# Noise of Thermal AgitationNovember 1944 Radio News Article Radio-Electronic Engineering Department (insert)

Admittedly, I did not do any follow-up research on this, but there is reason to believe that prior to this Radio News magazine article, there was not a general agreement on what formula to use for thermal noise in an electrical system. Here is a statement made by author S.J. Mallory, "At first, however, there was no general agreement concerning the magnitude of this basic Johnson noise power level. Some engineers used the quantity KTB, others used 2KTB and still others used 4KTB." We of course all use KTB nowadays for thermal noise power - aka Johnson noise. It's a good read on the subject of sources that determine the noise floor of a system. There's also this kind of Johnson noise.

## Noise of Thermal Agitation

Fig. 1 - Minimum detectable incoming signal must have rms value several limes first stage thermal noise so output may be intelligible. Magnified resistor shows random electron motion.

By S. J. Mallory

Calculation of the "Johnson noise" thermal noise voltage developed in the input resistor of a high-gain amplifier.

In 1928, J. B. Johnson published an article describing an ever-present source of noise in electrical conductors. The noise is caused by free electrons moving about in the conductor. This discovery is of great importance in the field of communications because it is a factor which provides as a definite barrier, the lower limit of detectable signal. Signals of lower level than this basic "Johnson Noise" are lost in the noise.

There are also other noise sources prevalent in receivers, for example, the various noises occurring in vacuum tubes, but first circuit Johnson noise is usually predominant in well constructed receivers. If the first tube input resistance is small, it is possible for vacuum tube noise to exceed first circuit noise. These other miscellaneous noises are added to the basic Johnson noise to increase the level of the minimum detectable signal. The total noise across the output terminals of a receiver, converter or i.f. amplifier is a measure of the performance of the unit because ordinarily any signal of a level lower than this output noise level cannot be distinguished.

Thermal Noise Voltage

The noise in a simple resistance made up from a length of conductor will be discussed first. The conductivity of metals is a result of the presence of free electrons which move about at a velocity that depends on their temperature. At any instant there are more electrons moving in one direction than in any other; this produces a difference of potential across the conductor ends. This voltage varies from instant to instant in accordance with the predominant motion of the electrons. The magnitude of the mean square voltage developed per cycle of band width is directly proportional to the absolute temperature of the resistance and to the magnitude of the resistance involved. The voltage is independent of the size, shape and material of the conductor. The existence of this random voltage is evidenced as noise in the first circuits of amplifiers and in other circuit elements which normally operate at low signal levels. The noise is called the noise of thermal agitation, or Johnson noise, after its discoverer.

Typical receiver with high-gain i.f. amplifier. First stage noise must be considered in design.

Fig. 2 - (A) Equivalent circuit for calculating Johnson noise. (B) Source of noise, with internal impedance R1 working into load resistance R2 input resistance of amplifier.

Fig. 3 - Did not appear in the magazine article.

The noise of thermal agitation is quite uniformly distributed over the entire frequency spectrum from zero to frequencies much higher than normally used for radio. Therefore, a 5000 cycle band lying in the audio range would contain as much Johnson noise power as a 5000 cycle band in the microwave region, provided that the resistance is constant over the frequency band under consideration.

Let R in Fig. 2(A) be the internal resistance of a signal generator. Thus, if R is constant over the frequency bandwidth B, J. B. Johnson found that the square of the effective value of the noise voltage components over the frequency band is:

E2 = 4KTRB ................ (1)

where,

K = 1.38 X 10-23, Boltzmann's constant.

T = Absolute temperature, deg. Kelvin.

R = Resistance in ohms.

B = The effective bandwidth in cycles per second. The effective bandwidth may be determined as follows: An energy response curve (output current squared vs. frequency) is plotted for the amplifier under consideration. The total area under the curve is divided by the height of the curve at resonance, thus obtaining the width of an equivalent rectangle. This width is the effective bandwidth in cycles per second.2

A temperature of 290° K, corresponding to 17° C. or about 63° F., is frequently used.

Noise Power Reference Level

Other noises in amplifiers may be controlled to a great extent by design considerations, but it became recognized that the noise of thermal agitation was omnipresent. After all other inherent noises were eliminated, Johnson noise remained and it seems determined to stay as long as conductors are used.

As the use of extremely high-gain i.f. amplifiers became more widespread, Johnson noise became a more widely recognized factor in amplifier design. Gradually, the ratings of high-gain amplifiers have become known in terms of noise performance. The noise power output level is quoted as a certain number of db. above the power level of Johnson noise. At first, however, there was no general agreement concerning the magnitude of this basic Johnson noise power level. Some engineers used the quantity KTB, others used 2KTB and still others used 4KTB. One engineer using the 4KTB base might rate an amplifier as having a noise performance of 11 db. above theoretical, a second engineer using the 2KTB base would rate the identical equipment as having a noise figure of 8 db., and the third engineer might consider 5 db. the correct value.

2KTB is the maximum noise power available in a load resistance R when it is used to terminate a network of equal resistance and temperature. Under certain conditions when the load resistance is different than the generator resistance, the thermal noise level may be less than 2KTB. To avoid using a reference level which was not an ideal condition and to avoid the confusion prevalent with the existence of several different reference levels, in 1941 it was generally agreed upon that KTB should be used as the basis for determining the noise performance of receivers.

First Stage Noise Power

Fig. 4 - Electronically-regulated power supply. manufactured by the Communication Measurements Lab. The output voltage variations are amplified twice before being applied to 6B4 tubes. Voltage range may be adjusted by changing R2 and R3.

Fig. 5 - A commercial application of the circuit given in Fig. 8. A power supply employing this circuit was designed and manufactured by the Harvey Radio Labs., Inc.

In Fig. 2 (B) R1 is considered to be the internal resistance of a signal source, and R2 the input resistance of the first tube of a high gain amplifier. (In such an amplifier the gain will raise the level of the input signal and noise to a point where it will override the inherent tube noise and other noise sources within the amplifier.) E1 and E2 are the sources of noise in R1 and R2 respectively.

In order to determine the effect of thermal noise on the first circuit of an amplifier, the value of the maximum available noise power in R2 must be found as it is this noise power in the first circuit input resistance that is a measure of the output noise of the amplifier. The noise voltage generated in R1 over a bandwidth B is:

E12 = 4KT1BR1 .................. (2)

Neglecting for the moment the noise power generated in R2, the maximum noise power available from R1 is that power which would be absorbed by a matched output circuit. Therefore, when R1 = R2

P01 = E12/4R1 = KT1B ....... (3) watts noise power delivered to R2 by R1. But, when the temperatures T1 and T2 of R1 and R2 respectively are equal, then R2 contributes a like amount of noise power to itself. Therefore the total noise power in R2 is:

PNM = P01 + P02 = 2KTB .... (4)

Since a major portion of communications circuits are designed around the premise that connected circuits shall have matched impedances, the quantity 2KTB was originally used as the basic minimum amount of noise in an ideal receiver. An ideal receiver is one with negligible output noise when the input is shorted. Later it was found that a judicious mismatch would result in a noise power somewhat less than 2KTB. It was concluded that theoretically the ultimate in mismatch would result in a minimum noise power of KTB. But since this ultimate in mismatch occurs when R2 of Fig. 2(B) is infinite, the minimum noise, KTB, has never been realized. The quantity KTB is the thermal noise in the first circuit of an ideal receiver.

To compare the noise in a circuit under matched conditions with the noise under mismatched conditions, the signal-to-noise ratios for these two conditions is calculated. For the matched case in Fig. 3, let R1 = R2. The signal power in R2 is:

PS = ES2/4R2 ............ (5)

The noise power is given by (4).

Therefore, ......... (6)

the signal-to-noise power ratio for the matched case.

For the mismatched case it is assumed that R2 is very large, so large in fact that there is no appreciable voltage drop across R1. It is also assumed then that the voltages appearing across R2 are the signal voltage ES and the noise voltage √(4KTBR2). The signal power in R2 will be:

PS = ES2/R2 ................ (7)

and the noise power,

PN = (4KTBR2)/R2 ........ (8)

Therefore, the signal-to-noise power ratio for the ultimate of mismatch will be:

PS/PN = ES2/(4KTBR2) .......... (9)

The signal-to-noise ratio in case of voltage-regulator tube, VR 105-30. Resistor R2 is made variable in order that the output voltage may be placed at the desired value. The position of this setting, however, does not in any way interfere with the operation of the stabilizing unit. Condenser C aids the regulating action when changes in Eout, appear rapidly.

The first commercial power supply schematic is shown in Fig. 5. This circuit is much the same as that given in Fig. 8. The 6SJ7 is the sharp cut-off pentode tube that amplifies the output voltage fluctuations applied at its grid through the series combination of R1, R2, and R3. The setting of R2 enables the operator to vary the output voltage between 200 and 300 volts. R1 and R5 provide the proper screen-grid voltage to the 6SJ7.

Any variations in the plate current of the pentode cause the voltage across R. to vary. Since this resistor is connected to the grids of the parallel 6Y6's, their grid voltage will change and thus control will be had over the output current and voltage. R7 and R8 are used to isolate the two 6Y6's from each other and prevent any tendency toward oscillation. Meter M1 and R9 form a voltmeter to keep constant track of the output voltage. The rest of the circuit is similar to what has been described already and requires no further explanation. Notice how well fused the set is and the use of separate switches for the plate and filament transformers.

The next commercial circuit (Fig. 7) has a slight variation over the foregoing power supplies. Here all voltage fluctuations are applied to the grid of one of the triodes of the 6SN7GT tube. After amplification, the variations are sent through the other triode section of the 6SN7 and on to the 500,000 ohm resistor, R1 connected between the control grid and plate of each of the three 7C5's in parallel. From here on the action differs in no way from the explanation given previously for

40 either one series tube or two in parallel. With the three 7C5's, a maximum current of 250 milliamperes may be drawn from the supply at a plate voltage of 200 to 400 volts. It is to be noticed here that input voltage changes in the a.c. line immediately affect the grid of one triode section of the 6SN7 and this acts in conjunction with any output variations that affect the other triode. The result is better regulation than if the line voltage instabilities were left to affect the output voltage and then ironed out.

Fig. 6 - A vacuum-tube voltage regulator circuit using the Gm of the tube as its basis of operation.

Fig. 7 - Power supply manufactured by Technical Apparatus Co. employs a double triode to amplify input and output variations. Unit supplies 200-400 volts d.c. at 40 watts.

Fig. 8 - Improved voltage regulator employing the same basic circuit as Fig. 2.

For the final commercial diagram, Fig. 4, we again find a double triode (a 6SL7), this time used differently, however. The output voltage of one triode is fed directly to the grid of the other triode and from thence to the 6B4G regulator tubes. This double amplification gives rise to a very sensitive arrangement whereby even small amounts of ripple may be eliminated. The duo-triode used in the previous power supply described above did not have the output of one amplified by the other. Rather both were connected into different circuits (the output and the input) and their voltage changes worked in conjunction with each other. In the present setup, both triodes deal mostly with variations in Eout only, and they are unable to counteract input line changes directly. Resistor R1 transmits the amplified regulating voltages to the grids of the 6B4 tubes. Note that direct coupling is used throughout most of these amplifiers. This insures that even slow changes in output voltage are effective in this regulating action. Condenser-coupled amplifiers would not be suitable because of their poor low-frequency response.

The above power units are indicative of what may be expected in present day electronically-regulated supplies. Each unit is now being sold in the radio market. Now let us see how they may be designed. Fig. 8 will be used as illustration for the procedure.

The first point that must be decided concerns the amounts of current and voltage that the power supply will be called upon to deliver. A thirty per cent tolerance should be allowed between the required load and that which the unit can supply. This will permit a comfortable leeway and prevent overheating of the power transformer. With the current and voltage values determined, the proper electrolytic condensers, filter choke (or chokes) and power transformer can then be chosen. These are connected as shown in Fig. 5. One pi filter section will work satisfactorily since the degenerative amplifier will remove any remaining ripple by its regulator action.

The design of the regulator section is attacked next. The choice of the series tube (or tubes) will depend on the amount of current that the power supply will have to supply. Generally, a power-amplifier tube is used because it will allow greater plate dissipation and current flow than ordinary amplifier tubes. If one tube should prove insufficient, then several may be placed in parallel. Experimentally, the amount of wattage being dissipated in the tube may be determined by connecting it into the circuit and measuring the voltage dropped across it. This figure, multiplied by the current through the tube, gives the wattage in the tube. The foregoing generally must be done because the desired information is not obtainable from tube manufacturers.

For the feedback tube, a sharp cut-off pentode is desirable, this giving better and more clear-cut control than the remote cut-off pentodes. A glance through the tube manual will reveal several suitable tubes. The pentode, however, is not the only possible tube. This was made clear in two of the commercial circuits just examined. Here duo-triodes were used quite successfully. The main purpose of the feedback amplifier is to have large variations in plate current from small changes in grid voltages. The greater this action, the more sensitive the circuit will be to output voltage variations. The amplifier (or amplifiers) does not contain any condenser coupling, this automatically reducing low-frequency changes from having the effect they should have. Direct coupling is used throughout.

The remaining problem of designing R2 R3, and R4 is decided in the following manner. Eg, that portion of the output voltage applied to the grid of the 6J7, is related to Eout by this simple equation:

where: Rx is the bottom portion of R3 between the center arm and R4. This voltage, when subtracted from the 105 volts of the VR tube, gives the actual volt-age on the grid of the 6J7. Generally, this value is such as to keep the grid negative with respect to the cathode. In the present case, this would mean that Eg would have a value less than the VR's 105 volts. A characteristic chart of the tube used will indicate quickly the amount of variation desirable. From this value, the proportion that R3 will be of the total resistance of R2 plus R3, plus R4 can be easily computed. R2 and R4 are placed' on either side of R3 to insure that the voltage applied to the 6J7 will be below a certain maximum value and above a certain minimum.

When actually constructed, these units are rather large in size. A roomy chassis will insure good ventilation with adequate space between components, especially transformers and power chokes. A well-designed panel, such as shown in the photograph will aid the operator to quickly reach the controls easily and safely. Any radio manufacturer will confirm the fact that good layout is just as important as circuit design, and it is always given a great deal of thought. Note that even fuses are placed on the panel, each possible of quick replacement. For the man who uses these units every day, these little considerations may determine the choice of one unit over another.

Posted November 1, 2019