AgilentFundamentals of RF and MicrowaveNoise Figure Measurements Application Note 57-1

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Table of Contents

1. What is Noise Figure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4The Importance of Noise in Communication Systems . . . . . . . . . . . . . .5Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6The Concept of Noise Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Noise Figure and Noise Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

2. Noise Characteristics of Two-Port Networks . . . . . . . . . . . .9The Noise Figure of Multi-stage Systems . . . . . . . . . . . . . . . . . . . . . . . .9Gain and Mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10The Effect of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

3. The Measurement of Noise Figure . . . . . . . . . . . . . . . . . . . . . . . .12Noise Power Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12The Y-Factor Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13The Signal Generator Twice-Power Method . . . . . . . . . . . . . . . . . . . . . .14The Direct Noise Measurement Method . . . . . . . . . . . . . . . . . . . . . . . . .14Corrected Noise Figure and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Frequency Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16LO Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16LO Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Unwanted Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

Noise Figure Measuring Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Noise Figure Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Spectrum Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Network Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Noise Parameter Test Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Power Meters and True-RMS Voltmeters . . . . . . . . . . . . . . . . . . . .18

4. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

6. Additional Agilent Resources, Literature and Tools . . . . . .31

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Chapter 1. What is Noise Figure?

IntroductionModern receiving systems must often process very weak signals, but the noise added by the system components tends to obscure those very weak signals.Sensitivity, bit error ratio (BER) and noise figure are system parameters that characterize the ability toprocess low-level signals. Of these parameters, noise figure is unique in that it is suitable not only for characterizing the entire system but also the systemcomponents such as the pre-amplifier, mixer, and IFamplifier that make up the system. By controlling thenoise figure and gain of system components, thedesigner directly controls the noise figure of the over-all system. Once the noise figure is known, systemsensitivity can be easily estimated from system band-width. Noise figure is often the key parameter thatdifferentiates one system from another, one amplifierfrom an other, and one transistor from another. Suchwidespread application of noise figure specificationsimplies that highly repeatable and accurate measure-ments between suppliers and their customers are veryimportant.

The reason for measuring noise properties of net-works is to minimize the problem of noise generatedin receiving systems. One approach to overcome noiseis to make the weak signal stronger. This can beaccomplished by raising the signal power transmittedin the direction of the receiver, or by increasing theamount of power the receiving antenna intercepts, forexample, by increasing the aperture of the receivingantenna. Raising antenna gain, which usually means alarger antenna, and raising the transmitter power, areeventually limited by government regulations, engi-neering considerations, or economics. The otherapproach is to minimize the noise generated withinreceiver components. Noise measurements are key toassuring that the added noise is minimal. Once noisejoins the signals, receiver components can no longerdistinguish noise in the signal frequency band fromlegitimate signal fluctuations. The signal and noise getprocessed together. Subsequent raising of the signallevel with gain, for example, will raise the noise levelan equal amount.

This application note is part of a series about noisemeasurement. Much of what is discussed is eithermaterial that is common to most noise figure meas-urements or background material. It should proveuseful as a primer on noise figure measurements. Theneed for highly repeatable, accurate and meaningfulmeasurements of noise without the complexity ofmanual measurements and calculations has lead tothe development of noise figure measurement instru-ments with simple user interfaces. Using these instruments does not require an extensive background in noise theory. A littlenoise background may prove helpful, however, inbuilding confidence and understanding a more com-plete picture of noise in RF and microwave systems.Other literature to consider for additional informationon noise figure measurements is indicated through-out this note. Numbers appearing throughout thisdocument in square brackets [ ] correspond to thesame numerical listing in the References section.Related Agilent Technologies literature and webresources appear later in this application note.

NFA simplifies noise figure measurements

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The Importance of Noise inCommunication SystemsThe signal-to-noise (S/N) ratio at the output of receiv-ing systems is a very important criterion in communi-cation systems. Identifying or listening to radiosignals in the presence of noise is a commonly experi-enced difficulty. The ability to interpret the audioinformation, however, is difficult to quantify because itdepends on such human factors as language familiari-ty, fatigue, training, experience and the nature of themessage. Noise figure and sensitivity are measurableand objective figures of merit. Noise figure and sensi-tivity are closely related (see Sensitivity in the glos-sary). For digital communication systems, aquantitative reliability measure is often stated interms of bit error ratio (BER) or the probability P(e)that any received bit is in error. BER is related tonoise figure in a non-linear way. As the S/N ratiodecreases gradually, for example, the BER increasessuddenly near the noise level where l’s and 0’sbecome confused. Noise figure shows the health of thesystem but BER shows whether the system is dead oralive. Figure 1-1, which shows the probability of errorvs. carrier-to-noise ratio for several types of digitalmodulation, indicates that BER changes by severalorders of magnitude for only a few dB change in sig-nal-to-noise ratio.

Figure 1-1. Probability of error, P(e), as a function of carrier-to-noise ratio,C/N (which can be interpreted as signal-to-noise ratio), for various kinds of digital modulation. From Kamilo Feher, DIGITAL COMMUNICATIONS:Microwave Applications, ©1981, p.71. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ

The output signal-to-noise ratio depends on twothings—the input signal-to-noise ratio and the noise figure. In terrestrial systems the input signal-to-noiseratio is a function of the transmitted power, transmit-ter antenna gain, atmospheric transmission coeffi-cient, atmospheric temperature, receiver antennagain, and receiver noise figure. Lowering the receivernoise figure has the same effect on the output signal-to-noise ratio as improving any one of the other quantities.

In satellite systems, noise figure may be particularlyimportant. Consider the example of lowering thenoise figure of a direct broadcast satellite (DBS)receiver. One option for improving receiver noise fig-ure is to increase the transmitter power, however, thisoption can be very costly to implement. A betteralternative is to substantially improve the perform-ance of the receiver low noise amplifier (LNA). It iseasier to improve LNA performance than to increasetransmitter power.

DBS receiver

In the case of a production line that produces satellitereceivers, it may be quite easy to reduce the noise figure 1 dB by adjusting impedance levels or carefullyselecting specific transistors. A 1dB reduction innoise figure has approximately the same effect asincreasing the antenna diameter by 40%. But increas-ing the diameter could change the design and signifi-cantly raise the cost of the antenna and supportstructure.

Sometimes noise is an important parameter of trans-mitter design. For example, if a linear, broadband,power amplifier is used on a base station, excessbroadband noise could degrade the signal-to-noiseratio at the adjacent channels and limit the effective-ness of the system. The noise figure of the poweramplifier could be measured to provide a figure ofmerit to insure acceptable noise levels before it isinstalled in the system.5

6 8 10 12 14 16 18 20 22 24 26

10 – 3 10 – 4 10 – 5 10 – 6 10 – 7 10 – 8

10 – 9 10 – 10

BPSK

4-PSK(QAM

)

8-PSK

16-PSK

ClassI OPR

8-APK16-APK

or 16QAM

Carrier to Noise Ratio - (dB)

Prob

ality

of E

rror

- P(

e)

Sources of NoiseThe noise being characterized by noise measurements consists of spontaneous fluctuations caused by ordi-nary phenomena in the electrical equipment. Thermalnoise arises from vibrations of conduction electronsand holes due to their finite temperature. Some of thevibrations have spectral content within the frequencyband of interest and contribute noise to the signals.The noise spectrum produced by thermal noise isnearly uniform over RF and microwave frequencies.The power delivered by a thermal source into animpedance matched load is kTB watts, where k isBoltzmann’s constant (1.38 x 10-23 joules/K), T is thetemperature in K, and B is the systems noise band-width. The available power is independent of thesource impedance. The available power into amatched load is directly proportional to the bandwidth so that twice the bandwidth would allow twice the power to be delivered to the load. (see Thermal Noise in the glossary)

Shot noise arises from the quantized nature of currentflow (see Shot Noise in the glossary). Other randomphenomena occur in nature that are quantized and produce noise in the manner of shot noise. Examples are the generation and recombination of hole/electronpairs in semiconductors (G-R noise), and the division of emitter current between the base and collector intransistors (partition noise). These noise generatingmechanisms have the characteristic that like thermalnoise, the frequency spectra is essentially uniform, producing equal power density across the entire RF and microwave frequency range.

There are many causes of random noise in electricaldevices. Noise characterization usually refers to thecombined effect from all the causes in a component. The combined effect is often referred to as if it allwere caused by thermal noise. Referring to a device ashaving a certain noise temperature does not meanthat the component is that physical temperature, butmerely that it’s noise power is equivalent to a thermalsource of that temperature. Although the noise tem-perature does not directly correspond with physicaltemperature there may be a dependence on tempera-ture. Some very low noise figures can be achievedwhen the device is cooled to a temperature belowambient.

Noise as referred to in this application note does notinclude human-generated interference, although suchinterference is very important when receiving weak sig-nals. This note is not concerned with noise from igni-tion, sparks, or with undesired pick-up of spurioussignals. Nor is this note concerned with erratic dis-turbances like electrical storms in the atmosphere.Such noise problems are usually resolved by techniqueslike relocation, filtering, and proper shielding. Yetthese sources of noise are important here in onesense—they upset the measurements of the sponta-neous noise this note is concerned with. A manufac-turer of LNAs may have difficulty measuring the noisefigure because there is commonly a base station neabyradiating RF power at the very frequencies they areusing to make their sensitive measurements. For thisreason, accurate noise figure measurements are oftenperformed in shielded rooms.

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The Concept of Noise FigureThe most basic definition of noise figure came into popular use in the 1940’s when Harold Friis [8]defined the noise figure F of a network to be the ratioof the signal-to-noise power ratio at the input to thesignal-to-noise power ratio at the output.

Thus the noise figure of a network is the decrease ordegradation in the signal-to-noise ratio as the signalgoes through the network. A perfect amplifier wouldamplify the noise at its input along with the signal,maintaining the same signal-to-noise ratio at its inputand output (the source of input noise is often thermalnoise associated with the earth’s surface temperatureor with losses in the system). A realistic amplifier,however, also adds some extra noise from its owncomponents and degrades the signal-to-noise ratio. Alow noise figure means that very little noise is addedby the network. The concept of noise figure only fitsnetworks (with at least one input and one outputport) that process signals. This note is mainly abouttwo-port networks; although mixers are in generalthree-port devices, they are usually treated the sameas a two-port device with the local oscillator connected to the third port.

It might be worthwhile to mention what noise figuredoes not characterize. Noise figure is not a quality fac-tor of networks with one port; it is not a quality fac-tor of terminations or of oscillators. Oscillators havetheir own quality factors like “carrier-to-noise ratio”and “phase noise”. But receiver noise generated in thesidebands of the local oscillator driving the mixer, canget added by the mixer. Such added noise increasesthe noise figure of the receiver.

Noise figure has nothing to do with modulation ordemodulation. It is independent of the modulation for-mat and of the fidelity of modulators and demodula-tors. Noise figure is, therefore, a more general conceptthan noise-quieting used to indicate the sensitivity ofFM receivers or BER used in digital communications.

Noise figure should be thought of as separate from gain. Once noise is added to the signal, subsequentgain amplifies signal and noise together and does notchange the signal-to-noise ratio.

Figure 1-2(a) shows an example situation at the input of an amplifier. The depicted signal is 40 dB above the

7

F = Si/Ni (1-1)So/No – 40

– 60

– 80

– 100

– 1202.6 2.65 2.7

Frequency (GHz)(a)

Inpu

t Pow

er L

evel

(dBm

) – 40

– 60

– 80

– 100

– 1202.6 2.65 2.7

Frequency (GHz)(b)

Outp

ut P

ower

Lev

el (d

Bm)

F =

=

=

Si/Ni

So/No

Si/Ni

GSi/(Na + GNi)

Na + GNi

GNi

(1-2)

noise floor: Figure 1-2(b) shows the situation at theamplifier output. The amplifier’s gain has boosted thesignal by 20 dB. It also boosted the input noise levelby 20 dB and then added its own noise. The outputsignal is now only 30 dB above the noise floor. Sincethe degradation in signal-to-noise ratio is 10 dB, the amplifier has a 10 dB noise figure.

Figure 1-2. Typical signal and noise levels vs. frequency (a) at an amplifier’sinput and (b) at its output. Note that the noise level rises more than the signal level due to added noise from amplifier circuits. This relative rise innoise level is expressed by the amplifier noise figure.

Note that if the input signal level were 5 dB lower (35 dB above the noise floor) it would also be 5 dBlower at the output (25 dB above the noise floor), andthe noise figure would still be 10 dB. Thus noise fig-ure is independent of the input signal level.

A more subtle effect will now be described. The degra-dation in a network’s signal-to-noise ratio is depend-ent on the temperature of the source that excites thenetwork. This can be proven with a calculation of thenoise figure F, where Si and Ni represent the signaland noise levels

available at the input to the device under test (DUT),So and No represent the signal and noise levels avail-

able at the output, Na is the noise added by the DUT,and G is the gain of the DUT. Equation (1-2) shows thedependence on noise at the input Ni. The input noiselevel is usually thermal noise from the source and isreferred to by kToB. Friis [8] suggested a referencesource temperature of 290K (denoted by To ), which isequivalent to 16.8˚ C and 62.3˚ F. This temperature isclose to the average temperature seen by receivingantennas directed across the atmosphere at the trans-mitting antenna.

The power spectral density kTo, furthermore, is theeven number 4.00 x 10-21 watts per hertz of band-width (–174 dBm/Hz). The IRE (forerunner of theIEEE) adopted 290K as the standard temperature fordetermining noise figure [7]. Then equation (1-2)becomes

which is the definition of noise figure adopted by the IRE.

Noise figure is generally a function of frequency but it is usually independent of bandwidth (so long as themeasurement bandwidth is narrow enough to resolvevariations with frequency). Noise powers Na and Ni ofequation (1-2) are each proportional to bandwidth.But the bandwidth in the numerator of (1-2) cancelswith that of the denominator—resulting in noise figurebeing independent of bandwidth.

In summary, the noise figure of a DUT is the degrada-tion in the signal-to-noise ratio as a signal passesthrough the DUT. The specific input noise level fordetermining the degradation is that associated with a290K source temperature. The noise figure of a DUT isindependent of the signal level so long as the DUT islinear (output power vs. input power).

The IEEE Standard definition of noise figure, equa-tion (1-3), states that noise figure is the ratio of thetotal noise power output to that portion of the noisepower output due to noise at the input when the inputsource temperature is 290K.

While the quantity F in equation (1-3) is often called“noise figure”, more often it is called “noise factor” orsometimes “noise figure in linear terms”. Modernusage of “noise figure” usually is reserved for thequantity NF, expressed in dB units:

NF = 10 log F (1-4)

This is the convention used in the remainder of this application note.

Noise Figure and Noise TemperatureSometimes “effective input noise temperature”, Te, is used to describe the noise performance of a device rather than thenoise figure, (NF). Quite often temperature units are used for devices used in satellite receivers. Te isthe equivalent temperature of a source impedance into a perfect(noise-free) device that would produce the same added noise, Na. It isoften defined as

It can be related to the noise factor F:

Te = To(F-1), where To is 290K (1-6)

The input noise level present in terrestrial VHF and microwave com-munications is often close to the 290K reference temperature used innoise figure calculations due to the earth’s surface temperature.When this is the case, a 3 dB change in noise figure will result in a 3dB change in the signal-to-noise ratio.

In satellite receivers the noise level coming from the antenna can befar less, limited by sidelobe radiation and the background sky temper-ature to values often below 100K. In these situations, a 3 dB changein the receiver noise figure may result in much more than 3 dB signal-to-noise change. While system performance may be calculated using noise figure without any errors (the 290K referencetemperature need not correspond to actual temperature), systemdesigners may prefer to use Te as a system parameter.Figure 1-3. Degradation in the S/N ratio vs Te of a device for various valuesof temperature for the source impedance. Noise figure is defined for asource temperature of 290K.

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F = (1-3)Na + kToBGkToBG

Te = (1-5)Na

kGB

10

9

8

7

6

5

4

3

2

1

00 25 50 75 100 125 150

Te(K)

S/N

Deg

rada

tion

(dB

)

Ts = 5K

Ts = 30K

Ts = 100K

Ts = 50K

Ts = 290K(Gives Noise Figure)

Chapter 2. Noise Characteristics ofTwo-Port Networks

The Noise Figure of Multi-stage SystemsThe noise figure definition covered in Chapter 1 canbe applied to both individual components such as asingle transistor amplifier, or to a complete systemsuch as a receiver. The overall noise figure of the sys-tem can be calculated if the individual noise figuresand gains of the system components are known. Tofind the noise figure of each component in a system,the internal noise added by each stage, Na, must befound. The gain must also be known. The actual meth-ods used to determine noise and gain are covered inChapter 3: The Measurement of Noise Figure. Thebasic relationship between the individual componentsand the system will be discussed here.

Figure 2-1. The effect of second stage contribution.

For two stages see Figure 2-1, the output noise willconsist of the kToB source noise amplified by bothgains, G1G2, plus the first amplifier output noise, Na1,amplified by the second gain, G2, plus the secondamplifiers output noise, Na2. The noise power contri-butions may be added since they are uncorrelated.Using equation (1-3) to express the individual amplifi-er noise contributions, the output noise can beexpressed in terms of their noise factors, F.

With the output noise known, the noise factor of the combination of both amplifiers can be calculatedusing equation (1-1). This is the overall system noisefigure of this two-stage example.

The quantity (F2-1)/G1 is often called the second stagecontribution. One can see that as long as the firststage gain is high, the second stage contribution willbe small. This is why the pre-amplifier gain is animportant parameter in receiver design. Equation (2-2)can be re-written to find F1 if the gain and overallsystem noise factor is known. This is the basis of cor-rected noise measurements and will be discussed inthe next chapter.

This calculation may be extended to a n-stage cascadeof devices and expressed as

Equation (2-3) is often called the cascade noise equation.

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BG1, Na1 BG2, Na2

InputNoisekTo B

R

1st Stage

2nd Stage

kTo B kTo BG1kTo BG1G2

Na = (F-1) kToBG Noise Input x System Gain

Na1

Na2

Na1 G2

TotalNoiseAdded

TotalNoisePowerOutput

[ ]No = (2-1)kToBG1G2 F1 +F2 – 1

G1

Fsys = (2-2) F1 +F2 – 1

G1

Fsys = (2-3) F1 + + + …F2 – 1G1

F3 – 1G1G2

Fn – 1G1G2…Gn-1

Gain and MismatchThe device gain is an important parameter in noise calculations. When an input power of kToB is used inthese calculations, it is an available power, the maxi-mum that can be delivered to a matched load. If thedevice has a large input mismatch (not unusual forlow-noise amplifiers), the actual power delivered tothe device would be less. If the gain of the device isdefined as the ratio of the actual power delivered tothe load to the maximum power available from thesource we can ignore the mismatch loss present at theinput of the device since it is taken into account inour gain definition. This definition of gain is calledtransducer gain, Gt. When cascading devices, however,mismatch errors arise if the input impedance of thedevice differs from the load impedance. In this casethe total gain of a cascaded series of devices does notequal the product of the gains.

Available gain, (Ga), is often given as a transistor parameter, it is the gain that will result when a givensource admittance, Ys, drives the device and the out-put is matched to the load. It is often used whendesigning amplifiers. Refer to the glossary for a morecomplete description of the different definitions ofgain.

Most often insertion gain, Gi, or the forward transmis-sion coefficient, (S21)2, is the quantity specified ormeasured for gain in a 50 ohm system. If the measure-ment system has low reflection coefficients and thedevice has a good output match there will be littleerror in applying the cascade noise figure equation (2-3) to actual systems. If the device has a poor outputmatch or the measurement system has significant mis-match errors, an error between the actual system andcalculated performance will occur. If, for example, theoutput impedance of the first stage was different fromthe 50 ohm source impedance that was used when thesecond stage was characterized for noise figure, thenoise generated in the second stage could be altered.Fortunately, the second stage noise contribution isreduced by the first stage gain so that in many appli-cations errors involving the second stage are minimal.When the first stage has low gain (G≤F2), second stageerrors can become significant. The complete analysis of mismatch effects in noise calculations islengthy and generally requires understanding thedependence of noise figure on source impedance. Thiseffect, in addition to the gain mismatch effect, will be discussed in the next section (Noise Parameters). It isbecause of this noise figure dependence that S-parame-ter correction is not as useful as it would seem inremoving the errors associated with mismatch [4]

Noise ParametersNoise figure is, in principle, a simplified model of theactual noise in a system. A single, theoretical noise element is present in each stage. Most actual amplify-ing devices such as transistors can have multiplenoise contributors; thermal, shot, and partition asexamples. The effect of source impedance on thesenoise generation processes can be a very complex rela-tionship. The noise figure that results from a noisefigure measurement is influenced by the match of thenoise source and the match of the measuring instru-ment; the noise source is the source impedance forthe DUT, and the DUT is the source impedance forthe measuring instrument. The actual noise figureperformance of the device when it is in its operatingenvironment will be determined by the match of othersystem components.

Designing low noise amplifiers requires tradeoffsbetween the gain of a stage and its correspondingnoise figure. These decisions require knowledge ofhow the active device’s gain and noise figure changeas a function of the source impedance or admittance.The minimum noise figure does not necessarily occurat either the system impedance, Zo, or at the conju-gate match impedance that maximizes gain.

To fully understand the effect of mismatch in a sys-tem, two characterizations of the device-under-test(DUT) are needed, one for noise figure and anotherfor gain. While S-parameter correction can be used tocalculate the available gain in a perfectly matched sys-tem, it can not be used to find the optimum noise fig-ure. A noise parameter characterization uses a specialtuner to present different complex impedances to theDUT. [29]

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The dependence of noise factor on source impedancepresented by the tuner is described by

where the Gs is the source reflection coefficient thatresults in the noise factor F. In the equation, Fmin isthe minimum noise factor for the device that occurswhen Gs= Gopt. Rn is the noise resistance (the sensitiv-ity of noise figure to source admittance changes). Fmin,Rn, and Gopt are frequently referred to as the “noiseparameters”, and it is their determination which iscalled “noise characterization”. When Gs is plotted ona Smith chart for a set of constant noise factors, F, theresult is “noise circles”. Noise circles are a convenientformat to display the complex relation between sourceimpedance and noise figure.

Figure 2-2. Noise circles

The available gain, Ga, provided by a device when it isdriven by a specified source impedance, can be calcu-lated from the S-parameters of the device [35, 40] and the source reflection coefficient, Gs, using equation (2-5). S-parameters are commonly measuredwith a network analyzer.

When the source reflection coefficient, Γs, is plotted on a Smith chartcorresponding to a set of fixed gains, “gain circles” result. Gain circlesare a convenient format to display the relation between source imped-ance and gain.

The Effect of BandwidthAlthough the system bandwidth is an important fac-tor in many systems and is involved in the actual sig-nal-to-noise calculations for demodulated signals,noise figure is independent of device bandwidth. Ageneral assumption made when performing noisemeasurements is that the device to be tested has anamplitude-versus-frequency characteristic that is con-stant over the measurement bandwidth. This meansthat noise measurement bandwidth should be lessthan the device bandwidth. When this is not the case,an error will be introduced [34]. The higher endAgilent NFA series noise figure analyzers have vari-able bandwidths to facilitate measurement of narrow-band devices, as do spectrum analyzer-basedmeasurement systems.(PSA with the noise figure measurement personality has a bandwidth that can be reduced to 1 Hz.)

Most often the bandwidth-defining element in a sys-tem, such as a receiver, will be the IF or the detector.It will usually have a bandwidth much narrower thanthe RF circuits. In this case noise figure is a validparameter to describe the noise performance of theRF circuitry. In the unusual case where the RF cir-cuits have a bandwidth narrower than the IF or detec-tor, noise figure may still be used as a figure of meritfor comparisons, but a complete analysis of the sys-tem signal-to-noise ratio will require the input band-width as a parameter.

11

F = Fmin + (2-4)4Rn Zo

|Γopt – Γs|2

|1 + Γopt|2 (1– |Γs|

2) ( )

F min = 1.1dBF = 1.2 dBF = 1.6 dBF = 2.1 dBF = 3.1 dBF = 4.1 dB

Ga = (2-5)|1– S11Γs|

2(1– |S22 + |2)S12S21Γs

1– S11Γs

(1– |Γs|2)|S21|

2

Chapter 3.The Measurement of Noise Figure

Noise Power LinearityThe basis of most noise figure measurements dependson a fundamental characteristic of linear two-portdevices, noise linearity. The noise power out of adevice is linearly dependent on the input noise poweror temperature as shown in Figure 3-1. If the slope ofthis characteristic and a reference point is known, theoutput power corresponding to a noiseless inputpower, Na can be found. From Na the noise figure oreffective input noise temperature can be calculated asdescribed in Chapter 1. Because of the need for lin-earity, any automatic gain control (AGC) circuitrymust be deactivated for noise figure measurements.

Figure 3-1. The straight-line power output vs. source temperature character-istic of linear, two-port devices. For a source impedance with a temperatureof absolute zero, the power output consists solely of added noise Na from thedevice under test (DUT). For other source temperatures the power output isincreased by thermal noise from the source amplified by the gain character-istic of the DUT.

Noise SourcesOne way of determining the noise slope is to applytwo different levels of input noise and measure theoutput power change. A noise source is a device thatwill provide these two known levels of noise. The mostpopular noise source consists of a special low-capaci-tance diode that generates noise when reverse biasedinto avalanche breakdown with a constant current[5]. Precision noise sources such as the Agilent SNS-series have an output attenuator to provide a lowSWR to minimize mismatch errors in measurements.If there is a difference between the on and off stateimpedance an error can be introduced into the noisefigure measurement [23]. The N4000A noise sourcehas a larger value of attenuation to minimize thiseffect.

When the diode is biased, the output noise will begreater than kTcB due to avalanche noise generationin the diode [11, 12, 13, 15, 20, 21]; when unbiased,the output will be the thermal noise produced in theattenuator, kTcB. These levels are sometimes called Thand Tc corresponding to the terms “hot” and “cold”.The N4001A produces noise levels approximatelyequivalent to a 10,000K when on and 290K when off.Diode noise sources are available to 50 GHz fromAgilent.

SNS-Series Noise Source

To make noise figure measurements a noise source must have a calibrated output noise level, representedby excess noise ratio (ENR). Unique ENR calibrationinformation is supplied with the noise source and, inthe case of the SNS-Series, is stored internally onEEPROM. Other noise sources come with data on afloppy disk, or hard-copy. ENRdB is the ratio,expressed in dB of the difference between Th and Tc,divided by 290K. It should be noted that a 0 dB ENRnoise source produces a 290K temperature changebetween its on and off states. ENR is not the “on”noise relative to kTB as is often erroneously believed.

12

DUTPZS, TS POUTPUT

slope = kGaB

Source Temperature (K)

Pow

er O

utpu

t (W

)

TS

Na

0

ENRdB = 10 log (3-1))Th – Tc

To(

Tc in equation (3-1) is assumed to be 290K when it iscalibrated. When the noise source is used at a differ-ent physical temperature, compensation must beapplied to the measurement. The SNS-Series noisesources contain a temperature sensor which can beread by Agilent’s NFA analyzers. The temperaturecompensation will be covered in the next section ofthis chapter.

In many noise figure calculations the linear form ofENR will be used.

Noise sources may be calibrated from a transfer stan-dard noise source (calibrated traceable to a top levelNational Standards laboratory) or by a primary physi-cal standard such as a hot/cold load. Most noisesources will be supplied with an ENR characterizedversus frequency.

Hot and cold loads are used in some special applica-tions as a noise source. Ideally the two loads need to bekept at constant temperatures for good measurementprecision. One method immerses one load into liquidnitrogen at a temperature of 77K, the other may bekept at room temperature or in a temperature con-trolled oven. The relatively small temperature differ-ence compared to noise diode sources and potentialSWR changes resulting from switching to differenttemperature loads usually limits this method to cali-bration labs and millimeter-wave users.

Gas discharge tubes imbedded into waveguide struc-tures produce noise due to the kinetic energy of theplasma. Traditionally they have been used as a sourceof millimeter-wave noise. They have been essentiallyreplaced by solid-state noise diodes at frequenciesbelow 50 GHz. The noise diode is simpler to use andgenerally is a more stable source of noise. Althoughthe noise diode is generally a coaxial device, integral,precision waveguide adapters may be used to providea waveguide output.

R/Q 347B waveguide noise sources

The Y-Factor MethodThe Y-Factor method is the basis of most noise figuremeasurements whether they are manual or automati-cally performed internally in a noise figure analyzer.Using a noise source, this method allows the determi-nation of the internal noise in the DUT and thereforethe noise figure or effective input noise temperature.

With a noise source connected to the DUT, the outputpower can be measured corresponding to the noisesource on and the noise source off (N2 and N1). Theratio of these two powers is called the Y-factor. Thepower detector used to make this measurement maybe a power meter, spectrum analyzer, or a specialinternal power detector in the case of noise figuremeters and analyzers. The relative level accuracy isimportant. One of the advantages of modern noise fig-ure analyzers is that the internal power detector isvery linear and can very precisely measure levelchanges. The absolute power level accuracy of themeasuring device is not important since a ratio is tobe measured.

Sometimes this ratio is measured in dB units, in thiscase:

The Y-factor and the ENR can be used to find thenoise slope of the DUT that is depicted in Figure 3-1.Since the calibrated ENR of the noise source repre-sents a reference level for input noise, an equation forthe DUT internal noise, Na can be derived. In a modern noise figure analyzer, this will be automatically determined by modulating the noisesource between the on and off states and applyinginternal calculations.

13

ENR = 10 (3-2)

ENRdB10

Y = (3-3)N2

N1

Y = 10 (3-4)

Ydb10

Na = kToBG( –1) (3-5)ENRY –1

From this we can derive a very simple expression for the noise factor. The noise factor that results is thetotal “system noise factor”, Fsys. System noise factorincludes the noise contribution of all the individualparts of the system. In this case the noise generated inthe measuring instrument has been included as a sec-ond stage contribution. If the DUT gain is large(G1>>F2), the noise contribution from this secondstage will be small. The second stage contribution canbe removed from the calculation of noise figure if thenoise figure of the second stage and the gain of theDUT is known. This will be covered in the section oncorrected noise figure and gain. Note that the devicegain is not needed to find Fsys.

When the noise figure is much higher than the ENR, the device noise tends to mask the noise source out-put. In this case the Y-factor will be very close to 1.Accurate measurement of small ratios can be difficult.Generally the Y-factor method is not used when thenoise figure is more than 10 dB above the ENR of thenoise source, depending on the measurement instru-ment.

This equation can be modified to correct for the condition when the noise source cold temperature, Tc, is not at the 290K reference temperature, To.

This often used equation assumes that Th is unaffect-ed by changes in Tc as is the case with hot and coldloads. With solid-state noise sources, Th will likely beaffected by changes in Tc. Since the physical noisesource is at a temperature of Tc, the internal attenua-tor noise due to Tc is added both when the noisesource is on and off. In this case it is better to assumethat the noise change between the on and off stateremains constant (Th-Tc). This distinction is mostimportant for low ENR noise sources when Th is lessthan 10 Tc. An alternate equation can be used to correct for this case.

The Signal Generator Twice-power MethodBefore noise sources were available this method waspopular. It is still particularly useful for high noise fig-ure devices where the Y-factors can be very small anddifficult to accurately measure. First, the outputpower is measured with the device input terminatedwith a load at a temperature of approximately 290K.Then a signal generator is connected, providing a sig-nal within the measurement bandwidth. The genera-tor output power is adjusted to produce a 3 dBincrease in the output power. If the generator powerlevel and measurement bandwidth are known we cancalculate the noise factor. It is not necessary to knowthe DUT gain.

There are some factors that limit the accuracy of thismethod. The noise bandwidth of the power-measuringdevice must be known, perhaps requiring a network analyzer. Noise bandwidth, B, is a calculated equiva-lent bandwidth, having a rectangular, “flat-top” spec-tral shape with the same gain bandwidth product asthe actual filter shape. The output power must bemeasured on a device that measures true power sincewe have a mix of noise and a CW signal present.Thermal-based power meters measure true powervery accurately but may require much amplificationto read a low noise level and will require a band-width-defining filter. Spectrum analyzers have goodsensitivity and a well-defined bandwidth but thedetector may respond differently to CW signals andnoise. Absolute level accuracy is not needed in thepower detector since a ratio is being measured.

The Direct Noise Measurement MethodThis method is also useful for high noise figuredevices. The output power of the device is measuredwith an input termination at a temperature of approx-imately 290K. If the gain of the device and noise band-width of the measurement system is known, the noisefactor can be determined.

Again with this method the noise bandwidth, B, mustbe known and the power-measuring device may needto be very sensitive. Unlike the twice-power method,the DUT gain must be known and the power detectormust have absolute level accuracy.

14

Fsys = (3-6)ENRY –1

Fsys = (3-9)Pgen

kToB

Fsys = (3-10)No

kToBG

Fsys = (3-7)ENR – Y( –1)

Tc

To

Y –1

Fsys = (3-8)ENR ( )

Tc

To

Y –1

Corrected Noise Figure and GainThe previous measurements are used to measure thetotal system noise factor, Fsys, including the measure-ment system. Generally it is the DUT noise figure that is desired. From the cascade noise-figure equation itcan be seen that if the DUT gain is large, the measure-ment system will have little effect on the measure-ment. The noise figure of high gain DUTs can bedirectly measured with the previously discussedmethods. When a low gain DUT is to be measured orthe highest accuracy is needed, a correction can beapplied if we know the gain of the DUT and the noisefigure of the system. Using equation (2-2) and re-writ-ing to solve for F1 gives the equation for the actual DUT noise factor.

Both the gain of the DUT and the measurement sys-tem noise factor, F2, can be determined with an addi-tional noise source measurement. This step is called asystem calibration. With a noise-figure analyzer thiscalibration is usually performed before connecting theDUT so that all subsequent measurements can use thecorrections and the corrected noise figure can be dis-played. The necessary calculations to find the gainand the corrected noise figure are automatically per-formed internally. When manual measurements aremade with alternative instruments, a calibrated noisefigure measurement can be performed as follows:

1.Connect the noise source directly to the measure-ment system and measure the noise power levels corresponding to the noise source “on” and “off”. These levels; N2 and N1 respectively, can then be used to calculate the measurement system noise factor F2 using the Y-factor method.

2.The DUT is inserted into the system. The noiselevels N2 and N1 are measured when the noise source is turned on and off. The DUT gain can be calculated with the noise level values.

The gain is usually displayed in dB terms: Gdb=10 log G

3.The overall system noise factor, Fsys, can be calculated by applying the Y-factor method to the values N2 and N1.

4.The DUT noise factor, F1, can be calculated withequation (3-11). The DUT noise figure is 10 log F1.

JitterNoise can be thought of as a series of random events,electrical impulses in this case. The goal of any noisemeasurement is to find the mean noise level at the output of the device. These levels can be used, withappropriate corrections, to calculate the actual noise figure of the device. In theory, the time required tofind the true mean noise level would be infinite. Inpractice, averaging is performed over some finite timeperiod. The difference between the measured averageand the true mean will fluctuate and give rise to arepeatability error.

Figure 3-2. Noise jitter

For small variations, the deviation is proportional to 1/ (t) so that longer averaging times will produce bet-ter averages. Because the average includes moreevents it is closer to the true mean. The variation isalso proportional to 1/ (B). Larger measurementbandwidths will produce a better average becausethere are more noise events per unit of time in a largebandwidth; therefore, more events are included in theaverage. Usually noise figure should be measured witha bandwidth as wide as possible but narrower thanthe DUT.

15

F1 = Fsys – (3-11)F2 – 1

G1

NoiseSignalAmplitude

Variation (dB)

Mean

Time

G1 = (3-12)N2 – N1

N2 – N1

' '

' '

' '

Frequency ConvertersFrequency converters such as receivers and mixers usually are designed to convert an RF frequency band to an IF frequency band. While the noise figurerelationships discussed in this application note applyto converters as well as non-converters, there aresome additional characteristics of these devices thatcan affect noise figure measurements. In addition toDUTs that are frequency converters, sometimes thenoise measurement system uses mixing to extend themeasurement frequency range.

LossAmplifiers usually have a gain associated with them,while passive mixers have loss. All the equations fornoise figure still apply; however, the linear gain valuesused will be less than one. One implication of this can be seen by applying the cascade noise figure equation;the second stage noise contribution can be major (See equation 2-2). Another is that passive mixers, ifmeasured using the Y-factor technique, can have small Y-factors owing to their high noise figures. This mayincrease measurement uncertainty. High ENR noisesources can be used to provide a larger Y-factor.

LO NoiseReceivers and mixers have local oscillator (LO) signalsthat may have noise present. This noise can be con-verted in the mixer to the IF frequency band andbecome an additional contribution to the system’snoise figure. The magnitude of this effect varies wide-ly depending on the specific mixer type and howmuch noise is in the LO. It is possible to eliminatethis noise in fixed frequency LO systems with a band-pass filter on the LO port of the mixer. A filter thatrejects noise at fLO+/-fIF, fIF, and fRF while passing fLOwill generally eliminate this noise. There may also behigher order noise conversions that could contributeif the LO noise level is very high. A lowpass filter canbe used to prevent noise conversions at harmonics ofthe LO frequency.

LO LeakageA residual LO signal may be present at the output (IF)of a mixer or converter. The presence of this signal isgenerally unrelated to the noise performance of theDUT and may be acceptable when used for the intend-ed application. When a noise figure measurement ismade, this LO signal may overload the noise measure-ment instrument or create other spurious mixingproducts. This is most likely to be an issue when themeasuring system has a broadband amplifier or otherunfiltered circuit at it’s input. Often a filter can beadded to the instrument input to filter out the LO sig-nal while passing the IF.

Unwanted ResponsesSometimes the desired RF frequency band is not theonly band that converts to the IF frequency band.Unwanted frequency band conversions may occur ifunwanted frequencies are present at the RF port in addition to the desired RF signal. Some of these are: the image response (fLO + fIF or fLO – fIF depending onthe converter), harmonic responses (2fLO ± fIF, 3fLO ±fIF, etc.), spurious responses, and IF feed-throughresponse. Often, particularly in receivers, theseresponses are negligible due to internal filtering. Withmany other devices, especially mixers, one or more ofthese responses may be present and may convertadditional noise to the IF frequency band.Figure 3-3. Possible noise conversion mechanisms with mixers and converters. (1) IF feedthrough response, (2) double sideband response,

(3) harmonic response.

Mixers having two main responses (fLO + fIF and fLO –fIF) are often termed double side-band (DSB) mixers.fLO + fRF is called the upper side-band (USB). fLO – fIFis called the lower side-band (LSB). They convertnoise in both frequency bands to the IF frequencyband. When such a mixer is part of the noise meas-urement system, the second response will create anerror in noise figure measurements unless a correc-tion, usually +3dB, is applied. Ideally filtering is usedat the RF port to eliminate the second response sothat single side-band (SSB) measurements can bemade.

When a DSB mixer is the DUT we have a choice whenmeasuring the noise figure. Usually the user wants tomeasure the equivalent SSB noise figure. In passive mixers that do not have LO noise, the equivalent SSBnoise figure is often close in value to the conversionloss measured with a CW signal. There are two waysto make this measurement; an input filter can beused, or the +3dB correction can be applied. Thereare accuracy implications with these methods thatmust be considered if precision measurements are tobe made; an input filter will add loss that should becorrected for, the +3dB correction factor assumesequal USB and LSB responses.

Converters used in noise receivers, such as radiome-ters and radiometric sensors are often designed tomake use of both main responses, in which case it isdesirable to know the DSB noise figure. In this case,no correction or input filter is used; the resultingnoise figure measured will be in DSB terms.

16

DeviceInput

DownconvertedNoise

(1) (2) (3)Noise fromnoise source

Frequencyf lF f LO-f lF f LO + f lF 3f LO-f lF 3f LO+ f lFf LO 3f LO

Noise Figure Measuring Instruments

Noise Figure AnalyzersThe noise figure analyzer represents the most recentevolution of noise figure measurement solutions. A noise figure analyzer in its most basic form consists of a receiver with an accurate power detector and a cir-cuit to power the noise source. It provides for ENRentry and displays the resulting noise figure valuecorresponding to the frequency it is tuned to.Internally a noise figure analyzer computes the noisefigure using the Y-factor method.

A noise figure analyzer allows the display of swept frequency noise figure and gain and associated fea-tures such as markers and limit lines. The AgilentNFA series noise figure analyzers, combined with theSNS-Series noise sources offer improvements in accu-racy and measurement speed, important factors inmanufacturing environments. The NFA is specificallydesigned and optimized for one purpose: to makenoise figure measurements. Combination productsthat must make other measurements usually compro-mises accuracy to some degree.

NFA Series noise figure analyzer

Features: • Flexible, intuitive user interface makes it easy to characterize amplifiers and frequency-converting devices

• Measurement to 26.5 GHz in a single instrument eliminates the need for a separate system downconverter

• Accurate and repeatable results allow tighter specification of device performance.

Spectrum AnalyzersSpectrum analyzers are often used to measure noise figure, because they are already present in the test racks of many RF and microwave production facilitiesperforming a variety of tasks. With software and a controller they can be used to measure noise figureusing any of the methods outlined in this productnote. They are particularly useful for measuring highnoise figure devices using the signal generator ordirect power measurement method. The variable reso-lution bandwidths allow measurement of narrow-banddevices. The noise figure measurement personality onboth PSA and ESA-E Series spectrum analyzers pro-vides a suite of noise figure and gain measurementssimilar to the NFA Series noise figure analyzers.

One of the advantages of a spectrum analyzer-basednoise figure analyzer is the multi-functionality. It can, for example, make distortion measurements on an amplifier. Also it can locate spurious or stray signals and then the noise figure of the device can be meas-ured at frequencies where the signals will not inter-fere with noise measurements.

Spectrum analyzers generally require the addition ofa low noise pre-amplifier to improve sensitivity. Theuser must take care not to overload the system withbroadband noise power or stray signals. The dedicat-ed noise figure analyzer is generally faster and moreaccurate than spectrum analyzer solutions; however,for measurements below 10 MHz, a spectrum analyzerplatform would be the recommended solution.

17

PSA Spectrum Analyzer with Noise Figure Capability

Network AnalyzersLike spectrum analyzers, network analyzers are com-mon multi-use instruments in industry. Products areavailable that offer noise figure measurements inaddition to the usual network measurements. Anadvantage is that they can offer other measurementscommonly associated with devices: such as gain andmatch. Because network measurements are usuallymade with the same internal receiver architecture,there can be some performance limitations when usedin noise figure applications. Often the receiver is ofthe double side-band (DSB) type, where noise figure isactually measured at two frequencies and an internalcorrection is applied. When a wide measurementbandwidth is used this may result in error if thedevice noise figure or gain is not constant over thisfrequency range. When narrow measurement band-width is used to measure narrow-band devices, theunused frequency spectrum between the upper andlower side-band does not contribute to the measure-ment and a longer measurement time is needed toreduce jitter (see Jitter in this chapter).

Network analyzers have the ability to measure the S-parameters of the device. It has been consideredthat S-parameter data can reduce noise figure meas-urement uncertainty by offering mismatch correction.Ideally this mismatch correction would provide amore accurate gain measurement of the device so thatthe second stage noise contribution can be subtractedwith more precision. Unfortunately, the mismatch alsoeffects the noise generation in the second stage whichcannot be corrected for without knowing the noiseparameters of the device. The same situation occursat the input of the device when a mismatch is presentbetween the noise source and DUT input. (see noiseparameters in Chapter 2 of this note) [4]. Networkanalyzers do not, by themselves, provide measurementof the noise parameters. The measurement of noiseparameters generally requires a tuner and software inaddition to the network analyzer. The resulting meas-urement system can be complex and expensive. Errorcorrection in a network analyzer is primarily of benefit for gain measurements and calculation ofavailable gain.

Noise Parameter Test SetsA noise parameter test set is usually used in conjunc-tion with software, a vector network analyzer and anoise analyzer to make a series of measurements,allowing the determination of the noise parameters ofthe device [29] (see Noise Parameters in Chapter 2).Noise parameters can then be used to calculate theminimum device noise figure, the optimum sourceimpedance, and the effect of source impedance onnoise figure. The test set has an adjustable tuner topresent various source impedances to the DUT.Internal networks provide bias to semi-conductor devices that may be tested. A noise sourceis coupled to the test set to allow noise figure measure-ments at different source impedances. The correspon-ding source impedances are measured with thenetwork analyzer. From this data, the complete noiseparameters of the device can be calculated. Generallythe complete device S-parameters are also measuredso that gain parameters can also be determined.Because of the number of measurements involved,measurement of the full noise parameters of a deviceis much slower than making a conventional noise fig-ure measurement but yields useful design parameters.Noise parameters are often supplied on low-noisetransistor data sheets. Noise parameters are generallynot measured on components and assemblies that areintended to be used in well matched 50 (or 75) ohmsystems because the source impedance is defined inthe application.

Power Meters and True-RMS VoltmetersAs basic level measuring devices, power meters andtrue-RMS voltmeters can be used to measure noise figure with any of the methods described in this notewith the necessary manual or computer calculations.Being broadband devices, they need a filter to limittheir bandwidth to be narrower than the DUT. Such afilter will usually be fixed in frequency and allow meas-urements only at this frequency. Power meters aremost often used to measure receiver noise figureswhere the receiver has a fixed IF frequency and muchgain. The sensitivity of power meters and voltmetersis usually poor but the receiver may provide enoughgain to make measurements. If additional gain isadded ahead of a power meter to increase sensitivity,care should be taken to avoid temperature drift and oscillations.

EPM Series Power Meter

18

4. Glossary Symbols and abbreviationsB Noise BandwidthBER Bit Error Ratio|bs|2 Power delivered by a generator to a non

reflecting loadC/N Carrier to Noise RatioDBS Direct Broadcast by SatelliteDSB Double SidebandDUT Device Under TestENR dB Excess Noise RatioF Noise FactorF1 First Stage Noise FactorFM Frequency ModulationFmin Minimum Noise FactorFsys System Noise Factor1/f Flicker NoiseGp Power GainGass Associated GainGa Available GainGi Insertion GainGt Transducer GainG/T Gain-to-Temperature RatioIEEE Institute of Electrical and ElectronicsEngineersIF Intermediate FrequencyIRE Institute of Radio EngineersK Kelvins (Unit of Temperature)k Boltzmann’s ConstantLNA Low Noise AmplifierLSB Lower SidebandM Noise MeasureMu Mismatch UncertaintyNa Noise AddedNF Noise FigureNoff =N1 (see Y Factor)Non =N2 (see Y Factor)N1 Nout for Tc (see Y Factor)N2 Nout for Th (see Y Factor)Ni Input Noise PowerNo Output Noise PowerRF Radio FrequencyRMS Root Mean SquareRn Equivalent Noise Resistancern Equivalent Noise Resistance, normalizedRSS Root Sum-of-the-SquaresS/N Signal to Noise RatioSSB Single Sideband|S21|2 Forward Transmission CoefficientSi Input Signal PowerSo Output Signal PowerTa Noise TemperatureTC, Tc Cold Temperature (see Tc)Te Effective Input Noise TemperatureTH, Th Hot Temperature (see Th)Tne Effective Noise Temperature

Toff Off Temperature (see Toff)Ton On Temperature (see Ton)Top Operating Noise TemperatureTo Standard Noise Temperature (290K)Ts Effective Source Noise TemperatureUSB Upper SidebandGopt Optimum Source Reflection CoefficientGs Source Reflection CoefficientGL Load Reflection Coefficient

Glossary TermsAssociated Gain (Gass). The available gain of adevicewhen the source reflection coefficient is theop-timum reflection coefficient Gopt corresponding withFmin.

Available Gain (Ga). [2, 35, 40] The ratio, at a specific frequency, of power available from the outputof the network Pao to the power available from thesource Pas.

Ga = ____ (1)

For a source with output |bs|2 and reflection coefficient Gs

where

An alternative expression for the available output power is

These lead to two expressions for Ga

NOTE: Ga is a function of the network parameters and ofthe source reflection coefficient Gs. Ga is independent of theload reflection coefficient GL. Ga is often expressed in dB

Ga(dB) = 10 log ____ (8)

19

Pao

Pas

Pao

Pas

Bandwidth (B). See Noise Bandwidth.

Boltzmann's Constant (k). 1.38 x10-23 joules/kelvin.

Cascade Effect. [8]. The relationship, when severalnetworks are connected in cascade, of the noise char-acteristics (F or Te and Ga) of each individual networkto the noise characteristics of the overall or combined network. If F1, F2, . . ., Fn (numerical ratios, not dB) are the individual noise figures and Ga1, Ga2, …,Gan(numerical ratios) are the individual available gains, the combined noise figure is

the combined available gain is

In terms of individual effective input noise tempera-tures Te1, Te2, …, Ten the overall effective input noisetemperature is

NOTE: Each Fi, Tei, and Gai above refers to the value forthe source impedance that corresponds to the outputimpedance of the previous stage.

Diode Noise Source. [11, 12, 13, 15, 20, 21] A noisesource that depends on the noise generated in a solidstate diode that is reverse biased into the avalancheregion. Excess noise ratios of well-matched devicesare usually about 15 dB (Tne ≈10000K). Higher excessnoise ratios are possible by sacrificing impedancematch and flat frequency response.

Double Sideband (DSB). See Single-sideband (SSB).

Effective Input Noise Temperature (Te). [17] Thenoise temperature assigned to the impedance at theinput port of a DUT which would, when connected toa noise-free equivalent of the DUT, yield the same out-put power as the actual DUT when it is connected to anoise-free input port impedance. The same tempera-ture applies simultaneously for the entire set of fre-quencies that contribute to the out put frequency. Ifthere are several input ports, each having a specifiedimpedance, the same temperature applies simultane-ously to all the ports. All ports except the output areto be considered input ports for purposes of definingTe. For a two-port transducer with a single input anda single output frequency, Te is related to the noisefigure F by

Te = 290(F–1) (1)

Effective Noise Temperature (Tne). [1] (This is aproperty of a one-port, for example, a noise source.) The temperature that yields the power emerging fromthe output port of the noise source when it is connect-ed to a nonreflecting, nonemitting load. The relation-ship between the noise temperature Ta and effectivenoise temperature Tne is

(l)

where Γ is the reflection coefficient of the noisesource. The proportionality factor for the emergingpower is kB so that

(2)

where Pe is the emerging power, k is Boltzmann’s constant, and B is the bandwidth of the power measurement. The power spectral density across themeasurement bandwidth is assumed to be constant.

Equivalent Noise Resistance (rn or Rn). See NoiseFigure Circles.

Excess Noise Ratio (ENR). [1] A noise generator property calculated from the hot and cold noise temperatures (Th and Tc) using the equation

ENR dB =10 log _______ (1)

where To is the standard temperature of 290K. Noisetemperatures Th and Tc should be the “effective” noisetemperatures. (See Effective Noise Temperature) [25].The ENR calibration of diode noise sources assumesTc=To.

A few examples of the relationship between ENR and Th may be worthwhile. An ENR of 0 dB corresponds to Th = 580K. Th of 100°C (373K) corresponds to an ENRof –5.43 dB. Th of 290K corresponds to an ENR of –∞ dB.

20

Th – Tc

To

Flicker Noise and 1/f Noise. [33, 39] Any noisewhose power spectral density varies inversely withfrequency. Especially important at audio frequenciesor with GASFET’s below about 100 MHz.

Forward Transmission Coefficient (S21)2. The ratio,at a specific frequency, of the power delivered by theoutput of a network, to the power delivered to the input of the network when the network is terminated by a nonreflecting load and excited by a nonreflectinggenerator.

The magnitude of this parameter is often given in dB.

|S2l|2 (dB) = 10 log |S2l|

2 (1)

Gain to Temperature Ratio (G/T). [32, 41] A figureof merit for a satellite or radio astronomy receiversystem, including the antenna, that portrays the oper-ation of the total system. The numerator is the anten-na gain, the denominator is the operating noisetemperature of the receiver. The ratio is usuallyexpressed in dB, for example, 10log(G/T). G/T is oftenmeasured by comparing the receiver response whenthe antenna input is a “hot” celestial noise source tothe response when the input is the background radia-tion of space (≈3K).

Gas Discharge Noise Source. [25, 26] A noise source that depends on the temperature of an ionizednoble gas. This type of noise source usually requires several thousand volts to begin the discharge but only about a hundred volts to sustain the discharge.Components of the high turn-on voltage sometimes feed through the output to damage certain small, frail,low-noise, solid-state devices. The gas discharge noisesource has been replaced by the avalanche diodenoise source in most applications. Gas dischargetubes are still used at millimeter wavelengths. Excessnoise ratios (ENR) for argon tubes is about 15.5 dB(l0000K).

Gaussian Noise. [6] Noise whose probabilitydistribution or probability density function is gauss-ian, that is, it has the standard form

where s is the standard deviation. Noise that is steadyor stationary in character and originates from thesum of a large number of small events, tends to begaussian by the central limit theorem of probabilitytheory. Thermal noise and shot noise are gaussian.

Hot/Cold Noise Source. In one sense most noise figure measurements depend on noise power meas-urements at two source temperatures—one hot andone cold. The expression “Hot/Cold,” however, fre-quently refers to measurements made with a cold ter-mination at liquid nitrogen temperatures (77K) oreven liquid helium (4K), and a hot termination at373K (100°C). Such terminations are sometimes used as primary standards and for highly accuratecalibration laboratory measurements.

Insertion Gain (Gi). The gain that is measured byinserting the DUT between a generator and load. Thenumerator of the ratio is the power delivered to the load while the DUT is inserted, Pd. The denominator,or reference power Pr, is the power delivered to theload while the source is directly connected. Measuringthe denominator might be called the calibration step.

The load power while the source and load are directlyconnected is

where the subscript “r” denotes the source characteris-tics while establishing the reference power, i.e., duringthe calibration step. The load power while the DUT is inserted is

or

21

In equations (3,4, and 5) the subscript “d” denotes thesource characteristics while the DUT is inserted. The S-parameters refer to the DUT. The source characteris-tics while calibrating and while the DUT is insertedare some times different. Consider that the DUT, forexample, is a microwave receiver with a waveguideinput and an IF output at 70 MHz. During the calibra-tion step, the source has a coaxial output at 70 MHz,but while the DUT is inserted the source has a wave-guide output at the microwave frequency. Using theabove equations, insertion gain is

In those situations where the same source at the samefrequency is used during the calibration step and DUTinsertion, |bd|2= |br|2 and Gsr= Gsd. This is usuallythe case when measuring amplifiers.

Instrument Uncertainty. The uncertainty caused by errors within the circuits of electronic instruments.For noise figure analyzers/meters this includes errorsdue to the detector, A/D converter, math round-offeffects, any mixer non-linearities, saturation effects,and gain instability during measurement. This uncer-tainty is often mistakingly taken as the overall measurement accuracy because it can be easily foundon specification sheets. With modern techniques,however, it is seldom the most significant cause ofuncertainty.

Johnson Noise. [19] The same as thermal noise.

Minimum Noise Factor (Fmin). See Noise FigureCircles.

Mismatch Uncertainty (Mu). Mismatch uncertainty is caused by re-reflections between one device (thesource) and the device that follows it (the load). The re-reflections cause the power emerging from thesource (incident to the load) to change from its valuewith a reflectionless load.

An expression for the power incident upon the load,which includes the effects of re-reflections, is

where |bs|2 is the power the source delivers to a non-reflecting load, Γs is the source reflection coefficient,and Γl is the load reflection coefficient. If accurateevaluation of the power incident is needed when |bs|2

is given or vice versa, then the phase and magnitudeof Γs and Γl is needed—probably requiring a vectornetwork analyzer.

When the phase of the reflection coefficients is notknown, the extremes of |1– ΓsΓl|2 can be calculatedfrom the magnitudes of Γs and Γl, for example, Ps andPl. The extremes of |1– ΓsΓl|2 in dB can be found fromthe nomograph (Figure 4-l).

Mu=20 log(1±PsPl)

The effect of mismatch on noise figure measurements is extremely complicated to analyze. Consider, for example, a noise source whose impedance is not quite 50 ohms.

22

Figure 4-1. This nomograph gives the extreme effects of re-reflections when only the reflection coefficient magnitudes are known. Mismatch uncertainty lim-its of this nomograph apply to noise figure measurement accuracy for devices that include an isolator at the input.

The source takes part in re-reflections of its own generated noise, but it also reflects noise originating in the DUT and emerging from the DUT input (noiseadded by a DUT, after all, is a function of the sourceimpedance). The changed source impedance also caus-es the DUT’s available gain to change (remember that available gain is also a function of source impedance).The situation can be complicated further because thesource impedance can change between the hot state and the cold state. [23] Many attempts have beenmade to establish a simple rule-of-thumb for evaluat-ing the effect of mismatch—all with limited success.One very important case was analyzed by Strid [36] tohave a particularly simple result. Strid considered theDUT to include an isolator at the input with sufficientisolation to prevent interaction of succeeding deviceswith the noise source. The effect of noise emergingfrom the isolator input and re-reflections between theisolator and noise source are included in the finalresult. The result is that the error in noise figure is

where Fact is the noise figure for a reflectionless noisesource, Find is the measured noise figure, S11 is thereflection coefficient looking into the DUT, for exam-ple, into the isolator input, and Gsh is the reflectioncoefficient looking back into the noise source when inthe hot or on condition. Strid also assumed that theisolator and Tcold are both 290K. Note that the resultis independent of the DUT noise figure, Y factor, andthe noise source reflection coefficient for Tcold.

Mismatch uncertainty may also occur while character-izing the noise contribution of the measurement sys-tem and also at the output of DUT during gainmeasurement. Gain measurement mismatch effectscan be calculated by evaluating the differencebetween available gain and insertion gain.

Mismatch uncertainty is often the most significantuncertainty in noise figure measurements. Correctionusually requires full noise characterization (see NoiseFigure Circles) and measurement of phase and amplitude of the reflection coefficients.

N1 See “Y Factor".

N2 See “Y Factor”.

Noff Same as N1. See “Y factor”.

Non Same as N2. See “Y factor.”

Noise Added (Na). The component of the outputnoise power that arises from sources within the net-work under test. This component of output noise isusually differentiated from the component that comesfrom amplifying the noise that originates in the inputsource for the network. Occasionally the noise addedis referred to the input port, the added noise power atthe output is divided by G.

Noise Bandwidth (B). [18, 26] An equivalent rectan-gular pass band that passes the same amount of noisepower as the actual system being considered. Theheight of the pass band is the transducer power gain atsome reference frequency. The reference frequency isusually chosen to be either the band center or the fre-quency of maximum gain. The area under the equiva-lent (rectangular) gain vs. frequency curve is equal tothe area under the actual gain vs. frequency curve. Inequation form

where Go is the gain at the reference frequency. For a multistage system, the noise bandwidth is nearlyequal to the 3 dB bandwidth.

Noise Figure and Noise Factor (NF and F). [7] At aspecified input frequency, noise factor is the ratio of(1) the total noise power/hertz at a correspondingoutput frequency available at the output port whenthe noise temperature of the input termination isstandard (290K) at all frequencies, to (2) that portionof the output power due to the input termination.

The output noise power is often considered to havetwo components—added noise from the device, Na, andamplified input noise, for example, the output powerfrom the input termination amplified by the DUT,kToBG. Then noise figure can be written

F= ____________ (1)

Note: Characterizing a system by noise figure is mean-ingful only when the impedance (or its equivalent) ofthe input termination is specified.

Noise figure and noise factor are sometimes differenti-ated by [31]

Noise Figure = 10 log (Noise Factor) (2)

so that noise figure is in dB and noise factor is thenumerical ratio. Other times the terms are used inter-changeably. There should be no confusion, however,because the symbol “dB” seems to be invariably usedwhen 10 log (NF) has been taken. No “dB” symbolimplies that the numerical ratio is meant.

23

Na + kToBG

kToBG

Noise Figure Circles. [9, 18] This refers to the contours of constant noise figure for a network whenplotted on the complex plane of the source imped-ance, admittance, or reflection coefficient seen by the network. The general equation expressing the noise factor of a network as a function of source reflectioncoefficient Γs is

where Γopt is the source reflection coefficient thatresults in the minimum noise figure of the network, Fmin is the minimum noise figure, Zo is the referenceimpedance for defining Γs (usually 50 ohms) and Rn iscalled the equivalent noise resistance. SometimesRn/Zo, is given as the single parameter rn, called thenormalized equivalent noise resistance. Loci of con-stant F, plotted as a function of Γs, form circles on thecomplex plane. Noise figure circles with available gaincircles are highly useful for circuit designer insightsinto optimizing the overall network for low noise fig-ure and flat gain.

Noise Measure (M). [14] A quality factor thatincludes both the noise figure and gain of a networkas follows

If two amplifiers with different noise figures andgains are to be cascaded, the amplifier with the lowestM should be used at the input to achieve the smallestoverall noise figure. Like noise figure and availablepower gain, a network’s noise measure generallyvaries with source impedance [9]. To make the deci-sion as to which amplifier to place first, the sourceimpedances must be such that F and G for each ampli-fier are independent of the order of cascading.

Noise measure is also used to express the overallnoise figure of an infinite cascade of identical net-works. The overall noise figure is

Sometimes Ftot of equation (2) is called the noise measure instead of M in equation (1). Care should beexercised as to which definition is being used becausethey differ by 1.

Noise Temperature (Ta). [1] The temperature thatyields the available power spectral density from asource. It is obtained when the corresponding reflec-tion coefficients for the generator and load are com-plex conjugates. The relationship to the availablepower Pa is

where k is Boltzmann’s constant and B is the band-width of the power measurement. The power spectraldensity across the measurement band is to be con-stant. Also see Effective Noise Temperature (Tne)

Noise temperature can be equivalently defined [26] asthe temperature of a passive source resistance havingthe same available noise power spectral density asthat of the actual source.

Nyquist’s Theorem. See Thermal Noise.

Operating Noise Temperature (Top). [7] The temper-ature in kelvins given by:

where No is the output noise power/hertz from theDUT at a specified output frequency delivered intothe output circuit under operating conditions, k isBoltzmann's constant, and Gs is the transducer powergain for the signal. NOTE: In a linear two-port trans-ducer with a single input and a single output frequen-cy, Top is related to the noise temperature of the inputtermination Te, and the effective input noise tempera-ture Te, by:

Top=Ta+Te(2)

Optimum Reflection Coefficient (Γopt). See NoiseFigure Circles.

Partition Noise. [26, 39] An apparent additionalnoise source due to the random division of currentamong various electrodes or elements of a device.

24

Power Gain (GP). [2, 35, 40] The ratio, at a specific frequency, of power delivered by a network to an arbi-trary load Pl to the power delivered to the network bythe source Ps.

GP= _____ (1)

The words “power gain” and the symbol G are oftenused when referring to noise, but what is probablyintended is “available power gain (Ga)”, or “transducerpower gain (Gt)”, or “insertion power gain (Gi)”. Foran arbitrary source and load, the power gain of a net-work is given by

GP= |S21|2 _________________________ (2)

where

Γl = S11 + _________ (3)

NOTE 1: GP is function of the load reflection coefficientand the scattering parameters of the network but isindependent of the source reflection coefficient.

NOTE 2: The expression for GP is the same as that for Gaif Gl is substituted for Gs, and S11 is substituted forS22.

GP is often expressed in dB

GP(dB) = 10 log ____ (4)

Root Sum-of-the Squares Uncertainty (RSS). Amethod of combining several individual uncertainties of known limits to form an overall uncertainty. If a particular measurement has individual uncertainties ±A, ±B, ±C, etc, then the RSS uncertainty is

URSS = (A2+B2+C2+ …)1/2 (1)

The RSS uncertainty is based on the fact that most ofthe errors of measurement, although systematic andnot random, are independent of each other. Since theyare independent they are random with respect to each other and combine like random variables.

Second-Stage Effect. A reference to the cascadeeffect during measurement situations where the DUT is the first stage and the measurement equipment is the second stage. The noise figure measured is the combined noise figure of the DUT cascaded to the measurement equipment. If F2 is the noise factor ofthe measurement system alone, and Fsys is the com-bined noise factor of the DUT and system, then F1, thenoise factor of the DUT, is

F1 = Fsys – _______ (1)

where G is the gain of the DUT.

NOTE: F2 in equation (1) is the noise factor of themeasurement system for a source impedance corre-sponding to the output impedance of the DUT.

Sensitivity. The smallest signal that a network canreliably detect. Sensitivity specifies the strength of thesmallest signal at the input of a network that causesthe output signal power to be M times the outputnoise power where M must be specified. M=1 is verypopular. For a source temperature of 290K, the rela-tionship of sensitivity to noise figure is

Si = MkToBF (1)

In dBm

Si (dBm) = –174 dBm + F(dB) + 10 log B + 10 log M (2)

Thus sensitivity is related to noise figure in terrestrialsystems once the bandwidth is known.

Shot Noise. [6, 39] Noise is caused by the quantizedand random nature of current flow. Current is notcontinuous but quantized, being limited by the small-est unit of charge (e=1.6 x 10-19 coulombs). Particlesof charge also flow with random spacing. The arrivalof one unit of charge at a boundary is independent ofwhen the previous unit arrived or when the succeed-ing unit will arrive. When dc current Io flows, theaverage current is Io but that does not indicate whatthe variation in the current is or what frequencies areinvolved in the random variations of current.Statistical analysis of the random occurrence of parti-cle flow yields that the mean square current varia-tions are uniformly distributed in frequency up to theinverse of the transit time of carriers across thedevice. Like thermal noise, the noise power resultingfrom this noise current, produces power in a loadresistance that is directly proportional to bandwidth.

in2 (f) = 2eIo A2/Hz (2)

This formula holds for those frequencies which haveperiods much less than the transit time of carriers across the device. The noisy current flowing through aload resistance forms the power variations known asshot noise.

25

Pl

Ps

1 – |Γl|2

|1 – |ΓlS22|2(1 – |Γl|2)

Pl

Ps

S12S21Γl

1 – ΓlS22

F2 – 1G

Single-sideband (SSB). Refers to using only one ofthe two main frequency bands that get converted toan IF. In noise figure discussions, single-sideband isderived from the meaning attached to modulationschemes in communication systems where energy onone side of the carrier is suppressed to more optimal-ly utilize the radio spectrum. Many noise figure meas-urements are in systems that include downconversion using a mixer and local oscillator at fre-quency fLO to generate an intermediate frequency fIF.The IF power from the mixer is usually increased byan amplifier having bandwidth B. Some of these downconverting systems respond only to signals over band-width B centered at fLO + fIF. These are single-side-band measurements at the upper sideband (USB).Some other systems respond only to signals overbandwidth B centered at fLO – fIF. These are single-sideband measurements at the lower sideband (LSB).Other systems respond to signals in both bands. Suchmeasurements are called double-side-band (DSB). SSBsystems usually use pre-selection filtering or imagerejection to eliminate the unwanted sideband.

Confusion often arises when DSB noise figure measurement results for receivers or mixers are to beinterpreted for single-sideband applications. Thecause of the confusion is that the definition of noisefigure (see the notes under Noise Figure in this glos-sary) states that the numerator should include noisefrom all frequency transformations of the system,including the image frequency and other spuriousresponses, but the denominator should only includethe principal frequency transformation of the system.For systems that respond equally to the upper side-band and lower sideband, but where the intended fre-quency translation is to be for only one sideband, thedenominator noise power in the definition should behalf the total measured output power due to the inputnoise (assuming gain and bandwidth are the same inboth bands). Double-sideband noise figure measure-ments normally do not make the distinction. Since thenoise source contains noise at all frequencies, all fre-quency transformations are included in both thenumerator and denominator. Thus, if the final appli-cation of the network being measured has desired signals in only one sideband but responds to noise in both sidebands, the denominator of DSB measurements is too large and the measured noise figure is too small—usually a factor of about two (3 dB).

There are occasions when the information in both side-bands is desired and processed. The measured DSB noise figure is proper and no correction shouldbe performed. In many of those applications, the sig-nal being measured is radiation so the receiver iscalled a radiometer. Radiometers are used in radioastronomy.

Noise figure measurements of amplifiers made withmeasurement systems that respond to both sidebandsshould not include a 3 dB correction factor. In thiscase, the noise figure measurement system is operat-ing as a radiometer because it is using the informa-tion in both sidebands.

Spot Noise Figure and Spot Noise Factor. A termused when it is desired to emphasize that the noise figure or noise factor pertains to a single frequency as opposed to being averaged over a broad band.

Standard Noise Temperature (To). [7] The standardreference temperature for noise figure measurements. It is defined to be 290K.

TC, Tc, or Tcold. The colder of two noise source tempera-tures, usually in kelvins, used to measure a network’snoise characteristics .

TH, Th,or Thot. The hotter of two noise source tempera-tures, usually in kelvins, used to measure a network’snoise characteristics .

Toff. The temperature, usually in kelvins, of a noisesource when it is biased off. This corresponds to Tcold.

Ton. The temperature, usually in kelvins, of a noisesource when it is biased on. This corresponds to Thot.

Thermal Noise. [19, 26, 30] Thermal noise refers to the kinetic energy of a body of particles as a result of its finite temperature. If some particles are charged (ionized), vibrational kinetic energy may be coupledelectrically to another device if a suitable transmis-sion path is provided. The probability distribution ofthe voltage is gaussian with mean square voltage

where k is Boltzmann’s constant (1.38 x10-23

joules/kelvin), T is the absolute temperature inkelvins, R is the resistance in ohms, f is the frequencyin hertz, f1 and f2 specify the band over which thevoltage is observed, and h is the Planck’s constant(6.62 x10-34) joule seconds.

26

For frequencies below 100 GHz and for T = 290K, l>p(f)>0.992, so p(f)=1 and equation (l)becomes

The power available, that is, the power delivered to a complex conjugate load at absolute zero, is

The units of kTB are usually joules/second, which arethe same as watts.

The available power spectral density is kTwatts/hertz. Although this development appears tomake equation (3) more fundamental than (4),Nyquist [30] first arrived at the value of power spectral density (equation (4)) and then calculatedthe voltage and current involved (equation (3)). The expression for the voltage generator is

en2 df = 4 RkT df (5)

Equation (5) is frequently referred to as Nyquist’sTheorem. This should not be confused with Nyquist’swork in other areas such as sampling theory and sta-bility criteria where other relations may also bereferred to as Nyquist’s Theorem. When T is equal tothe standard temperature To (290K), kTo = 4 x10-21

W/Hz = –174 dBm/Hz.

A brief examination of kTB shows that each of the factors makes sense. Boltzmann’s constant k gives the average mechanical energy per particle that can be coupled out by electrical means, per degree of temperature. Boltzmann’s constant is thus a conver-sion constant between two forms of expressing energy—in terms of absolute temperature and in termsof joules.

The power available depends directly on temperature.The more energy that is present in the form of highertemperature or larger vibrations, the more energy that it is possible to remove per second.

It might not be apparent that bandwidth should bepart of the expression. Consider the example of atransmission band limited to the 10 to 11 Hz range.Then only that small portion of the vibrational energyin the 10 to 11 Hz band can be coupled out. The sameamount of energy applies to the 11 to 12 Hz band(because the energy is evenly distributed across thefrequency spectrum). If, however, the band were 10 to12Hz, then the total energy of the two Hz range, twiceas much, is available to be coupled out. Thus it is rea-sonable to have bandwidth, B, in the expression foravailable power.

It should be emphasized that kTB is the power avail-able from the device. This power can only be coupledout into the optimum load, a complex-conjugateimpedance that is at absolute zero so that it does notsend any energy back.

It might seem like the power available should depend on the physical size or on the number of charge carri-ers and therefore the resistance. A larger body, con-tains more total energy per degree and more chargedparticles would seem to provide more paths for cou-pling energy. It is easy to show with an example thatthe power available is independent of size or resist-ance. Consider a system consisting of a large object ata certain temperature, electrically connected to asmall object at the same temperature. If there were anet power flow from the large object to the smallobject, then the large object would become cooler andthe small object would become warmer. This violatesour common experience—and the second law of ther-modynamics. So the power from the large object mustbe the same as that from the small object. The samereasoning applies to a large resistance and smallresistance instead of a large and small object.

This brings up the point that if a source of noise is emitting energy it should be cooling off. Such is gener-ally the case, but for the problems in electrical equip-ment, any energy removed by noise power transfer is so small that it is quickly replenished by the environment at the same rate. This is because sources of noise are in thermal equilibrium with their environment.

0

27

Transducer Power Gain (Gt). [2, 35, 40] The ratio, at a specific frequency, of power delivered by a network to an arbitrary load Pl to the power availablefrom the source Pas

For a source of strength |bs|2 and reflection coeffi-cient Gs, and for a load reflection coefficient G1.

where the S parameters refer to the DUT. An equivalent expression for P1 is

where

Transducer gain is then

Transducer gain is a function of the source and loadreflection coefficients as well as the network parame-ters.

The term “transducer” arises because the result compares the power delivered to an arbitrary loadfrom an arbitrary generator through the DUT with thepower delivered to the load through a lossless trans-ducer which transfers all of the available generatorpower to the load.

Transducer gain is often measured in dB

White Noise. Noise whose power spectral density(watts/hertz) is constant for the frequency range ofinterest. The term “white” is borrowed from the lay-man’s concept of white light being a composite of allcolors, hence containing all frequencies.

Worst Case Uncertainty. A conservative method ofcombining several individual measurement uncertain-ties of known limits to form an overall measurementuncertainty. Each individual uncertainty is assumedto be at its limit in the direction that causes it to com-bine with the other individual uncertainties to havethe largest effect on the measurement result.

Y Factor. The ratio of N2 to N1 in noise figure meas-urements where N2 is the measured noise power out-put from the network under test when the sourceimpedance is turned on or at its hot temperature andN1 is the measured power output when the sourceimpedance is turned off or at its cold temperature.

28

5. References

[1] Accuracy Information Sheet, United States National Bureau of Standards (NBS), enclosure returned with noise sources sent to NBS for calibration.

[2] Anderson, R.W. S-Parameter Techniques for Faster, More Accurate Network Design, Hewlett-Packard Application Note 95-1.

[3] Beatty, Robert W. Insertion Loss Concepts,Proc. of the IEEE, June, 1964, pp. 663-671.

[4] Boyd, Duncan Calculate the Uncertainty of NF Measurements. “Microwaves and RF”, October, 1999, p.93.

[5] Chambers, D. R. A Noise Source for Noise FigureMeasurements, Hewlett-Packard Journal, April, 1983, pp. 26-27.

[6] Davenport, Wilbur B. Jr. and William L. Root. An Introduction to the Theory of Random Signals and Noise, McGraw-Hill Book Co., Inc, New York, Toronto, London,1958.

[7] Description of the Noise Performance of Amplifiersand Receiving Systems, Sponsored by IRE subcommittee 7.9 on Noise, Proc. of the

IEEE, March,1963, pp. 436-442.

[8] Friis, H.T. Noise Figures of Radio Receivers, Proc. of the IRE, July, 1944, pp. 419-422.

[9] Fukui, H. Available Power Gain, Noise Figure and Noise Measure of Two-Ports and Their Graphical Representations, IEEE Trans. on Circuit Theory, June, 1966, pp. 137-143.

[10] Fukui, H. (editor) Low Noise Microwave Transistors and Amplifiers, IEEE Press and John Wiley & Sons, New York,1981. (This book of reprints contains many of the articles referenced here.)

[11] Gupta, M-S. Noise in Avalanche Transit-Time Devices, Proc. of the IEEE, December, 1971, pp. 1674-1687.

[12] Haitz, R.H. and F.W. Voltmer. Noise Studies in Uniform Avalanche Diodes, Appl. Phys. Lett, 15 November, 1966, pp. 381-383.

[13] Haitz, R.H. and F.W. Voltmer. Noise of a SelfSustaining Avalanche Discharge in Silicon: Studies at Microwave Frequencies, J. Appl. Phys., June 1968, pp. 3379-3384.

[14] Haus, H.A. and R.B. Adler. Optimum Noise Performance of Linear Amplifiers, Proc. of

the IRE, August, 1958, pp. 1517-1533.

[15] Hines, M.E. Noise Theory for the Read Type Avalanche Diode, IEEE Trans. on Electron devices, January, 1966, pp. 158-163.

[16] IRE Standards on Electron Tubes. Part 9, Noise in Linear Two-Ports, IRE subcommittee 7.9, Noise, 1957.

[17] IRE Standards on Electron Tubes: Definitions ofTerms, 1962 (62 IRE 7.52), Proc. of the IEEE, March, 1963, pp. 434-435

[18] IRE Standards on Methods of Measuring Noise in Linear Twoports, 1959, IRE Subcommittee on Noise, Proc. of the IRE, January, 1960, pp. 60-68. See also Representation of Noise in Linear Twoports, Proc. of the IRE, January,1960, pp. 69-74.

[19] Johnson, J.B. Thermal Agitation of Electricity inConductors, Physical Review, July, 1928, pp. 97-109.

[20] Kanda, M. A Statistical Measure for the Stability of Solid State Noise Sources, IEEE Trans. on Micro. Th. and Tech, August, 1977, pp. 676-682.

[21] Kanda, M. An Improved Solid-State Noise Source, IEEE Trans. on Micro. Th. and Tech, December, 1976, pp. 990-995.

[22] Kuhn, N.J. Simplified Signal Flow Graph Analysis, “Microwave Journal”, November 1963, pp. 59-66.

[23] Kuhn, N.J. Curing a Subtle but Significant Cause of Noise Figure Error, “Microwave Journal”, June, 1984, p. 85.

[24] Maximizing Accuracy in Noise Figure Measurements, Hewlett Packard Product Note 85719A-1, July 1992, (5091-4801E).

[25] Mumford, W.W. A Broadband Microwave Noise Source, Bell Syst. Tech. J., October,1949, pp.608-618.

[26] Mumford, W.W. and Elmer H. Scheibe. Noise Performance Factors in Communication Systems, Horizon House-Microwave, Inc., Dedham, Massachusetts, 1968.

29

[27] NBS Monograph 142, The Measurement of Noise Performance Factors: A Metrology Guide, U.S. Government Printing Office, Washington, D.C.,1974.

[28] NBS Technical Note 640, Considerations for the Precise Measurement of Amplifier Noise, U.S. Government Printing Office, Washington, D.C.,1973.

[29] Noise Parameter Measurement Using the HP 8970B Noise Figure Meter and the ATN NP4 Noise Parameter Test Set, Hewlett Packard Product Note HP 8970B/S-3, December, 1998, (5952-6639).

[30] Nyquist, H. Thermal Agitation of Electric Charge in Conductors, Physical Review, July,1928, pp.110-113.

[31] Oliver, B.M. Noise Figure and Its Measurement,Hewlett-Packard Journal, Vol.9, No. 5 (January, 1958), pp.3-5.

[32] Saam, Thomas J. Small Computers RevolutionizeG/T Tests, “Microwaves”, August, 1980, p. 37.

[33] Schwartz, Mischa. Information Transmission, Modulation and Noise, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1959.

[34] Slater, Carla Spectrum-Analyzer-Based System Simplifies Noise Figure Measurement, “RF Design”, December, 1993, p.24.

[35] S-Parameter Design, Hewlett Packard Application Note 154, March, 1990, (5952-1087).

[36] Strid, E. Noise Measurements For Low-Noise GaA FET Amplifiers, Microwave Systems News, November 1981, pp. 62-70.

[37] Strid, E. Noise Measurement Checklist Eliminates Costly Errors, “Microwave Systems News”, December, 1981, pp. 88-107.

[38] Swain, H. L. and R. M. Cox Noise Figure Meter Sets Record for Accuracy, Repeatability, and Convenience, Hewlett-Packard J., April, 1983, pp. 23-32.

[39] van der Ziel, Aldert. Noise: Sources, Characterization, Measurement, Pentice-Hall, Inc., Englewood Cliffs, New Jersey, 1970.

[40] Vendelin, George D., Design of Amplifiers and Oscillators by the S-Parameter Method, Wiley-Interscience, 1982.

[41] Wait, D.F., Satellite Earth Terminal G/T Measurements, “Microwave Journal”, April, 1977, p. 49.

30

6. Additional Agilent Resources, Literature and Tools

10 Hints for Making Successful Noise FigureMeasurements, Application Note 1341, literature number 5980-0288E

Noise Figure Measurement Accuracy, Application Note 57-2, literature number 5952-3706

Calculate the Uncertainty of NF MeasurementsSoftware and web-based tool available at:www.agilent.com/find/nfu

User guides for Agilent noise figure products available at:www.agilent.com/find/nf

Component Test web site: www.agilent.com/find/component_test

Spectrum analysis web sites: www.agilent.com/find/psa_personalitieswww.agilent.com/find/esa_solutions

31

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Remove all doubt

Our repair and calibration services will get

your equipment back to you, performing

like new, when promised. You will get full

value out of your Agilent equipment

throughout its lifetime. Your equipment

will be serviced by Agilent-trained

technicians using the latest factory

calibration procedures, automated repair

diagnostics and genuine parts. You will

always have the utmost confidence in

your measurements.

Agilent offers a wide range of additional

expert test and measurement services for

your equipment, including initial start-up

assistance onsite education and training,

as well as design, system integration,

and project management.

For more information on repair and

calibration services, go to

www.agilent.com/find/removealldoubt