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100-1004 Application Note 1004

Application Note 1004

Understanding Noise Equivalent Power in RadiometricDetectors and Instruments

Sid Levingston

03/20/2007

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Noise Equivalent Power, or NEP, is a basic indicator of detector performance. NEP is thenoise 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:

o f freqeuncy someat Rv

Hz Voltage Noise NEP ,=

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:

( )

TauThermal ic Pyroelectr f C R

f

f

f

f

f

f

f R R

f Rv

therm

f f

pole

poletherm

therm

f i

=⋅⋅⋅

=

+⋅

+

⋅⋅

=

π 2

1

11

22

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:

( )

( )2

21 f f

f i

C R f

R R f Rv

⋅⋅⋅⋅+

⋅

=π

RF

VoCDRD

en

Vjin

CF

Figure 1, Current Mode Radiometer with Noise Sources

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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 noiseresulting 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 theamplifier 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:

( )( )( )

( )Noise Amp

C R f

C C R f e f V

f f

D f f n

e2

2

21

21

⋅⋅⋅⋅+

+⋅⋅⋅⋅+⋅

=π

π

( )( )

Noise JohnsonC R f

RT K f V

f f

f

J 2

21

4

⋅⋅⋅⋅+

⋅⋅⋅

=π

( )( )

NoiseCurrent C R f

I q R f V

f f

bias f

i2

21

2

⋅⋅⋅⋅+

⋅⋅⋅

=π

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

( )( )( )( )

( )2222

21

2421

f f

bias f f D f f n

noise

C R f

I q R RT K C C R f e f V

⋅⋅⋅⋅+

⋅⋅⋅+⋅⋅⋅++⋅⋅⋅⋅+⋅

=π

π

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

( )( )( )( )

i f

bias f f D f f n

R R

I q R RT K C C R f e f NEP

⋅

⋅⋅⋅+⋅⋅⋅++⋅⋅⋅⋅+⋅

=2421

222π

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

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( )( )

( )( )( )( )

( )

( )( )( )

i

bias

f

D f

f

n

i f

bias f f D f f n

D f f

R

I q R

T K C C f

R

e

f NEP

R R

I q R RT K C C R f e f NEP

C C R f

⋅⋅+⋅⋅

++⋅⋅⋅⋅

≈

⋅

⋅⋅⋅+⋅⋅⋅++⋅⋅⋅⋅⋅

≈

+⋅⋅⋅⋅<<

24

2

242

21

2

2

222

2

π

π

π

It can now be seen that for a given detector with a characteristic responsivity andcapacitance, reducing NEP can be accomplished by: increasing R f , decreasing I bias,

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 I bias contribution starts todominate. 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 2for increasing R feedback. NEP improvement is negligible for values bigger than 100G_.

NEP vs. R Feedback

0.1

1

10

100

1000

1.E+06 1.E+08 1.E+10 1.E+12 1.E+14

R Feedback

N E P

( n W / r o o t H z )

NEP

Figure 2, NEP vs. R Feedback

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At Spectrum Detector our low noise detectors use a 100G_ feedback resistor and we takegreat 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, anultra low NEP detector. The parameters for this detector are:

( )

K T and Hz f at

Hz

nW f NEP

W A R R pF C pF C fA I

Hz nV e i f f Dbiasn

2985

9.01055.

1081064.1101.8

55.1012.02225013

6

323138

11

==

=⋅

⋅+⋅+⋅

≈

======

−

−−−

µ

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 sourcesand does not contribute to NEP. The theoretical lowest NEP for the detector is limited bythe 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, thefeedback 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:

( )( )

Noise AmpC R f

e f V

f f

n

e2

21 ⋅⋅⋅⋅+=

π

( )( )

Noise JohnsonC R f

RT K f V

f f

f

J 2

21

4

⋅⋅⋅⋅+

⋅⋅⋅

=π

( ) ( ) NoiseCurrent C R f

I q R

f V f f

bias f

i2

21

2

⋅⋅⋅⋅+

⋅⋅⋅

=π

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

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( )( )

+⋅

⋅⋅⋅⋅+

⋅⋅⋅+⋅⋅⋅+=

D

f

f f

bias f f n

noise R

R

C R f

I q R RT K e f V 1

21

24

2

22

π

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

( )( )2

22

21

24

f f

bias f f n

noise

C R f

I q R RT K e f V

⋅⋅⋅⋅+

⋅⋅⋅+⋅⋅⋅+=

π

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

Germanium for R

R

R

I q R

T K

R

e

NEP

Silicon for R

I q R

T K

R

e

NEP

D

f

i

bias

f f

n

i

bias

f f

n

+⋅

⋅⋅+

⋅⋅+

≈

⋅⋅+

⋅⋅+

≈

1

24

24

2

2

2

2

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:

K T and Hz f at

Hz

pW NEP

W

A R R R fA I

Hz

nV e i f Dbiasn

2985

26.05.0

1081064.11069.1

50.101025013

322628

68

==

=

⋅+⋅+⋅

≈

=====

−−−

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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:

K T and Hz f at

Hz

pW NEP

W

A R R R fA I

Hz

nV e i f Dbiasn

2985

0.3105

101

9.0

1081064.11069.1

90.1010525013

4

6322628

64

==

=

⋅+⋅

⋅+⋅+⋅≈

==⋅===

−−−

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. buildsand 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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