Texas Instruments Incorporated
Data Acquisition
7
Analog Applications Journal
2Q 2006 www.ti.com/aaj High-Performance Analog Products
We can easily calculate the noise under-
neath the curve in Figure 2 for different
input voltage noise bandwidths in the 1/f
region. The first order of business in this
calculation is to determine the input
noise density at 1 Hz. Once we find that
value, the following simple formula will
provide the rms noise under the curve.
where C is equal to the input noise den-
sity at 1 Hz.
As an example, the amount of rms
noise produced by the amplifier shown
in Figure 2 from 0.1 Hz to 6000 Hz is:
With this calculation, and with the
amplifier noise gain G = 1, the SNR at
the output of the amplifier for the 1/f noise is:
When we think about noise at these low frequencies,
we may jump to the conclusion that we should take this
formula down to a very low frequency, such as 0.0001 Hz
(0.0001 Hz = 1 cycle per 2.8 hours). However, at frequen-
cies lower than 0.1 Hz, which is one cycle every 10 seconds,
it is very possible that other things such as temperature,
aging, or component life are changing in the circuit. Realis-
tically, low-frequency noise from the amplifier will probably
not appear at this sample speed; but changes in the circuit,
such as temperature or power supply voltage, may.
The amplifier table of specifications (Figure 2) also
gives the input noise density value. This specification is
always at a higher frequency, in the area where the input
voltage noise is relatively constant. For this region of the
curve, multiplying the square root of the bandwidth and
the noise density derives the noise across this bandwidth.
For example, if the noise of the amplifier is 17 nV/√Hz at
10 kHz, the noise from the amplifier across the bandwidth
of 6 kHz to 100 kHz is:
where BW is equal to the bandwidth of interest.
So how do we get from the manufacturer’s graph to an
RTO noise value? We calculate the area beneath the noise
curve and multiply that times the noise gain of the amplifier.
In this example, the noise gain of our circuit is +1 V/V. We
determine the noise that the amplifier contributes in both
regions and then add the two values together using the
square root of the sum of the squares. Figure 3 shows the
formula for this calculation and illustrates the two regions.
Figure 3 separates the noise into two parts. In region e
1
,
we gain the 1/f noise of the amplifier by the dc gain of the
amplifier circuit, which is +1 V/V. The specifications for
amplifier noise are in nanovolts per square root of hertz.
So the analysis is complete when we multiply the average
noise over the region by the square root of the bandwidth
of that region. For CMOS amplifiers, the 1/f region is usually
from 0.1 Hz to 100 Hz up to 1000 Hz. Since this noise
value is multiplied by the square root of the bandwidth, its