Direct-Current Motors: PMDC Motor Performance

This section is intended to give the PMDC motor designer a method to calculate PMDC motor performance given the material magnetic and electrical properties and physical dimensions.

The basic construction of a PMDC motor is as shown in FGR. 78.To calculate the performance for this motor, one must predict the air gap flux and calculate the no load speed and current and the stall torque and current. A straight line drawn between the no-load speed and the stall torque represents the speed-torque curve of the motor. A straight line drawn between the no-load current and the stall current represents the current-torque performance curve. Examples of such curves are shown in Ills. 79 and 80.

FGR. 78 PMDC motor construction.

FGR. 79 PMDC motor performance curves.

FGR. 80 PMDC motor performance varying with voltage.

Predicting Air Gap

A typical approach to predicting air gap flux is described here. The magnets are generally attached to a magnetically soft steel housing. When they are charged, they set up a nearly constant flux in the air gap between the magnets and the armature. In order to determine the motor performance, you need to know the amount of air gap flux linking the armature conductors, the number of conductors, the number of poles, and the current in the armature. Fgr. 81 shows the direction of the flux due to the magnets. You can determine the flux in the air gap by the following procedure:

_ Finding the permeance coefficient of the magnetic circuit

_ Determining the flux density in the magnet

_ Finding the total flux

_ Factoring out the leakage flux

FGR. 81 Flux due to magnets.

The remaining flux interacts with the armature conductors and produces the motor torque.

The permeance coefficient is determined by the geometry of the cross section of the motor:

... where s= flux leakage factor (typically 1.05 to 1.15) Rf = reluctance factor (typically 1.1 to 1.3) Lmr = radial length (thickness) of the magnet

Lg = length of the air gap in the radial direction

Am = area of the magnet

Ag = area of the air gap

The area of the magnet is not necessarily the same as the area of the air gap, because the magnets typically overhang the armature.

Ag is usually smaller than Am.

The permeance coefficient determines the load line of the magnet on its normal demagnetization curve. These curves are commonly supplied by the magnet manufacturer. A typical curve is shown in FGR. 82; it’s the upper left quadrant of the hysteresis loop shown in other sections of this handbook. The H and B axes must be appropriately scaled for this technique to be accurate.

FGR. 82 Finding magnet load point.

To plot the load line, take the arctangent of the permeance coefficient, calculate the angle ?= tan-1 Pc , and plot the line as shown in FGR. 82.The flux density in the magnet Bm can be found by finding the intersection of the load line and the normal curve and reading the induction from the right vertical axis. The flux in the magnet is found by multiplying the flux density by the area of the magnet:

The air gap flux can then be found by dividing the magnet flux by the leakage factor:

This is a ballpark approach used by Ireland (1968) and Puchstein (1961). The effects of magnet overhang should be included as they add some additional flux.

The method used here also predicts the permeance coefficient, but the effect of slots in the armature on the magnetic air gap length is accounted for using Carter's coefficient.

The effects of magnet overhang are predicted by calculating the permeance coefficient of each section of the overhang using methods developed by Roters (1941).

The magnetic air gap length Lgl is determined by Carter's method, but first you must calculate the circumferential width of the armature slot, as follows.

Given the following dimensions ( Ills. 83 and 84):

Ra = outside radius of the armature lamination

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

Nat = number of armature teeth

Rpole = radius of the permanent-magnet pole face

FGR. 83 PMDC motor cross-section.

To find the tooth pitch angle, in degrees, between two consecutive tooth centers:

Tooth pitch, the circumferential distance between two tooth centers, in inches:

Width of the armature slot along the circumference, in inches:

FGR. 84 Permeances for stack, corner and end.

The magnetic length of the air gap is longer than the mechanical length because the lines of flux fringe through the armature slot toward the armature teeth. The following is a mathematical method for determining Carter's coefficient KC, which accounts for this fringing.

Effective magnetic air gap length, in inches:

Next, calculate the leakage factor s. This is determined by finding the total permeance factors, including the leakage permeance factors at the ends of the magnet and along the edges of the magnet.

Now that you know the magnetic air gap length, determine the magnetic areas of the air gap between the armature and the magnet.

Geometric mean radius of the magnet, in inches:

...where Rhi is the radius of the inside of the housing.

The effective magnet area is the arc distance of the pole at the geometric mean radius times the mechanical stack length, in square inches:

Lstk = length of armature lamination stack The average air gap radius is the distance from the center of the armature to the center of the air gap, in inches:

Air gap area over the stack, in square inches:

The permeance factor (or permeance path) for the area between the magnet and the armature stack is the ratio of the area of the air gap over the stack to the magnetic air gap length, in units of inches (these units may not make sense at this time, but they cancel out when calculating the leakage factor, s):

To account for the flux at the corner, as shown in Ills. 4.84 and 4.85, the effective air gap length at the corner, in inches, has been empirically determined to be

The effective radius of the air gap at the corner is the distance from the center of the armature to the pole face minus 65 percent of the magnetic air gap length, in inches:

The circumferential length of the gap at the corner, as shown in FGR. 86, is the arc length of the radius of the air gap at the corner swung along the arc of the magnet, in inches:

Area of the gap at the corner, in square inches...

Permeance factor of the gap at the corner:

FGR. 86 Permeances of corner and mean flux path.

Area of the magnet at the corner, in square inches:

The flux at the end of the magnet segment will now be accounted for. The over hang length per end Lmoe contributes flux to the armature stack depending on the lengths of the armature radius and the air gap.

[...]

Armature Calculation

Now that you know the air gap flux, you must solve the magnetic circuit by deter mining the flux densities in the sections of the armature and shell and then the mmf drops. First, analyze the geometry of the armature laminations. This analysis gives you the following:

Gross slot area. This tells you 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.

Weights 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.

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

FGR. 87 Armature lamination with round-bottom slots.

Armature Slot Calculations

To start the calculations, first determine the following from a 88:

Ra = outside radius of the armature lamination

Ral = 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

To find the tooth pitch angle, in degrees, between two consecutive tooth centers:

Tooth pitch, the circumferential distance between two tooth centers, in inches:

Width of the armature slot along the circumference, in inches:

Width of the tooth tip along the circumference, in inches:

The method used to calculate areas and paths of the armature is the same as that used for the series motor. This information can readily be calculated by any modern computer-aided design (CAD) program.

FGR. 88 Armature lamination with flat-bottom slots.

Magnetic Circuit

Special consideration is taken to account for saturation and for the air gap.

The air gap is the area between the pole and the tooth tips on the armature. The mechanical Lg and magnetic Lgl air gap lengths were calculated earlier.

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

FGR. 89 Housing magnetic paths.

Flux is required for an electric motor to produce torque. In order to produce flux, mmf must be supplied. In the case of a PMDC motor, the mmf is supplied by the magnet. 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 magnetic field intensity in ampere-turns per inch can be determined. The mmf drop in ampere-turns for each specific area can be determined by multiplying the field intensity by the magnetic length for each specific area.

For the previously determined flux in the air gap fg, kline, the flux densities for the various paths are as follows:

If Bat >100 kline/in^2 , it needs to be corrected using the Tricky method presented in the Plot of Flux per Pole Versus Ampere-Turns Excitation subsection.

..., in the armature teeth:

..., in the armature yoke:

For the following calculation, assume that the flux is in the return path in the housing. Flux density in the housing, in kilolines per square inch:

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 Hgap, (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.

[...]

Average flux density in the magnet, in gauss:

Magnetizing intensity due to the magnet, using Mx as the slope of the magnet curve, in ampere-turns per inch:

The mmf due to the magnet is the field intensity times the radial length. MMF in ampere-turns,

If the total mmf drops in the magnetic circuit _total don’t equal within reason the mmf supplied by the magnet, then choose another total magnet flux ft and reiterate the procedure. Remember during the iteration process to use the same value for line current (preferably locked rotor current) when you recalculate the mmf drops due to armature reaction and brush shift.

Once the total mmf drops equal the supplied mmf, calculate the average permeance coefficient.

Average reluctance factor (previously estimated at 1.5):

Output

The number of armature conductors can be found by using all of the conductors in the slots of the armature.

Or, to account for the fact that the conductors are not spanning the entire flux path, two distribution factors are introduced. The winding distribution factor Kw1 accounts for the spread of the winding as compared to a full 180°.

The pole arc distribution factor Kw^2 accounts for the spread of the pole arc as com pared to a full 180°.

The effective number of armature conductors Zac can be calculated by multiplying the actual number of armature conductors times the winding distribution factor and the pole arc distribution factor.

Effective turns are also a function of the number of coils shorted by the brushes.

These formulas for Kw take that effect into account in many small motors.

Zae may be substituted in the following equations for Za.

The locked rotor current is found as follows:

… where: Et = terminal voltage

Rat = terminal resistance of the armature

Rb = brush resistance

Torque developed by the motor at locked rotor, in oz _ in:

Tdev = (2.256 × 10^-7)

Since fg is relatively constant for a PMDC motor, the only variable in this torque equation is the current I. The developed torque can be described by defining a torque constant Kt, (oz _ in)/A:

The developed torque equation now becomes… where I_load is the armature current at the particular load.

The actual output locked-rotor torque is less the internal motor friction. Internal motor friction Tfi is determined from test data on a similar motor.

Locked rotor torque, in oz _ in:

FGR. 90 Kf curve.

The true no-load speed can be calculated using the preceding values for terminal voltage, number of conductors, and air gap flux.

True no-load speed, in rpm:

The true no-load speed is the speed the motor would run at if it had no internal friction or windage. It’s the speed at which the generated voltage of the armature equals the terminal voltage.

The no-load current is determined by dividing the friction and windage torque by the torque constant.

No-load current, in amps:

… where Kf, (oz _ in)/rpm, is the open circuit damping coefficient which was previously determined from a similar motor. It’s determined by driving the motor and measuring the reaction torque.

The actual no-load speed Snl can be calculated as follows:

Now that the end points of the speed-torque and current-torque curves are known, the motor performance can be plotted as in FGR. 79.

For specific load points, the speed, torque, and current can be taken from the curves and more performance parameters can be calculated. The output power is a function of the load speed S_load and the load torque T_load.

Output power:

Input power, in watts:

Efficiency of the motor at the load conditions, in percent:

Losses. The power lost in copper windings as heat is the square of the line current times the armature resistance.

Power lost in the windings as heat, in watts:

The loss which has not been accounted for, the stray loss, is the input power - (output power + losses).

Current Densities in the Brushes and Conductors. Current densities are in units of amperes per square inch. Using the diameter of the bare wire Daw, bare the current density in the armature wire is found as follows:

Current density in the brushes, in amperes per square inch:

… where Wbr is the width of the brush, measured in the direction parallel to the armature shaft.

Brushes should be limited to about 75 to 100 A/in^2 , depending on vendor specification.

Motor Constants. Following are some of the motor constants and figures of merit for a PMDC motor.

Loaded acceleration, in radians per second squared:

where Tload = motor output torque, oz _ in

Tfe = external friction torque, oz _ in

Ja = armature inertia, oz _ in _ s^2

Jload = load inertia, oz _ in _ s^2

The torque constant Kt has already been determined.

Back-emf constant Kbe,Volts/krpm (English units):

Back-emf constant Kbm,Volts/(rad _ s) (metric units):

Mechanical time constant, in seconds:

Zero-impedance (leads short-circuited) damping coefficient, in ounce-inches per radian per second:

...where Rbnl is the no-load brush resistance (empirically determined).

Motor constant Km:

Magnetic-to-electric loading ratio for the load point:

...where fg is in lines. M should be greater than 50 for good commutation.