Question 8.6: Estimate Vonoise,RMS for the circuit seen in Fig. 8.15. Veri......

The only element in this circuit that generates noise is the resistor. It generates thermal noise. The resistor’s noise voltage spectral density is \sqrt{4k T R} . The output

V_{o n o i s e}(f)=\sqrt{4k T R}\,\frac{1/j\omega C}{1/j\omega C+R}=\frac{\sqrt{4k T R}}{1+j\frac{f}{f_{3d B}}} units, V/\sqrt{Hz}

or

V_{\ o n o i s e}^{2}(f) =\frac{4k T R}{\mid 1 +j\frac{f}{f_{3d B}}\mid^{2}}=\frac{4k T R}{(\sqrt{1+({f}/{f_{3d B})^{2}}})^{2}}=\frac{4k T R}{1+(f/f_{3d B})^{2}} units, V^{\,2}/Hz

where f_{3d B}=1/2\pi R C. This single-pole roll-off was the reason we discussed noise-equivalent bandwidth (NEB) earlier, Ex. 8.2. Using Eq. (8.17), the output RMS noise voltage is

V_{o n o i s e,R M S}=\ \sqrt{f_{3d B}\cdot\frac{\pi}{2}}\ \cdot\ \sqrt{V_{L F,n o i s e}^{2}}                (8.17)

V_{o n o i s e,R M S}=\sqrt{\frac{1}{2\pi R C}\cdot \frac {\pi}{2}\cdot 4kTR}=\sqrt{\frac{k T}{C}}           (8.24)

The RMS value of the thermal noise in this circuit is limited by the size of the capacitor and independent of the size of the resistor. This result is very useful. This “Kay Tee over Cee” noise is frequently used to determine the size of the capacitors used in filtering or sampling circuits used in, for example, analogto-digital converters (ADCs).

The SPICE netlist and output are seen below.

Using Eq. (8.24) at 300 °K, we get V_{ o n o i s e,RMS}^{2}=4.14\times 10^{-9}\ V^{2}. This is close to what SPICE gives above for onoise_total (close but not exact; we stopped the simulation at 1 GHz not infinity). The output RMS noise is 64 μV (keeping in mind that the peak-to-peak value of the thermal noise will be larger than this).

The input-referred noise, inoise_total, in this simulation example is somewhat meaningless. Changing the stop frequency in the simulation from 1 GHz to 10 GHz has little effect on the output RMS noise but does cause the input-referred noise to increase. As indicated in Eq. (8.19) and the associated discussion, the input-referred noise spectral density increases indefinitely. Integrating this spectral density from DC to infinity results in an infinite RMS input-referred voltage. Since the low-frequency gain here is one, we would specify V_{i n o i s e,R M S}=V_{o n o i s e,R M S}.

V_{i n o i s e}^{2}(f)=\frac{V_{o n o i s e}^{2}(f)}{|A(f)|^{2}}=\frac{V_{L F,n o i s e}^{2}}{1+(f/f_{3dB})^{2}}\cdot\frac{1+(f/f_{3d B})^{2}}{A_{D C}^{2}}=\frac{V_{L F,n o i s e}^{2}}{A_{D C}^{2}} (8.19)