## Wednesday, February 18, 2015

### HP Prime & TI-84+: Thermal Noise (Johnson-Nyquist Noise)

Introduction

The following program calculates the RMS (root-mean-square) voltage of a resistor at a certain temperature.  Thermal noise, also known as Johnson-Nyquist Noise is generated when the resistor has a temperature above absolute zero.  The equation to calculate the voltage is:

V = √(4*k*T*R*B) where

k = Boltzmann’s Constant = 1.3806488 * 10^-23 J/K
T = temperature in degrees Kelvin (K)
R = resistance in ohms (Ω)

B = noise bandwidth (Hz)

This program also calculates the noise power in decibels.  Yes, the power for most calculations for this application will be a negative quantity.   (I did a double take when I first learned about noise power.)   The equation for the noise power is:

P = −198.599167802+10*LOG(T*B)

where -198.599167802 = 10*LOG(k*1000) (see above)

HP Prime:  THNOISE

EXPORT THNOISE(R,T,B)
BEGIN
// 2015-2-18
// R = resistor (ohms)
// T = temp (°K)
// B = bandwidth (Hz)
LOCAL V,P;
// Boltzmann′s constant:
// can get retrieved by pressing Shift, Units, Const, 2, 2
V:=√(4*1.3806488ᴇ−23*T*R*B);
// volts
P:=−198.599167802+10*LOG(T*B);
// dBs
RETURN {V,P};
END;

TI-84+:  THNOISE

Disp "R=RESIS. (OHM)"
Disp "T=TEMP. (°K)"
Disp "B=BANDWIDTH (HZ)"
Prompt R,T,B
√(4*1.3806488ᴇ−23*T*R*B)→V
−198.599167802+10*LOG(T*B)→P
Disp "V=VOLTAGE"
Pause V
Disp "P = POWER"
Pause P

Example:

R = 1040 Ω
T = 300 K
B = 19000 Hz

Results:
V » 5.721708 * 10^-7 V
P » -131.040419 dB

Sources:

John A. Ball.  "Algorithms for RPN Calculators"  John Wiley & Sons, Inc.  New York.  1978.  pg. 267

Johnson-Nyquist Noise.  http://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise  Retrieved 2015-2-15

This blog is property of Edward Shore - 2015.