# How to Calculate Hysteresis for Your Uncertainty Budget

## Introduction

Hysteresis is a source of uncertainty that affects many types of measurement equipment and their associated measurement results. With a quick search on the internet, you will find a lot studies dedicated to hysteresis. If you take a look at manufacturer manuals and datasheets, you may find specifications related to hysteresis. However, you won’t see hysteresis commonly included in laboratory’s uncertainty budgets.

When I read many of the studies available, most of them were related to electrical hysteresis and magnetism. It wasn’t until searched for mechanical or elastic hysteresis that I begin to find the information beneficial to most calibration and testing laboratories.

However, you will not find many papers specifically studying the effects on measurement devices.

Therefore, I decided to create a guide on hysteresis for calibration and testing labs, so you could consider it in your uncertainty budgets (if it is applicable).

In this guide, you will learn the following information about hysteresis:

1. What is hysteresis
2. Why is hysteresis important
3. When should you include hysteresis
4. How to reduce hysteresis
5. Formulas to calculate hysteresis
6. How to perform a hysteresis test
7. How to calculate hysteresis

## What is Hysteresis

According to the Oxford English Dictionary, hysteresis is the phenomenon where the value of a physical property lags behind changes in the effect causing it.

According to NASA, hysteresis is the response of a structure to loading and unloading that is commonly associated with energy loss under load cycling and hence damping with the structure.

This is the definition that I like the most.

For you, it means that your measurement result when increasing load, force, pressure, etc is not the same as your result when decreasing load, force, pressure, etc. If you are not taken this into consideration when performing tests or calibrations, then you will add additional uncertainty to your results.

If this is occurring with your test and measurement equipment and affecting your measurement results, you may want to consider hysteresis in your uncertainty budgets.

## Why is Hysteresis Important

Hysteresis is important because it contributes to uncertainty in measurement results.

If you are not aware of the effects of hysteresis and how to reduce them, you could obtain erroneous results when performing measurements. Worse yet, you could be passing this error on to your customer’s measurement results without either one of you knowing about it.

In some cases, it may not be a big deal for you. However, when you incorporate risk and cost into the evaluation, your thoughts may change.

Try assigning a cost to your errors. If your measurement errors resulted in a monetary loss (associated with failures, repairs, downtime, or nonconforming work), would you think differently about hysteresis?

Would your perspective change if your measurement errors affected human health or life?

While this is not typically the case, you should be aware of the risks and consequences of measurement errors and the negative impacts they can cause. This is why hysteresis is important.

In case you are curious, just do a Google search on hysteresis. You will be amazed by how many studies have been published on the topic.

## When Should You Include Hysteresis

Consider hysteresis in your uncertainty analysis when it is commonly known to affect your type of measurement system or have an impact on your measurement results.

Hysteresis can occur in various types of measurement equipment and systems, including:

• Electrical measurement equipment,
• Mechanical measurement equipment, and
• Thermodynamic measurement equipment.

Review your equipment’s manufacturer manuals and specifications to find out whether or not hysteresis is thought to have an effect on your measurement results. If it is not specified by the manufacturer, you may find more information by reading your test methods or calibration procedures.

If hysteresis is mentioned (and given its own specification), it is most likely significant and should be included in your uncertainty analysis. If not, you may be able to omit it from your uncertainty budget.

Just make sure that you do some research before deciding whether or not to include it in your uncertainty analysis.

## How to Reduce Hysteresis

Hysteresis is unavoidable, but it’s effects can be minimized.

To reduce the effects of hysteresis, it is common practice to exercise your equipment before use or calibration. This means that you should load and unload your equipment several times before you use it.

Most manufacture manuals, calibration procedures, and test methods recommend this practice in an effort to reduce the effects of hysteresis.

Take a look at the excerpts below. I have compiled statements from several manuals, methods, and procedures that recommend exercising equipment prior to use.

#### NAVAIR 17-51MP-006: Hydraulic and Pneumatic Pressure Gauges and Switches

If you are using equipment that may be affected by hysteresis, you can minimize the effect by exercising the equipment before use or calibration. Following the statements above, it appears that it is best practice to exercise equipment for a minimum of 3 cycles (loading and unloading) before it is used.

## Formulas to Calculate Hysteresis

When you need calculate hysteresis for your uncertainty budgets, there are two methods that you can use.

### Method 1: Maximum Difference Between Upscale and Downscale

This method is the most common. You will find it published in many documents, including:

This equation takes into account the differences between loading and unloading across the entire measurement range. The benefit to this method is that you will find the maximum hysteresis error no matter where it occurs in the range.

However, the problem with this method is you may confound bias and linearity which will cause you to overstate your estimated uncertainty.

### Method 2: Difference Between Upscale and Downscale at Midscale and Zero.

This method is less common than the first method. I found it in the MSL Technical Guide 25: Calibrating Balances several years ago and decided to give it a try.

Today, MSL has replaced this formula in their Technical Guide 25, but I still prefer to use it sometimes. I like the method because it takes changes at zero into account.

However, it only evaluates hysteresis at midscale. This can be a problem since the maximum difference does not always occur at midscale. Make sure that you take that into account when using this equation.

## How to Perform a Hysteresis Test

Performing a hysteresis test is not very difficult as long as you have the equipment. Just follow the instructions below.

Otherwise, you may already have the data in one of your calibration reports. If so, there is no need to perform this test. You can proceed to analyzing data.

### Method 1: Maximum Difference Between Upscale and Downscale

1. Select an item for hysteresis testing (e.g. scale, pressure gauge, etc.)
2. Select equipment for making comparisons.
3. Exercise equipment as necessary before testing.
4. With no load applied, zero the unit under test and record the result.
5. Incrementally increase the load (e.g. 10% increments) on the unit under test and record the result after each step.
6. Incrementally decrease the load (e.g. 10% increments) on the unit under test in the reverse order and record the result after each step.
7. Analyze the results.

### Method 2: Difference Between Upscale and Downscale at Midscale and Zero.

1. Select an item for hysteresis testing (e.g. scale, pressure gauge, etc.)
2. Select equipment for making comparisons.
3. Exercise equipment as necessary before testing.
4. With no load applied, zero the unit under test and record the result (yZ1).
5. Load the unit under test to 50% of full-scale and record the result (yH1).
6. Load the unit under test to 100% of full-scale.
7. Reduce the load on the unit under test to 50% of full-scale and record the result (yH2).
8. Reduce the load to zero and record the result (yZ2).
9. Analyze the results.

## How to Calculate Hysteresis

If you have measurement results from in-house testing or a calibration report, you are ready to calculate hysteresis. Just select the method that you would like to use for analysis and follow the instructions below.

### Method 1: Maximum Difference Between Upscale and Downscale

#### 1. Create a table.

Create a table in Microsoft Excel similar to the one in the image below.

#### 2. Enter your upscale results.

Enter the results of your upscale test in the upscale column.

#### 3. Enter your downscale results

Enter the results of your downscale test in the downscale column.

## Conclusion

Hysteresis is a source of uncertainty that is considered to be important by many but not included in most uncertainty budgets. Many methods and procedures give instructions to help reduce hysteresis, but you most likely will not be able to eliminate it. Therefore, you should give it consideration in your uncertainty analysis and include it in your uncertainty budget if it is applicable.

This guide has given you a lot of information about hysteresis; what is it, why is it important, and when to include it in your uncertainty budget. Additionally, you should have learned how to perform a hysteresis test and how to analyze the results for uncertainty analysis.

With this information, you should be able to easily calculate hysteresis and add it in your uncertainty budgets.