Page 35 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Amplified Detection Circuitry

28 Chapter Two

resistor, but arranged differently. The opamp is connected with resistive feed-
back provided by the load resistor and the noninverting input is grounded. Neg-
ative feedback tries to force the two amplifier inputs to the same potential, so
the inverting input becomes a “virtual earth.” The impedance seen looking into
this node is low.
Because the amplifier’s input current is very low, a few tens of picoamperes,
the bulk of the photocurrent has to flow as shown through R L to the amplifier’s
output. With the polarities shown the output voltage must therefore become
negative to “pull” current out of the photodiode anode. With the capacitance
connected to the virtual earth, changes in photocurrent barely change the
voltage on C p. If the voltage does not change then neither does the charge Q =
CV, and its apparent capacitance is greatly reduced. In the ideal case the diode
capacitance is effectively shorted out, making it invisible to the photocurrent
and feedback resistor. Consequently, the slow response time of the follower is
significantly improved.
It is as though the capacitance now experiences a feedback resistance reduced
by the value of the effective loop gain of the amplifier. The limiting bandwidth
of the receiver (f limit) is then given by the original bandwidth of the bias box or
follower circuit (1/2pR LC p) multiplied by the open loop amplifier gain at the lim-
iting bandwidth. Hence we can write:

f limit = A flimit (2.2)
(2p RC p )
L
where A flimit is the closed loop amplifier gain at the limiting frequency. The
majority of conventional, frequency-compensated opamps use an internal RC
combination to give a dominant frequency pole at around f 1 = 20Hz (Fig. 2.7).
Above this frequency the gain drops off at a rate of -20dB/decade, reaching
0dB (unity gain) at the frequency corresponding to the “gain-bandwidth
product” (GBW) or “unity gain frequency.” The gain is therefore an approxi-
mately inverse function of frequency over much of the useful frequency range
and GBW/f 1 = DC gain. In the figure, 20·log(4MHz/20Hz) = 106dB, and the
gain at any frequency f is approximately GBW/f. Hence we can write:

GBW
f limit = (2.3)
( f limit 2p R C p )
L
Rearranging the above for the upper frequency limit or bandwidth of the
photodiode-transimpedance amplifier configuration we obtain:
12
Ê GBW ˆ
f limit = Ë 2p RC p ¯ (2.4)
L

This approximate expression should not be relied on for exacting accuracy, but
it is useful for this situation. Typically the limiting frequency will be about one-
half the value calculated from Eq. (2.4). For example, the LMC7101 FET opamp

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