This entry is a continuation of my current to voltage converter project. In this entry we'll be taking about component selection and noise in the circuit. The discussion leans heavily on this Burr Brown application note.

### Sources of Noise

Below is figure 4 from the above document, which is the noise model that I'll be using in this analysis.

There are multiple sources of noise to consider:

- Incident light shot noise,
- Thermal (Johnson) noise on Rf,
- Op-amp input current noise,
- Photo-diode dark current shot noise
- Op-amp input voltage noise,

The lower the technical noise sources are, the less laser power is needed in order to have the measurement be limited by the shot noise on the laser and the more power we have available to use in the experiment. We will need to choose components that minimize the total technical noise in the circuit. The first three contributions in the list are voltage noise sources and thus follow the voltage gain transfer function given by:

Which looks something like this for the op-amp we've chosen and a reasonable bandwidth limiting capacitor `Cf`

:

The other items on the noise budget are current noise sources, whose transfer function is just the trans-impedance of the circuit and is given by:

#### Component Choices

As with most optimization tasks, the component choices are coupled. Different combinations of component choices will be optimal for different end goals. However we can start by defining the bandwidth of the circuit in terms of the op-amp selected

##### The bandwidth limiting capacitor: Cf

We can generate a formula for the "ideal" feedback capacitor with knowledge of a few parameters:

- The Gain-Bandwidth Product (GBW) of the op-amp
- The capacitance of the photo-diode (plus parasitic capacitances in the circuit)
`Cd`

- The feedback resistor
`Rf`

This formula makes sure that the asymptotic voltage loop gain (solid blue line), `1+Cd/Cf`

, is equal to the op-amp's open loop gain (dashed red line) at the pole frequency `fp`

(green line), guaranteeing stable behavior.
This is demonstrated below:

Of course, in real life we will want a slightly larger `Cf`

to make sure we aren't flirting with disaster.
How much margin we require will depend on how well we have characterized the real circuit and the accuracy of the component values chosen.

##### The Op-amp

The choice in op-amp is the most application critical decision for the whole circuit, with the only exception maybe the photo-diode which is another important factor in the maximum achievable bandwidth.
We want a gain bandwidth product that is high enough to reach the desired bandwidth, but in general a higher bandwidth usually means a higher noise floor.
So we want to pick a low-noise op-amp with high enough `GBW`

to reach the design goal (~1 MHz) with enough wiggle room to account for the loop-gain of the circuit.
Therefore we want to be looking for an op-amp with a `GBW`

around 10 MHz or higher.

Next attribute consideration we need to consider is the input current and voltage noise.
Once again there is a trade-off to consider, in general as you decrease the input voltage noise of an op-amp the current noise increases, and therefore the optimal choice of the op-amp will depend on the feedback resistance.
The larger `Rf`

is required to be by the design goal, the more significant the current noise contribution.
So the optimal selection parameter is the ratio of the current `i`

and voltage `v`

noise and the trans-impedance designed `Rf`

, `v/i = Rf`

.

In terms of the shape of the input noise, we want the 1/f "flicker noise" to intersect the white noise at the lowest possible frequency, while still roughly obeying the optimal ratio condition specified above.

##### Our Design

For our design goals we chose to use the AD8675 op-amp.
It had a gain-bandwidth product of 10 MHz, a current white noise density of 0.3 pA/rtHz, a voltage noise density of 2.8 nV/rtHz (`v/i = 10 kOhm`

), and the 1/f voltage noise gives way to white noise at around 20 Hz (the current 1/f noise is not reported).
Below is the expected technical noise:

For the second stage we decided to limit the bandwidth with an RC filter with the pole at the highest achievable bandwidth (`Rf = 1 kOhm`

).
We decided to use the AD829 as a line driver with voltage gain of 2, which has a lower voltage noise (but higher current noise and worse 1/f properties), and will not add any appreciable noise to the output signal.