Friday, November 4, 2011

How to do an Accuracy Calculation on Agilent's 53200 Series Counters

Calculating and understanding the accuracy of Agilent’s 53200A series of universal counters is not easy and can be a source of frustration. In this post I will walk you through an example frequency accuracy calculation using Agilent’s 53230A universal counter. Each step will include an explanation of the various terms so in the future the reader can perform an accuracy calculation on any of the 53200A series of counters with any input signal. I will also refer to data sheet pages where calculations, specs, and notes can be obtained. Let’s start by setting the initial conditions of the signal we are measuring and the counter we are using.

Signal being measured: 1 MHz sine wave with 1 Vpp amplitude and 1 mV of RMS noise

Counter hardware and settings: 53230A with TCXO timebase set for frequency measurements in ‘Auto’ mode (default) with a 1 s gate time. The input amplitude range is 5 V with a trigger level of 0 V on a positive edge event. The counter was calibrated at the factory upon purchase and it has been ~90 days since the initial 30 day warm-up period (page 15 note 1 of data sheet).

The basic accuracy calculation is as follows (page 16 datasheet):
Accuracy = +/- [(k*Random Uncertainty) + Systematic Uncertainty + Timebase Uncertainty]

For a definition of random, systematic, and timebase uncertainty refer to page 16 of the data sheet.
Let’s start by calculating the random uncertainty (RU) error which is the hardest of the three. The variable ‘k’ is the standard deviation or sigma multiplier for establishing the confidence interval of a Gaussian or normal distribution, RU has a Gaussian distribution. For our calculation we will choose a ‘k’ of 3 which gives us a confidence interval of 99.7%. The equation for calculating the RU is (data sheet page 16):
RU = [1.4*(TSS^2 + TE^2)^1/2] / RE*gate

TSS --> Single-shot time resolution spec found on page 19 of data sheet. For the 53230A this spec is 20 ps.

TE --> Threshold error (page 19) is the amplitude noise on the signal being measured that causes error at the trigger level point on the signal edge. The equation to calculate TE for 5 V input range is: (500μV^2 + EN^2 + VX^2)^1/2 / SR
  •  The ‘500μV’ term is amplitude noise added by the counter
  • The ‘EN’ term is the amplitude noise on the signal being measured (page 20). In our initial conditions we said this was 1 mV. EN is sometimes hard to calculate so if you are working with a clean signal it is safe to assume EN is zero.
  • The ‘SR’ is the slew rate of the signal at the counter’s set trigger point. The slew rate or the rate of change at any point on a given waveform is found by taking the derivative of the waveform function and then solving for the time when the set trigger amplitude occurs. Fortunately on page 20 the data sheet provides an SR formula for a sine wave and a square wave. The formulas are the result of solving for the derivative of each waveform and using the point of max SR as the trigger point. Choosing a trigger point on the waveform that has the highest slew rate leads to better measurement accuracy. In our example we have a sine wave which has max SR at the 50% amplitude point. A sine wave with no DC offset will then have its max slew rate at the 0V crossing and that is why we choose 0 V trigger point for our initial conditions. So we can use data sheet calculation for our example: SR = 2*pi*F*V0 to p
    • ‘F’ is the frequency of the signal being measured.
    • ‘V0 to pk’ is the delta amplitude from 0 to peak. For our signal this is 0.5 V (half of 1 Vpp).

RE --> When the 53230A and 53220A are in ‘Recip’ mode they return the average of all the measurements within the set gate time as the measurement result. When they are in ‘Auto’ mode, which is the default mode, they use a proprietary algorithm to achieve better resolution from a set of measurements in a specific gate time compared to just simply averaging the measurements. ‘RE’ stands for Resolution Enhancement. When in ‘Auto’ mode the RE factor can be more than 1 because the resolution enhancement algorithm is being implemented. When using the 53210A, 53220A, and the 53230A in ‘Recip’ or ‘TST’ modes RE = 1.   
Let’s calculate SR, TE, and RE
SR = 2*pi*F*V0 to pk = 2*3.1416*1e6*0.5 = 3.1416e6
TE = (500μV^2 + EN^2 + VX^2)^1/2 / SR = ((500e-6^2 + 1e-3^2)^1/2) / 3.1416e6 = 3.5588e-10

The data sheet on page 19 gives clear instructions on what to do for RE when Tss >> TE, we simply use the following equation RE = √(FIN * Gate/16) and check it against the max value RE table shown below:
Gate time >= 1 s, RE max of 6
Gate time 100 ms, RE max of 4
Gate time 10 ms, RE max of 2
Gate time =< 1 ms, RE = 1

If your gate time falls somewhere in between the gate time values in the table just use an RE value that falls between the table values, for instance if your gate time is 600 ms use an RE value of 5. Now here is where things get confusing, since the data sheet does not really provide much guidance on what to do for cases when Tss > TE, Tss < TE, and Tss << TE (which is the case we have for our example). The safe thing to do here is to just use the same procedure that was used for Tss >> TE. Since the resolution enhancement algorithm results in increased resolution the higher TE is, using the above calculation method ensures a safe result every time. Hopefully a future version of the data sheet will provide a better guidance for calculating RE. Let’s calculate the RE value for our example measurement:

 RE = √(FIN * Gate/16) = √(1e6 * 1/16) = 250, since the result of the equation is higher than the table value we use the table value so RE = 6

Now we have the variables we need to determine the random uncertainty in our accuracy calculation.
RU = [1.4*(TSS^2 + TE^2)^1/2] / RE*gate = [1.4*(20e-12^2 + 3.5588e-10^2)^1/2] / 6*1
RU = 8.3170e-011

Notice above, since TSS and TE are random error components we apply the root sum of squares (RSS) method to them. Also since they are random the higher the gate time the lower the random uncertainty is due to more averaging.
From page 16 of the data sheet the SU is determined as follows:
If RE ≥ 2: 10 ps / gate (max), 2 ps / gate (typ)
If RE < 2 or REC mode (RE = 1): 100 ps / gate

We can see SU is based off of the gate time and the calculated RE value. For each range of RE values there are two different ways to calculate SU, ‘max’ or maximum and ‘typ’ or typical. The typical spec is the performance of most of the units tested. The maximum is the warranty spec of the counter. For our calculation we will use the maximum:
SU = 10 ps / gate = 10e-12 / 1 = 10e-12

The last term we need to calculate is the Timebase Uncertainty (TU). On page 15 you will find the TU equation, the various time base specs, and four important notes. Understanding time base specs can be a little tricky. It is important to note that crystal oscillators are as much a mechanical device as they are an electrical device. That is why factors such as temperature, loss of power, and movement have such a dramatic effect on time base uncertainty. The TU calculation is:
TU = ( Aging + Temperature + Calibration Uncertainty )

The following is an overview of each of the three parts that make up the TU calculation:
Aging --> The 53200A series of counters must be on for 30 days after you receive it from the factory before the time base aging specs take effect. This is a settling time that is required by the physics of the crystal oscillator. The aging specs apply from the date of the counter’s last calibration. Since the 53230A for this example has been around 90 days after the initial settling time and we have the TCXO timebase our aging spec is +/- 0.6 ppm. This was calculated by multiplying the 30-day aging spec by 3 (90 days). Use the 30-day aging spec times the number of months since calibration for about 5 months at which time you will want to switch to the 1-year aging spec since it will be the lesser of the two. Notice the second half of note one on page 15, after the first year you use half of the 1-year and 30-day aging specs.

      Temperature --> the first spec “0 °C to 55 °C relative to 25 °C” is used if the counter was operated at temperatures more than +/- 5 °C from the temperature it was calibrated at (25 °C ideally). Since we are not sure what temperatures our example counter experienced during shipping we will use this in our calculation. The TCAL spec can be ignored for our accuracy calculation since it is included in the aging spec.

           Calibration Uncertainty --> This term is only used if the counter still has the factory calibration. It was added because after the counter is shipped from the factory we do not know what kind of handling the counter may experience during shipping. For instance the package may be dropped which can affect the calibration. Once the counter has been calibrated again this term can be ignored. Since in our example the factory was the last place the 53230A was calibrated we will use this spec. The 53200A series of counters should be calibrated onsite for best performance. If you send the 53200A counter off site for calibration you should add the “Initial factory calibration” spec into any accuracy calculation.
      We can now calculate the TU:

      TU = (Aging + Temp + CU) = 0.6 ppm + 1 ppm + 0.5 ppm = 6e-7 + 1e-6 + 5e-7 = 2.1 e-6
      Finally we now have all the information and data needed to calculate the basic accuracy of our example 1 MHz signal:

Accuracy = +/- [(k*RU) + SU + TU] = (3 * 8.317e-011) + 10e-12 + 2.1e-6 = 2.10025951e-6 or in parts and rounded 2.1003 ppm

Since the error calculation is in parts the result 2.10026 ppm means that if our signal was exactly 1 MHz the counter would output a reading between 1,000,002.10026  and  999,997.899740 as the measured frequency.

Important Accuracy Notes:
Notice in for our example that the timebase error is multiple orders of magnitude higher than the other error components. Even with the OCXO option this will still be true. That means in the future, if you are not using a highly stable external time base reference, you can just use the TU as a very close approximation of the counter’s measurement accuracy.

The accuracy of any of the 53200A counters can be increased by multiple orders of magnitude by using a rubidium or GPS based external frequency standard.

Better measurement accuracy is achieved by calibrating the 53200A series counter onsite after the initial 30 day warm-up / settling period and all future calibrations are carried out onsite.

Triggering at the amplitude point of maximum slew rate on the signal you are measuring provides better measurement accuracy.