The Radio Link--A tutorial--Part V

As was described earlier, thermal noise is generated by any lossy system, not just an isolated resistor. An interesting consequence of the thermal nature of the noise is that if we observe any passive system with a single electrical port, whatever noise there is appears to be generated by the equivalent resistance at the port. We will not attempt to prove this assertion, but we can illustrate it with several examples.

Example

The circuit shown in Figure 2.13 consists of three resistors all in thermal equilibrium with their surroundings at temperature T. We wish to find the average power (v2(t )) in the noise voltage v(t ).

Method 1

Replace each resistor with a noise model as shown in Figure 2.12. The result is shown in Figure 2.14. This circuit can now be solved by standard circuit theory methods, say, superposition or the node-voltage method. We obtain

The mean-square value of v(t ) is given by

where we have used the fact that the average of a sum is the sum of the averages. Now when the voltages from two distinct resistors are multiplied and averaged, we have, for example,

where we have taken advantage of the statistical independence of the voltages v1(t ) and v2(t ) and also of the fact that thermal noise has a zero average value. Applying Equation (2.46) to Equation (2.45) gives

Method 2

As an alternate approach to finding the average power in v(t), let us replace the three resistors in Figure 2.13 with a single equivalent resistor R. To do this we first combine resistors R1 and R2 in parallel and then combine the result in series with R3. The result is

The average thermal noise power generated by the equivalent resistor R is given by Equation (2.39) as

As expected, Equation (2.50) expresses the same result as Equation (2.48), but with considerably less effort. Note that the average noise power is proportional to the measurement bandwidth B. To avoid the vagueness of a result that contains an unspecified bandwidth, we can specify the noise power spectrum rather than the average noise power. Comparing Equation (2.32) and Equation (2.39) shows that the noise power spectrum at the output of the circuit is

Example

Figure 2.15 shows a one-port passive circuit that contains a reactive element as well as a resistor. As before, the circuit is at thermal equilibrium with its surroundings at temperature T. We wish to find the power spectrum Sv(f ) of the output voltage v(t ) and also to find the average power Pv = (v2(t)). As in the previous example, we will solve the problem two ways. The second solution will illustrate the "equivalent resistance" assertion.

Method 1

Begin by replacing the resistor with its noise equivalent as shown in Figure 2.12. The result is the circuit shown in Figure 2.16. The voltage vR(t ) is the resistor noise voltage. The power spectrum of this voltage is SR(f) = 2kTR. The circuit of Figure 2.16 looks like an ideal voltage source driving an RC lowpass filter. We can, therefore, use Equation (2.36) to find the power spectrum of the noise at the output. First, the frequency response of the lowpass filter can be found by using a voltage divider. We obtain

Next, the magnitude squared of the frequency response of the filter is

It is interesting to note that the noise at the output of the RC circuit is not white. At low frequencies the capacitor acts as an open circuit, and the noise power spectrum is essentially that of the resistor acting alone. At high frequencies the capacitor impedance becomes very small compared to the resistor and most of the noise voltage is dropped across the resistor rather than across the capacitor. The power spectrum reflects the capacitor voltage and decreases monotonically as frequency increases.

Method 2

The impedance of the circuit of Figure 2.15 as seen from the terminals is

If we write

then the resistive part of this impedance is given by

Now the thermal noise at the output of the circuit appears to be generated by the equivalent resistance at the port. Substituting in Equation (2.32) gives

which is precisely the same as the answer given in Equation (2.54). To find the average power at the circuit terminals, we can calculate the area under the power spectrum. This will give a finite result since the noise generated by the RC circuit is not white. We find

There is an interesting interpretation to Equation (2.59). If we multiply and divide by 4R, we obtain

where f3dB is the 3 dB bandwidth of an RC lowpass filter. Now refer to Figure 2.16. It appears that the average power of the noise generated by the resistor is being measured through an RC filter that sets the measurement bandwidth. If we compare Equation (2.60) with Equation (2.39), we see that the measurement bandwidth is given by

What sort of bandwidth this is will be the subject of a subsequent discussion.

Not all sources of noise are passive. Allowing amplification of the noise can produce interesting effects. An example will illustrate the idea.

A white noise source is often characterized by a so-called noise temperature . This is the temperature at which a resistor would have to be to produce thermal noise of power (in a given bandwidth) equal to the observed noise power. The noise temperature of the circuit of Figure 2.17 is 290,000 K even though the circuit is physically at room temperature. Noise temperature is often used to characterize the noise observed at the terminals of an antenna. Antenna noise is usually a combination of noise picked up from electrical discharges in the atmosphere (static); radiation from the Earth, the sun, and other bodies that may lie within the antenna beam; cosmic radiation from space; and, of course, thermal noise from the conductive material out of which the antenna is made. At frequencies below about 30 MHz atmospheric noise is the principal component of antenna noise, and the noise temperature can be as high as 1010  K. Atmospheric noise becomes less important at frequencies above 30 MHz. A narrow-beam antenna at 5 GHz pointing directly upward on a dark night may pick up noise primarily from cosmic radiation and oxygen absorption in the atmosphere. The noise temperature in this case may be only 5 K.

Figure 2.20 shows a voltage source with a source resistance R  driving a load resistance RL . This simple model might represent the output of an antenna or the output of a filter or the output of an amplifier. Two common situations arise in practice. In some circuits RL  >> R  and the load voltage is approximately equal to the source voltage. If the source and load voltages happen to be noise, it is meaningful to characterize these voltages by their mean-square values or, to use another term for the same quantity, by their normalized average power. A second important situation is the one in which we wish to transfer power, rather than voltage, from the source to the load. For a fixed value of source resistance R , maximum power transfer occurs when RL = R . Under this condition we have

This quantity is called the available power.

Next:  Characterizing Two-Ports

Introduction to Wireless Systems By Bruce A. Black, Philip S. DiPiazza, Bruce A. Ferguson, David R. Voltmer, Frederick C. Berry, Published Jun 7, 2011 by Prentice Hall, is reprinted with permission by Pearson Publishing.