Mike Howell
EASA Technical Support Specialist

Aside from managing his family’s brewery in England, J.P. Joule did some pretty amazing work in physics dur­ing the mid-nineteenth century. Joule discovered that the rate at which heat is produced by a steady current in any part of an electric circuit is proportional to the resistance and the square of the current. So, the I²R loss of a conductor is not creatively named.

How does this apply to motor stator windings? The stator I²R loss is typically the largest contributing factor to the stator winding tempera­ture rise and the largest detractor of efficiency.

The motor’s job is to convert electri­cal energy to mechanical energy, and the efficiency of a motor tells us how effectively it does its job. We normally discuss machine output and losses in units of power (e.g., horsepower, kilowatts) which is the rate of energy transfer or conversion. When you sub­tract the output power from the input power, you’re left with the losses and the ratio of the output power to input power is the efficiency.

Squirrel cage induction motor losses are segregated into five types (see Table 1). These losses are converted to heat, vibration and noise. The heat must be dissipated by the motor to prevent over­heating. Given that the stator I²R loss is a significant percentage of the total machine loss, it is important to consider how changes to a stator winding during repair may affect the machine.

For a three phase machine, the sta­tor I²R loss (P_SIR) in watts is calculated as follows:

P_SIR = 1.5 I²R = 3 I²R₁

where
I    is the current per line terminal, in amperes
R    is the DC resistance, in ohms, be­tween any two line terminals
R₁    is the DC resistance, in ohms, per phase

Rewinding stator for same rating
In the case of rewinds where the machine ratings do not change, the current should be constant and the variable of concern is winding resis­tance. As you can see from the I²R loss calculation, increasing the winding resistance will increase the stator I²R loss and thus the stator temperature rise will increase and the machine’s efficiency will decrease.

Winding resistance is a function of the conduc­tor temperature, mate­rial and dimensions. It is directly proportional to the temperature and conductor length while inversely proportional to the conductor area. Copper is the primary conductor material for stator windings because it is approximately 70% more electrically conductive than alu­minum for a given length and area.

Additionally, it is about 75% more thermally conductive than aluminum meaning it can transfer heat at a much greater rate. Essentially, this allows the manufacturer to build a smaller, more efficient machine. The DC resistance, in ohms, per phase (R₁) of the stator winding is calculated as follows: 
 where

ρ is the conductor material resistiv­ity
MLT    is the mean length of turn (MLT)
N   is the number of turns in series per phase
A_c  is the conductor cross sectional area
c    is the number of parallel cir­cuits per phase

For a given winding configuration, the stator I₂R loss can be reduced by shortening the MLT and/or increasing the conductor cross sectional area. Conversely, you want to avoid any increases in MLT and/or decreases in conductor cross sectional area.

For an example of calculating the stator winding resistance per phase (R₁), let’s look at a 15hp, 230V, 6-pole stator winding. The stator has 54 slots and a standard 2-layer lap winding connected 2Y with 54 coils, 9 turns per coil and 2 #14 AWG strands per turn. The coil geometry is similar to the round coil shown in Figure 1 and we will use that estimation for calculat­ing the mean length of turn (MLT). We will assume that the current will divide evenly through each parallel circuit per phase. So, the total effective area through which the current will flow is equal to the conductor cross sectional area (A_c) multiplied by the number of parallel circuits per phase (c).  See Figure 2.

Reducing the MLT
The MLT is the average length of one coil turn and approximate calcula­tions can be performed per Figure 1. For random wound machines, the MLT may be reduced by shortening the straight section of the coil where it exits the slot to the minimum required to avoid mechanical strain on the slot liner. Whichever coil shape is used, the coil end turns should be no longer than those of the original winding.

For form wound machines, ad­ditional consideration must be made regarding mechanical and electrical clearances coil to coil and between the coil and stator support components such as stationary rings and finger plates. The coil end turns should be no longer than the original unless consideration for proper mechanical and electrical clear­ances dictates otherwise. If an increase of MLT is required, an effort should be made to offset it with a corresponding increase in copper cross section to avoid an increase in winding resistance.

Increasing the stator copper cross section
With random wound machines, many winders can look at a motor nameplate and make a pretty good guess as to how full the manufacturer wound the stator slot. Some will tend to be very tight, some will vary from machine to machine and others will al­most always be loose (see Figure 3). Whenever practical, the sta­tor copper cross section should be increased. Loose wires will vibrate in the slots causing chafed insulation and fatigue fractures. Heat transfer also suffers with low slot fill because copper and electrical insulation are better thermal conductors than air.

With form of wound medium voltage machines, a much larger percentage of the slot area is filled with insulat­ing materials (see Figure 4). When rewinding older machines, significant increases in copper cross section can be achieved due to ad­vances made in insulating materials without sacrificing winding life. 

Newer designs present more problems given that manufacturers taking advantage of new technol­ogy are producing medium voltage windings with thinner insulating materials than typically used by many repairers. In this situation, it is sometimes difficult to maintain the same cross sectional area without adopting higher voltage stress across insulating components or using non-standard conductor sizes.

Typically, the current density in the stator winding conductor for continuous duty machines may range from 1.6 A/mm² (1200 CM/A) to 7.8 A/mm² (250 CM/A). It is im­portant that the repairer compare replacement windings to the origi­nal winding and not an arbitrarily chosen “acceptable value” (e.g., 400 CM/A). However, when rewinding a stator for a higher output rating, simply maintaining the same cur­rent density with the new winding is not sufficient for maintaining the same stator I²R loss, since power is proportional to current but losses are proportional to current squared.

Rewinding stator for increased output rating
When a stator is rewound for a high­er output power, the stator line cur­rent will increase approximately in di­rect proportion. For simplicity, we are assuming no other operating charac­teristics are changed (e.g., voltage, frequency, poles). Since the heat produced by the stator wind­ing varies directly with the stator I²R loss, it is evident that the winding resistance cannot be reduced in direct proportion to the increase in current to avoid additional heat production by the stator winding. In other words, if we want the stator heating that results from the stator I²R loss to re­main unchanged or improve with the power increase, then the new stator I²R loss has to be equal to or less than the original stator I²R loss. Figure 5 gives a corresponding per­cent reduction in winding resistance based on a given percent increase in output power.

The target maximum new winding resistance can be calcu­lated as follows:
 where

R₁    original DC resistance, in ohms, per phase
R₂    target maximum DC resistance, in ohms, per phase after rede­sign
I₁    original current per line termi­nal, in amperes
I₂    expected current per line termi­nal, in amperes after redesign

Let’s look at an example. A mem­ber recently had a 400hp, 4kV, 6-pole induction machine that was being overloaded in a quarry, and the end user had requested a redesign to 500hp. Fortunately, the magnetic circuit had been conservatively de­signed and the required increase in flux would not cause saturation. There would be some unavoidable additional heating due to the in­crease in core loss, rotor I²R loss and stray load loss. The stator current would increase by approximately 25% but because heat produced by the stator winding varies directly with the stator I²R loss, a 37% reduc­tion in winding resistance would be required to prevent the stator from producing additional heat. For this particular design, the resistance was reduced by two methods.

First, the span was increased which slightly increased the MLT but this allowed a reduction in turns that reduced the total length of the coil by 17%. Next, the reduction in turns per coil allowed for an increase in conduc­tor area of 28%.  The end result was a stator winding I²R loss increase of only 1% with a 25% increase in cur­rent (see Table 2). 

When increasing output power, rotor I²R and stray load losses will increase due to increased load and contribute to additional heating of the motor. Additionally, the core loss will increase when the magnetic flux is increased. Reducing the winding re­sistance as much as practical to reduce the stator I²R loss will minimize the effect of the output power increase on the thermal life of the stator winding.

So what?
So, how do we translate changes to the stator I²R loss to the stator winding temperature rise and the machine’s efficiency rating? We’ll use the relationships we’ve described above to provide estimates. Since the amount of heat produced by the stator winding is proportional to the stator I²R loss, a common approach for estimating change in temperature rise is to use the ratio of the losses. Using this method, if you have a stator winding with a 50°C rise and you increase the stator I²R loss by 5%, then the expected temperature rise would be approximately (50°C)(1.05) = 53°C. Depending on where the temperature falls in relation to the insulation system thermal rating, the winding life could be reduced. Conversely, reducing the stator I²R loss can increase the thermal life of the winding.

How might the same 5% increase in stator I²R loss affect efficiency? For illustration purposes we’ll assume the machine efficiency to be 95% (5% loss­es) with the input power at 100 kW and the output power at 95 kW. If the stator I²R loss is 40% of the 5% losses, it is 2kW. An increase of 5% raises it to 2.1kW. Now, for the same output power of 95 kW, the input power has to be raised by 0.1kW to 100.1 kW. The efficiency would be reduced from 95% to 94.9%. With most rewinds, it is possible to maintain or even decrease the stator I²R loss. This is consistent with the EASA/AEMT rewind study conducted at the University of Not­tingham in 2003 which found that, if 'best repair practices' were followed, efficiency could be maintained, and in some cases even be improved.

Additional stator winding losses
Additional losses exist in the sta­tor winding but are only significant in large form-wound machines. For induction motors, these losses are included in the stray load losses. These losses will be explored in a future article to obtain a general understanding of why they exist and to adopt some simple practices that will prevent increasing them during repairs.

Also see Beyond I2R: Additional copper losses in stator windings

Categories: AC motors, Stators/windings, Design/theory/application

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