Input-referred noise of an amplifier

Output noise divided by gain

The input-referred noise of an amplifier is the noise source that, placed at the input of a noiseless copy of it, gives the same output noise. Frequency by frequency, it is the output noise density divided by the gain magnitude. It sets the smallest signal the amplifier can tell apart from its own noise.

In ngspice it is inoise_spectrum: onoise_spectrum divided by the gain from the source named on the noise line. In the lab that source is VIP on the + input. At 1 kHz OTA-A's output noise is 56.39 µV/√Hz and its gain 136.7, so its input noise is 412.6 nV/√Hz.

Which transistors make the noise

With noise ... 1 ngspice lists every noise source of every device. The figure divides each by the gain and groups them: flicker of all transistors, and channel thermal noise of all transistors.

The input pair dominates. Its flicker noise sets the low-frequency noise, and its thermal noise most of the floor. The load mirror comes second, scaled by g_m3/g_m1 = 0.57. The tail and the bias diode are 80 times below the total at 1 kHz: their noise is a common-mode current, which the mirror cancels.

The hand formula for the thermal floor adds the pair and the load: v_n,in² = 2 · 4kT (γ_{n}/g_m1 + γ_{p} g_m3/g_m1²). With γ = 2/3 for both and the run's g_m1 = 165.0 µS and g_m3 = 93.63 µS, it gives 14.49 nV/√Hz. The run's thermal part is 22.65 nV/√Hz.

The gap is γ. The pair's thermal noise implies γ = 1.99 and the load's γ = 1.00, close to the 2.00 and 0.99 that E7.1 measured on single devices. With those, the formula gives 22.70 nV/√Hz. The rest of the 24.15 at 10 MHz is flicker noise of the pair, still 8.23 nV/√Hz there.

Three ways to lower it

Each lever acts on one row of the table. A larger input pair lowers its flicker noise. A larger g_m1 lowers the thermal floor. A smaller g_m3/g_m1, or a larger load, lowers the load's share.

Four times the pair area should cut the pair's flicker density by about two. The total at 1 kHz falls by 1.8, from 412.6 to 229.5 nV/√Hz, because the load's flicker noise does not shrink with the pair. The longer load helps less, because the load was already the smaller term.

XM1 x inp tail 0 sky130_fd_pr__nfet_01v8 W=10 L=1
XM2 out inn tail 0 sky130_fd_pr__nfet_01v8 W=10 L=1
XM3 x x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=1
XM4 out x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=1

XM1 x inp tail 0 sky130_fd_pr__nfet_01v8 W=20 L=2
XM2 out inn tail 0 sky130_fd_pr__nfet_01v8 W=20 L=2
XM3 x x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=2
XM4 out x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=2

A density is not yet a number you can compare with a signal. Integrate it over the band you use: from 10 Hz to 100 kHz, OTA-A has 41.92 µV rms of input noise. Flicker noise makes most of it.

Try it

Run it and read the densities from 10 Hz to 10 MHz and the band total vin_band, the rms input noise from 10 Hz to 100 kHz. a_1k is the gain the noise was divided by.

Now make the input pair W=20 L=2 and run again: vin_1k should fall to about 229 nV/√Hz. Then lengthen the load to L=2 and compare vin_1k and vin_1meg with the figure.

Check yourself

* two ideal amplifiers, gain 10 and gain 20, each with 10 k at its input
V1 a 0 DC 0 AC 1
R1 a b 10k
E1 o1 0 b 0 10
V2 c 0 DC 0 AC 1
R2 c d 10k
E2 o2 0 d 0 20
.control
noise v(o1) V1 dec 10 1 1meg
noise v(o2) V2 dec 10 1 1meg
ac dec 10 1 1meg
let ratio = noise3.inoise_spectrum/noise1.inoise_spectrum
meas ac ratio FIND ratio AT=1k
.endc
.end