Nyquist Noise Formula Cascading Noisy Components Noise Figure

Nyquist Noise Formula Cascading Noisy Components Noise Figure
Due to thermal agitation of charges in resistors, attenuators,
mixers, etc., such devices produce noise voltages and currents.
For example, in a resistor the charges move randomly due to
thermal agitation:

As you know, applying a voltage across a resistor makes it
warm. Conversely, heat in a resistor produces voltage and
current in the resistor. We’ll call these two quantities “noise
voltage” and “noise current,” respectively. By this reasoning, we wouldn’t expect much electrical noise to be generated in an inductor or capacitor. The famous Nyquist noise formula states that the rms noise voltage from a noisy resistor is

where k = Boltzman’s constant = 1.38×10^{?23} J/K,
T = temperature in K,
B = bandwidth (taken has 1 Hz in the text), and
R = resistance (?).
From this formula, we can produce an equivalent circuit model
for a “noisy resistor”:

We can now ask: What is the maximum available noise power
from this noisy resistor? To determine this, we’ll attach a perfect
(i.e., noiseless) resistor R to this circuit

The available noise power P_{n} may now be computed as

Again, this P_{n} is the (maximum) available noise power from a noisy resistor. From the definition of noise power density P_{n} = NB in we find

which is the (maximum) available noise power density from a
noisy resistor.
Note that this available noise power density in (14.21) is NOT
dependent on the value of R! However, after careful thought this
is perhaps not too surprising since we’re dealing with the
maximum power that is available. Noise Temperature
This last formula (14.21) is so simple that it is often convenient
to use temperature as a measure of noise power density as

where T_{e} is the effective noise temperature. This is commonly done, even if the noise is not thermal in origin!
In the case of receivers, amplifiers, mixers and attenuators, the
noise temperature is found by dividing the equivalent input
noise power density N_{input} by k as

But, with NEP = N/G then

Note that if we are considering anything but a resistor, T_{n} is an effective temperature and has nothing to do with the physical environment. It is also common to define an equivalent noise temperature for an antenna. Antennas actually produce very little noise themselves. Instead, they receive noise signals from natural and manmade sources

Cascading Noisy Components
When we connect parts of a receiver together, it’s important to
know the overall output noise power density as well as which
subsections contribute most to this noise. Then we can design
those portions of the circuit to reduce the output noise power.
If sources of noise in a receiver are “uncorrelated,” then noise
power from one section can simply be added to the next.
• Uncorrelated signals: random thermal variation is an
example.
• Correlated signals: power supply fluctuations that
simultaneously affect many subsystems is an example.
shows an example of cascading noisy components:

This sample receiver consists of four subsystems: an antenna
and three cascaded amplifiers.
With uncorrelated signals, the output noise power can found by
adding the amplified noise powers from each stage:
P_{n.out} = P_{n3} + P_{n2} ?G _{3}+ P_{n1} ?G_{2} G_{3} + P_{na} ?G_{1}G_{2} G_{3}
Dividing by the bandwidth of the system, we find
,
Consequently, from this last expression and using the definition
(14.5), we find

where N_{a} is the noise power density from the antenna. We can deduce from this expression that the output noise power density N is the sum of the amplified noise power densities (a sum since the noise contributions are uncorrelated).
Notice that the noise power density from the last stage (N3)
appears directly at the output. However, the noise power
densities of all other stages are multiplied by the gain of
succeeding stages. In terms of an effective receiver noise temperature T_{r}, we can begin with:

and G = G_{1}G_{2} G_{3} to produce

Or

Using (14.23) again, but only for each stage, we find that

Notice that the noise temperatures of stages 2 and 3 are
proportionally reduced by the gains of earlier stages.
Consequently, the receiver noise temperature could be
dominated by the first stages in the chain of receiver subsystems
if the gains of the following stages are appreciable.
As an example of this, we’ll soon compute the noise temperature
of the NorCal 40A. Noise Figure
An alternative to noise temperature that is often used to quantify
the noisiness of electrical components is the noise figure F.

Or

where T_{0} is a reference temperature, often 290 K.
For example, at 45 MHz from the SA602AN datasheet (p. 417)

Or
T_{n}=627k
which is the effective noise temperature of this active, double
balanced mixer. Noise Temperature of the NorCal 40A Receiver
As an application of this discussion on noise, we’ll estimate the
noise temperature of the NorCal 40A receiver, but only for the
components shown in (i.e., excluding the antenna):

• For the two mixers, the SA602AN datasheet specifies a gain
of approximately 18 dB (?G =10^{18/10} = 63.1) and a noise figure F = 5 dB (?T_{m} =627K).
• What about the filters? We’ll assume a physical temperature
of 290 K and a loss in the pass band of 5 dB (? L =10^{5/10} = 3.2).
To compute the noise temperature of the filters, we need to assume that the losses in the passband are due to resistances in the filter. (Perhaps not completely true, but this will provide a worst-case scenario.) In such a case, the filter in the passband acts as an attenuator. From Section 14.4 in the text, the noise temperature of an attenuator T_{a} is given as
T_{a}=T(L-1)[k]
where T is the physical temperature and L is the loss.
Using (14.27) for the two filters in we find
T_{a} =290(3.2?1)=638K
Now, we are in a position to compute the noise temperature of
the NorCal 40A. From (14.29), we start with T1 and extend to a
fourth stage

Noting that 1 3 G = G =1/ L, then the noise temperature of the
NorCal 40A is approximately

Now, with 638 f a T_{f} = T_{a} =638 K, L = 3.2, T_{m} = T_{n} =627K and G = 63.1 then

Or

From this last result we can deduce a very important fact: the
receiver noise is wholly dominated by the noise generated by the
RF Mixer (2,006 K) and the RF Filter (638 K).
Actually, once the receiver is connected to the antenna, you’ll
see that the noise temperature of 2,778 K is much, much smaller
than the noise temperature of the antenna