System development and methods
2.1 Construction of optical tweezers
2.1.7 PSD circuit: performance and electronic noise control
factors deter-mining the cir-cuit performance
The circuit performance of a transimpedance amplifier in terms of gain, bandwidth, stability and signal-to-noise ratio (SNR) depends on a compli-cated and nonlinear interplay of several factors (Graeme 1996, Burr-Brown 1994), the most relevant of which are:
• desired bandwidth (BW)
• desired gain factor (G)
• feedback resistor (RF)
• feedback capacitance (CF)
• gain-bandwidth product of the op-amp (GBP)
• op-amp voltage noise (Vnoise(amp))
• op-amp current noise (Inoise(amp))
• op-amp input capacitance (CAmp)
• photodiode junction capacitance (CPD)
• photodiode shunt resistance (RSh)
In the case of a photodiode transimpedance amplifier, the three circuit de-sign variables are:
• the feedback loop
• the op-amp
• the photodiode
The circuit response, gain and stability are strongly affected by the non-ideal characteristics of these three elements. Their interplay in the actual circuit can be properly analyzed by using the model shown in Fig. 2.16.
FIGURE 2.16 Simplified model for stability and noise analysis of the transimpedance
transimpedance gain
voltage noise
a) Feedback loop. In the feedback loop of the current-to-voltage converter, noise and bandwidth as well as gain are largely determined by the values of resistance (RF) and capacity (CF). An approximated estimation of the gain of the current-to-voltage converter is given by the transfer function:
F
VOUT is the voltage at the output of the operational amplifier in V;
IPC is the photocurrent produced by the photodiode in A;
ℜ is the responsivity of the photodiode in A/W;
P0 is the incident light power in W.
For a resistance value of RF = 10 MΩ = 107Ω the gain is
G = 20 log10(107) = 140 dB (2.8) The value of the noise originating from the feedback resistor, VNOISE(R)(unit Vrms), can be quickly calculated by using the relation
BW T is the temperature in Kelvin,
BW the system bandwidth in Hz.
Since the experimentally measured bandwidth of the amplifier is 110 kHz (see Paragraph 2.2.2), the noise caused by the feedback resistor is 0.134 mVrms. Increasing the size of the resistor does not only increase the output noise in a square-root manner (Eq. (2.9)), but also increases the output sig-nal in a linear manner (Eq. (2.7)). Thus, sigsig-nal-to-noise ratio tends to in-crease with the square root of the resistance,
RF
SNR∝ (2.10)
Since the gain-bandwidth product (GBP) is a constant (for the op-amp LT1114 GBP = 4.5 MHz typ.), increasing the feedback resistance improves the gain and the SNR of the circuit, but limits the bandwidth.
b) op-amp. As already mentioned in Paragraph 2.1.5, the quadruple op-amp LT1114 is used in the current-to-voltage converter stage. Some relevant parameters of this integrated circuit are shown in Table 2.3.
Amplifier characteristics Value
input noise voltage (Vn) 1.8 ·10-8 V√Hz input current noise (In) 8.0 · 10-15 A√Hz input bias current (Ib) 1.5 · 10-10 A gain bandwidth product (GBP) 4.5 · 106 Hz input offset voltage (Voff) 6.0 · 10-5 V input capacitance (Ca) 4.0 · 10-12 F
TABLE II.3 Specifications of the op-amp LT1114.
op-amp current noise
op-amp voltage noise
The current noise generated by the op-amp flows through the feedback re-sistor and experiences the same gain as the signal current, leading to the additional noise component
BW amp I
R amp
VNOISE( )= F NOISE( ) [Vrms] (2.11) where RF is the feedback resistance, INOISE is the total op-amp current noise, and BW is the system’s bandwidth. For usual values of feedback resistance, current noises are negligible if op-amps with noise levels in the pA range are chosen.
The contribution of op-amp voltage noise to the total noise is relevant only at low resistance levels (up to about 10 KΩ). Between 10 KΩ and 10 MΩ, resistor noise (expressed by Eq. (2.9)) is the dominant factor. At higher val-ues of RF, the op-amp noise becomes relevant again [Graeme 1996, Burr-Brown 1994].
photodiode
where tr(amp) is the rise time of the amplifier, defined as the time during which the signal rises from 10% to 90% of its final value. tr(amp) is with tPDr being the rise time of the photodetector, approximated by
dB
where f3dB is the bandwidth of the photodetector.
Since f3dB = 20 MHz, one obtains tPDr = 1.75⋅10-8 sec. Taking RF = 10 MΩ and CF = 1 pF, it follows that the amplifier bandwidth is determined mainly by the time constant of the feedback loop τ = RF CF≅ 10-5 sec.
c) Photodiode. As seen in Paragraph 2.1.4, the PSD is operated in voltaic mode, which means that no external voltage is applied to the photo-diode (zero biasing). This implies that the dominating factor of the photodi-ode’s current noise (called dark current or leakage current) is a diffusion current, and varies with the square root of the temperature. Such noise is also called thermal or Johnson noise, and its value is given by
( )
where Rsh is the parasitic resistance (or shunt resistance) of the photodiode.
Its value typically exceeds 100 MΩ. The r.m.s. noise value is then given by
( )
PD I R BW NEP R BWVNOISE = jn F =ℜ⋅ ⋅ F [Vrms] (2.17) with
RF the feedback resistance,
ℜ the responsivity of the photodiode,
NEP the noise equivalent power.
NEP is defined as the amount of light power which generates a photocurrent equal to the noise current. For a photodiode with large active area, the NEP value is about 10-11 W√Hz. For the system used in this work, the calculated VNOISE(PD) is about 50 mVrms.
Circuit capacitances have profound performance effects in terms of stabil-ity, bandwidth and noise. It is known that the input capacitance of an op-amp (CAmp) can cause instability when the op-amp is used with a feedback resistor. Particularly, the circuit will oscillate and display gain peaking. Sta-bility is usually achieved by adding a capacitor CF across the feedback re-sistor RF (see Fig. 2.16).
The photodiode junction capacitance (CPD) must also be taken into account when considering stability. CPD is a by-product of the width of the depletion region between the p-type and n-type material of the photodiode. A wider depletion region, as found in PIN photodiodes, increases the magnitude of the junction capacitance. In the PSD S5981, the total capacitance from the four quadrants is 60 pF.
Considering CPD and Camp, the optimized value for CF can be found through the equation [Graeme 1996]: Applying Eq. (2.17), an optimal value of CF≅ 0.5 pF is found for preventing oscillations in the current-to-voltage converter. In practice, the parasitic ca-pacitance (about 0.1 pF) originating from on-board capacitive coupling and from parasitic capacitances is enough to avoid circuit oscillations. There-fore, it was decided not to add a feedback capacitance to the circuit, thus avoiding further bandwidth limitation without compromising stability.
Electromagnetic coupling. With its very high impedance, a current-to-voltage converter is extremely sensitive to noise coupling from low and ra-dio frequency electromagnetic sources. These sources require excellent shielding, grounding, and “smart” relative arrangement of all components for preventing them to become dominant noise contributors. A very
com-Hz). This noise is coupled via the mutual capacitances that exist between any two objects. In order to avoid such coupling, the PSD is placed as close as possible to the op-amp’s input terminals using short leads (Fig. 2.13). A further improvement is achieved by a metallic housing which is earth-grounded (Fig. 2.14).