# Improving Measurement Accuracy by Controlling Mismatch Uncertainty

If you use a power meter, signal generator, noise-figure
analyzer, spectrum analyzer or network analyzer, reflections
reflections, you can keep them under control with some simple
practical measures.

 * The phrase 'uncertainty of measurement' has a technical definition. It is the “result of the evaluation aimed at characterizing the range within which the true value of a measurand is estimated to lie, generally with a given confidence.” The phrase 'uncertainty' has not been used in this rigorous manner in this article. I did not want to get bogged down in the correct use of technical terminology, though it is wholly justified in many instances. The name of the mismatch uncertainty effect precedes the development of the precise language of metrology under the International Standards Organization (ISO). I hope readers will forgive the occasional misuse of terms.

The term “mismatch uncertainty”* seems to have been invented
just to deter the curious. The theory behind this common
phenomenon is actually straightforward. For many measurements,
the mismatch-uncertainty term is one of the largest in the
error budget. This is true for average power measurements, for
example. In other cases, mismatch uncertainty is much smaller
than other sources of inaccuracy.

However, the amount of mismatch uncertainty, calculated as
described in the following section, is sometimes only the tip
of the iceberg. Reflections from adapters or damaged cables and
connectors can mean that the real uncertainty contribution in a
practical measurement set-up is much larger than the
theoretical value. Keeping the unnecessary mismatch-uncertainty
contribution under control should be a very high priority. The
end of this article has a checklist of measures to take to
achieve this goal.

Theory

This explanation of mismatch uncertainty uses a signal flow
graph (Figure 1 ) for the signal generator
and a load, such as a power sensor. Looking at the load first,
the sensor is passive and can be described simply by its
reflection coefficient,
G
l (gamma 1). This is the ratio of the
reflected wave, bl , to the incident wave,
al . These are complex numbers and have both
amplitude and phase.

Figure 1:  Signal flow graph for the load

(1)

The quantities al and bl are
normalized, so that is the power incident on the load
(Pi ) is

Pi = |al |
(2)

and the power reflected from the load (Pr ) is

Pr = |bl |
(3)

The power dissipated in the load is the difference between
incident and reflected power

Pd = |al | –
|bl | (4)

Figure 2:  Signal flow graph for the generator

The signal-flow graph for a generator (Figure 2 ) is
nearly as simple as for the load. Here the amplitude generated
in the signal generator is bs . The emerging wave is
the sum of the internally generated wave and the incident wave
reflected by the generator,
G
g ag , and obeys the equation:

bg = bs + G g ag
(5)

The generator and load are shown connected in Figure
3
. Power from the generator is reflected by the load, and
then re-reflected from the generator, combining with the power
created in the generator, to produce a new incident wave.

Figure 3:  Signal flow graph for the generator and load

In Figure 3 we can see that:

bl =
G
l al = ag
(6)

and

bg = bs + G g ag =
al
(7)

Solving Equations 6 and 7 for al and
bl

al = bs + G g G l al
(8)

such that

(9)

Since

bl =
G
l al (10)

we have

(11)

We can now calculate the power incident on the load and the
power dissipated in the load. The power indicated by a power meter usually means the power that would be dissipated in a resistive load equal to the system characteristic impedance, Z0 . In a power sensor, the calibration factor takes into account the fact that the sensor reflects some of the power. This can be found from the above equation by making G l = 0. Then, bl becomes equal to bs , and the incident power becomes:

Pi = |al |² = |bs |²      (12)

defining |bs |² as the power that would be dissipated by the generator in a load equal to Z0 ohms (resistive).

The power dissipated when
G
l ≠ 0 is:

(13)

Equation 13 consists of three distinct terms:
|bs |, 1-|G l |, and 1/(|1-G g G l |). Taking these in
turn:

• |bs | is the power that would be dissipated in
a load equal to the characteristic impedance of the cable,
such that that |G l |=0.

• 1-|G l | is known as
the mismatch loss, since it accounts for the power reflected
from the load (|G l | is
the magnitude of the reflection coefficient and is often
written r l ).

• 1/(|1-G g G l |) is an interaction
between the load mismatch and the source mismatch and is the
cause of mismatch uncertainty.

These equations tell us all we need to know about mismatch
uncertainty. If we knew the exact values of G l and
G
g there would be zero mismatch uncertainty.
However, we may only be told the amplitudes of the reflection
coefficients, but not their phases, which means that we cannot
calculate exactly the value of the dissipated power.

The phases of the reflection coefficients could be measured
on a vector network analyzer. To use the information, we would
need to know the exact electrical length of all cables and
adapters in the system, and they would need to remain constant
over time, temperature and probably many connection and
disconnection cycles. This is unlikely, and it would require
more network-analyzer measurements to verify the stability of
the phases. It comes as no surprise that most people do not
make the required accurate measurements, but allow for the fact
that they have imprecise knowledge of their measurement system
by applying a further term to the error budget.

To calculate the mismatch uncertainty limits, we need to
know the maximum and minimum values of the term Mu ,
where

(14)

It is fairly straightforward to show that the limits of the
mismatch uncertainty in terms of
r
g and
r
l , the magnitudes of the complex generator

(15)

and

(16)

Expressing this quantity in decibels

Mu = 20log10 (1+/-r g r l )dB
(17)

It is worthwhile noting here that the terms 'voltage
reflection coefficient', 'return loss', 'SWR', and 'VSWR' are
all used to describe the magnitude of the reflection on a
transmission line. Table 1 shows the numerical
relationship between these parameters. Similarly, G is measured using a vector network
analyzer and s-parameter test set as s11 or
s22 , where s11 is the complex reflection
coefficient at the input and s22 the complex
reflection coefficient at the output of a two-port network.
r is |s11 |. You can also
measure the scalar reflection coefficient, |r |, on a scalar network analyzer.

 Voltage ReflectionCoefficient Return Loss (dB) VSWR (or SWR) |r | 20log10( |r |) (1+|r |)/(1-|r |) 0 -∞ 1.00 0.05 -26.0 1.11 0.1 -20.0 1.22 0.15 -16.4 1.35 0.2 -13.9 1.50 0.25 -12.0 1.67 0.3 -10.4 1.86 0.35 -9.1 2.08 0.4 -7.9 2.33 0.45 -6.9 2.64 0.5 -6.0 3.00 0.55 -5.2 3.44 0.6 -4.4 4.00 0.65 -3.7 4.71 0.7 -3.1 5.67 0.75 -2.5 7.00 0.8 -1.9 9.00 0.85 -1.4 12.3 0.9 -0.91 19.0 0.95 -0.45 39.0 1 0 ∞

Table 1:   Conversion between voltage reflection coefficient, return loss, and VSWR

The upper and lower limits of measurement uncertainty are
slightly different, but for practical purposes, the largest
deviation can be taken for both limits. Figure 4 shows
the value of the largest of these limits for reflection
coefficients up to r =0.5.

Figure 4:  Mismatch uncertainty (dB) versus the two reflection coefficients

Example 1
A 75-ohm cable used between a 50-ohm impedance signal generator and 50-ohm power meter.

The following situation should never happen, but it does. You pull out an unmarked cable with BNC connectors from a drawer, or borrow one from a colleague. Unknowingly, you connect a 75-ohm cable into your 50-ohm test-system. In a system of four components (Figure 5 ), that do not individually vary with frequency, the power dissipated in the load varies with frequency. Figure 6 shows the simulated power dissipated in the load resistor for a transmission line with a 1 ns delay. In practice, this cable would be about 200 mm (8-inches) long.

Figure 5:  50-ohm signal generator and 50-ohm power sensor connected by a 75-ohm cable

At low frequencies, say below 10 MHz, the system behaves as if the source and load were connected directly together. The load sees half the source voltage. However, as the frequency increases, the power dissipated in the load reduces at first and then increases again in cyclic fashion. When the two-way transit time of the cable is equal to one cycle of the generator frequency, the cycle begins again. This is at 500 MHz with a 1ns-long cable delay. The peak-to-peak variation is about 0.7 dB, and can be calculated from the mismatch uncertainty limits.

Figure 6:  Power dissipated in the load versus generator frequency
Note: Vertical axis is in dB

The system has a characteristic impedance of 75 ohms, the same as the transmission line, or cable. The reflection coefficient of the 50-ohm source and load is given by:

Such that |r l | = |r g | = 0.2, and the mismatch uncertainty limits from Equation 17 are:

+0.34 1dB > Mu > -0.355 dB

A relatively short piece of coaxial cable of the incorrect impedance has generated serious frequency variation in the power measurement.

Example 2
A power meter connected to a signal generator

Consider a signal-generator output-power measurement using a power meter at 2.4GHz. This is the RF frequency for Bluetooth and IEEE 802.11b wireless LAN radio systems. You might, for example, use an Agilent E4433A signal generator, E1416A power meter and 8481A power sensor for the measurements. The output SWR of the E4433B, at 2.4GHz, is 1.9:1 (with an electronically switched attenuator) or 1.35:1 (with a mechanically switched attenuator).

The SWR of an 8481A sensor, at 2.4GHz, is 1.18:1. To find the voltage reflection-coefficient from the SWR data, the equation in Table 1 is inverted, giving:

The 1.9:1 VSWR of the signal generator is equivalent to |r g |= 0.310, and the 1.18 VSWR of the power sensor is equivalent to |r l | = 0.0826. The mismatch uncertainty, is +0.219, -0.225 dB.

If the mechanical attenuator version of the signal generator is acceptable as a replacement for the electronically switched standard attenuator, the VSWR comes down to 1.35:1, and the reflection coefficient |r g | = 0.149. The mismatch uncertainty is reduced by half, to +0.106 dB, -0.107 dB. Note that the manufacturer's accuracy specification cannot include mismatch uncertainty because the load SWR is unknown and variable. The electronically switched attenuator is likely to be more reliable in an automatic test system; however, the mechanically switched version shows less mismatch uncertainty and allows a high maximum output-power level.

Simple Techniques for Controlling Measurement Uncertainty

Controlling mismatch uncertainty is as simple as reducing the
reflection coefficient on any transmission lines that are part
of the test arrangement. Assuming that equipment with the
lowest practical SWR has been selected as a first requirement,
many other simple measures can be taken to ensure that the
performance of the test system does not become degraded. At
lower frequencies, say less than 300 MHz, you can minimize the
length of the transmission lines to keep the changes of phase
with frequency small. This is not a viable method for higher
frequencies, because even short lengths of cable form
significant fractions of a wavelength, as in Example
1
.

The use of good quality cables intended for many connection
cycles is highly recommended. This is particularly important
for the connection to the unit-under-test (UUT), as this
connection may be repeatedly made and broken. Some
manufacturers produce cables with measured VSWR and loss values
at frequencies up to say 18GHz. This is a good indication of
the intended purpose of these cables. They should be far more
reliable than general-purpose test cables intended for use at
lower frequencies. Where possible, the equipment should be rack
mounted, or fixed immovably to the test bench. Semi-rigid
cables can then be used. However, it is important not to use
semi-rigid cables for connection to the UUT. When the cable is
connected and disconnected, it may often be flexed and will
soon become damaged. In this regard, you should not go below
the minimum bend radius, specified by the cable manufacturer,
and the cable should not be kinked.

Connector choice is also important. The connection to the
UUT will probably not be under your control, but for the other
cables, choose threaded types like N-type, SMA, or APC3.5 in
preference to bayonet type, such as BNC, as they provide more
repeatable results. When tightening the screw-type connectors,
use a torque wrench to avoid over- or under-tightening the
connector; then there will be little variation in tightness
when another operator takes over.

The use of adapters to convert between different families of
connectors may be unavoidable, but should be minimized.
Adapters should convert directly and shouldn't be stacked. For
example, don't convert from N-type to BNC, and then BNC to SMA.
Use the proper N-type to SMA adapter. Also, be wary of mating
dissimilar connectors. APC3.5 and SMA look very similar but
have different mechanical interfaces. The use of a precision
adapter or connection saver is recommended between APC3.5 and
SMA connectors.

There are two kinds of N-type connectors, 50-ohm and the
rare 75-ohm types. A male, 75 ohm, N-type connector connected
to a 50-ohm, female, N-type connector will often result in an
open circuit because the center pin of a 75-ohm connector is
smaller in diameter than the 50-ohm version. If a 50-ohm, male,
N-type connector is inserted into a 75-ohm, N-type, female
connector then the male connector will cause irreparable damage
to the female connector. This is one reason why 75-ohm N-types
are rare! BNC connectors also come in 50-ohm and 75-ohm
varieties but usually, mixing the two kinds usually causes no
damage, although premature wear is possible and the SWR will
not be a good as it could be.

The best way of checking the performance of cables and
adapters is to use a vector network analyzer and record the
results for comparison at the next regular audit of the test
station. The ultimate connectors are hermaphrodite, meaning
there is only one sex of connector. This means that male-male
and female-female adapters are never required. Examples of this
kind of connector are APC-7 and the older General Radio
connector. Finally, many cables become stiffer when used at low
temperatures. This operation can strain the connectors, so a
limited operating temperature range is advisable. Finally,
precision connectors should be regularly cleaned and
gauged”measured with a micrometer to ensure that they have not
been mechanically damaged. A damaged connector can instantly ruin the
mated part.

In summary:

• Select test equipment for lowest SWR
• Keep cable length as short as possible
• Use good quality cables
• Select appropriate connectors
• Keep the connectors clean
• Measure (gauge) the connectors regularly
• Replace faulty, worn, or damaged cables and connectors
promptly
• Don't make your own cables for use at high frequencies
unless you test them first
• Minimize the number of adapters
• If possible, use semi-rigid cables for permanently
connected cables
• Follow the cable manufacturer's recommendation for
• Fix the measurement equipment to the bench if possible
(or rack it up)
• Don't over-tighten connectors and don't allow them to
become loose”use a torque wrench
• Don't mix mating, but dissimilar, families
• Avoid extremes of temperature.
Advanced Techniques for Controlling Measurement Uncertainty

When the performance of a test arrangement is simply not good
enough for the job, there are a number of techniques that allow
an improvement in accuracy. These include adding an attenuator
to one end of the transmission line to improve the VSWR, using
an isolator to reduce the reflections from a load, using a
leveling loop, and making a ratio measurement rather than an
absolute measurement.

The use of an attenuator to improve the flatness of a
transmission line depends on the fact that the return loss of
the attenuator is better than the original source or load. The
attenuator is usually placed at the end of the line with the
worst return loss. Clearly, the generator level will need to be
increased to keep the signal-level constant at the load, which
may limit the applicability of this method to the mid-range of
power levels. Attenuators are usually broadband devices. In a
similar fashion, you can use an isolator to reduce the
reflected energy on the line. Isolators are applied at high
power levels, where the economic cost of the power lost in an
attenuator would be high, and at very low power levels, where
the signal would be masked by thermal noise. They are
narrow-band devices and are likely to be more expensive than
attenuators.

A leveling loop uses low-frequency feedback to improve the
effective source match to the line. This requires a
two-resistor power splitter or a directional coupler. The output of the generator is measured on a
power meter and the generator is adjusted so that the indicated
power is at the level you need. This technique depends on
having a power meter that is more accurate than the signal
generator, and an accurately matched two-resistor
power-splitter or directional coupler.

Conclusion

As the measurement frequency increases, so does the importance
of maintaining a low SWR on the transmission line. You can
never completely eliminate mismatch uncertainty, but simple
practical measures allow you to keep in to a minimum. Further
information on measurement uncertainty can be found in an
application note.

Anthony Lymer received his BS (with honors) in Electrical and Electronics Engineering from the University College of North Wales, Bangor, U.K. in 1975. He became a development engineer at the Marconi Research Laboratories in Great Baddow, U.K. where he developed and field-tested a data-transmission scheme for VHF/UHF mobile radio systems. From 1977-1980 Tony was a research student at the University of Bath, U.K. and the Marconi Research Laboratories developing a single-sideband UHF mobile radio system using phase-lock loops. Tony joined Agilent (Hewlett-Packard) in January 1982 where he became a senior development engineer at the company's Queensferry Telecomms Division. Mr. Lymer is a Corporate Member of Institution of Electrical Engineers and a Chartered Engineer.

## 1 comment on “Improving Measurement Accuracy by Controlling Mismatch Uncertainty”

1. asodvijwoef
August 17, 2015

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