Mike Howell
EASA Technical Support Specialist

The March 2013 Currents article titled “Stator I²R loss: considerations for rewinds and redesigns” describes the stator I²R loss, its calculation and how to control it during rewinds. This follow up will provide a brief review and then explore the additional stator copper losses mentioned in that article.

Stator I²R loss
Under typical loading conditions, the stator I²R loss usually is the largest loss component of three-phase AC rotating machines.  The stator I²R loss (P_SIR) in watts per phase is calculated as follows:

P_SIR = I²R

where
I is the current, in amperes, per phase
R is the DC resistance, in ohms, per phase

So, for a given winding configuration, the stator I²R loss can be reduced by shortening the coil length and/or increasing the conductor cross sectional area. Any such changes must consider properly dimensioned insulation, adequate clearances and provisions for installation. Conversely, avoid increases in coil length and/or decreases in conductor cross sectional area whenever possible. Also, note that end winding geometry is often sized for optimum cooling and changes may increase temperature rise.

Additional stator copper losses

Additional losses exist in the stator winding but are only significant in large form-wound machines. It’s difficult to quantify limits for what size machine might be of concern because several variables are involved. However, as an example, these losses are more likely to be a significant concern in a 4000 hp (3000 kW) machine than in a 400 hp (300 kW) machine. Let’s look at the source of these losses and how they’re typically addressed in design. When current flows in the stator winding conductors, a magnetic field is established that encloses the conductors and crosses the slot. As shown in Figure 1, this cross-slot leakage flux density increases from 0 at the bottom of the slot to a maximum value B at the top of the slot. This causes the additional copper losses. We can segregate these losses into strand eddy current loss and circulating current loss and calculate a factor to account for each in calculating the total stator copper loss (P_Cu).

Then, the total stator copper loss (P_Cu) in watts per phase is calculated as follows:

P_Cu = P_SIR + P_SIR  (k_s + m_c k_c)

where
P_SIR is the stator I²R loss, in watts, per phase
k_s is the per unit strand loss (eddy current)
k_c is the per unit circulating current loss
m_c is a reduction factor for use of a transposition

Strand eddy current loss and k_s

The cross-slot leakage flux induces a voltage in the stator conductors that is higher in the top of the conductor than in the bottom. The potential difference between the top and bottom causes eddy currents to flow as shown in Figure 2. 

When the conductor is a single deep strand as shown in the top of Figure 2, there will be a large difference in cross-slot leakage flux from top to bottom and thus a large potential difference resulting in a large eddy current. However, if the conductor is laminated with each strand insulated from one another as shown in the bottom of Figure 2, then the difference in cross-slot leakage flux from top to bottom of each strand will be smaller, thus a smaller potential difference resulting in small eddy currents. Generally, as the number of depth-wise or vertical strands per turn is increased, the strand loss factor (k_s) will decrease, thereby decreasing the total copper loss.

The strand loss factor (k_s) should typically be limited to a maximum of about 0.1 per unit of the stator I²R loss. To achieve this, the strand height will typically be in the range of 1.6 to 3.2 mm (0.064 to 0.128 inches). Additionally, for manufacturing reasons, the strand width to height ratio should be limited to around 4 for most designs. 

Approximate calculation of k_s can be done as follows, assuming the coil pitch is between 2/3 and 1, dimensions are in inches, and power frequency is 60 Hz. For dimensions in other units or a different frequency, refer to [1] for a slightly different approach.

 

where
L_g is the gross stator core length
n_d is the number of air ducts
w_d is the width of each air duct
MLT is the stator coil mean turn length
w_oc is the net width of copper in the slot
w_s is the width of the stator slot
d_oc is the net height of copper in the slot
d_c is the average strand height

For the service center, the most important point to remember is not to reduce the number of vertical strands per turn without careful consideration of the effect on the winding losses.

Circulating current loss and k_c
If there are multiple strands depth-wise per turn, insulated from one another, the strand eddy current loss can be reduced as previously discussed. However, common practice is to then join all those strands at the coil terminals when brazing connections. Since there is a potential difference between those strands due to the cross-slot magnetic flux density, circulating currents will flow within the coil whose only effect is to cause additional losses, reducing machine efficiency and heating the winding. Generally, as the number of turns per coil is reduced and the height of each turn is increased, the circulating current loss factor (k_c) will increase.

The circulating current loss factor (k_c) should typically be limited to a maximum value of 0.1 per unit of the stator I²R loss. To achieve this, designs utilizing 6 or more turns and two coil sides per slot usually do not require special manufacturing techniques. For machines with 5 or fewer turns and a relatively deep slot, it is common to see k_c values approach very high levels and, in these cases, failure to mitigate the circulating currents will result in decreased efficiency and overheating of the stator winding. Approximate calculation of kc can be done as follows, assuming the coil pitch is between 2/3 and 1, dimensions are in inches, and power frequency is 60 Hz. For dimensions in other units or a different frequency, refer to [1], for a slightly different approach.

where
n_t is the number of vertical strands per turn
* all other variables defined above

When k_c exceeds 0.1, the circulating current loss can be greatly reduced by using transpositions. Calculation of the reduction factor (m_c) for use of a transposition is extensive and beyond the scope of this article. However, some of the transposition types used will be discussed along with how they affect the product m_c · k_c, which determines the circulating current loss.

Some designs maintain the strand insulation through some of the brazed connections thereby increasing the length (and resistance) of the conductor. This effectively reduces the circulating currents to some degree in accordance with Ohm’s Law. For large machines with 5 or fewer turns, a more effective approach to reduce these losses is to transpose the conductors at one or more points in the coil such that the individual strands effectively occupy different radial positions in the slot. This reduces the induced voltage between strands thereby reducing the circulating currents.

There are a variety of transposition methods with the most common being transpositions performed in the end connections between coils, transpositions within the coil extension of multi-turn coils by executing a 180° twist during the looping process (see Figures 4 and 5), and lastly, transposition of each strand to occupy all radial locations throughout the straight portion of the coil, and sometimes beyond, referred to as a Roebel transposition (see Figures 6 and 7). Transpositions formed within the connections and transpositions utilizing the 180° twist can be performed such that mckc is reduced to an acceptable level (well below 0.1). The Roebel type transposition, demonstrated in a patent figure (see Figure 3) by German engineer Ludwig Roebel, reduces m_c · k_c to practically zero.

The 180° twist, or inverted turn transposition is performed during the looping process as shown in Figure 4. Two twist bar assemblies are fabricated to slide over the turn package such that the operator can perform the twist manually away from the loop bar and towards the magnet wire reels. Some wire racks are built to roll 180° to make the process easier. Optimal locations for the inverted turn transposition (mc is minimized) are given in Table 1. A five-turn coil with a 180° twist transposition is shown in Figure 5 with a transposition after 2 ½ turns. For manufacturing consideration, the transposition is typically reinforced with additional insulation for mechanical and electrical reasons. Also, when the ratio of radial turn depth becomes large relative to the coil width, the transposition can enter too far into the space between coils causing issues.

Stator half-coils or bars manufactured with a 360° Roebel transposition like that shown in Figure 6 require that each strand has two offset bends. Other transpositions (e.g., 180°, 540°) are used depending on the size of the machine and number of poles, but the 360° is very common. The bar is formed in two halves that nest together with crossover points on the top and bottom of the bar. The crossover points shown in Figure 6 would be reinforced later in the process with inserted insulation pieces.

Figure 7 shows the path of one strand along the straight portion from a side view where one can observe that the strand occupies all radial positions within the slot. The strand top view shows the location of the offsets required for the transposition. Below the bar, the select section views show what the cross section of the coil would look like at several points along the length of the bar with the one strand shaded black. It is evident that the strand begins in the top right at the left side of the bar and then takes a counter clockwise rotation through each of the positions in the bar returning to the top. Some general dimensioning guidelines for the 360° Roebel values R, C, and T as shown in Figure 7 are calculated as follows, where n_p is the number of strands per coil side.

Typical manufacturing processes will require a minimum distance T of around 1.5 to 2.0 inches (38 to 50 mm). Tolerances are usually established for the offset bend radius as well. Note that one disadvantage to the Roebel transposition is that the coil height must be about 1 strand height larger than the total vertical strands per coil to accommodate the crossovers. However, the loss in copper area is recovered by the elimination of circulating currents.

Example of managing additional losses
To demonstrate the application of conductor lamination and transposition, several options for a generator winding are provided in Table 2. There is a significant reduction in circulating current losses by using a transposition (per Table 1). Additionally, further lamination of the conductor significantly reduces the strand loss. However, note that as the number of strands is increased, so is the I2R loss due to the additional strand insulation. Thus, there is a greater gain from Design 1 to Design 2 than from Design 2 to Design 3. Also note that Design 3 with a transposition results in per unit (pu) strand and circulating current loss of less than 0.1.

Practical takeaways for the service center
While it is not necessary to calculate these losses for most rewinds, it is prudent for very large machines, especially generators. For the three stator copper loss components, here are some general tips to keep in mind. 

I²R loss
The stator I²R loss can be reduced by shortening the coil length and/or increasing the conductor cross sectional area. Any such changes should consider properly dimensioned insulation, adequate clearances and provisions for installation. Conversely, avoid increases in coil length and/or decreases in conductor cross sectional area whenever possible. 

Strand eddy current loss and k_s
Do not reduce the number of vertical strands per turn without careful consideration of the effect on the winding losses. For example, if a machine is received for rewind with unusually thick strands, e.g., 0.15 inches (3.8 mm), where temperature rise has been an issue, this is a candidate for improvement. Large calculated values of ks (greater than 0.1) should be evaluated.

Circulating current loss and k_c
When rewinding very large machines, especially generators, be on the lookout for transpositions. You are much more likely to find Roebel bar transpositions in single turn, half-coil designs, but two-turn Roebel bars do exist. 

Inspect connections for continued insulation and strand transpositions. Inspect individual coils for inverted turn transpositions. The inverted turn transposition can sometimes be identified by a bulge in one of the four coil arms (see Figure 5) and can be confirmed by removing insulation or using a multimeter once the coil start and finish leads are exposed and strands are separated. 

Transpositions should be replaced as-found unless a more effective design is determined procedurally. Large calculated values of k_c (greater than 0.1) should be evaluated. For example, if k_c is calculated to be 0.8 but a transposition was not identified during winding removal, it’s almost a certainty that it was overlooked.

Also, note that transpositions are not only used in three-phase AC machines. They can also be found in some single-phase machines and in DC armature windings.

References
[1] I. Summers, "Reduction of Armature Copper Losses," in Winter Convention of the AIEE, New York, 1927. 

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

Tags: Motor design

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