229-A2.2

Activity: Noise Analysis

• Instructions: This activity is structured as a tutorial with a design problem at the end. Should you have any questions, clarifications, or issues, please contact your instructor as soon as possible.
• At the end of this activity, the student should be able to:

1. Understand and observe the effects of noise in electronic circuits.

Simulating Resistor Noise

Figure 1: A simple RC circuit.

Consider a simple RC circuit shown in Fig. 1. The noise generated by the resistor is:

${\overline {v_{R}^{2}}}=4kTR\Delta f$

(1)

Thus, for a resistor $R=1\mathrm {k\Omega }$ and a temperature $T=300K$, and with $k=1.38\times 10^{-23}\mathrm {\tfrac {J}{K}}$, the spot noise is equal to:

${\frac {\overline {v_{R}^{2}}}{\Delta f}}=4kTR=1.66\times 10^{-17}\mathrm {\frac {V^{2}}{Hz}}$

(2)

Thus, for a capacitor $C=1\mathrm {\mu F}$, the total output integrated noise, shaped by the transfer function of the RC circuit is:

${\overline {v_{o,T}^{2}}}=\int _{0}^{\infty }4kTR\left|{\frac {1}{1+sRC}}\right|^{2}df={\frac {kT}{C}}=4.144\times 10^{-15}\mathrm {V^{2}}$

(3)

Or equivalently,

$v_{o,\mathrm {rms} }={\sqrt {\overline {v_{o,T}^{2}}}}=64.4\mathrm {nV}$

(4)

If we change the value of the resistor to $R=10\mathrm {k\Omega }$, the resistor spot noise changes to:

${\frac {\overline {v_{R}^{2}}}{\Delta f}}=4kTR=1.66\times 10^{-16}\mathrm {\frac {V^{2}}{Hz}}$

(5)

But the total integrated noise should remain the same since the bandwidth of the RC circuit is reduced by the same amount. To verify our calculations, we can use SPICE to simulate the noise of the RC circuit. Using this netlist and this Python script, we get the following ngspice results:

******
** ngspice-31 : Circuit level simulation program
** The U. C. Berkeley CAD Group
** Copyright 1985-1994, Regents of the University of California.
** Please get your ngspice manual from http://ngspice.sourceforge.net/docs.html
** Please file your bug-reports at http://ngspice.sourceforge.net/bugrep.html
******
ngspice 1 -> source RC_noise.sp

Circuit: * noise simulations

Doing analysis at TEMP = 27.000000 and TNOM = 27.000000


No. of Data Rows : 9001

No. of Data Rows : 1
onoise_total = 4.143579e-15
Doing analysis at TEMP = 27.000000 and TNOM = 27.000000


No. of Data Rows : 9001

No. of Data Rows : 1
onoise_total = 4.143808e-15
ngspice 2 ->

We can use our Python script to plot the output noise spectral density, as shown in Fig. 2, and the root spectral density, seen in Fig. 3.

Figure 2: The power spectral density of the RC circuit.

Figure 3: The root power spectral density of the RC circuit.

As expected, the total integrated output noise is independent of the value of the resistor, and only depends on the value of the capacitor.