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TOPIC: The Rationale for Equivalency of Force Constant and Back EMF Constant in Linear Motors
Scope:
Customers have inquired as to why the Airex linear motor specification values for back-EMF (KE) and force
constant (KF) are equal. This question arises because some linear motor manufacturers define these parameters
as unequal. This TechTip Series provides insight into how Airex defines and tests these values and will clarify
outstanding questions around the equivalency of the linear motor force constant and back-EMF constant.
Discussion:
The force constant (KF) is a measure of how much force a motor develops per unit of applied current, while the
back-EMF constant (KE) is a measure of how a motor behaves as a generator when the coils move relative to the
magnetic field. To start, there are two decisions regarding methodology explaining why Airex has set the force
constant and back-EMF constant to be equal in its work with linear motors.
First, when analyzing an ironless linear motor application, Airex operates in metric “SI” units (previously “MKS”)
in lieu of English units. As a result, the values for KE (volts/meter/second) and KF (newton/ampere) have the
same numerical value. Airex applies this methodology because it eliminates various scaling factors and requires
additional constants. Since the two parameters are developed in the same physical configuration and magnetic
field, they are, therefore, based on the “same principles”. Secondly, Airex rates both KE and KF using “minimum
expected” peak values instead of root-mean-square (RMS) values. Both of these methods help illuminate the
equality Airex imposes on force constant and back-EMF constant.
Next, in order to further demonstrate the rationale of these chosen methods, it is necessary to recognize that an
Airex linear motor uses a distributed coil. Furthermore, the magnetic field is fairly constant over a relatively wide
area of each magnet, in relation to coil width, such that the “minimum peak” value is valid over a particular
commutation cycle step width. The resulting voltage generated is called the back-EMF and is a direct
consequence of the “Generator Equation” as depicted below. The force constant is a direct result of the “Motor
Force Equation”, also shown below, where 1 ampere of current is applied to the motor coil and with the coil
physically located within its commutation step location. Additionally, the force constant is treated as a “minimum
peak” value. These two basic principles, the “Motor Force Equation” and “Generator Equation” are at work in all
motors regardless of number of phases or construction type.
Motor Force Equation:
(1)
F = Blin
Where:
F = Motor force, Newton (N)
B = Magnetic Flux Density, Tesla (T)
l = Length of conductor, meter (m)
i = Current, Ampere (A); equal to 1 for force
constant
n = number of turns (dimensionless)
Generator Equation:
(2)
V = Blvn
Where:
V = Generated voltage, volts/meter/second
(V/m/s)
v = relative velocity between coil and
magnetic field, meters/second (m/s)
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m 2 kg
V
=>
Since the fundamental units defining the volt are
and the fundamental units for a newton are
s3 A
mkg
N => 2 whereas kg = kilogram, S = second and A = ampere, it is, therefore, clear that the value of
s
KE in (V/m/s) must have the same numerical value as KF in (N/ampere). Both are directly related by the
strength of the magnetic field, the length and number of coil conductors exposed to the magnetic field.
As an example, consider a motor phase with a 1 newton/ampere KF, which will have a KE of 1 volt per meter
per second. When moving at a speed of 1m/s, the motor will develop a peak voltage of 1 volt (peak to peak
= 2 volts, approximately RMS = .7 volts) while the center of the coils within each motor phase passes directly
over the center of the magnets. A current of 1 ampere applied to a phase of the same motor will produce a
force of 1 newton. Since an Airex linear motor contains distributed coils, the force constant for a single
phase will be the same as the “line to line” force constant for a Delta connected motor. The WYE connection
will yield a force constant and back-EMF constant equal to the “square-root of 3” times the force constant of
either a single phase or the Delta connection.
To summarize, Airex uses methodology which apportions KE and KF to be set as equal. Moreover, the effect
of KE and KF is clear from the physical effort exerted when manually “pushing” the motor coil through the
magnet track with the motor phase leads short circuited together. The faster one pushes the motor through
the magnet track, the more current flows through the short-circuited motor phases and the more resistant
the motor is to the motion. This linear motor evaluation, as described here, provides empirical evidence
and explains why Airex sets equal values for both the force constant and the back-EMF constant.
Reference:
Please contact an Airex representative on this or any TechTip as well as for assistance with motor sizing and
selection.
Airex Corporation
15 Lilac Lane
Somersworth, NH 03878
T: 603-841-2040
F: 603-692-4317
E: info@airex.com
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