 + X^2)$, with equal coefficients, so every contour is a circle, centred on the line $X = -\gamma_i e/i$, each checked at eight points around it. The circles nest towards the optimum and widen away from it; the resistance axis cuts them, which is why the earlier essays’ bowls, drawn along that axis, had a bottom at all.

At 0.8j the optimum is 4 kΩ with 5.33 kΩ of reactance. The resistance falls as $\sqrt{1 - \gamma_i^2}$ and the reactance grows as $\gamma_i$, so as the correlation approaches a pure imaginary unit the best source becomes nearly all reactance.

What each part of the correlation is worth #

The two parts of γ are worth very different amounts. The real part moves the floor’s excess linearly, from 0.128 dB at −0.9 through 0.606 at −0.5 and 1.138 at zero to 1.612 at +0.5 and 1.957 at +0.9. A strongly negative real correlation nearly removes the amplifier’s noise from the figure, because at the optimum the current generator’s voltage across the source cancels most of the voltage generator.

The imaginary part lowers the floor only as $\sqrt{1-\gamma_i^2}$ — 1.002 dB at 0.5j, 0.718 at 0.8j, 0.389 at 0.95j — and only for a source that carries the reactance it calls for. A resistive source cannot use an imaginary correlation at all: with X held at zero the cross term’s imaginary half vanishes, and the best resistive source’s floor is the independent 1.138 dB at every $\gamma_i$. A part whose generators are correlated in quadrature looks, to every resistive measurement, exactly like a part whose generators are independent.

When the flicker is shared #

The drift essay’s part had its voltage noise’s flicker corner at 10 Hz and its current noise’s at 1 kHz. Suppose the two flicker components are one fluctuation seen twice — the same slow wander in a device’s bias, appearing as a current at the input and, through an internal resistance, as a voltage. Then the flicker parts are fully correlated and the white parts are not, and the overall correlation coefficient is

$$\gamma(f) = \pm\frac{\sqrt{f_e f_i}/f}{\sqrt{(1 + f_e/f)(1 + f_i/f)}},$$

0.070 at a kilohertz, 0.287 at 100 Hz, 0.704 at 10 Hz and 0.953 at 1 Hz in magnitude, tending to one far below both corners.

With the shared part switched off, the figure reproduces the drift essay’s floors exactly: 3.10 dB at 100 Hz, 7.21 at 10 Hz, 15.11 at 1 Hz, 24.80 at a tenth of a hertz. Shared with opposite signs, the floor is 2.41, 3.55, 3.94 and 4.00 dB at the same frequencies — it stops rising and levels off near four decibels, because as the correlation approaches −1 the shrinking $1 + \gamma$ cancels the growing product $ei$. Shared with the same sign it is 3.70, 9.17, 17.95 and 27.80 dB, worse than independent at every frequency.

So the question the drift essay asked — how much of the low-frequency floor a correlated part could recover — has an answer that depends entirely on a sign. At one hertz a part with opposite-signed shared flicker has a floor eleven decibels below an otherwise identical part with independent flicker, and one with same-signed shared flicker is three decibels above it. Which a given device has is a fact about its structure — how the shared fluctuation reaches the two generators — and no specification that quotes the two densities separately can say.

Measuring the correlation with three resistors #

A data sheet gives $e$ and $i$ and almost never a correlation, but the correlation’s real part is measurable with nothing more than resistors. The output noise density with a source resistance R, after subtracting the resistor’s own $4kTR$, is $e^2 + 2\gamma_r eiR + i^2R^2$ — a quadratic in R with three unknown coefficients. Three resistances give three readings and fix all three.

With perfect readings the three return γ = −0.3 exactly. With each reading wrong by one per cent, in the worst of the eight sign combinations, the recovered correlation is off by 0.262 with the middle reading at 300 Ω, 0.098 at 1 kΩ, 0.048 at 6.67 kΩ, 0.079 at 30 kΩ and 0.706 at 300 kΩ. The least error, ±0.048, is with the middle resistance at 6.14 kΩ: the cross term is largest relative to the other two near $e/i$, where the bowl’s bottom is, so that is where the measurement is most sensitive to it. The optimum source is also the best place to measure what moves it.

The imaginary part is another matter. It enters only through a reactance, so it is invisible to any number of resistive readings, and measuring it needs at least one source with a known reactance — a capacitor in series with a resistor, read the same way. The standard four-parameter noise characterisation of a radio-frequency amplifier is exactly this measurement made at many source impedances; at low frequencies, where the parts are cheaper and the measurements easier, it is almost never made.

The bowl moves, its shape does not #

The real correlation’s effect has a cleaner statement in the noise factor than in decibels. Subtract the floor from the noise factor at any resistance and what is left is

$$F(R) - F_{\min} = \frac{(e - iR)^2}{4kTR},$$

with no γ in it at all. In noise factor the bowl is rigidly translated: every source resistance pays the same excess over the floor that it paid with independent generators, and only the floor itself has moved. A designer who has already characterised how far a source may stray from the optimum — the width the bowl essay measured — keeps that characterisation unchanged when the correlation is discovered.

In decibels the picture is slightly different, because a decibel is a ratio: a fixed excess on a lower floor is a larger fraction of it, so a negatively correlated part’s bowl looks narrower in decibels and a positively correlated one’s wider. That is a property of the unit, not of the part, and it is one more reason the noise factor rather than the noise figure is the quantity to do arithmetic in.

The limit where the amplifier vanishes #

At γ = −1 the two generators are one fluctuation with opposite signs, and the floor’s excess is zero: the noise figure at the optimum is exactly 0 dB. The reason is visible in the input’s noise, $e + iR$. With perfect negative correlation the current generator’s voltage across the source is the voltage generator’s mirror image, and at $R = e/i$ the two cancel sample for sample. The amplifier is still noisy — its output into a short circuit or an open circuit shows both generators in full — but from that one source it adds nothing.

No real part reaches that limit, because some of each noise source always comes from a cause the other does not share, and the shared flicker of the figure above levelled off at four decibels rather than zero for exactly that reason: the white parts stay independent. But the limit explains the shape of every figure here. The cross term is a cancellation, the optimum is where the cancellation is most complete, and the correlation’s magnitude is the fraction of the noise that is available to cancel.

The same numbers as a noise temperature #

Designers of very quiet amplifiers quote noise temperatures rather than figures, because at the bottom of the scale a temperature is easier to read: $T = T_0(F - 1)$ with $T_0$ = 290 K. The independent part’s floor of 1.138 dB is a noise temperature of 87 K. With a correlation of −0.3 it is 61 K; with +0.6, 140 K. At one hertz the drifting part’s independent floor of 15.1 dB is about 9,000 K; the opposite-signed shared flicker brings it to 430 K, and the same-signed to over 17,000. Read this way, the correlation is not a refinement of a noise figure. It is the difference between an amplifier that adds thirty times the noise of its source’s resistor and one that adds half as much again, at the frequency where a sub-hertz measurement lives.

Four numbers, and why radio frequency already uses them #

The four quantities the two-generator model needs once correlation is admitted — $e$, $i$ and the two parts of γ — are, rearranged, the four noise parameters every radio-frequency transistor is specified by: a minimum noise figure, the source impedance at which it occurs (a resistance and a reactance), and a resistance that sets how fast the figure rises away from it. The noise circles drawn above are the contours that specification produces, and an engineer matching a low-noise amplifier at a gigahertz reads them every day.

At low frequencies the convention is two numbers, and the reasons are mostly historical: sources there were taken to be resistive, and a resistive source cannot see an imaginary correlation, so specifying one seemed pointless. The figures above say why that leaves two things out. A real correlation moves the floor, which a resistive source does see; and the low-frequency sources that matter most — capacitive transducers, piezoelectric sensors, coils — are not resistive at all.

How a part comes by its correlation #

Where a correlation comes from decides its size and sign, and the simplest mechanism shows the shape of the argument. If a noise current $i_b$ flows inside the device through an internal resistance $r$ before reaching the point where the device’s other noise is referred, then moving it to the input as an equivalent current generator also requires an equivalent voltage generator $i_b r$ — the same fluctuation in both. The correlation’s magnitude is then the fraction of each noise source that $i_b$ supplies, and its sign is fixed by the direction in which the internal resistance sits relative to the input. That is a property of the device’s topology, not of its process, and it is the same for every part of a design; what varies from part to part is the share.

Two consequences follow for the drifting part. The share that is correlated is largest where the shared fluctuation dominates both generators, which for flicker means below both corners — so a correlation that is negligible in the white region can be nearly total at a hertz, as the figure above assumed. And the sign matters most exactly where the drift essay found the floor rising fastest. A designer choosing between two parts with identical published densities for a sub-hertz measurement is choosing, without being told, between floors that can differ by fourteen decibels.

What it leaves out #

A correlation that depends on frequency in phase as well as size. The flicker sharing here is real at every frequency. A shared fluctuation that reaches one generator through a reactance — a capacitance at the input, a time constant in a bias network — arrives with a phase, and then the correlation is complex and frequency-dependent, and the optimum source’s reactance moves with frequency as its resistance did in the drift essay.

More than two generators. The two-generator model is itself a reduction. A device with several internal noise sources has a two-generator equivalent at each frequency, with a correlation that is whatever the reduction produces; that is what makes the model general, and also why its correlation is rarely a simple number.

Still open: the transformer, the reactive source, and a correlation that can be chosen #

A transformer for a band. A low-impedance source is brought up to an amplifier’s optimum by a step-up transformer, whose turns ratio is one number for a whole band while the optimum drifts with frequency. Which ratio suits a band, how much of the frequency-by-frequency optimum one ratio recovers, and what the transformer’s own winding resistance costs — counted, as the loss in front always is, more than once — is the practical question both this essay and the drift essay lead to; one turns ratio for a band answers it for uncorrelated generators.

A source whose reactance is the transducer. A capacitive sensor is a reactance before it is a resistance, and an imaginary correlation of the right sign would make a capacitive source quieter than any resistive one of the same resistance. Whether a part’s correlation can be matched to a transducer’s reactance across a band, as the drift essay asked of its resistance, is the question for a sensor amplifier.

A correlation imposed rather than found. Feeding back a fraction of an amplifier’s own input current noise into its voltage noise — a deliberate internal resistance in the right place — creates a negative real correlation by design. How much of the floor such a circuit can remove, and at what cost in the amplifier’s other properties, is the engineering form of this essay’s figures.