Direct-Current Motors: Lamination, Field, and Housing Geometry

Universal (Series-Wound) Motor Construction

FGR. 46 Universal motor construction.

FGR. 47 Universal motor.

FGR. 48 Four-pole dc series motor.

Figures 46 and 47 show the general construction of a two-pole-wound field universal motor. Fgr. 48 shows a larger four-pole series motor designed to run on direct current.

The stator (field) and armature laminations are stacked, insulated, and wound with magnet wire. The armature ( FGR. 47) is wound in consecutive continuous loops or coils. Each coil end is connected to a commutator bar. The field is wound with magnet wire and the coils are connected in series such that they will provide opposite magnetic polarity. The field ( FGR. 47) is connected in series with the armature through the brush-commutator connection.

FGR. 49 Armature lamination.

Armature Geometry. We start by analyzing the physical dimensions of the armature lamination, which gives us the following:

Gross slot area. This tells us how much cross-sectional area is available to accommodate the armature windings (before adding insulation and slot pegs).

Net slot area. This is the net available area to accommodate the copper wire after the slot insulators and pegs are subtracted from the gross slot area.

Magnetic path length. For the armature, this is the distance that the flux will travel and over which the mmf will be dropped.

Magnetic path area. This is the cross-sectional area of the magnetic path as seen from the flux's point of view. The smaller the path area, the higher the flux density, and the more mmf will be dropped along the path.

Weight of steel. This must be known to calculate the inertias of the armature assembly.

Inertia. This is used in determining start-up acceleration or load matching.

Slot constants. This is used in estimating reactance. This is largely a function of lamination geometry.

Using trigonometry, we break the armature lamination up into several sections in order to calculate the gross slot area. The radii used in the calculations are readily available from most lamination drawings supplied by the manufacturer.

Using FGR. 49, determine the following dimensions:

Ra = outside radius of the armature lamination

Ra1 = inside radius of the armature teeth

Rs = radius of the shaft

Wast = width of the armature slot top in straight-line distance

Ra2 = radius of the circle on which the arc segments of the bottoms of the slots are centered

Nat = number of armature teeth

Lstk = length of the armature stack

?pole = angle of pole arc

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FGR. 50 Armature lamination Slot.

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FGR. 51 Armature magnetic paths and reactance dimensions.

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FGR. 52 Armature lamination slot.

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FGR. 53 Finite element analysis showing flux paths through stator teeth.

The magnetic path area of the shaft is accounted for in the armature yoke magnetic path calculations.

Armature Conductors

FGR. 54 Armature lamination slot.

In order to determine the resistance of the armature as seen by the brushes, the following parameters need to be set:

Sac = number of teeth over which one coil is wound

Tpca = number of turns of wire per armature coil

CPS = number of coil sides in each armature slot

The length of one turn in the winding must be determined. There are three different methods of determining the length of a turn.

Armature End-Turn Length-Method 1. This method uses the geometry of the lamination to estimate the average end-turn length. The accuracy of this method is dependent on the capabilities of the winding machinery. Using geometry and the width of the armature yoke for a magnetic shaft, the mean end-turn length is found by averaging the distances around the lamination at the radii of the top of the slot and the bottom of the slot. The mean end-turn length of one side of one turn, in inches, is:

In order to understand the other methods of determining end turn length, the formulas for armature resistance are discussed first.

Armature Resistance. The mean turn length is the sum of the mean end-turn length for each of two ends plus twice the stack length, in feet (because wire resistances are given in ohms per foot):

Length of copper per armature coil, in feet:

Resistance of one armature coil, in ohms:

...where ?a is the armature wire resistance per foot and Fwsa is the armature winding stretch factor, usually 1.02 to 1.10.

The number of coils to be considered for armature resistance depends on the number of commutating coils, so the number of active coils for the armature is

... where Nat is odd, and

... where Nat is even.

The net armature resistance ( FGR. 55) is the combination of half the number of active coils in parallel with half the number of active coils, in ohms:

Armature End-Turn Length-Method 2. This method assumes that you already have a motor and know Rat, Tpca, Fwsa, and ?a. From this, the Lmeta can be calculated and then used for calculations for other motors with the same lamination. The following four formulas determine the mean end-turn length for one end of one turn.

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Armature End-Turn Length-Method 3. Measure what looks like the average end turn on an existing motor.

FGR. 55 Schematic of armature current paths.

Armature Slot Fill. The maximum armature slot fill depends on the winding equipment. A full slot could be anywhere from 40 to 55 percent. The slot fill is calculated by assuming that the wires are lying side by side as if they were square. Using the outside diameter of the armature wire as Daw (this depends on insulation thickness), the percentage of armature slot fill is:

Armature Copper Weight. Using pound-feet from the wire tables, the weight of the armature copper W , is:

Armature Inertia.

Armature Balance. ISO Standard 1941 gives guidelines for acceptable balance and vibration limits for electric machines based on their usage. In general, the lower the unbalance, the longer the motor life. Unbalance affects bearing wear by causing the shaft to pound against the bearing, breaking down the surface and pumping out the lubricant. It affects brush life adversely by causing the brushes to instantaneously lose contact with the commutator surface, resulting in electrical arcing, wear, and higher levels of EMI. If this occurs near a resonant speed, the condition becomes severe and the motor instantaneously ceases operation because the brush lifts completely away from the commutator, causing an open circuit.

Armatures should be aligned in the magnetic field such that there is a net magnetic force pulling the armature toward the commutator end of the motor. This reduces end bounce and improves brush life.

Field (Stator)

The geometry of the stator lamination requires breaking it down into several small areas in the same manner as used for the armature.

Using Ills. 56, 57, and 58 and the XY coordinate system as shown, this sub section uses the given dimensions to determine the field slot area.

The following dimensions are needed from the lamination drawing. Some manufacturers distribute drawings with only a few dimensions on them, but will give these detailed drawings if requested.

FGR. 56 Example of a lamination drawing.

FGR. 57 Stator (field) lamination.

Lbpt = distance between opposite pole tips Xptc, Yptc = coordinates of pole tip center

Rpt = radius of pole tip Xossc, Yossc = coordinates of center of arc of outside of stator slot

Ross = radius of outside of stator slot Xssc, Yssc = coordinates of center of radius of stator slot center

Rss = radius of stator slot

Rosw = radius of outside of stator wall

Risw = radius of inside of stator wall, which blends with arc of outside of stator slot

Depending on which stator dimension is given, the X coordinate for the center of the pole tip, in inches, is:

Or, the distance between opposite pole tips, in inches, is:

FGR. 58 Stator (field) slot area.

The pole arc angle is twice the angle from the center of the lamination to the tip of the pole closest to the other pole. The pole arc angle, in degrees, is Angle from the center of the stator to the center of the pole tip, in degrees:

Distance from the center of the stator to the pole tip center, in inches:

Using stator triangle TRS1, which has the hypotenuse parallel to Lptc, the height is:

Angle between the height and the hypotenuse, in degrees:

For later use, the angle from the center of the stator to the center of the stator slot, in degrees, is

The triangle TRS3 has the height of the difference of Y coordinates of the stator slot center and the outside of the stator slot center, in inches:

LTRM4 = Rss (4.208) And the length, in inches:

LTRS3 = Xssc - Xossc

Angle between the hypotenuse and the height of TRS3, in degrees:

Length of the hypotenuse of TRS3, in inches:

The small TRS2 has the leg from the center of the stator slot with its length, in inches:

Angle between the length of TRS2 and the hypotenuse, in inches:

Height of the triangle TRS2, in inches:

Length of the hypotenuse of TRS2, in inches:

Area of TRS2, in square inches:

The arc segment SEG5 at the end of the stator slot traverses the angle, in degrees:

Area of SEG5, in square inches:

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The net slot area available for the field winding on one pole is the GSA minus the area of the insulation, in square inches:

...where Tssi is the thickness of the stator slot insulation in inches.

Constants for Stator Reactance Calculations. As with the armature, the geometry of the stator slot must be known in order to calculate the reactance. Note that the field slot can be turned to resemble the armature slot. Use Ills. 4.58 and 4.59 to determine the following values which will be needed later.

FGR. 59 Dimensions for stator reactance.

The values of a and b, as similar to the armature, in inches:

Since the stator slot has a different shape than the armature slot, the value d3 for the stator is the average of d1 and d2, in:

Length of the line across the wide opening of the slot e, in:

There is no c dimension as in the armature slot dimensions.

Length of the interpolar air gap, which is the distance between the armature and the stator yoke at their closest position, in inches:

...where YST1 is the Y coordinate of the center of radius RST1.

The points on the pole tip A,B, and C are used as the starting points of three arcs which are drawn with the edge of the armature as the center. These arcs were derived from flux mapping using curvilinear squares, which won’t be discussed here. C is the point where line e intersects the pole. A is the point where the radius of the pole tip would be parallel to the armature tooth tip. B is the point on the slot side of the pole tip which is halfway between A and C. The point B' is the point that the average leakage flux would travel to on the stator yoke.

Arc length BB' is the length of the arc of radius RBB' (which is the distance from A' to B) shortened by the radius RST1.Arc length AC is the arc around the pole tip of about 90°.

Magnetic Paths in the Field (Stator)

Using the quarter-stator drawing in FGR. 60, the portion of the stator is divided up into seven magnetic paths. In order to determine the widths and lengths of these paths, the dimensions of four triangles need to be determined. All magnetic path widths are multiplied by 2 because this analysis is done on a per-pole basis and the dimensions being used are from a drawing of half of one pole. For the initial calculations, use Lstke = 1.0.

Width of the stator yoke, in inches:

... magnetic length of the stator yoke, in inches:

... Magnetic area of the stator yoke, in square inches:

Angle of triangle TRM1 between the length and the hypotenuse, in degrees:

FGR. 60 Stator magnetic path lengths and widths.

Width of stator path 1, in inches:

Length of triangle TRM1, in inches:

Height of TRM1, in inches:

Magnetic length of stator path 1, in inches:

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Field Conductors

In order to determine the resistance and slot fill of the field, the following parameters need to be determined by the designer:

Tpcf = turns of wire per field coil Dfw = diameter of field wire The length of one turn of the field winding must be determined. As with the armature winding, there are three different methods of determining end-turn length for the field.

Field End-Turn Length-Method 1. The mean end-turn length for the field is the average of the length of the shortest turn which is wound tightly across the pole and the length of the turn on the outer edges of each slot. The shortest turn is the straight distance across the pole, and the longest turn is the length of the arc swung across the outer edges of the slots. The field winding mean end-turn length, in inches, is

... where p is 180° converted to radians. The end-turn length will vary slightly with slot fill, winding equipment, and the overhang of the field slot insulation relative to the stack.

In order to use method 2 of determining end-turn length, which is similar to the armature method, you first need to know how the field resistance is determined.

Resistance of Field Windings. The field resistance is the total resistance of two field coils in series. The mean turn length of one field coil, in feet (because wire resistances are given in ohms per foot), is

Length of copper per field coil, in feet:

Resistance of one field coil, in ohms:

... where ?f is the field wire resistance per foot, found from the wire tables for the known wire gauge, and Fwsf is the winding stretch factor, usually 1.02 to 1.10.

The net field resistance is the series combination of two field coils, in ohms:

Field End-Turn Length-Method 2. As with the armature, this method assumes that you already have a motor and know Rf, Tpcf, Fwsf, and ?f (from wire tables). From this, Lmetf can be calculated and then used for calculations for other motors with the same lamination.

The following four formulas give the mean end-turn length for one end of one field turn, in inches.

Field End-Turn Length-Method 3. Measure what looks like the average end turn on existing motor with the same lamination.

Field (Stator) Slot Fill. The maximum field slot fill depends on the winding equipment. A full slot could be anywhere from 80 to 100 percent. These percentages are greater than for the armature because there is only one coil in the field slot and it’s possible to use the slot insulation to extend the area of the slot. The field slot fill is calculated by assuming that the wires are lying side by side as if they were square.

Using the outside diameter of the field wire as Dfw (found in the wire tables taking into account insulation thickness), the percentage of winding slot fill is

Field Copper Weight. Using pound-feet from the wire tables, the weight of the field copper Wtfc,lb, is:

Turns Ratio. The field-to-armature turns ratio is the ratio of the total number of field turns to the total number of series-connected turns in one path of the armature.

...where P is the number of poles and Nca is the number of active coils in the armature.

This ratio represents the mmf produced by the field in relation to the mmf produced by the armature. If the ratio is too high, the field produces strong mmf, which induces a higher voltage in the commutating coil. This high voltage causes arcing problems during commutation. If the ratio is too low, the armature mmf weakens the pole too much and a loss of torque results. A good range for this ratio is 1.00 to 1.30.

Magnet Circuit

This subsection derives a curve for the relationship of flux across the air gap versus the related mmf drops. This curve tells you how many ampere-turns are necessary to produce a specified air gap flux. Special consideration is taken to account for saturation and for the air gap.

The air gap is the area between the pole on the stator and the tooth tips on the armature. The air gap is not uniform because of the spaces between the tooth tips and because the pole is not perfectly round. Fgr. 61 shows that the air gap is larger at the pole tips than at the center.

First, take into account the longer air gap at the pole tips. On the manufacturer's drawing, the pole arc changes to a straight line at a specified angle ?ref, which is usually about 45°. When measuring from the center of the armature, at 90°-?ref, the air gap starts to get longer and the length continues to increase out to the pole tip. (Longer refers to the magnetic length.) The aver age length of the air gap over this part of the arc is, in inches:

The next formula gives an effective mechanical air gap length by using a weighted average over half the pole arc.

The effective mechanical air gap length, in inches, is

The magnetic length must account for the spaces between the armature teeth.

Carter's coefficient KC accounts for the flux that fringes from the pole to the sides of the armature tooth tips ( FGR. 62). Carter's coefficient can be determined by first using an available graph of the width of armature slot along the circumference divided by the effective mechanical air gap length versus sC, where sC is an arbitrary variable name. Using sC:

Another method for determining Carter's coefficient (with no graph available) is by the following two formulas:

FGR. 61 Nonuniform air gap.

Effective magnetic air gap length, in inches:

The magnetic width of the air gap is the length of the arc along the center of the magnetic air gap, in inches:

Magnetic area of the air gap, in square inches:

Fgr. 63 shows the analogous electrical circuit for the magnetic circuit of a universal motor. Each resistor is associated with an mmf drop, similar to an emf (or voltage) drop in an electrical circuit. The resistor in the middle of the armature slot represents an mmf drop that occurs only at high saturation. The Trickey factor Ktr accounts for this mmf drop and is discussed later.

FGR. 62 Fringing of flux at air gap.

Plot of Flux per Pole versus Ampere-Turns Excitation. Flux is required for an electric motor to produce torque. In order to produce flux, mmf must be supplied.

Each combination of armature and stator laminations will produce a different amount of flux for a given supplied mmf curve of air gap flux (in kilolines fgap) versus mmf in ampere-turns of excitation can be determined by determining the sum of all mmf drops across the magnetic circuit (lamination set) for each of several values of flux. In order to determine the mmf drop across a specific area, the flux density must be determined. From the BH curve for the material being used, the mmf drop in ampere-turns can be determined for the specific area.

For a given fgap, klines, the flux densities for the various paths are as follows: Bgap,

..., in the armature tooth tips:

FGR. 63 Schematic of magnetic circuits.

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In order to calculate the total mmf drops in the magnetic circuit, you need to read the magnetic field intensity H, (A _ turn)/in, from the BH curve for each section. The mmf drop for the section is the intensity times the length of the section.

Field intensity H , (A _ turn)/in, in the air gap:

MMF drop across the air gap, in ampere-turns:

For the following equations, the value of H is read from the graph for each different flux density B.

Special consideration must be taken to account for the fact that when the flux density reaches saturation in the armature teeth, the flux will set up on the sides of the teeth. At about 110 kline/in^2 ..., the armature teeth saturate, and any additional flux will exist along the sides of the teeth as well as in the teeth themselves.

The mmf drop due to the flux on the sides of the tooth can be considered by adding it to the mmf drop in the tooth itself. Also, as Bat becomes greater than 110,000 ..., the initial calculation of Bat becomes Bat, apparent.

FGR. 64 Flux in air beside teeth (Trickey factor).

The Trickey factor Ktr must be determined:

The Trickey factor is the number of tooth widths in the air beside the teeth that the flux occupies at saturation. Note that it accounts for the air between the laminations by dividing by the stacking factor.

A little extra effort is required to determine the field intensity H for a saturated tooth. Using the BH curve for the material and the Bat, apparent which is over 110 kline/in^2 ..., determine the apparent field intensity H_temp in the armature teeth. A new apparent flux density is Batal.

The value _total represents the field ampere turns per pole necessary to provide the specified air gap flux fgap. This means that the product of the number of field turns Tpcf and the field current Iline must equal _total. A curve of air gap flux versus ampere-turns of excitation can be plotted by varying fgap. Remember that each curve is specific to the lamination set and that it does not account for any loss in mmf due to armature reaction.

Reactances. Reactances are largely a function of lamination geometry. In order to calculate reactances, the slot dimensions and lengths must be known. The following subsections examine methods for calculating field and armature reactances.

Field Leakage Reactance. The first formula is from Puchstein (1961) for field leakage reactance, in ohms.

...where Lmtf is the mean turn length in inches of the field and fc is the number of lines per ampere-conductor per inch of mean turn length. A good value to use for fc is 6.

The next formula is from Trickey for field leakage reactance.

where tpcf = field turns per pole Ksf1 = slot constant for part containing coil Ksf2 = slot constant for pole tip flux Sfc = span of field coil, in You also need to know the following in order to calculate armature reactances:

?p =?pole/pole pitch =?pole/180° Nas = number of armature slots = Nat Zs = armature conductors in a slot = Tpca(CPS) Za = total armature conductors = Tpca(CPS)Nat f = line-current frequency, Hz The span of the field coil is the arc length from the center of the field winding in the slot to the same position in the other slot, in inches.

Using either of the preceding two methods for field leakage reactance, find the total reactance.

Note: Trickey's method does not consider end-turn leakage.

Armature Reactances. In order to calculate reactances, the slot constants of the armature slots must be determined. Fgr. 66 shows the dimensions which are needed to calculate slot constants. You also must use the actual value for Lstke. These dimensions were calculated previously.

Armature Leakage Reactances Under a Pole, ?

Armature Slot Leakage Reactance, ?

... where Ksa = slot constant for round-bottom armature slots. Use the following formulas.

FGR. 66 Armature magnetic paths and reactance dimensions.