Page 253 - Organic Electronics in Sensors and Biotechnology

230 Cha pte r S i x

In fact this last equation isn’t strictly correct because we have over-
†
looked the capacitance C of the photodiode. This can be taken into
d
account by replacing the shunt resistance R by the complex imped-
sh
ance Z = R | | Z , where Z = 1/(j2πfC ). The capacitance introduces
sh sh C C d
a frequency dependence into the gain, and after integrating over all
frequencies (see the Appendix), the root mean squared noise voltage
at the output becomes
⎛ R ⎞ 2 π 2
σ out = B ⎜ 1 + f ⎟ + 4 CR B 2 (6.54)
σ
2
2
Va Va d f
⎝ R sh⎠ 3
The transimpedance circuitry introduces one other noise source into
the final amplified signal: thermal noise from the feedback resistor,
which appears at the output with a root mean squared value:
σ out = 4 kTR B (6.55)
Rf B f
Combining all three noise contributions, we obtain a total root mean
squared noise voltage at the output of
V = ( ) +( ) +( ) 2
2
2
σ
σ
σ
out
out
out
N Ia Va Rf
(6.56)
⎛ R ⎞ 2 4π 2
2
2
+
V = B 12 1 ⎜ + f ⎟ σ 2 + CR B σ 2 + R σ + 4kTR
2
2
2
/
N Va d f Va f Ia B f
⎝ R R sh⎠ 3
Hence, dividing through by the amplified signal iR , we obtain for the
f
signal-to-noise ratio
2
/
V B 12 ⎛ R f ⎞ σ 2 4π 2 2 2 4 kT
SNR = N = ⎜ 1 + ⎟ Va + C B σ 2 + σ + B (6.57)
2
iR i ⎝ R sh⎠ R 2 3 d d Va Ia R
f f f
It follows from inspection of Eq. (6.57) that to maximize the signal-to-
noise ratio, we must
• Maximize the photocurrent i by collecting as much of the
incident light as possible and ensuring the photodiode has a
high quantum efficiency.
• Select an op-amp with small values of σ Va
Ia and σ , i.e., one
in which the input noise current and input noise voltage are
† Note that we didn’t need to worry about the capacitance when dealing with the
offset voltage because V has a constant dc value.
OS