# Application Note 1004 Understanding Noise Equivalent Power in

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```					100-1004                                     Application Note 1004

Application Note 1004

Understanding Noise Equivalent Power in Radiometric
Detectors and Instruments

Sid Levingston
03/20/2007
100-1004                                                                     Application Note 1004

Noise Equivalent Power, or NEP, is a basic indicator of detector performance. NEP is the
noise floor of a detector, normalized to a 1Hz bandwidth. NEP is expressed in Watts per
square root bandwidth. To derive NEP, two parameters must be measured: the detector
responsivity at a specified frequency, and the detector voltage noise at the same
frequency. NEP can then be calculated as:

Noise Voltage           Hz
NEP =                                        at some freqeuncy, f o
Rv

If the electrical pole of the detector circuit shown in figure 1 is much less than the unity
gain bandwidth of the amplifier, the responsivity for the detector is given by:

f
Ri ⋅ R f ⋅
f therm
Rv( f ) =
2                         2
 f                f             
1+ 
f           ⋅ 1+ 


f              
 therm            pole          

1
f pole =                          f therm = Pyroelectric Thermal Tau
2 ⋅π ⋅ Rf ⋅ C f

If we specify that the frequency of measurement is higher then the thermal frequency and
lower than the electrical pole, then the responsivity can be simplified to:

Ri ⋅ R f
Rv( f ) =
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

CF

RF          Vj
in

Vo
RD              CD

en

Figure 1, Current Mode Radiometer with Noise Sources
100-1004                                                                                               Application Note 1004

The noise in the circuit has three sources we need to consider. There is the noise voltage
of the amplifier, the Johnson noise voltage of the feedback resistor, and the noise
resulting from the noise current of the amplifier flowing through the feedback resistor. A
fourth noise source is the loss tangent noise of the detector, but its contribution is much
smaller than the main three and can be ignored. Note that the noise voltage of the
amplifier will be amplified by the noise gain of the amplifier. For a Pyroelectric detector,
the detector shunt resistance is much larger than the feedback resistor so that the noise
gain is dominated by the ratio of the detector capacitance to the feedback capacitance.
This is not always true for photodiodes, so care must be taken when deriving NEP for
those cases. The dominant noise sources for a Pyroelectric sensor, normalized to a 1Hz
bandwidth are given by:

)
en ⋅ 1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ ( f + C D )
2
C
Ve ( f ) =                                                            Amp Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

4 ⋅ K ⋅T ⋅ Rf
VJ ( f ) =                                             Johnson Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

R f ⋅ 2 ⋅ q ⋅ I bias
Vi ( f ) =                                             Current Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

The total noise will be the rms sum of these three sources:

Vnoise ( f ) =
2
(
en ⋅ 1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ ( f + C D ) + 4 ⋅ K ⋅ T ⋅ R f + R 2 ⋅ 2 ⋅ q ⋅ I bias
C         )   2
)               f

1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

If we divide this by the responsivity, the equation for NEP becomes:

NEP( f ) =
2
(
en ⋅ 1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ ( f + C D ) + 4 ⋅ K ⋅ T ⋅ R f + R 2 ⋅ 2 ⋅ q ⋅ I bias
C         )      2
)            f

R f ⋅ Ri

Assuming a large feedback resistor and that the detector is in the flat bandwidth region,
this will simplify further to:
100-1004                                                                                        Application Note 1004

)
1 << (f ⋅ 2 ⋅ π ⋅ R f ⋅ ( f + C D )
2
C

NEP ( f ) ≈
2
(
en ⋅ (f ⋅ 2 ⋅ π ⋅ R f ⋅ ( f + C D ) + 4 ⋅ K ⋅ T ⋅ R f + R 2 ⋅ 2 ⋅ q ⋅ I bias
C         )  2
)                f

R f ⋅ Ri

2
en
(           C         )2 4 ⋅ K ⋅ T + 2 ⋅ q ⋅ I bias
⋅ (f ⋅ 2 ⋅ π ⋅ ( f + C D ) +  )
Rf                                Rf
NEP ( f ) ≈
Ri

It can now be seen that for a given detector with a characteristic responsivity and
capacitance, reducing NEP can be accomplished by: increasing Rf, decreasing Ibias,
decreasing the amplifier noise, or decreasing Cf. Unfortunately, not all of these
parameters can be independently changed. Since the frequency response and amplifier
stability of the circuit are controlled by the feedback capacitor and feedback resistor,
neither can be changed without verifying that performance hasn’t been compromised. As
the feedback resistor is increased, the feedback capacitor must be decreased to maintain
bandwidth. At some point the feedback capacitor can not be reduced due to circuit
parasitic capacitance. NEP improvement will stop as the Ibias contribution starts to
dominate. The amplifier noise is also of no consequence for feedback resistors larger than
100K_, so reducing it has no benefit. The NEP for a typical detector is plotted in figure 2
for increasing R feedback. NEP improvement is negligible for values bigger than 100G_.

NEP vs. R Feedback
R Feedback
1.E+06               1.E+08           1.E+10             1.E+12             1.E+14
1000
NEP (nW/root Hz)

100

10                                                                                        NEP

1

0.1

Figure 2, NEP vs. R Feedback
100-1004                                                                         Application Note 1004

At Spectrum Detector our low noise detectors use a 100G_ feedback resistor and we take
great care to ensure that we use the lowest input bias current amplifiers available. We test
each device to ensure it meets our standards. Spectrum Detector sells the SPH-42, an
ultra low NEP detector. The parameters for this detector are:

nV                                                                                    µA
en = 13                I bias = 250 fA C D = 22 pF         C f = 0.12 pF   R f = 1011   Ri = .55
Hz                                                                                   W

8.1 ⋅ 10 −38 + 1.64 ⋅ 10 −31 + 8 ⋅ 10 −32       nW
NEP ( f ) ≈                        −6
= 0.9
.55 ⋅ 10                           Hz

at     f = 5 Hz and T = 298 K

The measured NEP of a typical SPH-42 detector is 0.82nW/root (Hz). Note that the
amplifier noise (the first term) is several orders of magnitude less than the other sources
and does not contribute to NEP. The theoretical lowest NEP for the detector is limited by
the Johnson noise of the feedback resistor and would be 0.74nW/root (Hz).

When deriving NEP for a photodiode the shunt resistance of the device comes into play.
This is because for a Germanium device, the shunt resistance can be on the order of
10K_. Silicon diodes have shunt resistances on the order of 100M_. Since the current
responsivity for these devices is one million times higher than a Pyroelectric, the
feedback resistors are proportionally smaller. This means the feedback capacitor is larger
to maintain a similar bandwidth. The resistor ratios now dominate the capacitor ratios.
The dominate noise sources will be:

en
Ve ( f ) =                                        Amp Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f     )
2

4 ⋅ K ⋅T ⋅ Rf
VJ ( f ) =                                      Johnson Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

R f ⋅ 2 ⋅ q ⋅ I bias
Vi ( f ) =                                      Current Noise
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2

The total noise will be the rms sum of these three sources times the voltage gain provided
by the feedback resistor and shunt resistor:
100-1004                                                                                        Application Note 1004

2
e n + 4 ⋅ K ⋅ T ⋅ R f + R 2 ⋅ 2 ⋅ q ⋅ I bias 
f                        Rf 
Vnoise ( f ) =                                                     ⋅ 1 +


1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f )
2
    RD 


For a Silicon device, the feedback to shunt resistor ratio is much less than 1, and this
simplifies to:

2
en + 4 ⋅ K ⋅ T ⋅ R f + R 2 ⋅ 2 ⋅ q ⋅ I bias
f
Vnoise ( f ) =
1 + (f ⋅ 2 ⋅ π ⋅ R f ⋅ C f   )
2

Rv will be the same as with the simplified Pyroelectric equation, so NEP for the two
cases are:

2
en            4 ⋅ K ⋅T
2
+            + 2 ⋅ q ⋅ I bias
R    f           Rf
NEP ≈                                                          for Silicon
Ri

2
en            4 ⋅ K ⋅T
2
+            + 2 ⋅ q ⋅ I bias
R    f           Rf                           Rf 
NEP ≈                                                ⋅ 1 +

       for Germanium
Ri                        RD 


While it appears at first glance that we can increase the feedback resistor to improve the
Silicon device, there is a practical limit set by the detector maximum voltage output and
bandwidth requirements. With the Germanium device, the shunt resistance is a device
parameter we cannot control, so it cannot be arbitrarily increased. Spectrum Detector
sells the SSI-A-45 Silicon Detector Instrument. The parameters for this detector are:

nV                                                                             A
en = 13                     I bias = 250 fA RD = 10 8               R f = 10 6   Ri = .50
Hz                                                                         W

1.69 ⋅ 10 − 28 + 1.64 ⋅ 10 − 26 + 8 ⋅ 10 −32        pW
NEP ≈                                                         = 0.26
0.5                               Hz

at     f = 5 Hz and T = 298 K
100-1004                                                                               Application Note 1004

The measured NEP of a typical SSI-A-45 Silicon Detector Instrument is 0.27pW/root
(Hz).

Spectrum Detector sells the SGI-A-45 Germanium Detector Instrument. The parameters
for this detector are:

nV                                                                       A
en = 13          I bias = 250 fA RD = 5 ⋅ 10 4            R f = 10 6    Ri = .90
Hz                                                                      W

1.69 ⋅ 10 − 28 + 1.64 ⋅ 10 − 26 + 8 ⋅ 10 −32         10 6           pW
NEP ≈                                                    ⋅ 1 +
 5 ⋅ 10 4    = 3.0

0.9                                               Hz

at   f = 5 Hz and T = 298 K

The measured NEP of a typical SGI-A-45 Germanium detector is 2.6pW/root (Hz).

Understanding the sources of noise in Radiometric Detectors and Instruments provides a
standard of performance based on solid theoretical limits. Spectrum Detector, Inc. builds
and sells Radiometers that perform to those theoretical limits. This ensures that our
customers are getting the best device available.

updated 11/28/2007

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