How to determine noise figure: part 2

The relationship between noise and temperature prompted a precursor of the IEEE to promulgate an alternative definition of noise figure in 1959.

In part 1 of this series, we described the work of the Danish-American radio engineer Harald Friis, who described noise factor F of a device or system as the ratio of the input-power signal-to-noise ratio (SNR) to the output-power SNR. If S_IN, S_OUT, N_IN, and N_OUT are the input and output signal levels and the input and output noise levels, respectively, then also note that noise figure (NF) is generally expressed in decibels:

Equation 2.

Finally, we noted that if S_IN, S_OUT, N_IN, and N_OUT are all decibel values, then NF in dB is

Equation 3.

You mentioned that we would be looking at another aspect of NF.
Yes. We can look at noise figures from the perspective of a fundamental noise cause — temperature. In the late 1920s, Schottky and Johnson hypothesized and measured the thermal noise phenomenon, in which an open-circuit resistor develops a noise voltage v_N that varies with temperature. Shortly after, Nyquist showed that:

where T equals the temperature in Kelvin, R equals the ohms resistance, B equals hertz’s measurement bandwidth, and k equals Boltzmann’s constant (1.38 ´ 10^-23 Joules/Kelvin). Essentially, a resistor becomes a voltage source of level v_N in series with a source resistance with a value of R_S, as shown in Figure 1a.

In RF/microwave circuits, we are interested in the noise power level P_N that a noise voltage source can deliver to a load—the input to a device under test (DUT), for example (Figure 1b). The worst case will occur when the DUT input impedance RL matches RS in accordance with the maximum power transfer theorem. If we set both equal to R, we can calculate the noise current:

Consequently, the maximum noise power transfer into the DUT in watts is

Substituting Equation 4 for v_N gives us:

The resistances R cancel out, and a bandwidth of 1 Hz is typically used for comparison purposes. Hence, the noise power spectral density PN per hertz depends only on T. A common reference temperature level is 290K (17°C or about 62°F). You can calculate that P_N equals -204 dBW/Hz or -174 dBm/Hz at that temperature.

So, how does this apply to the noise figure?
First, we can rewrite Equation 1 with G representing the DUT gain (in linear terms, not decibels) and N_ADD equaling the noise power added by the DUT (again in linear terms, not decibels):

Now, we note that the input noise N_IN equals P_N = kTB from Equation 5, so we can rewrite Equation 6 as follows:

Equation 7.

Then, as was the case for the method described in part 1, we can calculate the noise figure in dB:

Equation 8.

This temperature approach to noise figure was promulgated in 1959 by the Institute of Radio Engineers, a precursor to the IEEE.

Are there any other approaches to specifying noise performance?
Yes. We can rewrite Equation 3 as follows:

Consider Figure 2, where a noise source is connected to an impedance-matched power sensor and meter. The latter, which we assume contributes minimal noise, tells us the noise source adds noise power N_ADD.

We can now answer a key question: If we want to replace our noise source with a resistor, as in Figure 1a, at what temperature — call it T_E, for equivalent noise temperature (sometimes called effective noise temperature, or just noise temperature) — would we need to maintain the resistor for it to generate the same noise power N_ADD that our noise source is generating?

Why would we want to do that?
We wouldn’t want to do that physically, but T_E gives us another figure of merit for specifying the noise performance of a component. We can calculate T_E by substituting N_ADD for P_N in Equation 9:

Wow! We’ve got two equations for noise factor (Equations 1 and 7), three equations for noise figure (Equations 2, 3, and 8), and now we have an equation for T_E (Equation 10). When do we use each, and can we convert between them?
We’ll sort that out in part 3.