NTIA Report TR-04-410

Gain Characterization of the RF

Measurement Path

J. Wayde Allen

U.S. DEPARTMENT OF COMMERCE

Donald L. Evans, Secretary

Michael D. Gallagher, Acting Assistant Secretary

for Communications and Information

February 2004
 
CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. WHAT KIND OF GAIN? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3. MEASUREMENT UNCERTAINTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4. MEASUREMENT METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1 Measuring Ga Using a Signal Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.1 Single-Point Calibration Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

4.1.2 Two-Point Calibration Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Measuring Ga with a Noise Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.1 Uncertainties when Measuring Ga with a Noise Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Using the Network Analyzer to Determine Ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3.1 Uncertainties in Measuring Ga with a Network Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

iii
 
GAIN CHARACTERIZATION OF THE RF MEASUREMENT PATH

J. Wayde Allen ∗

In radio frequency (RF) measurements the gain (or loss) of the signal path connecting the measurement equipment to the measurement reference plane must be accounted for. This tutorial

paper discusses the various definitions of gain, and how to determine the gain using either a

calibrated signal generator, noise source, or network analyzer.

Key words: available power; cable loss; calibration; delivered power; gain; mismatch

1. INTRODUCTION

At the Institute for Telecommunication Sciences (ITS) systems such as the Radio Spectrum Measurement

System (RSMS) are often called upon to perform a wide range of radio frequency (RF) measurements. For

this reason, the RSMS is not a single system but rather a toolbox of equipment that can be configured

in a variety of ways. While this gives the RSMS a great deal of flexibility, it complicates the job of the

engineer who must provide an analysis of the inherent measurement uncertainties for a given test. The

uncertainties associated with each test configuration need to be dealt with on a case-by-case basis. The goal

here is to provide a collection of basic uncertainty analyses to be used as a starting point for determining

the overall test uncertainty. In a sense, this means creating a toolbox of uncertainty analysis methods that

can be adapted as needed. The measurement configurations and analyses that follow deal specifically with

the problem of accounting for the gain or loss in the RF path from where the signal is sampled to where it is

ultimately detected by the test instrumentation (power meter, spectrum analyzer, etc.). This arrangement

is shown schematically in Figure 1.

Since the test instrumentation can only respond to the signal delivered to it, the value indicated by the

test equipment must be corrected to remove the effects of the intervening two-port. This involves finding a

two-port gain in the form

Pi =

Pmeas

G

(1)

where:

Piis the power incident at the test plane,

Pmeas is power indicated by the test equipment, and

G is the gain of the intervening two-port.

Unfortunately, the term gain (G) can be defined in a number of ways, and often no information is provided

to indicate which definition is being used. This paper discusses the distinctions among different definitions

of gain, “available power gain” in particular, and describes a number of measurement methods.

∗The author is with the Institute for Telecommunication Sciences, National Telecommunications and Information Administration, U.S. Department of Commerce, Boulder, CO 80305
Intervening 2-port

(Cable, Amplifiers, etc.)

Test Equipment

(Power Meter, Spectrum Analyzer, etc.)

Test Plane

Figure 1. Typical measurement system configuration.

2. WHAT KIND OF GAIN?

When one talks about the gain of a two-port device it is important to specify what one really means. It is

not as simple as forming the ratio of the output power versus the input power. As can be seen in Figure 2,

there are a number of powers that can be chosen to form this ratio. Here the powers P1a and P1d refer to the

powers available and delivered to port one of the two-port, whereas powers P2a and P2d describe the powers

available and delivered from port two of the same device. This means that one could offer the following

definitions for the gain of the two-port:

G1 =

P2a

P1a

, (2)

G2 =

P2a

P1d

, (3)

G3 =

P2d

P1a

, (4)

G4 =

P2d

P1d

. (5)

Of these gain definitions, equation 2 is known as the “available gain” (Ga) of the two-port, equation 4 is

either the “signal gain” (Gs) or “transducer gain” (Gt), and equation 5 is commonly known as simply “power

gain” (G) [1]. To the best of my knowledge equation 3 is not named.

P1a P1d P2a P2d

2-Port

Figure 2. Generalized 2-port showing available and delivered powers.

2
bg

bg Γl

bg Γl Γg

bg Γl

2

Γg

bg Γl

2

Γg

2

bg Γl

3

Γg

2

(etc.)

Reflections from load (Γl) Reflections from generator (Γg)

Figure 3. Diagram of multiple reflections at the measurement plane.

Things are further complicated by several other possible gain definitions:

Associated Gain (Gass) The available gain of a device when the source reflection coefficient is the optimum

reflection coefficient (Γopt) [1, pg. 29].

Insertion Gain (Gi) The gain that is measured by inserting the DUT between a generator and a load.

The numerator 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 the load while the source is directly

connected [1, pg. 31].

Clearly it is important to specify the gain definition one is using.

Choosing the appropriate gain definition requires an understanding of the relationship between available

and delivered power. This is a direct result of what can be called the “microwave circuit theory analogue of

Thevenin’s or Norton’s Theorems” [2, pp 18–22] [3, pg 51]

b = bg + aΓg (6)

where:

b is the wave amplitude of the signal emerging from a source,

bg is the wave amplitude of the generator wave created by the source,

Γg is the reflection coefficient seen by energy incident on the source, and

a is the wave amplitude of energy incident on the source coming from the circuit the source is connected to.

By considering the multiple reflections of a transient signal across a reference plane as shown in Figure 3, we

can obtain a mathematical description of the mismatch term M. In this case, summing up all of the wave

amplitude terms directed to the right we get

b = bg + bgΓgΓl + bg(ΓgΓl)

2 + . . . (7)

3
which can be factored to give us

b = bg[1 + ΓgΓl + (ΓgΓl)

2 + . . .]. (8)

Recognizing that the factor consisting of an infinite sum of reflection coefficients is a geometric series that

converges to (1 − ΓgΓl)

−1 as long as |ΓgΓl

| < 1 allows us to write this as

b =

bg

1 − ΓgΓl

. (9)

Note that by definition the power delivered to a passive termination is

Pd = |b|

2 − |a|2

, (10)

and that

a = bΓl. (11)

This means equation 10 can be rewritten in terms of power as

Pd = |b|

2

[1 − |Γl|

2

]. (12)

Substituting equation 9 into equation 12 results in the general expression:

Pd =

|bg|

2

(1 − |Γg|

2)

(1 − |Γg|

2

)(1 − |Γl|

2

)

|1 − ΓgΓl|

2

= PaM (13)

where the available power Pa is given by

Pa =

|bg|

2

1 − |Γg|

2

, (14)

and the mismatch M is

M =

(1 − |Γg|

2

)(1 − |Γl|

2

)

|1 − ΓgΓl|

2

. (15)

With this in mind, let’s assume that we want to measure the power of some arbitrary signal source. Working

in terms of available power, the maximum power that this source can supply to a conjugate matched load is

Psource. Hence the maximum power available to the detector (power meter, spectrum analyzer, etc.) is:

Pout = GaPsource (16)

where Ga is the available gain of the intervening two-port. This is diagrammed in Figure 4.

2-Port Power

Detector

Source

M1 M2

(Psource)

Pout

Figure 4. Transfer of power to the detector using “available gain”.

It is still necessary to determine the power that will be “indicated” by the detector circuit due to the

impedance mismatch M2 and any calibration factors for the detector. However at this location in the measurement system the engineer usually has control over system design and can design the two-port connecting

the signal source to the detector so that either the mismatch term M2 is known or is constant.

If, on the other hand, we choose to work in terms of the power delivered at the input to this same two-port,

we need to know something about the mismatch M1 between the source and the two-port. This would be

described by

P1d = M1Psource, (17)

4
where:

P1d is the power delivered across the measurement plane,

M1 is the mismatch coefficient at the measurement plane, and

Psource is the power available power at the measurement plane.

The power delivered to the detector could then be found using the gain expression given in equation 3. The

power transfer equation in terms of delivered power would then be

Pout = M1G2Psource. (18)

This requires measuring values for both M1 and G2.

3. MEASUREMENT UNCERTAINTIES

Before going any further, it is worth noting that no measurement is complete unless there is some specification

describing how close the measured value comes to reporting the truth. For example, what if the gain of a

two-port was measured to be 50 dB one time and 46 dB the next? Given this information it appears that

there is a 4 dB difference between these measurements. However, if this is a measurement of the same

two-port under the same conditions do these measurements have a discrepancy of 4 dB? Perhaps, but what

if the measurement system had an uncertainty of ±2 dB? In this case, it is entirely possible that the system

may be measuring correctly and still show a 4 dB difference between values. What if the signal path had

roughly 1 dB of gain or loss? Does a measurement of 1 dB ±2 dB mean anything at all? We need to know

the measurement uncertainty in order to know the effective resolution of our measurements, and to ensure

that our data and measurement process can actually support the conclusions we draw from them.

Unfortunately experiment design and uncertainty analysis is too broad a topic to cover here. There are,

however, a number of sources for more detailed information. See [4] and [5]; the analyses that follow are

based on these two references. The basic idea is to establish some estimate of the measurement uncertainty

inherent in the measurement system (systematic uncertainty) and to combine this with any uncertainties

caused by statistical or random fluctuations. However, the use of the term “systematic uncertainty” has

recently been dropped in favor of the following definitions [4]:

Type A Uncertainties arrived at by statistical methods,

Type B Uncertainties which are evaluated by other means.

This paper uses the modern Type A and Type B uncertainty definitions.

Finally, it is worth pointing out the difference between uncertainties expressed in linear (fractional or percent)

and logarithmic (decibel) terms. It is common to express the uncertainty in terms of a plus and minus value

(±∆x). This works well when using fractional or percent change. However, since the decibel scale is

logarithmic the plus value does not imply the same magnitude of change as the minus value. Indeed, a

statement of ±3 dB indicates a plus 100% and minus 50% change in value. This means that uncertainties

expressed in dB need to be converted to linear (fractional or percent) uncertainty before being combined

using traditional error propagation equations.

To convert between fractional uncertainty and dB uncertainty consider the dB representation of a given

value x with respect to a reference value y

dB1 = 10 log(x

y

), (19)

5
and its fluctuation

dB2 = 10 log(x + ∆x

y

). (20)

From this the change in dB can be found as

dB2 − dB1 = ∆dB = 10 log(x + ∆x

y

) − 10 log(x

y

) = 10 log(x + ∆x

x

) (21)

or

∆x

x

= 10( ∆dB

10 ) − 1. (22)

Since the magnitude of the change depends on whether one is using plus or minus in the dB scale, the

convention taken through the rest of this paper is to assume the worst case change.

4. MEASUREMENT METHODS

4.1 Measuring Ga Using a Signal Generator

One common method for measuring gain is to use a calibrated signal generator to supply a source of known

available power at the input to the network. With this known input one then observes the power indicated

by the detector and forms the ratio. The basic test setup is described in Figure 5.

P1a P1d P2a P2d 2-Port Signal

Generator Power

Detector

Figure 5. Test setup for using a signal generator to measure gain.

However, is it sufficient to supply only a single known power, read the detector response and form the power

ratio (single-point calibration), or is it better to use two (or more) known levels from which the system gain

can be determined (two-point calibration)? Two possible gain curves are shown in Figure 6. Since the use

of a single known input and measured output power only identifies a single point on the curve it is clear that

this can only work for systems that have no zero point offset. In many real world measurement systems the

presence of detector offset, DC bias effects, or appreciable system noise can violate this condition. In these

cases, one must resort to the two-point method so that the zero point offset can be taken into account.

The following describes some of the key considerations between the single-point and two-point methods:

• Single-Point Method:

1. This is an absolute value measurement.

2. Assumes strict system linearity.

3. No zero point offset allowed.

4. Simple test.

• Two-Point Method:

1. Difference rather than absolute value measurement.

2. Identifies zero point offset (can be used to linearize the system).

3. More complicated since two points must be measured.

This is a good example of how the same hardware can be combined in different ways to produce very different

measurement methods.

6
Pin

Pout

Pin_1

Pout_1

Pin_2

Pout_2

Pout = G1 * Pin + 0

Pout = G2 * Pin + b

Figure 6. Two possible gain curves.

4.1.1 Single-Point Calibration Uncertainty

In this case we know the available power supplied at the input to the two-port (Pa1), and the detector circuit

indicates the power delivered to it from the two-port (P2d). Since the available gain is the ratio of the power

available at the output of the two-port to the power available at the input of the device, we make use of the

relationship

P2d = P2aM2 (23)

where:

P2d is the power delivered to the power detector,

P2a is the power available at the input to the detector, and

M2 is the mismatch between the two-port and the detector.

Solving for P2a and substituting this into equation 2 gives

Ga =

P2a

P1a

=

P2d

M2 ∗ P1a

. (24)

As long as we can obtain estimates for the variability of each of the terms in equation 24 (∆P2d, ∆M2, and

∆P1a), we can determine the Type B uncertainty by finding the following fractional uncertainties:

uP1a = |(

∂Ga

∂P1a

)( 1

Ga

)∆P1a| =

∆P1a

P1a

, (25)

uP2d = |(

∂Ga

∂P2d

)( 1

Ga

)∆P2d| =

∆P2d

P2d

, (26)

uM2 = |(

∂Ga

∂M2

)( 1

Ga

)∆M2| =

∆M2

M2

. (27)

7
Assuming that these uncertainties are independent we can combine them in quadrature to get a total Type B

uncertainty

uB =

q

u

2

P1a

+ u

2

P2d

+ u

2

M2

. (28)

In order to complete this analysis the Type A uncertainty is commonly found as the standard deviation

divided by the mean of a series of repeated gain measurements. Once this value (uA) is known the expanded

measurement uncertainty can be found as

U = k

q

u

2

A + u

2

B (29)

where k is defined by the National Institute of Standards and Technology (NIST) as the “coverage factor” [4].

The expanded uncertainty defines an uncertainty interval about which the measurement result is confidently

believed to lie, and the value of the coverage factor is chosen based on the desired level of confidence.

Typically, k is in the range of 2 to 3 with k = 2 defining an interval with approximately a 95% level of

confidence.

4.1.2 Two-Point Calibration Uncertainty

Supplying two known powers (Pin 1, Pin 2) at the input to the two-port, and measuring the output responses

(Pa out 1, Pa out 2) allows us to compute the gain as

Ga =

Pa out 2 − Pa out 1

(Pin 2 − Pin 1)

. (30)

This is simply a more general form of equation 24. One could even approach the uncertainty analysis for

this case as the combination of two single point measurements as was done in the prior section; however, of

particular interest is that this two-point method gives us the option of analysing the uncertainties as a ratio

of differences rather than a ratio of absolute values. This is important since power differences can often be

measured more accurately than absolute levels. In fact, one could actually determine the gain without having

to know the absolute values of the input powers Pin 1 and Pin 2, providing the difference between the levels

is known. However, let’s assume for this uncertainty analysis that we are given the absolute power values

for Pin 1 and Pin 2. Let’s also assume that we are measuring the output power difference (Pout 2 − Pout 1)

which can be written in terms of available powers (Pa out 1, Pa out 2) as:

Pdif f = Pa out 2 − Pa out 1 =

Pout 2 − Pout 1

M2

(31)

Assuming that the mismatch term M2 is constant, we only need to worry about the uncertainty in difference

itself (udif f ). Proceeding as for the single point case we have:

udif f = |(

∂Ga

∂Pdif f

)( 1

Ga

)∆Pdif f | =

∆Pdif f

Pdif f

, (32)

uP 1a = |(

∂Ga

∂P1a

)( 1

Ga

)∆P1a| =

∆P1a

P2a − P1a

, (33)

uP 2a = |(

∂Ga

∂P2a

)( 1

Ga

)∆P2a| =

∆P2a

P2a − P1a

. (34)

From this an estimate of the Type B uncertainty can be found as

uB =

q

u

2

dif f + u

2

P 1a + u

2

P 2a

. (35)

Again the Type A uncertainty (uA) is commonly found as the standard deviation divided by the mean of a

series of repeated gain measurements, and the expanded measurement uncertainty computed as

U = k

q

u

2

A + u

2

B (36)

where k = 2 is the most common coverage factor [4].

8
P1a P1d P2a P2d 2-Port Power

Detector

Noise

Source

Figure 7. Test setup using a noise source.

4.2 Measuring Ga with a Noise Source

For low noise systems with enough sensitivity, the use of a calibrated noise source rather than a signal

generator is an option. This can be particularly useful when performing broadband measurements where a

single, narrowband frequency calibration using a signal generator may not be practical or desired. In this

case, the calibrated signal is created by some device or object emitting energy according to the Nyquist

Theorem [2, pg. 203] as if it were a blackbody radiator. This could be a simulated blackbody radiator such

as a temperature controlled resistor [6][7], a gas discharge tube, a celestial radio source such as the supernova

remnant Cassiopeia A [8], or an avalanche diode. The hardware configuration for this measurement is shown

in Figure 7. This is very similar to the setup used for the single frequency case discussed in Section 4.1;

however, in this case the available power from the noise source is represented by the expression

Pnoise = kT B (37)

where:

Pnoise is the noise power,

k is Boltzman’s constant (k = 1.38 × 10−23 [Joules/Kelvin]),

T is the “effective” blackbody temperature of the noise source in Kelvins, and

B is the measurement bandwidth in Hertz.

Again it is very important to recognize that the noise power described by Equation 37 is a true “available”

rather than delivered power [1, pg. 4][2, pg. 203].

An added complication for analysis of the circuit in Figure 7 is that noise sources internal to the two-port

network need to be included. Writing this in terms of available gain and power we have

P2a = GaP1a + Na (38)

where:

P2a is the available noise power at port two,

Ga is the available gain of the two-port,

P1a is the available power at port 1, and

Na is the noise added by the two-port network.

By plotting this function in Figure 8, a few features of the model can be illustrated. First of all, if the

two-port could be connected to a blackbody noise source at absolute zero we would expect the internal noise

of the two-port to result in an available noise power of Na at port two of the device.

Finally, note that the measurement of Ga using a hot and cold noise source is really nothing more than using

two defined points in order to compute the slope of the equation for the line.

9
Na

P2a

P1a P1a_cold P1a_hot

P2a_hot

P2a_cold

P2a = Ga P1a + Na

-Pe

Figure 8. Graphical description of the noise model.

So for two noise sources of known values P1ahot and P1acold we have

P2ahot = P1ahotGa + Na (39)

and

P2acold = P1acoldGa + Na (40)

which can be solved for Ga to obtain

Ga =

P2ahot − P2acold

P1ahot − P1acold

. (41)

Quite often the calibration of the noise source is expressed as an “equivalent” blackbody temperature. This

is not usually the actual physical temperature of the device, but rather the temperature a true blackbody

radiator would need to have the same power output as the noise source as described by the Nyquist relation.

For devices such as noise diodes that can easily be turned on and off it is also common to use the diode

“on” temperature to represent P1ahot and the “off” temperature to represent P1acold . If this is done, the

“off” temperature usually assumes that of the connected two-port and detector. This has the advantage of

establishing the condition where there is no net power flow from the noise source to the connecting two-port

as long as these are both at the same temperature. In this case, the unbiased noise source is simply treated

as a passive resistor (blackbody) and accepted practice is to assume that the blackbody temperature of this

device is then approximately 290 Kelvin. This temperature is commonly represented using the notation To.

Rewriting equation 41 in terms of the noise source temperature T and To results in the following expression.

Ga =

P2ahot − P2acold

kB(T − To)

. (42)

Noting that the detector in Figure 7 can only respond to delivered power (P2d), and that there may be a

detector calibration factor to consider, the “indicated” power can be described by the expression

Pindicated = ηP2d = ηM2P2a (43)

10
where:

Pindicated is the power indicated by the detector,

η is the calibration factor for the detector, and

M2 is the mismatch between the detector and the two-port.

Letting Phot and Pcold represent the powers indicated by the detector with hot and cold noise sources

connected to the circuit, allows us to rewrite equation 42 as

Ga =

Phot

ηM2hot

−

Pcold

ηM2cold

kB(T − To)

. (44)

Here it is important to consider the design and operation of the measurement system. For example, the

powers Phot and Pcold could be measured independently, and assuming that values could be determined

for the detector calibration factor η and the mismatches M2hot and M2cold , using equation 44 to determine

the available gain is straightforward. However, the need to determine the quantities M2hot , M2cold , and

η combined with the implication that the system can make accurate power measurements relative to true

power is somewhat problematic. In most cases, the preferred approach would be to design the system to

measure the power difference (Phot − Pcold). It is important however, to note that

Phot − Pcold = η(M2hotP2ahot − M2cold P2acold ), (45)

and this is proportional to the numerator in equation 44 only if

M2hot = M2cold . (46)

4.2.1 Uncertainties when Measuring Ga with a Noise Source

To compute the uncertainties in the gain measurement described by equation 44 we need to find:

udif f the uncertainty in the difference Phot

ηM2hot

−

Pcold

ηM2cold

,

ubandwidth the uncertainty in the measurement bandwidth B,

uT the uncertainty in the noise source temperature T, and

uTothe uncertainty in the system temperature To.

Since Boltzman’s constant k is a universally accepted physical quantity it can be ignored as an uncertainty

source for this analysis.

Most test equipment, such as a spectrum analyzer, is designed primarily to measure power differences rather

than absolute powers. So it will be assumed that the system is carefully designed to meet the mismatch

requirements of equation 46. Furthermore, the following analysis assumes that the measurement detector

is well behaved so that not only is M2hot = M2cold , these mismatch terms and the calibration factor η are

constant. This simplifies the uncertainty analysis since it means the only variability comes from the difference

term, Phot − Pcold. This results in

udif f = |(

∂Ga

∂(Phot − Pcold)

)( 1

Ga

)∆(Phot − Pcold)| =

∆(Phot − Pcold)

(Phot − Pcold)

. (47)

11
Expressions for ubandwidth and uTo

are

ubandwdith = |(

∂Ga

∂B )( 1Ga

)∆B| =

∆B

B

(48)

and

uTo = |(

∂Ga

∂To

)( 1

Ga

)∆To| =

∆To

T − To

. (49)

However the noise source term is a bit more complicated. Assuming the calibration is done using a noise

diode the uT term is the result of a number of separate uncertainty mechanisms that must be accounted for.

This results in the expansion of the single uT term into:

uTcal the noise source calibration uncertainty,

uTamb ambient temperature effects on the noise source output,

uTVthe noise source power supply stability,

uTinterp uncertainty caused by interpolating between calibration points.

These can be written using the increments of the variable T as:

uTcal = |(

∂Ga

∂Tcal

)( 1

Ga

)∆Tcal| =

∆Tcal

T − To

, (50)

uTamb = |(

∂Ga

∂Tamb

)( 1

Ga

)∆Tamb| =

∆Tamb

T − To

, (51)

uTV = |(

∂Ga

∂TV

)( 1

Ga

)∆TV | =

∆TV

T − To

, (52)

uTinterp = |(

∂Ga

∂Tinterp

)( 1

Ga

)∆Tinterp| =

∆Tinterp

T − To

. (53)

where:

∆Tcal is the variation in the noise source temperature due to the calibration uncertainty,

∆Tamb is the variation in the noise source temperature caused by ambient temperature fluctuation,

∆TV is the variation in the noise source temperature caused by power supply variations, and

∆Tinterp describes how much variation can exist in the noise source output value (T) due to interpolating

between calibration points.

Assuming that these uncertainties are independent we can combine them in quadrature to obtain an estimate

of the total Type B uncertainty

uB =

q

u

2

dif f + u

2

bandwidth + u

2

To

+ u

2

Tcal

+ u

2

Tamb

+ u

2

TV

+ u

2

Tinterp

. (54)

12
The Type A uncertainty would commonly be found as the standard deviation of a series of repeated gain

measurements. Once values have been obtained for both the Type A (uA) and Type B (uB) uncertainties

the expanded uncertainty of the measurement can be computed as

U = k

q

u

2

A + u

2

B (55)

where the coverage factor k = 2 representing a 95% confidence limit is commonly used [4].

4.3 Using the Network Analyzer to Determine Ga

Laboratories equipped with vector network analyzers have the option of computing the available gain of

a two-port from measurements of the network’s scattering parameters (S11, S12, S21, S22) and reflection

coefficient of the signal source (Γs). In this case, the power available from the signal source at the input to

the two-port is described by equation 14, and the power at the output of the intervening two-port can be

described by either [1, pg. 29]

P2a =

|bg|

2

|S21|

2

(1 − |Γ2|

2

)

|(1 − ΓsS11)(1 − Γ

∗

2S22) − ΓsΓ

∗

2S12S21|

2

(56)

or

P2a =

|bg|

2

|S21|

2

|1 − ΓsS11|

2(1 − |Γ2|2)

(57)

where

Γ2 = S22 +

S12S21Γs

1 − S11Γs

. (58)

Since the available gain is given by the ratio of the available output power to the input power as described

in equation 2, we can compute the gain using either [1, pg. 29]

Ga = |S21|

2

(1 − |Γs|

2

)(1 − |Γ2|

2

)

|(1 − ΓsS11)(1 − Γ

∗

2S22) − ΓsΓ

∗

2S12S21|

2

(59)

or

Ga = |S21|

2

1 − |Γs|

2

|1 − ΓsS11|

2(1 − |Γ2|2)

. (60)

Here it can be readily seen how the available power gain depends only on the source impedance and network

parameters and is independent of the load connected at the output of the two-port.

4.3.1 Uncertainties in Measuring Ga with a Network Analyzer

In this case we are faced with five variables (Γs, S11, S12, S21, S22) that are complex numbers. The Sparameters are also correlated rather than independent variables, which complicates matters even more. In

a case like this, it is usually simplest to write a computer program to solve the gain equation and vary

the input variables to empirically determine the gain uncertainty. The difficult part to this approach is

deciding how to vary the input variables. One option is to estimate the uncertainties with which the network

analyzer can determine the real and imaginary components of S-parameters and reflection coefficients. This

information could be gleaned either from the specifications given by the manufacturer of the network analyzer,

13
from a statistical analysis of measuring a number of devices with known network parameters using the test

equipment at hand, or from participation in a measurement assurance or comparison program of some kind.

As far as writing the computer program goes, consider that each complex variable consists of a real and

imaginary component (a + jb) for which there are two possible uncertainty offsets. Using our estimate ∆x

for the determination of each real and imaginary component results in the values:

a

+ = a + ∆x, (61)

a

− = a − ∆x, (62)

b

+ = b + ∆x, (63)

b

− = b − ∆x. (64)

which means that for each complex variable in our gain equation there are four possible perturbations:

a

+ + jb+, (65)

a

+ + jb−, (66)

a

− + jb+, (67)

a

− + jb−, (68)

and that for our gain equations 59 and 60, which depend on five complex variables, there are 45 = 1, 024

possible combinations that need to be computed in addition to the “measured” gain value. Finding the

maximum difference between these “fluctuations” and the measured value determines the Type B uncertainty

(uB) in the measurement.

As usual, the Type A uncertainty (uA) would commonly be found as the standard deviation of a series of

repeated gain measurements. The expanded uncertainty would then be computed as:

U = k

q

u

2

A + u

2

B (69)

where the coverage factor k = 2 would commonly be used to indicate a 95% confidence interval [4].

5. SUMMARY

It is clear that when building a measurement system it is important to account for the gain (or loss) in the

RF measurement path, that there are a number of ways of expressing gain, and that the definition used

must be consistent with the manner in which the gain term will ultimately be used. Also, it is important

to note that how the measurement is performed has an effect on the results. A comparison of some of the

basic features of the gain calibration methods discussed can be found in Table 1.

It should also be clear that the uncertainties attainable by any given technique depend not only on the

equipment at hand, but also on how it is used. It should be noted that the uncertainty analyses presented

here are minimal. They are intended only as a baseline from which a more complete analysis for a specific

experiment can be derived.

14
Table 1. Comparison of Measurement Methods

Measurement Method Advantages Disadvantages

Signal Generator: singlepoint method

• Simple test to perform,

minimum time and equipment.

• Narrowband calibration.

• Suitable for low gain systems.

• Assumes strict system linearity.

• No zero point offset allowed.

• Requires absolute power

measurements which burdens the detector system.

Signal Generator: twopoint method

• Is a difference measurement. This puts less demand on the detector circuit.

• Zero point offset is allowed.

• Narrowband calibration.

• Suitable for low gain systems.

• Is a more complicated test.

Noise Source

• Wide bandwidth characterization.

• Noise sources tend to be

small and easily integrated

into a test system.

• Only suitable for high gain,

low noise systems.

• Not suitable if a narrowband calibration is needed.

Network Analyzer:

Gain computation from

S-parameters

• Gain is relatively simple to compute from

S-parameters.

• Results in a complete mathematical model of signal

propagation through the

two-port.

• Requires the use of complex

and expensive equipment.

• Requires measurement of

five complex variables.

15
6. REFERENCES

[1] “Fundamentals of RF and microwave noise figure measurements,” Agilent Application Note 57-1, July

1983.

[2] G. F. Engen, Microwave Circuit Theory and Foundations of Microwave Metrology, IEEE Electrical

Measurement Series 9, London, Peter Peregrinus Ltd., 1992.

[3] D. M. Kerns and R. W. Beatty, Basic Theory of Waveguide Junctions and Introductory Microwave

Network Analysis, International Series of Monographs in Electromagnetic Waves, London, Pergamon

Press, 1967.

[4] B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST

measurement results,” NIST Technical Note 1297, National Institute of Standards and Technology,

1994.

[5] J. R. Taylor, An Introduction to Error Analysis - The Study of Uncertainties in Physical Measurements,

Mill Valley, CA, University Science Books, 1982.

[6] W. C. Daywitt, “Design and error analysis for the WR10 thermal noise standard,” NBS Technical Note

1071, National Bureau of Standards, 1983.

[7] W. C. Daywitt, “A coaxial noise source for the 1 GHz to 12.4 GHz frequency range,” NBS Technical

Note 11074, National Bureau of Standards, 1984.

[8] J. D. Kraus, Radio Astronomy 2nd Edition. Powell, Ohio, Cygnus-Quasar Books, 1986.

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