Noise & shielding (IN) formulas

Open-circuit noise of resistance $R$ at absolute temperature $T$ over noise bandwidth $B$; $k=1.38\times 10^{-23}\text{J/K}$. Model: a noiseless $R$ in series with this voltage source. One-sided spectral density $4kTR$ in ${\text{V}}^2/\text{Hz}$.

Short-circuit noise current: a noiseless $R$ in parallel with this current source. Use it for resistors in parallel or for a node driven by a current source.

$Z(f)$ = impedance between two terminals of a passive network of resistors, capacitors and inductors, all at temperature $T$ and in thermal equilibrium. Only $\mathrm{Re}[Z]$ contributes. It follows that a network of resistors at one temperature has exactly the noise of its Thevenin resistance.

The maximum noise power a resistor can deliver, into a matched noiseless load $R_L=R$ (load voltage $\sqrt{4kTRB}/2$). It does not depend on $R$. At $T_0=290$ K, $kT$ corresponds to $-174$ dBm/Hz.

Total mean-square noise across $C$, integrated over all frequencies, when $C$ is shunted by any resistor $R$ at temperature $T$: the RC filter's noise bandwidth $1/(4RC)$ cancels $R$. More generally it follows from equipartition, $\frac{1}{2}C\overline{v_n^2}=\frac{1}{2}kT$, for a capacitor in any passive network in thermal equilibrium. It sets the noise floor of sample-and-hold and switched-capacitor circuits.

$q=1.6\times 10^{-19}$ C and $I_{DC}$ = DC current crossing the barrier. In a BJT the base and collector currents carry their own shot noise, $2q{I}_{B}B$ and $2q{I}_{C}B$. Valid at frequencies well below the reciprocal of the carrier transit time.

$K$ in ${\text{V}}^2$ (or ${\text{A}}^2$ for current noise). Equal power per decade: each decade adds $K\ln 10\approx 2.303K$.

$e_w$ = white-noise density (for example in $\text{nV}/\sqrt{\text{Hz}}$), $f_c$ = 1/f corner frequency, brick-wall band from $f_L$ to $f_H$. The same form applies to current noise.

Rms values of independent sources in the same bandwidth. It follows that resistors at the same temperature have, in series, the noise of $R_1+R_2$ and, in parallel, the noise of $R_1\parallel R_2$. Series resistors at different temperatures give $\overline{v_n^2}=4kB(R_1{T}_1+R_2{T}_2)$.

$H(f)$ = voltage transfer function and $|H_{\max }|$ its peak magnitude (the DC gain, for a low-pass filter). For a white input density $e_w$, the output noise is $|H_{\max }|e_w\sqrt{B_N}$.

About 1.57 times the 3-dB bandwidth. For an $n$-th order Butterworth low-pass, $B_N=f_{3\text{dB}}\frac{\pi /2n}{\sin (\pi /2n)}$: about $1.11f_{3\text{dB}}$ for $n=2$, approaching $f_{3\text{dB}}$ as $n$ grows. A single-tuned (parallel RLC) band-pass has the same ratio as the single RC: $B_N=(\pi /2)B_{3\text{dB}}$, with $B_{3\text{dB}}=f_0/Q$.

The voltage form needs both voltages across the same resistance. Use the rms signal, not the peak.

Defined with the source resistance at the standard temperature $T_0=290$ K. $F\ge 1$ ($NF\ge 0$ dB), and a noiseless amplifier has $F=1$. Equivalently, $F$ = total output noise power divided by the output noise power due to the source alone. $T_e=(F-1)T_0$ is the equivalent input noise temperature. In decibels, $NF={\mathrm{SNR}}_{\text{in,dB}}-{\mathrm{SNR}}_{\text{out,dB}}$.

Noise factors $F_i$ and available power gains $G_i$ as linear ratios, not dB. When $G_1$ is large the first stage dominates, so a low-noise, high-gain stage goes first. Same form for noise temperature: $T_e=T_1+T_2/G_1+\cdots$

$L=1/G\ge 1$ = power loss of the matched two-port as a linear ratio, $T$ = its physical temperature. At $T_0$ its noise figure in dB equals its loss in dB. Placed before the first amplifier (noise factor $F_2$), Friis gives $F=L+(F_2-1)L=L{F}_2$: the loss in dB adds directly to the system noise figure.

Multiply by $B_N$ for the mean-square noise and by the gain for the output noise. With low $R_s$, $e_n$ dominates (a bipolar-input op-amp usually has the lower $e_n$); with high $R_s$, $i_n{R}_s$ dominates (a JFET or CMOS input usually has the lower $i_n$). Add the feedback resistors' noise separately if it is not negligible.

$I_2$ = current of the aggressor circuit in the shared conductor. At high frequency or with fast edges the inductive term $L_{c}d{I}_2/dt$ usually dominates.

For a low-impedance victim the noise is proportional to $\omega$, $R$ and $C_{12}$, and leads $V_1$ by $90^{\circ }$. Equivalent to a current source $I_N=j\omega C_{12}{V}_1$ from the victim node to ground.

$M$ = mutual inductance; $B$ = sinusoidal flux density (an rms $B$ gives an rms $V_N$), taken as uniform over the loop area $A$; $\theta$ = angle between $B$ and the normal to the loop. Sign: $V_N=-d\Phi /dt$ (Lenz's law); usually only the magnitude matters.

Depth at which a field falls to $1/e$ (37 %) inside a conductor of permeability $\mu$ and conductivity $\sigma$. For copper: about 9.3 mm at 50 Hz and about 66 $\mu$m at 1 MHz.

$V_d$ = differential signal, $V_{cm}$ = common-mode voltage (for example a ground-potential difference). ${\mathrm{CMRR}}_{\text{dB}}=20{\log }_{10}\mathrm{CMRR}$. An imbalance in source resistances converts part of $V_{cm}$ into a differential error, so the real rejection can be worse than the datasheet CMRR.

A sinusoidal op-amp output of peak $V_p$ at frequency $f_{\max }$ is undistorted only if the slew rate meets this; equivalently the full-power bandwidth is $f_{FP}=\mathrm{SR}/(2\pi V_p)$.