Page 283 - Organic Electronics in Sensors and Biotechnology

260 Cha pte r S i x

Calculating the Output Noise Voltage
From Eq. (6.32), a noise component σ of frequency f and bandwidth
Va
Δf will perturb the potential of the inverting input of the op-amp by
an amount σ ( f ) = σ ( f ) Δf . Using Eq. (6.53) and taking into
Va Va
account the capacitance C of the photodiode, this noise component
d
will appear at the output amplified by an amount [1 + R /Z ( f )]
f sh
where
R / j2π fC R
f
f
|Z ( )|= R ||Z ( ) = sh d = = sh
sh sh Cd + 12π
R j / fC 1+4π 2 2 2 2
sh d fC R
d sh
The output noise variance per unit frequency due to this noise com-
ponent will therefore be
2
R
σ () = σ 2 1 + f f Δ
2
f
Va Va
Z ()
f
sh
which, after some rearrangement, yields
⎡ ⎛ R ⎞ 2 ⎤

σ () = σ 2 ⎢ 1 + f ⎟ + 4 π fC R 2 ⎥ Δf
2
2
2
2
f
⎢ ⎢
Va Va ⎜ ⎝ R ⎠ d sh ⎥
⎣ sh ⎦
Hence, integrating over the full measurement bandwidth, we obtain
for the total mean squared noise at the output
⎡ 2 ⎤
B ⎛ R ⎞

=
σ Va ∫ σ 2 ⎢ ⎢ 1 + f ⎟ + 4 π fC R 2 ⎥ ⎥ ⎥ df
2
2
2
2
Va ⎜
sh
d
0 ⎣ ⎝ R sh ⎠ ⎦
⎡ ⎛ ⎞ 2 2 ⎤
3⎥
⎢ ⎢
σ 2 = ⎜ 1 + R f ⎟ B + 4π C R B σ 2
2
2
Va ⎢ ⎝ R ⎠ 3 d f ⎥ Va
⎣ sh ⎦
References
1. S. Flyckt and C. Marmonier, Photomultiplier Tubes: Principles and Applications,
Philips Photonics, Brive, Francem, 2002.
2. S. Donati, Photodetectors, Prentice-Hall PTR, 1999.
3. M. McClish, R. Farrell, R. Myers, F. Olschner, G. Entine, and K. S. Shah, Nuclear
Instruments & Methods in Physics Research Section a––Accelerators Spectrometers
Detectors and Associated Equipment, 567:36 (2006).
4. “Avalanche Photodiode: A User’s Guide” http://optoelectronics.perkinelmer.
com.content/applicationnotes/appapduserguide.pdf
5. X. J. Wang, E. Perzon, J. L. Delgado, P. de la Cruz, F. L. Zhang, F. Langa, M.
Andersson, et al., Appl. Phys. Lett., 85:5081 (2004).