July 04, 2012

This study compares the total output noise voltage of the transimpedance circuit and the non-inverting amplifier circuit. The basic circuit for noise calculation purposes is shown to the right. The comparison demonstrates how the exact expressions for the output noise voltage reduce to approximate expressions involving single-pole like noise filtering characteristics. The different behaviour for these two circuits is a result of the different locations of the zero and pole frequencies of the noise-gain profile. Although it is easy to calculate the exact value of the total output voltage noise for any combination of parameters for this circuit, simplified expressions can be obtained for noise calculation in the two typical circuits discussed here, providing greater insight into the noise performance characteristics.

where

For this feedback network consisting of Ri, Ci, Rf, Cf, the zero and pole frequencies of the noise gain (1/β) are:

These expressions are exact and valid for any location of the zero and pole frequencies and for any op-amp GBW product. The spectral noise densities will be assumed constant over frequency. Therefore 1/f noise, which will be important in low frequency circuits, is not considered. Also, the op-amp current noise in the example below is extremely low and will be insignificant compared to the

In the circuits, R and C values are the TOTAL values. Both circuits have the same Rf of 50kohm. Ri for the non-inverting circuit is 1 Gohm a typical very high value for a FET-input op-amp. Ri for the non-inverting amplifier is 500ohm which provides a circuit gain of 101 or ~ 40dB. The Ci value is the total input capacitance at the inverting input to ground. For the transimpedance circuit, the value is 12 pF which would include the op-amp capacitances and the current source (e.g. a photodiode) capacitance plus strays. The transimpedance feedback shunt capacitance Cf of 1.35pF is higher than the non-inverting input value 0f 0.4 pF (strays only). This would correspond to adding an additional shunt capacitance for stability and phase compensation for this particular circuit.

The graph below shows the various gain plots of both circuits. The usual

These expressions are very similar to the single-pole noise-bandwidth result for the common RC filter of

This final expression shows that for the total output voltage noise contribution from

It is not difficult to show that with Ri included and as above assuming Fp<<BW and Fz<<BW, the noise bandwidth for

and the output voltage noise contribution from

Again, this final expression shows that

The plot below shows the exact output spectral voltage noise density for

The approximate and exact results for the resultant voltage output noises for the transimpedance circuit above are:

**en**voltage noise: 222.4 μV (approx) 218.4 μV (approx with Fz term)**218.9 μV(exact)**-
**Rf**thermal noise: 54.5 μV (approx)**53.5 μV (exact)**

The noise bandwidth in this case is simply that of a single-pole rolloff, as seen in the graph above, at approximately the non-inverting amplifier's f3db frequency, exactly as expected. Since Fz and Fp are at higher frequencies, the op-amps gain bandwidth limitation determines the circuit bandwidth and also the noise-bandwidth. Of course, Cf could be increased to lower Fp which eventually would lead to narrower circuit bandwidth and a lower noise-bandwidth, controlled by the location of Fp.

The approximate and exact results for the resultant voltage output noises are:

**en**voltage noise: 711 μV (approximate) 688 μV(exact)-
**Rf**thermal noise: 286 μV (approximate) 275 μV (exact)

For the transimpedance circuit, about 90% of the total (all frequency) noise is reached at a frequency of 20MHz, or 6X the transimpedance 3db bandwidth. This slow convergence to total noise is due to the gain peaking in the noise-gain profile. Notice how at ~ 1.8MHz, the

For the non-inverting amplifier, 90% of the total (all frequency) noise is reached at a frequency of 2 MHz or 3Xf3db = 3XGBW/N0. Both the