Direct-Current Motors: Series DC and Universal AC Performance

The process is the same for both for series dc and universal ac performance, except that in direct current there is no core loss in the field. The reactance values go to 0.

Transformer voltages go to 0, and the power factor goes to 1.The following describes the ac method.

Procedure:

1. From the dimensions of a known lamination set, calculate the magnetic circuit areas, lengths, and volumes.

2. Plot a flux f versus magnetomotive force (mmf) curve for the lamination set on a per-unit-length basis, including leakage flux.

3. Assume a field winding and an armature winding.

4. Select a line voltage.

5. Select a fundamental current

Ifund for the motor and build a matrix of Ifund versus the following:

Iload = load current

Fdl = field distortion mmf

Fbl = field distortion due to brush shift

Fnet = net mmf drop

ftemp = air gap flux per unit length

fact = actual air gap flux

Creal = real (or resistive) components of the phasor diagram less back-emf

Etf = field transformer voltage

Vim = imaginary (or reactive) components of the phasor diagram

qpf = power factor angle

PF = power factor

Vreal = real components of phasor diagram

Eg = back-emf or generated voltage

Sload = load speed

Tdev = developed torque

Tfw = friction and windage torque

TCL = torque loss due to core loss

Tload = output torque at calculated load speed Pout = output power, Watts

Pin = input power, Watts

Eff = motor efficiency

Performance Calculations

Using the lamination set of FGR. 91, and knowing the stack length Lstk, we calculate the magnetic circuit lengths, areas, and volumes.

Air Gap Flux Versus MMF Drop

We will create an f versus mmf curve per inch of stack, so for these calculations we will assume a 1.0-in stack height. Therefore, the magnetic area of the air gap in square inches is Agap = Wgap _ 1.0 (4.464) We must first make an assumption for the maximum value of air gap flux. We will use the following formula:

…where: fmax = maximum air gap flux, kline

Bmax = maximum flux density, line/in^2

AT, atm = total armature tooth area per pole, in^2

Starting with fg = 0 and iterating toward fmax, we can now create our graph. See Ills. 4.92 and 4.93 for an illustration of the flux paths. For any value of fg , the following formulas apply.

FGR. 91 Laminations of sample motor.

FGR. 92 Flux paths in the armature.

FGR. 93 Flux paths in the stator.

To determine the magnetic circuit variables in the stator, we must first calculate the leakage factor.

Calculation of Leakage Factor

The leakage factor is calculated based on several permeances. See FGR. 94 for a graphical representation of the key variables involved. First, calculate Rms, in, the mean stator radius:

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 for the area between the magnetizing field coil and the armature stack is the ratio of the area of the air gap over the stack to the magnetic air gap length, in inches:

...where µ0 is a constant defined as the permeability of free space, the value of which is constant for all of the permeance calculations and will cancel out when we calculate the leakage factor.

FGR. 94 Permeance paths in the universal motor.

Note that this value for the leakage factor works well at about half the maximum flux fmax. There will be no leakage when there is no flux (or fg = 0) and more leakage when the flux increases. The actual leakage factor will increase as flux increases. The method used by the author is to use linear interpolation from s = 0 to s1 as fg goes from 0 to fmax/2. A maximum leakage factor smax must then be assumed. smax can be determined from finite element analysis or from approximation. Normally, one is safe to assume a maximum leakage factor of 1.2 to 1.3. Linear interpolation can be used so that s = s1 to s = smax as fg increases from fmax/2 to fmax.

To calculate the magnetic circuit variables in the stator, we take each section of the stator as follows:

Friction and Windage

Next, we must calculate the torque loss due to friction and windage. Friction and windage is more complicated to determine for a universal motor than it’s for a PMDC or brushless dc (BLDC) motor. In order to calculate the friction and windage constants for a universal motor, the motor must be driven with another motor, usually a PMDC or BLDC motor. To calculate the friction and windage constants of the driver motor, the motor must be disconnected from any load sources, such as a dynamometer. Then, by recording speed versus current, the friction and windage constants can be determined as follows:

1. Multiply current, A, by the torque constant Kt , (oz _ in)/A, to get friction and windage torque, oz _ in.

2. Plot speed, rpm, versus friction and windage torque, oz _ in.

3. The slope of the best line fit going through this curve will be Kfe, (oz _ in)/rpm. If desired, Kf, oz _ in _ s, can be calculated by multiplying by 60/2p.

4. The intercept of the line is Tfi, oz _ in.

To determine the friction and windage constants for the universal motor, connect the driver motor's shaft to the test motor's shaft with a coupling. Repeat the test to obtain a curve like that shown in FGR. 97, record the speed versus current, and calculate the constants as follows:

FGR. 97 Friction and windage torque.

1. Multiply current, A, by the driver motor's torque constant to get the total friction and windage torque, oz _ in.

2. Calculate the driver motor's friction and windage torque from the following equation:

3. Subtract the driver motor's friction and windage torque from the total friction and windage torque to get the test motor's friction and windage torque.

4. The slope of the best line fit going through this curve will be Kfe, (oz _ in)/rpm, for the test motor. If desired, Kf , oz _ in _ s, can be calculated by multiplying by 60/2p.

5. The intercept of the line is Tfi, oz _ in, for the test motor.

6. Alternately, a polynomial may be used to determine the curve.

The load torque is then calculated by subtracting the friction and windage torque and the torque loss due to core loss from the total developed torque:

Power, in watts, can be calculated from the following two formulas:

The efficiency can be calculated from the following formula:

Table 2 shows the calculated results for the sample motor. Fgr. 98 shows the speed-torque curve of the calculated results.

FGR. 98 Calculation of speed-torque curve of sample motor: _ = speed,_= current.