Calculating Accuracy of Peak Power Measurements
by Richard Theiss, Boonton Electronics
Peak power meters have long been accepted as
accurate measurement standards. But just how accurate are these
power measurements? Calculating the accuracy of RF peak power
measurements requires more than just a glance at a specification
sheet.
RF peak power measurement accuracy is
dependent on a variety of factors contributed from both the
instrumentation and the test conditions specific to each DUT.
These factors include mismatch, the power level to the DUT, test
frequency, noise, the test environment, and the instrumentation
itself.
Peak power measurement and the accuracy of
these measurements have become increasingly more important,
especially in applications involving complex modulated signals
like TD-SCDMA. These complex wireless signals allow data to be
packed efficiently in the limited spectrum of communications
systems.
These signals look very much like random
noise that is time gated to a power meter when testing RF peak
power in the physical domain. They often are difficult to
capture, display, and analyze on a peak power meter, and various
tools on the market allow you to isolate very specific sections
of time in a complex modulated signal.
Designers and test engineers now are
interested in the average power, peak power, and peak-to-average
ratio within very specific time intervals, and these parameters
must be characterized precisely. Optimizing the test conditions
will yield the best results.
Uncertainty Contributions
The total measurement uncertainty of RF peak
power is calculated by combining the following terms:
1. Instrument Uncertainty
2. Calibrator Level Uncertainty
3. Calibrator Mismatch Uncertainty
4. Source Mismatch Uncertainty
5. Sensor Shaping Error
6. Sensor Temperature Coefficient
7. Sensor Noise and Zero Drift
8. Sensor Calibration Factor Uncertainty
The formula for worst-case measurement uncertainty is
UWorstCase
= U1 + U2 + U3 + U4 + ... UN
where: U1 through UN are all of the
worst-case uncertainty terms
The worst-case approach is a very
conservative method where the extreme conditions of the
individual uncertainties are added together. If the individual
uncertainties are independent of one another, the probability of
all being at the extreme condition is small.
For this reason, these uncertainties usually
are combined using the root-sum-of-squares (RSS) method. In this
method, each uncertainty is squared and added together, and the
square root of the summation is calculated, resulting in the
combined standard uncertainty. The formula is
UC
= (U12 + U22 + U32 + U42
+ ... UN2)0.5
where: U1 through UN are normalized
uncertainties based on each uncertainty's probability
distribution
This calculation yields what is commonly
referred to as the combined standard uncertainty with a level of
confidence of approximately 68%.
To gain higher levels of confidence, an
expanded uncertainty often is required. Using a coverage factor
of two will provide an expanded uncertainty 2 × UC
with a confidence level of approximately 95%.
Discussion of Uncertainty Terms
Following is a discussion of each term, its
definition, and how it is calculated.
Instrument Uncertainty
Instrument uncertainty represents the
amplification and digitization uncertainty in the power meter as
well as internal component temperature drift. In most cases,
this is very small since absolute errors in the circuitry are
calibrated out by the autocal process.
Calibrator Level Uncertainty
Calibrator output level uncertainty is the
uncertainty for a given calibrator setting. The figure is a
specification that depends upon the output level.
Calibrator Mismatch Uncertainty
Calibrator mismatch uncertainty is the
mismatch error caused by impedance differences between the
calibrator output and the sensor's termination. It is calculated
from the reflection coefficients of the calibrator (DCAL) and
sensor (DSNSR) at the calibration frequency with the equation
Calibrator Mismatch Uncertainty
= ±2 × DCAL × DSNSR × 100%
Source Mismatch Uncertainty
Source mismatch uncertainty is caused by
impedance differences between the measurement source output and
the sensor's termination. It is calculated from the reflection
coefficients of the source (DSRCE) and DSNSR at the measurement
frequency with the equation
Source Mismatch Uncertainty
= ±2 × DSRCE × DSNSR × 100%
The source reflection coefficient is a
characteristic of the RF source under test. If only the standing
wave ratio (SWR) of the source is known, its reflection
coefficient may be calculated from the source SWR using the
equation
DSRCE = (SWR - 1)/(SWR + 1)
The DSNSR is frequency dependent and
specified at various frequency levels. For most measurements,
this is the single largest error term, and care should be used
to ensure the best possible match between source and sensor.
Sensor Shaping Error
The sensor shaping error, sometimes called
linearity error, is the residual nonlinearity in the measurement
after an autocal has been performed to characterize the transfer
function of the sensor. Calibration is performed at discrete
level steps and extended to all levels.
Generally, the sensor shaping error is close
to zero at the autocal points and increases in between due to
imperfections in the curve-fitting algorithm. Also, remember
that the sensor's transfer function may not be identical at all
frequencies.
Sensor Temperature Coefficient
Sensor temperature coefficient is the cause
of the error that occurs when the sensor's temperature has
changed significantly from the temperature at which the sensor
was last autocalibrated. An example of the maximum uncertainty
due to temperature drift from the autocal temperature is
Temperature Error = ±0.04dB (0.93%) + 0.003dB
(0.069%)/°C
The first term of this equation is constant
while the second term must be multiplied by the number of
degrees that the sensor temperature has drifted from the autocal
temperature. Sensor temperature drift uncertainty may be assumed
to be zero for sensors operating exactly at the calibration
temperature.
Sensor Noise and Zero Drift
The noise contribution to pulse measurements
depends on the number of samples averaged to produce the power
reading, which is set by the averaging menu setting in the peak
power meter. In general, increasing filtering or averaging
reduces measurement noise.
Sensor noise typically is expressed as an
absolute power level. The uncertainty due to noise depends upon
the ratio of the noise to the signal power being measured. The
following expression is used to calculate uncertainty due to
noise
Noise Error = ±Sensor Noise (W)/Signal Power (W) × 100%
Noise error usually is insignificant when
measuring at high levels of 25 dB or more above the sensor's
minimum power rating.
Zero drift is the long-term change in the
zero-power reading that is not a random noise component.
Increasing filtering or averaging will not reduce zero drift.
For low-level measurements, this can be controlled by zeroing
the meter just before performing the measurement.
Zero drift typically is expressed as an
absolute power level, and its error contribution may be
calculated with the formula
Zero Drift Error = ±Sensor Zero Drift (W)/Signal Power (W) × 100 %
Zero drift error usually is insignificant
when measuring at high levels of 25 dB or more above the
sensor's minimum power rating.
Sensor Calibration Factor Uncertainty
Sensor frequency calibration factors (calfactors)
are used to correct sensor frequency response deviations. These
calfactors are characterized during factory calibration of each
sensor by measuring its output at a series of test frequencies
spanning its full operating range and storing the ratio of the
actual applied power to the measured power at each frequency.
During measurement operation, the power reading is multiplied by
the calfactor for the current measurement frequency to correct
the reading for a flat response.
The sensor calfactor uncertainty is due to
uncertainties encountered while performing this frequency
calibration, which includes both standards uncertainty and
measurement uncertainty, and is different for each frequency.
Both worst-case and RSS uncertainties typically are provided for
the frequency range covered by each sensor.
Refining the Uncertainty Contribution Model
Before the RSS calculation is performed, the
worst-case uncertainty values can be scaled or normalized to
adjust for differences in each term's probability distribution
or shape. The distribution shape is a statistical description of
how the actual error values are likely to vary from the ideal
value. Once normalized in this way, terms with different
distribution shapes can be combined freely using the RSS method.
Three distributions with different K
multilpliers are normal, rectangular, and U-shaped.
- Normal = 0.500; a distribution where the
variable becomes increasingly more frequent at intermediate
values.
- Rectangular or
0.577; a uniform
distribution in which each value of the variable occurs equally
often.
- U-shaped or
0.707; a distribution that
reveals polarization of the variable (typically very high or
very low).
The formula for calculating RSS measurement
uncertainty from worst-case values and scale factors is
URSS = [(U1K1)2 + (U2K2)2
+ (U3K3)2 + (U4K4)2 + ... (UNKN)2]
0.5
where: U1 through UN are worst-case uncertainty terms
K1 through KN are normalizing multipliers for each term based on its distribution shape
Again, this calculation yields what is
commonly referred to as the combined standard uncertainty, or UC,
with a level of confidence of approximately 68%. To gain higher
levels of confidence, an expanded uncertainty often is used.
A coverage factor of two will provide expanded uncertainty U = 2UC with a
confidence level of approximately 95%.
Example Peak Power Measurement Calculation
Here are the eight steps you need to follow
to complete a peak power measurement calculation. The test
conditions are outlined in Table 1.
Table 1. Equipment and Conditions for Peak Power Calculation
Step 1. Instrument Uncertainty for the
Boonton 4500B is ±0.20%.
U1 = ±0.20%
Step 2. Calibrator Level Uncertainty for the 4500B internal 1-GHz calibrator may be calculated from
the specification. The 0-dBm uncertainty is 0.065 dB or 1.51%.
To this figure, we must add 0.03 dB or 0.69% per 5-dB step from
0 dBm. The 13-dBm source level is rounded to three steps away.
U2 = ±[1.51% + (3 × 0.69%)]
= ±3.11%
Step 3. Calibrator Mismatch Uncertainty
is obtained using the published figure for DCAL and
calculating the value DSNSR from the SWR specification on the
datasheet for the Boonton 56518 sensor.
DCAL = 0.091 (internal 1-GHz calibrator's
reflection coefficient)
DSNSR = (1.15 - 1)/(1.15 + 1) = 0.070
(calculate reflection coefficient of 56518,
max SWR = 1.15 at 1
GHz)
U3 = ±2 × DCAL × DSNSR × 100%
= ±2 × 0.091 × 0.070 × 100%
= ±1.27%
Step 4. Source Mismatch Uncertainty is determined using the DUT's specification for DSRCE and
calculating the value DSNSR from the SWR specification on the
56518's datasheet.
DSRCE = 0.057 (source reflection coefficient
at 900 MHz)
DSNSR = (1.15 - 1)/(1.15 + 1) = 0.070
(calculate reflection coefficient of 56518,
max SWR = 1.15 at
0.9 GHz)
U4 = ±2 × DSRCE × DSNSR × 100%
= ±2 × 0.057 × 0.070 × 100%
= ±0.80%
Step 5. Sensor Shaping Error for a
peak sensor is 2% at all levels because the test frequency of
900 MHz is very close to the autocal frequency of 1 GHz.
U5 = ±2.0%
Step 6. Sensor Temperature Drift Error depends on how far the temperature has drifted from the
sensor calibration temperature and the temperature coefficient
of the sensor. In this example, the temperature has drifted by
11°C (49°C to 38°C) from the autocal temperature.
U6 = ±(0.93% + 0.069%/°C)
= [±0.93 + (0.069 × 11.0)]%
= ±1.69%
Step 7. Sensor Noise and Drift
contribution is a concern at low signal levels. The
signal level is 13 dBm or 20 mW. The noise-and-drift
specification for the sensor is 50 nW according to the sensor's
datasheet. Noise uncertainty is the ratio of these two figures.
U7 = ±Sensor Noise (W)/Signal (W)
= ±50.0-9/20.0-3 × 100%
= ±0.0003%
Sensor Zero Drift is combined in the
noise-and-drift specification
U7 = 0.00
Step 8. Sensor Calfactor Uncertainty
needs to be interpolated from the published uncertainty values
for the sensor of 1.99% at the 0.5-GHz sensor calibration point
and 0.00% at the 1-GHz sensor calibration point. The uncertainty
figure difference between 0.5 GHz and 1 GHz can be scaled by
one-fifth given that the 900-MHz test frequency falls closest to
1 GHz.
U8 = (1.99 - 0.00) × [(900 - 1,000)/(500 - 1,000)]
= 1.99 × 0.2 = ±0.40%
From this example, different error terms
dominate. Since the measurement is close to the calibration
frequency and matching is rather good, the shaping and level
errors are the largest. Expanded uncertainty of 5.16% translates
to an uncertainty of about 0.22 dB in the reading (Table 2).
Table 2. Typical Power Measurement Accuracy Calculation
For More Information
Measurement uncertainty calculation is a very
complex process, and the techniques shown here are somewhat
simplified to allow easier calculation. For more complete
information, consult these publications:
1. ISO Guide to the Expression of
Uncertainty in Measurement, 1995.
2. ANSI/NCSL Z540-2-1996, National
Conference of Standards Laboratories, Boulder, CO.
Conclusion
Understanding how different factors
contribute to the overall measurement accuracy of RF peak power
measurements is essential in optimizing the design and test of
many RF components and systems. Both test instrumentation and
test conditions are the factors to consider when calculating the
accuracy of those RF power measurements.
About the Author
Richard Theiss is a product manager and
senior applications engineer at Boonton Electronics with 20
years experience marketing and supporting high-performance test
instrumentation. Before joining the company in 2001, he was a
product manager for LeCroy's digital oscilloscope and PXI
digitizer groups. Mr. Theiss holds a B.S. in electrical
engineering from Boston University and an M.B.A. from Iona
College Hagan School of Business and has completed graduate
course work in Telecommunication Networks at Polytechnic
University. Boonton Electronics, a Wireless Telecom Group Co.,
25 Eastmans Rd., Parsippany, NJ 07054, 973-386-9696, e-mail:
rtheiss@boonton.com