Parent Category: 2015 HFE

*By Brian Avenell*

**Introduction**

The Y-factor technique is a popular and mature noise figure measurement method. A significant body of literature exists on the mechanics of making the Y-factor measurement. Not as well covered is the calculation of measurement uncertainty. In some of the literature, simplifications are made that may not necessarily apply to all measurement situations. At the other extreme, other literature discusses measurement uncertainty in a highly theoretical sense - covering a narrow portion of the overall uncertainty calculation that can leave the user unsure how to apply to the measurement under consideration.

Like any measurement, the accuracy of the noise figure result is essential. Noise figure measurement is especially prone to uncertainty due to its dependence on DUT gain, DUT noise figure value and the care taken in how the measurement is made. The derivation of noise figure measurement uncertainty is not a simple undertaking and for this reason, it is often a misunderstood topic.

One approach for the calculation of noise figure measurement uncertainty stems from Yield analysis or Monte Carlo analysis techniques that are used in RF circuit simulation software. In Yield analysis, all the parameters that comprise the final outcome are given a statistical probability distribution. A number of simulations are computed using a random selection of the parameter values. The final outcome of these simulations is itself randomly distributed. Based on these simulations, the mean of the final outcome’s distribution function gives the error and the resulting standard deviation is the measurement the uncertainty.

**The Y-Factor Technique in Brief**

Average noise power is given according to

N = kTB [Watts](1)

where k is Boltzmann’s constant, T is noise temperature, and B is the measurement bandwidth. Expressed in logarithmic units, this is the familiar -174 dBm/Hz. Equation (1) forms the basis of the Y-factor noise figure measurement whose setup is shown in Figure 1.

The Y-factor measurement procedure uses a calibrated noise source which outputs two noise levels, N_{icold} and N_{ihot}, corresponding to the noise generation diode being in its off (cold) and on (hot) states respectively. These two noise power levels are represented by the T_{C} and T_{H} cold and hot noise temperatures inside the noise source depicted in Figure 1. The noise source manufacturer specifies an excess noise ratio (ENR) from which the cold and hot noise temperatures can be derived^{[1]}. From equation (1), noise power is directly proportional to the noise temperature. The hot and cold noise temperatures are unit dependent and frequency dependent.

**Figure 1 • Y-factor measurement setup.**

Noise factor, F, is the ratio of input signal-to-noise to output signal-to-noise of the DUT. Noise figure is noise factor expressed in dB: Noise figure = 10*log(F). The more noise that the DUT adds, the higher the noise factor and noise figure values. The device added noise is given by^{[1]}:

(2)

where T_{o} is a standard temperature of 290 kelvin and F is the noise factor. Expanding equation (1), the noise power at the output of a DUT is given by:

(3)

where N_{i} is the noise power at the input of the DUT and G is the gain of the DUT.

The Y-factor measurement consists of four noise power measurements. In the first two measurements, the noise source is connected directly to the measurement receiver. The first measurement is made with the noise source cold and the second measurement is made with the noise source hot. This is considered the ‘Calibration’ phase – and it effectively characterizes the noise figure of the measurement receiver. For the next set of measurements, the DUT is connected between the noise source and measurement receiver. Subsequently, cold and hot noise measurements are made by the measurement receiver. This is considered the ‘Measurement’ phase.

The factor, Y, is expressed as the ratio of the measured hot noise power to the measured cold noise power. Y is given by^{[1]}:

(4)

The noise factor can then be extracted:

(5)

Equation (5) is applied for both the ‘Calibration’ and ‘Measurement’ steps. Manipulating Friis’ cascade noise figure equation results in the DUT’s noise factor, F_{DUT}:

(6)

The DUT Gain, G_{DUT},, can be extracted from the four noise power measurements used in the Y-factor measurement^{[1]}.

**Sources of Measurement Uncertainty**

The mechanics of calculating DUT noise figure using the Y-factor technique does not give any indication of the accuracy of the measured result. Noise figure measurements are prone to many sources of measurement uncertainty. Some of the larger sources of measurement uncertainty follows.

**Figure 2 • Signal Reflections Due to Source/Load Impedance Mismatch.**

**Impedance Mismatch**

Impedance mismatch itself leads to one of the larger sources of errors. However, when coupled with some of the other uncertainty sources listed below, impedance mismatch exacerbates those uncertainties as well. From transmission line theory, the voltage waveform consists of the combination of incident and reflected parts. For systems where the source and load impedances match, the reflected portion is zero and the entire incident signal is transmitted to the load. However, as depicted in Figure 2a, when there is a source and load impedance mismatch, the non-zero reflected portion of the returns back to the load.

The reflected signal then re-reflects back to the load at the source plane. The transmitted signal, as indicated in Figure 2b, is the vector combination of many of these re-reflected signals. When the impedances are characterized by their reflection coefficients, the resulting power transmitted to the load is:

(7)

The numerator in equation (7) is a power loss term. The denominator, however, represents the uncertainty in the power delivered to the load. The source and load reflection coefficients are vector terms whose phases are unknown in most circumstances.

In the Y-factor measurement technique, there are three impedance mismatch planes as shown in Figure 3.

Figure 3a shows noise source/measurement receiver mismatch during the calibration process and Figure 3B shows noise source/DUT input and DUT output/measurement receiver mismatch. For discrete transistor characterization, the mismatch problem is especially acute. In these cases, adding impedance transformer networks at the DUT ports is an important consideration

**Figure 3 • Impedance Mismatch Effects in the Y-Factor Measurement.**

Adding fixed attenuators at the DUT ports is another method of improving mismatch uncertainty. However, an input attenuator effectively degrades DUT noise figure and an output attenuator effectively degrades the receiver noise figure. As demonstrated below, both of these can adversely affect measurement uncertainty.

**Receiver Noise Figure**

Receiver noise figure as shown in Figure 4 can have a dramatic effect on measurement uncertainty.

In Figure 4a, the noise figure uncertainty is plotted versus DUT gain. Each trace represents a DUT noise figure value. In Figure 4b, each trace represents a measurement receiver noise figure value. In both cases, uncertainty increases for lower DUT gain and for higher DUT and measurement receiver noise figure.

**Figure 4 • Noise Figure Uncertainty vs. DUT Noise Figure and vs. Receiver Noise Figure.**

**Receiver Noise Measurement Variation**

The measurement of noise requires averaging to reduce the measurement variance. Figure 5 demonstrates the measurement receiver’s measured noise with two levels of trace averaging.

**Figure 5 • Noise Measurement Variation vs. Trace Averaging.**

The downside of trace averaging is that it results in longer measurement time. However, given that there are four noise measurements associated with the Y-factor technique, reducing noise variance by means of trace averaging is highly recommended in order to reduce overall noise figure measurement uncertainty.

Choosing the right combination of trace averaging and measurement time is a delicate art. The benefit of noise figure measurement utilities from instrument manufacturers is that this tradeoff has been optimized for the particular instrument.

**DUT Optimum Noise Match**

In some cases, gain block amplifiers and especially discrete transistor devices have noise figure values that depend on the impedance match presented to the input of the device. However, the calibrated noise source most often has a nominal impedance of 50 Ohms. A lossless impedance tuner, either with discrete components or a commercially available adjustable version, placed between the noise source and the DUT input port allows for optimum DUT noise match.

**Other Considerations**

Finally, the noise source ENR value will have its own measurement uncertainty. The noise source manufacturer or the calibration lab used to determine the ENR values will need to specify the ENR uncertainty. ENR temperature dependence should be accounted for. ENR values are defined for a temperature of 290 K (16.85 deg C). When the physical temperature is not 290 K, an adjustment must be made.

Spurious responses either generated from the DUT, the measurement receiver, or coupled in from the environment must also be considered. If the noise measurement is being made at a frequency where a spurious response resides, misleading results in the noise figure measurement will occur. First identifying whether a spurious response is at the measurement frequency should take place. A slight offset in the measurement frequency to move away from the spurious response is then needed. Shielding the DUT with a metal enclosure can further reduce the spurious signals from the environment coupling into the measurement setup.

**Measurement Uncertainty Calculation Procedure**

The measurement uncertainty calculation procedure used to generate the graphs in Figure 4 is based on a Monte Carlo analysis. In Monte Carlo, parameters used to compute an outcome are randomly distributed according to each parameter’s probability distribution function. The results of many simulations are in turn randomly distributed. From this result mean and standard deviation will yield the measurement error and measurement uncertainty.

For the noise figure measurement uncertainty, the Y-factor equations outlined above are used in the computation. Factors including DUT gain, DUT input and output reflection coefficients, DUT noise figure, measurement receiver noise figure, measurement receiver input reflection coefficient, noise measurement trace variation, noise source reflection coefficient, ENR uncertainty, and so on, are all given a probability distribution. Phases are given a uniform distribution and all other parameters are given a normal distribution.

The result for one simulation at a fixed DUT gain and DUT noise figure value is shown in Figure 6.

**Figure 6 • Noise Figure Measurement Histogram.**

The resulting histogram from 1000 Monte Carlo simulations shows an approximately normal probability distribution. From this the mean and standard deviation can be computed. Measurement uncertainty for the graphs in Figure 4 is computed as the mean error plus two standard deviations – which is equivalent to a 95% confidence level that the measured noise figure value is correct.

NI has a utility to compute the noise figure measurement uncertainty based on the Monte Carlo technique. Figure 7 shows the front panel of this measurement utility.

**Figure 7 • NI Noise Figure Uncertainty Calculator.**

Observe in Figure 7 that the user enters DUT and measurement system parameters. Under the Advanced Settings tab, the user can select which parameters are used in the Monte Carlo simulation. This feature allows the user to gauge the sensitivity of the individual parameters on the overall noise figure measurement uncertainty.

**Summary**

Any measurement is prone to a certain level of uncertainty in the resulting value. Noise figure uncertainty is especially complicated in that the individual parameters have inter-dependencies and the DUT gain and noise figure is a strong function of the end result. NI has provided a tool to ease the burden of the noise figure measurement uncertainty calculation.

**References**

[1] “Noise Figure Measurements: Theory and Application,” , National Instruments Application Note, http://www.ni.com/rf-academy//

**About the Author**

Brian Avenell is a RF/Microwave design engineer at NI. Brian holds a MSEE from the University of California, Santa Barbara. System and circuit level design work associated with test and measurement receivers and sources comprise Brian’s experience.

### March 2021

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