Mike Howell
EASA Technical Support Specialist

Locked-rotor testing of three-phase squirrel cage induction motors is used for design validation and qual­ity control; it also can be a valuable diagnostic tool. But, this testing isn’t a common task for most service centers. Two challenges service centers often face are dynamometer torque capac­ity and test panel electrical capacity. The work-around is usually reduced-voltage testing, which presents another challenge – how to extrapolate the test data to rated voltage with reasonable assurance of accuracy. If the extrapo­lation is too far off, we run the risk of either rejecting a good motor or accept­ing a bad one. 

The purpose of this article isn’t to provide detailed procedures for per­forming locked-rotor tests, but rather to present a practical approach for analyzing the reduced-voltage data us­ing tools that most service centers have access to at their facilities. Additionally, while this article will focus on locked-rotor test data, the methodology used can certainly be extended to other tests where similar conditions and relation­ships exist.

Background
It is typical for a reference text to state that the locked-rotor current var­ies directly with the voltage and that the locked-rotor torque varies directly with the square of the voltage. This is a reasonable approach for explaining the general concepts. However, for the purpose of extrapolating reduced-voltage data to rated voltage, this simplistic model is far too inaccurate for many machines, resulting in cal­culated locked-rotor current up to 25% lower than actual at rated voltage and locked-rotor torque up to 60% lower than actual at rated voltage [1].

One test procedure used for locked-rotor testing is included in IEEE Std. 112-2004. With regards to reduced-voltage testing, it states that a more exact method than the general model described above should be used for extrapolation and that the data should be plotted on log-log paper and cor­rected to rated voltage using a least square curve fit for maximum accuracy [2]. This is where service centers often run into trouble – they gather good reduced-voltage test data but struggle with the data analysis and sometimes draw the wrong conclusion accord­ingly.

Procedure
For this procedure to provide mean­ingful results, a locked-rotor test must be performed at a minimum of three different voltages. This is because we will use a statistic to indicate how well our data fits the model and since any two points define a straight line, we need at least three points to see how well our model will work. Also, as with any test, the closer the test voltage is to rated voltage, the better off we’ll be.

For the data analysis, we will use Microsoft Excel due to its prevalence, but most spreadsheet applications have very similar capabilities and worksheet setup would be similar. The parrot method will be used – that is, inexperienced users should be able to repeat what’s shown here and get a working spreadsheet. By inputting the sample data provided in the article, functionality can be verified. If more introductory info is needed for Micro­soft Excel, many free tutorials can be found on YouTube or other locations on the web.

Microsoft Excel worksheets are di­vided into rows and columns, similar to a table or grid. Columns are identi­fied by a letter from left to right (A, B, C…). Rows are identified by a number from top to bottom (1, 2, 3…). A cell is a specific location within the worksheet and identified by its column and row. For example, the upper most left cell would be referred to as A1. Cells can contain different types of input data or they can contain the result of a calculation.

Figure 1 shows one setup that will work for our analysis. In this setup, row 1 is used for header information so text has been entered in A1, B1 and C1 describing what will be input be­low (voltage, current and torque). Our locked-rotor test data will be input for each of the three tests into (A2,B2,C2), (A3,B3,C3) and (A4,B4,C4). In A5, we’ll input our rated voltage or the voltage we want to extrapolate our data to. In B6 and C6, we’ll enter functions that will use the test data entered to calculate our extrapolated locked-rotor current and locked-rotor torque at the rated voltage we entered. Note, in calculated cells, the formula entered must begin with “=” as this tells the application that you intend to perform a calculation. And, values used in the calculations are referred to by their respective cell names. In B7 and C7, we’ll enter the manufacturer’s published locked-rotor current and locked-rotor torque for the machine at rated voltage. In B8 and C8, we’ll cal­culate the difference between our calcu­lated values and the machine’s published values. In B9 and C9 we’ll calculate the coefficient of deter­mination (R²). The R² statistic is used to tell us how well our data fits our model. For our purposes, if R² is 0.990 or higher, the model is an accept­able fit. If R² is less than 0.990, repeated testing is recom­mended and increasing the number of test voltages to four or more may help. In B10 and C10, we’ll calculate the index or exponential relationship that defines how the locked-rotor cur­rent and locked-rotor torque vary with applied voltage. This index is simply the slope of the best-fit line.

Again, the sample test data used are entered in the beige-shaded cells in Figure 1. The machine rated values are entered in the blue-shaded cells and the calculated results are displayed in the green-shaded cells. The colors have no meaning other than clarification. Once the values shown in Figure 1 have been entered into the worksheet, the values shown in Figure 2 should be returned. Don’t become frustrated if you have problems at first – use web tutorials as previously men­tioned or feel free to contact the author for a template. We won’t cover topics like protecting cells here, but it’s important to note that if you type in a calculated cell after your formula is entered, the value you type will replace the formula. 

Now that the worksheet is set up, let’s discuss the sample test data and results. The sample data is for a 460 V rated induction motor with rated locked-rotor current and locked-rotor torque of 1880 A and 2130 ft-lb respectively. Three locked-rotor tests were performed at 155V, 175V and 210V. The linear regression model calculated values in B6 and C6 are both within 5% of the manufacturer’s published values and the R² values for torque and current both exceed 0.990. It’s reasonable to conclude in this case that the model is a good fit for the data and that the motor’s locked ro­tor torque and current are satisfactory when compared to the manufacturer’s specifications. If the current varied directly with the applied voltage, the index would be 1, but as shown in Figure 2, the index for this machine is 1.3377. Further, if the torque varied directly with the square of the voltage, the index would be 2, but as shown in Figure 2, the index for this machine is 2.8063.

Although we are using Microsoft Excel’s built-in functions to do our data analysis, Figure 3 is a log-log plot of the test data along with linear regression best fit lines and R² values provided here just to demonstrate graphically what was done. The equations and R² values are generated using Microsoft Excel’s trend line tool. The calculations used in our worksheet above give the same results – graphical representation is unnecessary. You can see that the ex­ponents shown in Figure 3 (1.3377 for current and 2.8063 for torque) are the same as calculated in our worksheet and are much higher than the general value of 1 for current and 2 for torque. If the general relationship of current varying directly with voltage and torque with the square of the voltage had been used, the extrapolated values would have been significantly lower than published, bringing the motor into question.

Need locked-rotor current only?
When looking to verify locked-rotor current only, it is possible to do so approximately without mechani­cally locking the rotor. If we consider a Y-equivalent circuit model for the three-phase winding, we know that the phase voltage will be equal to the line voltage divided by the square root of three. Likewise, if we apply single phase voltage to two line terminals, the phase voltage will be half of the applied voltage. For this reason, if we apply single-phase power to any two of the machine line terminals, the re­sulting current will be approximately 86% of what would be found by ap­plying three-phase power. This can be shown by:

So, if you perform this test and measure 100 A, the three-phase current would be approximately 100/0.86 = 116 A. If you don’t perform this test at rated voltage though, you need to apply the same procedure originally reviewed and do at least three reduced-voltage tests and extrapolate to rated voltage.

This test can also be used to es­timate locked rotor torque by using the input power and losses, but this is fairly difficult and beyond the scope of this article.

Why this works (optional reading)
First, you don’t need any of the information below to analyze your locked-rotor test data and extrapolate it to rated voltage. However, a general understand­ing of the principles involved can be use­ful. Due to the physical behavior of the induc­tion machine, it is well documented that power models can be used to accurately describe the relationship between applied voltage and the resulting locked-rotor current and locked-rotor torque. The model will typically be writ­ten as shown below where “y” is torque or current, x is applied voltage and “B” and “M” are parameters that define the relationship.

y = Bx^M

When you look at a power model graph, it can vary widely from a linear appearance to a steep curve (see Figure 4). The power model really isn’t easy to work with when trying to fit data, which is where the “log-log” paper rec­ommended in IEEE 112-2004 comes in.

Unfortunately, we do need to use a little math to see how this works but we’re not going through rigorous deri­vations or proofs. The term log here is  short notation for logarithm, which in mathematics is the inverse operation to exponentiation. When log-log paper is referenced, that just means to graph the logarithm of your values (y and x) instead of the values themselves, or to use a graph paper whose axes are logarithmic scales. We’re going to use the natural logarithm, denoted “ln” because it is easy to work with. Now, when you take the natural log of the power model equation, something interesting happens. No matter what the shape of the curve, it will resolve into a straight line in the form of the equation below. And, when you are working with straight lines instead of complex curves, the data analysis becomes much simpler.

ln y = M ln x +  ln B

You may remember the basic equa­tion for a straight line (y = mx + b). Well, the natural log of the power model equation works exactly the same way since the natural log of any number is just another number. Once we solve it, we are however left with ln(y) instead of y. It was previously mentioned that taking a logarithm is the inverse operation to exponen­tiation. Euler’s constant (denoted e) is the base of the natural logarithm. This means that if you raise e to the natu­ral log of any number, you’re simply left with that number. Put another way, eln(y) = y. So, when we solve the equation for ln(y) above, whether for torque, current or another parameter, we raise e to our answer and we get the desired quantity. Again, you don’t really need to know this information to use the tool described earlier in the article but it can be of use sometimes to understand at least some of what’s behind an automated calculation.

Bibliography
[1] J. Dymond, R. Ong and P. McKenna, "Locked-rotor and acceleration testing of large induction machines-methods, problems, and interpretation of the results," IEEE Transactions on Industry Applications, pp. 958-964, 2000.

[2] IEEE, Standard Test Procedure for Polyphase Induction Motors and Generators, IEEE 112-2004, 2004.

 

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Categories: AC motors, Rotors, Testing, Motor testing

Tags: Locked-rotor amps Motor testing Locked-rotor test

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