Dc servomotor users sometimes find themselves with the right motor envelope, but the wrong winding characteristics for the job at hand. Here is a quick way to calculate motor parameters that result from modifying a motor winding

Increasingly, equipment designers who use dc servomotors find that they need customized motor windings to match specific applications. For example, in a retrofit use, the motor’s envelope and footprint might already exist, but the performance characteristics no longer fit. Or, because of hardware economies, an OEM designer of short-run or single-unit customized machinery may prefer not to change motor physical characteristics, but needs to change performance characteristics to suit individual applications.

An armature-winding change affects many parameters, including the torque constant *K _{t}*, voltage constant

*K*, armature resistance Ra, and armature inductance

_{e}*L*.

_{a}It would help motor users much to have a way to quickly calculate new motor parameters and assess their effects on the servo system. A winding change is nothing more than a new combination of number of coil turns and magnet wire gage. The procedure is simple once you define the “load points.” On the basis of the applications, each load point is defined by the torque and speed the servomotor needs to do its tasks.

### Motor power

We must revisit some of the figures of merit that dominate dc motor design; the definition of motor parameters affected by winding changes; and their intrinsic relationship. To begin, let us define motor power output as the product of torque and speed measured at the motor shaft for a specific load point, that is:

*P _{out}* =

*Ts*(1)

where

*P _{out}* = Output power, W

*T* = Torque, Nm

*s* = Shaft speed, rad/sec

or

*P _{out}* =

*Tn*/7.04 (1a)

where

*P _{out}* = Output power, W

*T* = Torque, lb-ft

*n* = Shaft speed, rpm

Equation (1) shows that, to get the same continuous power output at a higher torque, you need only a proportional reduction in speed. Conversely, a higher speed at the same power output means a torque change inversely proportional to the speed increase.

Due to inefficiency, not all of the power input, *P _{in}*, to a motor becomes power output. The difference between

*P*and

_{in}*P*constitutes the motor losses. DC motor ratings depend on the motor’s ability to dissipate the heat created by the losses without exceeding its maximum operating temperature. There are limitations in power output capability in terms of torque, or speed, or both.

_{out}The new winding must not compromise the maximum safe temperature for the given motor size. Because the initial heat rise usually comes from armature current, this automatically sets a maximum current permissible in the armature windings for a given time. The maximum current also limits the amount of output-shaft torque the motor can produce. The designer also designates a maximum safe speed for the motor, usually depending on its rotor diameter or, if it is a brush-type motor, on number of commutator segments. In other words, there will be limits to the motor’s ability to meet the desired load points.

### Motor constants

Motor torque is directly proportional to the developed current in the armature windings thus:

*T* = *K _{t}I* (2)

where

*T* = Torque, Nm

*K _{t}* = Motor torque constant, Nm/A

*I* = Armature winding current, A or

*T* = [2.254310^{-7}(*zΦp/a*)]I (2a)

where

*T* = Torque, oz-in.

*z* = Effective number of series conductors per coil

*Φ* = Magnetic flux in webers, wb

*p* = Number of poles

*a* = Shaft speed, rpm

I = Current, A

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In Equation (2), the current I can have a maximum peak value *Ipeak* which produces a peak torque *Tpeak*. Sustaining *Ipeak* for excessive periods harms the motor because it develops high temperatures. Also, exceeding *Ipeak* demagnetizes the motor magnets.

Motor speed is directly proportional to the input voltage, *E*, applied to the motor terminals, thus:

*s* = *E/Ke* (3)

where

*s* = Shaft speed, rad/sec

*E* = Input voltage applied to motor terminals, V

*Ke* = Motor voltage constant, V/(rad per sec)

The voltage constant, *Ke*, is sometimes also called the back EMF constant. Also, if it is given in units of V/(krpm) (volts per thousand rpm), then shaft speed, s, is in krpm. *Ke* is fully dependent on motor structural design. Constants *Kt* and *Ke* have the same numerical value in the International System of Units (SI). In the English system they relate thus:

*Kt* = 1.3524*Ke* (4)

when

*Kt* is in oz-in./A

and

*Ke* is in V/(krpm)

Equation (4) shows that the voltage constant is also directly proportional to the effective number of series conductors per coil, the magnetic flux, and the number of poles. Here, we will focus on how to quickly calculate the new *Kt* and *Ke* caused by a winding change.

### Winding constants

Besides affecting *Kt* and *Ke*, a new winding affects other parameters such as resistance and inductance. Armature resistance changes because it relates to resistivity, length, and area of the wire:

*R* = *Qcul/A* (5)

where

*Qcu* = Resistivity of copper, Ω-m

*l* = Length of wire, m

*A* = Cross-sectional area of wire, m2

A winding change also causes a change in inductance, because inductance depends on total magnetic flux through a coil of a given number of turns, and current linked by the coil:

*L= NΦ/I* (6)

where

*L* = Inductance in henries, H

*N* = Number of turns in coil

*Φ* = Total magnetic flux in webers, wb

*A* = Cross-sectional area of wire, m2

Thus, you can write:

*Φ = NIA/lΦ*

where

*lΦ* = Length of wire in coil, m

That value of f in Equation (6) produces

*L = N2A/lΦ* (6a)

Equation (6a) shows that armature inductance is proportional to the square of the number of turns.

Armature resistance and inductance are figures of merit to servomotor users. The motor’s time constants change if the resistance or inductance — or both — change. The electrical time constant is the ratio of the winding’s inductance and resistance:

*te = L/R* (7)

where

*te* = Electrical time constant, sec The mechanical time constant is

*tm* = *RJ/KtKe* (8)

where

*J* = Rotor moment of inertia, kg-m2 and the entire equation is in SI units.

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### The method

Equations (2) through (8) show that many motor constants and other figures of merit depend on winding configuration. Consequently, a change in those parameters will affect the user’s power supplies and also the servo controls. The servo system designer benefits from finding an accurate value for these constants and motor parameters without having to wait for the motor designer to design the new winding.

Two simple, straightforward assumptions follow without loss of generality:

• The motor at hand, which will undergo the winding change, performs adequately at a desired full-load point.

• The slot fill of that motor winding is adequate.

If the first assumption is valid, then the motor operates at acceptable efficiency and the losses in the winding do not compromise the motor’s allowable maximum temperature.

If the second assumption applies, then the new winding calculations assure adequate slot fill, that is, good copper and iron utilization.

Let us assume that the servomotor user wants to change a motor winding so the motor will run at a faster new speed without changing power supply voltage. This immediately calls for a reduction in voltage constant, and consequently, in torque constant.

A look at Equations (2) and (3) shows that the new winding requires fewer turns because, as shown before, Ke and Kt are directly proportional to effective number of turns.

A straight reduction in the number of turns would leave some empty space in the slots. For the sake of efficiency, the slot area must be filled with a good percentage of copper, usually about 60%, but possibly nearly 100%. Larger-diameter magnet wire is needed to avoid “low slot fill.” The larger wire packs the empty area of the slot.

Magnet wire comes in only specific gages or diameters, so the motor designer has only a fixed number of selections to make. In the American Wire Gage system, the ratio of conductor cross-sectional areas between any two consecutive wire gages is always about 1.26. Also, gage number increases with decreasing wire diameter. For instance, the area of magnet wire gage 20 AWG doubles with a change of three wire gages to 17 AWG, because its area increases by the same factor, 1.26, for every consecutive gage change as in Table 1.

### Example

Consider, for example, that you have new requirement for a motor, calling for 2.5 times higher speed with no change in input voltage from the power supply. Also, the motor must operate at the same power output and efficiency as before. Furthermore, the present winding is 45 turns of 25 AWG per coil.

Equation (1) shows that the torque requirement must be 2.5 times less. According to Equation (3), the motor voltage constant, *Ke*, must also be reduced by a factor of 2.5. It follows that, to reduce *Ke* 2.5 times, you need only reduce the number of turns by 2.5. But that makes the slot 2.5 times less full. To get better slot fill, you must find a new gage wire.

The problem reduces to having to solve for the number of wire gages by which you must increase the original wire, so the new lower number of turns occupies the same slot area. In other words, solve for the value of *r* in this relationship:

*Kt*(1.26)*r* = *Kt*/2.5

where

*r* = Number of changes in AWG size

From which

1.26*r* = 1/2.5

or

*r*log (1.26) = log (0.4)

Solving for *r*,

*r* = -4

that is, four wire gage numbers smaller, which is four wire sizes larger.

Thus, the new winding will have

45/2.5 = 18 turns of wire of

25 - 4 = 21 AWG size.

Because armature resistance is a function of both the length and area of the conductor, the new armature resistance will be:

*Rnew* = *Rold*/(1.262)*r*

with

*r* = 4

From which

*Rnew*=*Rold*/6.35

Because armature inductance is proportional to the square of the number of turns, the new armature inductance will be

*Lnew*= *Lold*/(1.262)*r*

with

*r* = 4

which means that the new inductance is also reduced by a factor of 6.35.

Because wire area increases in the new winding, the conductor can survive higher current. The increase is proportional to the new wire area. From Equations (1) and (2) you can deduce that the new peak current can be

1.26*r* = 1.264

times larger than the old peak current. Also, the new maximum dc voltage (to maintain the same maximum power input) *decreases* in proportion to 1.264.

The preceding calculations show that the changes between old and new constants involve only the factor, 1.26, and the number of wire-gage step changes, *r*. The bulk of the work is merely in finding the value of *r* and using it to find the new motor parameter values.

A caution: Although this simple method applies to brush-type and brushless dc motors, be aware that brush resistance has been left out of the equations. However, the terminal resistance for a brush-type dc motor includes the resistance of the armature winding *and* of the brushes.

*Some brush-type dc motors are of the wound-field type. Be careful not to confuse the terminology. The term “armature” as used in this article strictly designates the windings that carry load current. Depending on the type of motor, armature windings reside in the rotating or stationary elements.

**P. Ramon Guitart** is Design Manager, DC/Servo Motors, at the Gallipolis, Ohio facility of Reliance Motion Control, Reliance Electric, Cleveland.