Op Amp Noise Analysis
This article explains
how to calculate the output noise of an op amp
circuit, backed up with simulations in LTspice^{®}.
Resistor Noise
A resistor produces
noise according to the equation
where
k = 1.38 x 10^{23}
(Boltzmann’s constant)
T = temperature in
°Kelvin (=273.15 + temperature in °C = 298°K for
standard room temp)
B = Bandwidth in Hertz
R = Resistance in Ohms
So a 1k Ohm resistor
produces
or
FIG 1 shows a circuit
to simulate this
FIG 1
This circuit can be
constructed in the normal way. To configure LTspice
to do noise analysis, from the menu bar choose
Simulate > Edit Simulation cmd then choose the
Noise tab and fill it in as shown in FIG 2
FIG 2
Note it is easier to
configure the output noise if this node has a label.
The output noise is calculated with respect to a
noiseless input voltage which has to be defined. In
this case it is the voltage source V3.
Running the simulation
results in a noise of slightly over 4nV/√Hz as
expected.
FIG 3
Op Amp Noise
OK, so how do we
calculate the noise figure of a circuit using an op
amp and resistors? Here, we need to consider the
voltage and current noise of the op amp, the noise
generated by the resistors as well as the gain in
the circuit.
FIG 4 shows a non
inverting op amp with unity gain.
FIG 4
The circuit can be
downloaded here:
Non Inverting Op Amp Noise
The LTC6241 has an
input noise voltage density of 710nV/√Hz and an
input noise current density of 0.56fA/√Hz at 1kHz.
Since we are driving both op amp inputs from a low
impedance source, the current density is of no
concern. However the input noise voltage density
needs to be considered.
To calculate the noise
of an op amp circuit, all inputs must be grounded.
As we will see later, this makes the inverting
configuration and the non inverting configuration
identical. The input noise voltage of the op amp can
be modelled as a voltage in series with the input as
shown in FIG 5
FIG 5
It is obvious that,
with the input to the circuit grounded
Output Voltage Noise =
Input Voltage Noise = 7nV/√Hz
and this is shown in
FIG 6
FIG 6
This circuit
demonstrates the noise of the op amp without the
effect of gain or external resistors. We can also
see from this circuit that LT Spice is working with
an input voltage noise of 7.2nV/√Hz instead of the
datasheet typical value of 7nV/√Hz.
We will now examine
the effect that gain and external resistors have on
the noise performance of the circuit. FIG 7 shows a
non inverting amplifier stage with gain of 2
FIG 7
This circuit can be
downloaded here:
Non Inverting Amplifier Noise
We mentioned earlier
that, in grounding the inputs to the circuit of FIG
7, both inverting and non inverting configurations
are identical and this is now easy to see. There are
2 methods of calculating the output noise of the
amplifier circuit.
Method 1
Method 1 uses a noise
model of the circuit of FIG 7 as shown in FIG 8.
FIG 8
Vn1 represents the
noise of the op amp itself (7.2nV/√Hz) and Vn2
represents the noise from the resistor R1. Now, we
can see that the noise generated by R1 is
effectively applied to the input of an inverting
amplifier, so is subjected to the gain of the
amplifier, in this case a gain of 1 (since noise as
such has no phase, a gain of 1 is the same as a
gain of +1). Vn3 represents the noise from resistor
R2 and is not subject to any gain. The noise of the
op amp itself is applied to the non inverting
input of the amplifier, so is subject to a gain of
+2.
To find the total
noise (in nV/√Hz) we need to square the
contributions of each noise source, sum them at the
output, then take the square root. This looks
complicated, but only involves simple (if a little
tedious) mathematics.
From the equation at
the top of the page, a 100 Ohm resistor produces
noise of 1.28nV/√Hz.
Therefore the total
noise resulting from op amp voltage noise and
resistor noise, seen at the output of the amplifier
circuit in FIG 8 is:
√{ (7.2nV/√Hz x 2)^{2}
+ (1.28nV/√Hz x 1)^{2} + (1.28nV/√Hz)^{2}}
= 14.5nV/√Hz.
We now need to
consider the effect of the op amp’s current noise.
Since our non inverting input is grounded (for the
purpose of noise analysis), the input noise current
(0.56fA/√Hz) can be modelled as flowing into the
inverting input. Since the non inverting input is
grounded and we have negative feedback around the
amplifier, the inverting terminal is at a virtual
earth. Therefore the current noise of the op amp
only flows through the feedback resistor. From Ohm’s
law, it produces a voltage of
(R2 x i_noise) = 100 x
0.56fA/√Hz = 56fV/√Hz.
Even though this is
negligible, to get an accurate calculation of the
total output noise, we need to square this voltage
and add it to the square of the voltage noise
calculated above.
Thus our total noise
is
√{ (14.5nV/√Hz)^{2}
+ (56fV/√Hz)^{2}} = 14.5nV/√Hz
FIG 9
An LTspice simulation
of the circuit shows the result to be 14.5nV/√Hz.
It was mentioned that
an alternative method of calculating the noise can
be implemented.
Method 2
Picturing where to put
the noise sources in the feedback components of the
amplifier can be tricky. Therefore Method 2 might
prove more intuitive and easier to visualise.
Considering again the circuit in FIG 7, if all the
inputs are grounded then the output will be at 0V
too. Thus the feedback resistor R2 is effectively in
parallel with R1. With this in mind, we end up with
the noise equivalent circuit shown in FIG 10
FIG 10
Here the feedback
resistor and the input resistor are in parallel and
the resistor noise is calculated by using the
parallel resistance value. However, with this
approach the noise is applied to the inverting input
of the op amp, but with no feedback resistor. All is
not lost however, as applying a voltage to the
inverting op amp input is the same as applying the
same voltage to the non inverting pin of the
amplifier since as stated earlier we are not
concerned about phase. Thus applying the noise
voltage to the non inverting input subjects the
noise to a gain of (1+R2/R1). Thus our voltage noise
is calculated as
√{ (7.2nV/√Hz x 2)^{2}
+ (0.907nV/√Hz x 2)^{2}} = 14.5nV/√Hz
Where 0.907nV/√Hz is
the noise of two 100 Ohms resistors in parallel.
This yields the same voltage noise as in Method 1.
Likewise, any noise
current flowing into the inverting terminal will
produce a voltage across the parallel combination of
R2 and R1. This is the same as applying a noise
voltage to the non inverting terminal and subjecting
it to a gain of (1+ R2/R1), thus the 'current' noise
is:
(100100) x
0.56fA/√Hz x 2 = 56fV/√Hz
This yields the same
current noise as in Method 1.
The total noise is
√{(14.5nV/√Hz)^{2}
+ (56fV/√Hz)^{2}} = 14.5nV/√Hz
We have discussed how
to calculate the output noise of an op amp and seen
how this is dependent on the voltage and current
noise specs of the amplifier as well as the gain and
the surrounding resistors.
It is worth repeating
the above with resistors of 1k, 10k and 100k and
these are tabulated below
R1/R2 
Calculated
Noise at 1kHz 
LTspice
Noise at 1kHz 
1k/1k 
15.47nV/√Hz 
15.51nV/√Hz 
10k/10k 
23.13nV/√Hz 
23.21nV/√Hz 
100k/100k 
58.37nV/√Hz 
59.36nV/√Hz 
100/1k 
80.33nV/√Hz 
80.34nV/√Hz 
1k/10k 
89.65nV/√Hz 
89.97nV/√Hz 
10k/100k 
155.75nV/√Hz 
156.55nV/√Hz 
This clearly shows
that the noise figure of the op amp is not done
justice if the gain setting resistors are large. To
ensure you get the noise performance you paid for,
keep the gain setting resistors as low as the
amplifier will allow.
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