To: file: \techniques\johnson

From: Jon Hagen

Date: 6-19-01

Subject: Johnson Noise

A resistor at temperature T is a noise generator. If connected to a second resistor of equal resistance, each will supply the other a noise power equal to kT watts per Hertz. One way to derive this formula is to consider a transmission line of length L. Let the ends be shorted, making the line is a one-dimensional cavity, capable of storing energy in each of the possible standing wave modes (modes that have V=0 at x =0 and at L=0, to satisfy the boundary conditions). If the line has even the slightest bit of loss, thermal energy in the line (elastic vibrations in the metal conductors) will eventually excite each mode to an average energy kT. These standing waves consist of equal power flow in both directions. The number of modes per Hertz is proportional to the length of the line. If the line is made longer, the (lineal) energy density in each mode is reduced but there will be more modes per Hz, so the energy density per Hertz does not change. Let us calculate the energy density:

The boundary conditions require that f = nc/2L where n = 1,2,3.... and c is the speed of light. Therefore the number of modes n in a frequency range, f = B, is given by n = 2LB /c. Assigning an energy kT to each mode, the energy density, in Joules per unit length, is u = 2BkT/c. The resulting power flow in each of the two directions is therefore

P= uc/2 = kTB Watts.

Now suppose that the terminations are not perfect shorts, but are resistors with a value, r, much less than Zo of the cable. The mismatch between the transmission line and the resistors will produce a large (near unity) voltage reflection coefficient, = (r-Zo)/(r+Zo), where Zo is the characteristic impedance of the line. The rate at which power flows into each "load" resistor will therefore be given by

P to load = kTB(1- 2).

Thermodynamics requires that the load resistors drive an equal amount of power back into the transmission line; otherwise the temperature of the resistors would increase and the temperature of the line would decrease, resulting in a heat to flow from a colder object to a warmer object, in violation of the 2nd Law. The load resistors, looking into the transmission line, see the same reflection coefficient, except for its sign. If these resistors are now regarded as Thévenin voltage generators, the power delivered to the line by each resistor will be given by

P from load = (1- 2)(Vrms )2 /4r

where Vrms is the generator's open-circuit voltage and r is its series resistance. Equating the power to the load with the power from the load, we find that the open circuit noise voltage of a resistor is given by

(Vrms )2 = 4kTBr.

A resistor will therefore supply a like valued resistor with a power given by kTB.