* 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.

Figure 1:  Signal flow graph for the load

(1)

The quantities a_l and b_l are normalized, so that is the power incident on the load (P_i) is

P_i = |a_l| (2)

and the power reflected from the load (P_r) is

P_r = |b_l| (3)

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

P_d = |a_l| - |b_l| (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 b_s. The emerging wave is the sum of the internally generated wave and the incident wave reflected by the generator, G_ga_g, and obeys the equation:

b_g = b_s + G_ga_g (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:

b_l = G_la_l = a_g (6)

and

b_g = b_s + G_ga_g = a_l (7)

Solving Equations 6 and 7 for a_l and b_l

a_l = b_s + G_gG_la_l (8)

such that

(9)

Since

b_l = G_la_l (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, b_l becomes equal to b_s, and the incident power becomes:

P_i = |a_l|² = |b_s|²      (12)

defining |b_s|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: |b_s|, 1-|G_l|, and 1/(|1-G_gG_l|). Taking these in turn:

• |b_s| 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_gG_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 M_u, 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 and load reflection coefficients, are:

(15)

and

(16)

Expressing this quantity in decibels

M_u = 20log₁₀(1+/-r_gr_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.

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

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.

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.

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 minimum bend-radius
• 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.

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.

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.