Calculating accuracy of peak power measurements.

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 TDSCDMA. 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

[U.sub.C] = ([U1.sup.2] + [U2.sup.2] + [U3.sup.2] + [U4.sup.2] + ... [UN.sup.2])[.sup.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 x [U.sub.C] 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 = [+ or -]2 x DCAL x DSNSR x 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 = [+ or -]2 x DSRCE x DSNSR x 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 = [+ or -]0.04dB (0.93%) + 0.003dB (0.069%)/[degrees]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 = [+ or -]Sensor Noise (W)/Signal Power (W) x 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 = [+ or -]Sensor Zero Drift (W)/Signal Power (W) x 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 [square root of 1/3] = 0.577; a uniform distribution in which each value of the variable occurs equally often.

* U-shaped or [square root of 1/2] = 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

[U.sub.RSS] = [(U1K1)[.sup.2] + (U2K2)[.sup.2] + (U3K3)[.sup.2] + (U4K4)[.sup.2] + ... (UNKN)[.sup.2]][.sup.0.5]