Oct 07, 2015

This note presents results for the complex

The noise-gain function (along with the op-amp open loop gain profile) determines circuit stability. For transimpedance circuits, the noise-gain is important because it determines how the op-amp input voltage noise source

It is also interesting to study the noise-gain transfer function with a series resistance Rs added to the circuit as shown below, both outside and inside the op-amp feedback loop which of course adds some complexity. The influence of real op-amp response and shunt resistance on Tz(f), noise-gain and total noise is discussed in full detail elsewhere

The frequencies of the

This noise gain transfer function is of first order with one zero and one pole. At high frequency, the noise-gain for this circuit approaches a constant value of NG=(1 + Cd/Cf). If Cd>>Cf as is common in photodiode circuits, this asymptotic high-frequency noise-gain "plateau" will be large and implies that the op-amp input noise source will continue to be amplified at frequencies well beyond the transimpedance f

The noise gain transfer function for this circuit is of 2nd order with two real poles, and two zeros (which may be complex).

where Tp1 is the same real pole as for the basic circuit above. The additional real pole is formed by Rs and Cd.

The two zeros for this circuit are:

For many cases of interest, Rf(Cf+Cd) << RsCd and the zeros are approximately:

where Tz1 is the same as for the basic circuit above and Tz2 is typically at very high frequency. Note that both the low and high frequency noise-gain values approach unity (0 dB).

With the series resistance Rs in the feedback loop with Cd essentially in a t-network, the zeros and poles are both more complicated and both may take on complex values indicating 2nd order resonant behaviour. The noise-gain response function is still 2nd order with two zeros and poles:

The poles are:

and the zeros are: (note they are at different frequencies than the zeros for the circuit above)

For many cases of interest, Rf(Cf+Cd) << RsCd and the zeros are approximately the same as those of the circuit above with Rs outside the feedback loop and are typically real. However, the poles can be complex for typical values. This occurs approximately for:

Note that as for the circuit above with Rs outside the feedback loop, the low and high frequency noise-gain both approach unity (0dB).

The two plots below show an example of the three circuits above for two values of Rs. The noise-gain (in db) is plotted versus log frequency. As Rs is increased, the two poles Fp1, Fp2 for the circuits with Rs move closer together as shown. For Rs=1000ohm, the poles for Cir3 (with Rs inside the feedback loop) are complex. The pole positions shown are the real pole frequencies for Cir2. The resonant-like peaking behaviour for Cir3 is clearly evident.The peak noise-gain for Cir2 is reduced as the two real poles more together and their influence overlaps resulting in attenuation of the "plateau" noise-gain:

Of course, with a real op-amp with a finite GBW product, the true high-frequency noise-gain will eventually become important and will roll-off the noise-gain response.