Mike Howell, PE
EASA Technical Support Specialist 

Many inquiries submitted to EASA technical support staff require a review of as-found winding data of three-phase machines to determine if the data is reasonable for the machine’s nameplate ratings and core size. The most common approach for accomplishing this review is to compare the as-found winding data to existing data from similarly rated and sized machines in EASA’s motor rewind database. If several existing records are very similar to the as-found data, especially from the same manufacturer, the review is straightforward. Other times, an inference, or educated guess is required. EASA will soon be releasing a new Motor Stats feature for both the AC Motor Verification & Redesign and Motor Rewind Data software to assist with the educated guesses.

Esson’s Rule
For well over a hundred years, one of the fundamental relationships used by manufacturers when sizing electric machines relates torque to rotor volume. This is often referred to as Esson’s Rule and can be written as follows 

P / N = C₀ x D²L

where P is the mechanical power, N is the rotor speed, and C₀ is referred to as the output coefficient. For our purposes, it is important to note that the output coefficient (C₀) depends in part on the magnetic flux density in the air gap, a value calculated for each machine in the motor rewind database. And, as shown in Figure 1, D is the rotor outside diameter and L is the rotor core length. 

It is reasonable when comparing machines to use stator inside diameter and stator core length in place of rotor outside diameter and rotor core length. Also, we typically use pole count in place of rotor speed, understanding that nameplate rated frequency differences must be considered. 

So, the inference we make using the motor winding database is that machines having similar mechanical power, pole count (and frequency), core length and stator bore diameter should have similar magnetic flux density in the air gap. And, while there are always exceptions, this is a reasonable approach for verifying as-found data and for developing winding data in the event as-found data is lost or incorrect. 

Motor Stats
If we wanted to know about the average height of 20-year-old males in a particular country, we could measure the height of a sample of 20-year-old males from their population and then use statistics to draw conclusions about the population based on the sample. 

Searching MotorDb will return a list of all motors in the database that match our search criteria. It would be helpful to know the average air gap flux density for this list of motors (our sample), and how likely that average is to represent the average for all motors in existence that match our criteria (the population). The Motor Stats data lets us do that. 

For example, let’s say that the population shown in Figure 2 represents all squirrel cage induction motors meeting the following criteria: 

• 100 hp (75 kW) ± 2% 
• 4 poles / 60 Hz 
• D = 7 inches (178 mm) ± 2% 
• L = 10 inches (254 mm) ± 2% 

Now, let’s say the sample shown in Figure 2 represents 25 motors in EASA’s motor rewind database that meet those criteria.

It is reasonable to assume that the air gap flux density for the population is normally distributed with a bell-shaped curve as shown in Figure 3. We base this assumption on our Esson’s Rule (D²L) discussion. 

If this is the case, the air gap flux density of most motors within the population will be reasonably close to the population mean (arithmetic average). We can calculate a confidence interval for any sample of data taken from the population. Confidence intervals for six samples are shown in Figure 3 and labeled A through F. If we take many samples from the population and calculate 95% confidence intervals for each sample, then in the long run, 95% of those intervals will contain the population mean. Also, as you might imagine, small confidence intervals are better than large confidence intervals. 

For our 100 hp (75 kW) example, EASA’s motor rewind database returned 25 motors, and the confidence interval for that sample is represented arbitrarily as Sample F of Figure 3. In our example, the Sample F confidence interval contains the population mean. Sample C in Figure 3 does not contain the population mean and when working with 95% confidence intervals, this will happen 5% of the time. 

The most convenient time to use the Motor Stats tool is when verifying or redesigning data using AC Motor Verification & Redesign. The time-saving benefit here is that you get the statistical summary without having to do a separate search in the database. Now, let’s look at a specific motor and compare it to the Motor Stats output. Figure 4 shows the winding data card, calculated densities, and Motor Stats output. If the as-found data provided a good slot fill, we can conclude it is reasonable for this machine since our calculated air gap flux density is close to the confidence interval for the mean, and both the tooth and back iron flux densities are below the maximum allowable values shown. 

The current density in the stator winding should also be evaluated to ensure it is reasonable for the assumed duty cycle of the machine. While the air gap flux density will typically fall within a reasonably small range, you will often find a wide variance with current density. For example, the current density for an intermittent duty submersible pump motor may be around 200 CMA (10 A/mm2), while a continuous duty premium efficiency motor with similar ratings might be around 800 CMA (2.5 A/mm2). Other machines will be higher or lower than these values. Slot fill should not be adjusted, especially reduced, just to hit some target arbitrary current density. 

The chord factor is the ratio of the voltage induced in a coil to the voltage that would be induced in the same coil if it were full pitch. Undesirable space harmonics are well controlled when the chord factor is in the 0.951-0.991 range, but there are designs with higher or lower values. Some two-pole motors have chord factors in the 0.707-0.866 range and for such machines, it is likely that the manufacturer has evaluated the consequences of space harmonics. Many two and four-pole generator stators have a 0.866 chord factor (2/3 pitch) to eliminate the third space harmonic. And a 0.966 chord factor (5/6 pitch) is found in many machines as it minimizes the fifth and seventh space harmonics.