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Building a Brushless Electric Motor
Sam Redmond and John Strong, Menlo School, Atherton, California 94027Dr. James Dann, Applied Science Research
October 23, 2013
Abstract
We designed and constructed an adaptable electric DC motor for Applied Science
Research in order to improve our electrical and mechanical engineering abilities and
familiarize ourselves with the Whitaker Lab. Motors are an omnipresent part of daily
life, and it is vital for any prospective engineer to have a deep understanding of how
these crucial components work. We designed and built two circuits, one frame, and
two axles to test our motor. We experimented with a variety of other techniques for
motor optimization. Our motor reached an RPM of 4545.6 RPM, a startup torque of
0.0625 N·m, a running torque of 0.0054 N·m, a power output of 0.976 W, an efficiency
of 1.94%, and a RPM to volt ratio of approximately 394.
1
Contents
1 Motivation and History 6
1.1 Why in ASR? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Big Picture Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Similar Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Theory of Operation 8
2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Differential and Integral Forms . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Maxwell’s Equations in Standard English . . . . . . . . . . . . . . . . 10
2.1.3 Relationship between Electricity and Magnetism . . . . . . . . . . . . 10
2.2 More Electromagnetic Equations . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Kirchhoff’s Junction Rule . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.4 Electrical Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Breadboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.5 Hall Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.6 Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Rotational Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 Ball Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Motor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.1 Poles and Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.2 Torque Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2
3 Design 17
3.1 Design Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.5 RPM vs. Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Component Specs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 Hall Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.4 Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.5 Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.6 Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Hall Chip, Transistor and Relay . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Hall Chip and Transistor . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 Poles and Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Axles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.1 Small Radius Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.2 Large Radius Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Results 28
4.1 RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Startup Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Running Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 A Note on Calculating Running Torque, Power, and Efficiency . . . . . . . . 32
4.5.1 Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5.2 Changing Velocity, Constant Acceleration . . . . . . . . . . . . . . . 33
4.5.3 Practical Choice of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 RPM vs. Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3
4.7 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.7.1 Circuit Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.7.2 Power vs. RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Conclusion 37
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.1 Flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.2 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.3 Equivalent Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.1 Non-Wobbling Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.2 Shorter Driveshaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.3 H Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.4 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Acknowledgements 40
A Appendix A - Arc Magnets 42
B Appendix B - H Bridge 42
B.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
B.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
C Appendix C - Power Circuit 44
D Appendix D - Extra Physics 44
D.1 Magnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
E Appendix E - Measuring Torque 46
F Appendix F - Python Code 46
List of Tables
1 Comparison of Linear and Rotational Motion . . . . . . . . . . . . . . . . . 15
4
2 Motor Specifications Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 28
List of Figures
1 Henry Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Relay Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Simple, Efficient Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Primary Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Torque Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6 RPM Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 Startup Torque Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8 Running Torque Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 RPM vs. Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10 Power vs. Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
11 H Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
12 Power Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5
1 Motivation and History
1.1 Why in ASR?
Building a DC motor is a perfect way to reinforce several concepts that we learned during
high school science classes, while still leaving room to explore several new fields. This
project reviewed concepts such as magnetic fields, directional forces, friction and resistance,
all applied in a hands-on manner. Not only did the project provide intellectual stimulation
regarding previously learned concepts but also allowed for the acquisition and development
of new skills in the machine shop and ASR lab. We learned to use a miter saw and drill
press to build the frame of our motor. The conceptual circuits of physic class are brought to
life with actual transistors, hall chips and resistors all plugged into solderless breadboards.
It is hard to think of a better project to start the ASR year off.
1.2 Big Picture Motivation
Motors play a large role in the modern world and are an integral part of industry. Without
motors, modern mass production would be impossible, making it exceedingly difficult to pro-
vide for a growing world population. Electric motors also have their hands in the operation
of aircraft and other modes of transportation. Almost all electrical appliances rely on electric
motors in some way or another. Many toys have even adopted motors in order to become
automated. Their importance is increasing as they appear to be a suitable replacement for
combustion engines in cars.
1.3 Similar Projects
1.3.1 Historical
When discussing the history of the electric motor, it is important to first talk briefly about
the Faraday motor (Dr. Dann’s personal favorite). It consists of a wire running through
a container of mercury with a permanent magnet at the center of the container. When
electricity is sent through the wire it creates a magnetic field which causes the wire to spin
around the magnet. Without any kind of axle, it was less of a motor and more of a device
to translate electrical energy into mechanical motion. Developed in 1821 by the British
scientist Michael Faraday, it was the first motor ever created; however, since it shares very
few similarities to the modern motor, it is fairly irrelevant to the ASR motor projects.
6
Figure 1: Joseph Henry’s oscillating beam motor was an early DC motor.
The oscillating beam motor, an early predecessor of the DC motor, was designed and
created by Joseph Henry of Yale in 1831. While his motor did not have an axle, a trend
with early motors, it is relevant to this paper because it was powered by direct current.
DC powers most of the ASR motor projects. Like a seesaw, it rocks from side to side in an
oscillating motion. When one of the electromagnets is down, its leads touch the power source,
causing current to flow through the magnet and the polarity to switch. The electromagnet
now repels from the permanent magnet below causing it to move upwards forcing the other
side of the pivot down. The other side then repeats the same cycle. The oscillating beam
motor?s simple mechanical circuit acts much like a reed switch. However, instead of sensing
the magnetic field to complete the circuit it senses the physical proximity to the permanent
magnet in order to know when to switch. This design was very crude and did not have any
practical purpose. Luckily, over the next hundred years motors evolved greatly. [1]
The motors most similar to the ones built during ASR are brushless DC motors. They
were developed in 1962 by T.G. Wilson and P.H. Trickey. They named their system “a DC
machine with solid state commutation.” The permanent magnets were placed on the axle
7
with electromagnets on the outside, essentially turning the brush motor inside out. They
were viewed as superior to the brush motor, since the spinning portions required no physical
contact with the outside, making them last much longer. Because of this new found reliability,
brushless motors were used in devices such as computer discs, robotics and airplanes. Aircraft
were greatly benefited by the brushless aspect since with the low humidities experienced
during flight, brush motors usually would wear out very quickly; however, since brushless
motors do not make physical contact with the surrounding solenoids they do not wear out as
quickly. These motors became more popular during the 1980s when the supply of permanent
magnet materials became available in higher quantities. Even though initial brushless motors
were much more reliable, they lacked power. Once permanent magnets were coupled with
high-voltage transistors, the problem was easily remedied. [2]
1.3.2 Modern
A great modern example of a brushless dc motor is the joint-lapped core motor developed by
the Mitsubishi Electric Company in 2002. It is a six pole concentrated winding motor which
was 3 percent more efficient than its 4 pole counterparts. Not only did the six pole motor
increase the efficiency but it also reduced the noise, which was a common problem amongst
winding motors. This motor is similar to the ones developed during class since they both
fall into the classification of brushless DC motors and both are optimized a great deal. This
means that the rotors contain permanent magnets which are surrounded by electromagnets
on the outside. Like all brushless motors there is an air gap separating the core and the
solenoids ensuring that the two do not touch and cause wear. The developers also focused
on improving the electromagnets’ strength through optimizing their form factor. Our group
also focused heavily on optimizing our electromagnets; however, we did so in a different
manner. [3]
2 Theory of Operation
Here, we extensively review the physics that underlie our motor. We only assume a basic
knowledge of physics. We’ll address the rest. Some results are stated without proof. These
results are proved in detail in AP Physics C, and proofs are available on demand from Sam.
8
2.1 Maxwell’s Equations
In 1861, James Clerk Maxwell published a set of four equations that unified physicist’s
understanding of electricity and magnetism. Other scientists, such as Gauss, Faraday, and
Ampere, had made some discoveries individually, but these are “Maxwell’s Equations” - not
“Gauss’s” or “Faraday’s” or “Ampere’s” because Maxwell was the one who demonstrated a
mathematical symmetry between all the other equations, unifying electricity and magnetism
into electromagnetism [4].
2.1.1 Differential and Integral Forms
Expressed as integral equations, Maxwell’s equations are:
∂Ω
~E · d~S =1
ε0
y
Ω
ρdV (1)
∂Ω
~B · d~S = 0 (2)
∮∂Σ
~E · d` = − d
dt
x
Σ
~B · d~S (3)
∮∂Σ
~B · d` = µ0
x
Σ
(~J + ε0
∂ ~E
∂t
)· d~S (4)
Expressed as differential equations, Maxwell’s equations are:
∇ · ~E =ρ
ε0
(5)
∇ · ~B = 0 (6)
∇× ~E = −∂~B
∂t(7)
∇× ~B = µ0
(~J + ε0
∂ ~E
∂t
)(8)
In these equations, µ0 is the permeability of free space, and ε0 is the permittivity of free
space. Both are universal constants (µ0 ≡ 10−7
4πV ·sA·m and ε0 ≈ 8.8542 ∗ 10−12 F
m), related by
9
c = 1√ε0µ0
, where c, the speed of light in free space, is defined to be 299, 792, 458ms
. The
concepts of relative permeability and relative permittivity in a material exist, and can be
used to define the speed of light in a material.
∇ is the gradient in 3D, so ∇· is the divergence and ∇× is the curl.
Ω is a closed volume with bounding surface ∂Ω and locally planar differential surface
d~S, directed perpendicularly out of the surface and with magnitude equal to the area of the
plane. Σ is an open surface with bounding contour ∂Σ and locally linear differential line d`
parallel to ∂Σ.~E is the electric field at a given point in space and ~B is the magnetic field at a given
point in space.
ρ is the charge density at a point in space, in units of Cm3 , related to total charge enclosed
in a surface by Qtot =t
ΩρdV . ~J is current density, in units of A
m2 , related to net current
though a surface bys
Σ~J · d~S
Why do we bother with both forms of the equations? The integral forms describe electro-
magnetic behavior in a global region of space, whereas the differential forms describe local
electromagnetic behavior at a point.
2.1.2 Maxwell’s Equations in Standard English
Gauss’s law states that the electric flux through a closed surface is proportional to the net
charge contained inside. Gauss’s law for magnetism states that the magnetic flux through
any closed surface is always zero (since there are no magnetic monopoles). Faraday’s law of
induction states that a changing magnetic flux (with time) will induce an electromotive force
(and thus an electric current) in a closed loop of wire that opposes the change in magnetic
flux. Ampere’s law states that an electric current establishes a magnetic field. Maxwell’s
correction argues that a changing electric field also induces a magnetic field.
2.1.3 Relationship between Electricity and Magnetism
Maxwell’s equations display a remarkable symmetry between electricity and magnetism,
suggesting that they are really the same force. Einstein’s general relativity offered an expla-
nation for this surprising connection - magnetic fields and electric fields are one and the same
- just viewed from different reference frames. Maxwell’s unification of Gauss’s, Faraday’s,
and Ampere’s Laws formed the defining equations of electromagnetism.
10
2.2 More Electromagnetic Equations
2.2.1 Lorentz Force
Lorentz’s force equation describes the force on a charged particle moving through a point in
space with an electric field and magnetic field. It states that
~F = q ~E + q~v × ~B
where F is the force on the particle, q is the particle’s charge (in coulombs), ~v is the particle’s
velocity, ~E is the electric field at the point in space, and ~B is the magnetic field at the point
in space.
The Lorentz force equation demonstrates that the force on a charged particle due to
an electric field is parallel to the electric field, and that force due to the magnetic field
is proportional the charge, velocity, and field strength, and is perpendicular to both the
particle’s velocity and the magnetic field.
2.2.2 Kirchhoff’s Junction Rule
Gustav Kirchhoff has two main circuit laws; however, none are remarkably insightful. The
first law, Kirchhoff’s Junction Rule, is another way to say charge is conserved. It states that
current into a junction is the same as current out of a junction in a circuit.∑Iin =
∑Iout
Kirchhoff’s second law, the Loop Rule, is a special case of Faraday’s law of induction
(Eq. 3,7) for when ∂ ~B∂t
= 0. As such, we won’t bother discussing it.
2.2.3 Ohm’s Law
In 1827, Georg Ohm noticed that the current between two points is proportional to the
voltage difference. He called the constant of proportionality the resistance of the circuit
between the two points. Mathematically,
V = IR
V has units of volts (A · Ω), and is sometimes known as electric potential. A voltage
difference can do work on a charged particle, given by W = qV . In this sense, volts are also
11
JC
. A common unit of energy in particle physics is the electronvolt (eV), which is the energy
an electron gains (or loses) moving through a potential difference of 1V. 1 electron volt is
approximately 1.6 ∗ 10−19J.
The voltage difference between two points in space can also be defined by
V =
∫ A
B
~E · d~l
A voltage difference, provides an electromotive force (E ) which will establish a current
in a circuit.
It’s worth noting that voltage difference is the relevant factor. There is no such thing
as absolute voltage (unless you define a universal reference point - as physicists have done,
letting V = 0 at r =∞).
2.2.4 Electrical Power
P =dWork
dt=V dQ
dt= IV
In a simple circuit, electrical power (in watts) is the product of current (in amps) and
voltage (in volts). Alternatively, P = I2R = V 2
R.
Note that this isn’t entirely accurate in the general case. In the presence of a nontrivial
external magnetic field, electric power is given by:∫S
( ~E × ~H) · ~dA
For simplicity, we’ll stick with P = IV for our calculations.
2.3 Circuit Elements
2.3.1 Resistor
A resistor is a simple circuit element that increases the resistance of a circuit. Resistors tend
to range from ohms to megaohms, but of course there exist larger and smaller resistances.
2.3.2 Breadboard
A breadboard is a simple piece of equipment that simplifies wiring up circuits - rows are
connected, and the power and ground columns are connected, reducing the necessary number
of alligator clips and solder points.
12
2.3.3 Inductor
An inductor, also known as an electromagnet or solenoid, is a coil of wire. When current
passes through an inductor, a magnetic field is established (due to Ampere’s Law). In an
ideal solenoid, there is no magnetic field outside of the bounding cylinder, and there is a
uniform field along the axis of the electromagnet. Without proof, the magnetic field of a
solenoid is
~B =
µni : inside
0 : outside
where n in the turn density of the solenoid (turns per meter) and i is the current that runs
through the solenoid.
An exceedingly important calculation is that of the magnetic field on the axis of the
solenoid away from the solenoid. Circular loops of wire act as magnetic dipoles, so the field
strength falls off as distance cubed.
Often, an electromagnet will have an iron core, because the relative permeability (µ =
kµ0) of iron is around 2,000 times that of air. Thus, an iron core focuses and magnifies the
magnetic field.
Electromagnets are used in practice because they can convert an electrical current to a
magnetic field. After all, electricity and magnetism are one and the same. However, due
to Faraday’s law of induction, as current to a solenoid is increased, and a magnetic field
is established that pierces the surface defined by turns of coil, a back electromotive force
is induced, opposing the change in magnetic flux through the solenoid, slowing the flow
of current. In fact, this characteristic (which is unfortunate for our motors, since there is
no instantaneous ’toggling’ of electromagnets) is used in households as a choke to prevent
sudden surges in current from destroying electronics such as a laptop or AC unit.
2.3.4 Transistor
A transistor is an amplification element. There is an incredible amount of research and
knowledge about transistors (since they underlie essentially every electronic device ever).
There are many different types of transistors. In the course of this project, we experimented
with bipolar junction transistors (BJTs). BJTs can be either PNP or NPN, which refers to
the polarity of silicon doping (positive-negative-positive or negative-positive-negative).
A way to think of transistors is to imagine a regular wire with a wall in the middle (the
wall being the middle negative or positive section). No current can flow through our wall.
However, when we drain the base by providing an opposite charge (such as electrons to the
13
positive section, or electron ’holes’ to the negative section), we lower the wall, and current
can flow through the main path. So, a small amount of current controls the height of the
wall (base), which in turn controls how much current can flow through.
In general, transistors have three pins out, the base (B), the collector (C), and the emitter
(E). We’ll focus on just NPNs for now.
If there is a voltage difference between the base and collector of more than 0.7V, current
will be able to flow from the collector to the emitter. However, it’s worth noting that
transistors aren’t exactly switches. There is some bleeding of current for different voltage
differences (in our analogy this means that the wall is partially, but not entirely, lowered).
For more information, refer to The Junction Transistor, by Morgan Sparks (Chris Sauer’s
HIWW Paper from last year).
2.3.5 Hall Chip
A Hall chip is a circuit element that exploits the Hall Effect in order to control flow of
electricity with an external magnetic field. When a current runs through a semiconducting
chip in the presence of an external magnetic field normal to the chip, negative electrons and
positive electron ’holes’ are deflected to opposite sides of the chip (depending on what type
of semiconductor it is) due to the magnetic force on the particle (see Section 2.2.1). This
differentiation of charge sets up an electric field across the chip (much like that between two
plates of a capacitor) which counterbalances the magnetic force until equilibrium is reached,
at which point current can flow normally again. The Hall voltage refers to the voltage
established by this separation of charge, and that voltage can be used in an external circuit.
A Hall chip typically has three pins. The first (Vcc) powers the Hall element. The second
pin (GND) goes to ground, and the third pin (OUT) is usually connected to two wires; one
supplies a current, the other reroutes the current when the Hall chip is off.
Internally, a Hall chip works by harnessing the Hall voltage to power the base of a
transistor. When there is an external magnetic field, the transistor is active (current can
flow through), and current can pass through OUT to GND, and does not enter the rest of
the circuit. When there is no magnetic field, and the transistor is inactive (no current can
flow through), current cannot go from OUT to GND and is instead rerouted out the second
wire, where it can be used in the rest of the circuit.
Essentially, a Hall chip allows a magnetic field to control whether or not there is current
in one wire (the ’reroute’ wire from OUT).
14
2.3.6 Relay
A relay is an electrical switch controlled by an electrical current. Current through and
inductor at one end of the relay controls the switch positions (on or off) of one or two
switches. When current runs through the relay’s inductor, a magnetic field is generated that
pulls the metal switches to a new pin out. Relays are effectively fancy switches controlled
by another circuit.
2.4 Motion
There are substantive parallels between linear mechanics and rotational mechanics, as seen
below.
Table 1: Comparison of Linear and Rotational Motion
Linear Motion Rotational Motion
Name Equation Equation Name
Position x θ Angular Position
Velocity dxdt
= v ω = dθdt
Angular Velocity
Acceleration dvdt
= a α = dωdt
Angular Acceleration
Mass (Linear Inertia) m I Rotational Inertia
Newton’s 2nd F = ma τ = Iα Newton’s 2nd
Momentum p = mv L = Iω Angular Momentum
Work W = Fd W = τω Work
Kinetic Energy K = 12mv2 K = 1
2Iω2 Kinetic Energy
Power P = Fv P = τω Power
All rotational motion is defined relative to an axis of rotation. For example, a pendulum’s
rotational axis is normal to its plane of motion at the point of rotation, and a merry-go-
round’s rotational axis is vertical at the center of the ride. The notion of lever arm distance
(r⊥)- the perpendicular distance from a point to the rotational axis, is thus a well defined
concept, given a specific rotational axis.
15
2.4.1 Rotational Inertia
Rotational inertia, also known as moment of inertia, quantifies an objects resistance to
change angular velocity about its axis of rotation. The contribution to an object’s rotational
inertia of a point mass m a distance r from the axis of rotation is r2dm. Integrating, we have
I =
∫V
ρ(~r) r2 dV
where ~r is the radius vector from the axis of rotation to dV, and ρ gives mass density (mass
per unit volume) for a point in space.
For the special case of a right cylinder of mass M with uniform density and radius R,
rotational inertia is I = MR2
2.
Rotational inertia has units of kg ·m2.
2.4.2 Torque
Torque induces a change in angular momentum about an axis of rotation. This change
usually manifests itself as an angular acceleration. It is calculated by
τ = ~r × ~F = r⊥F = rF⊥
Luckily, any force applied tangent to a cylindrical axle (perpendicular to the axis of
rotation) is also perpendicular to the axle’s radius, so usually for our purposes torque will
simply be the algebraic product of force and the radius of the axle. Torque has units of N ·m.
2.5 Tools
2.5.1 Ball Bearings
Ball bearings are tools that minimize friction by reducing a complex contact surface to a series
of point contacts where the internal spheres meet the external and internal external cylinders.
Since rolling resistance is typically less that kinetic friction, ball bearings significantly lower
friction.
16
2.6 Motor Theory
An ideal DC motor constantly applies positive torque to a rotating axle, combatting frictive
losses to maintain a high rotational speed. The force that drives rotation comes from the
magnetic interaction of electromagnets and permanent magnets. Without proof, like poles
repel, and similar poles attract. For more detail, see Appendix D.1. Usually, the axle
of a motor becomes a driveshaft, which serves to connect the motor to other mechanical
components.
2.6.1 Poles and Phases
A pole is a magnet embedded on a rotating axle. A phase is a stationary electromagnet
embedded into the motionless frame. It should be clear that poles and phases should be
arranged symmetrically about the axle.
2.6.2 Torque Curves
A torque curve refers to a graph of torque as a function of angle. Since this graph is naturally
periodic, we choose without loss of generality an reference angle θ = 0 such that the torque
on the axle is at a local maximum at θ = 0. Ideally, the torque curve would be linear, so
that the torque on the axle is constant. However, since axles have nonzero radii, the torque
curve from one electromagnet on one magnet will be sinusoidal. By switching the direction of
current flow in an electromagnet (making it ’push-pull’), this curve can approximate |sin(x)|.While this is an improvement, we nevertheless want a linear curve (for uniform torque, power,
etc.), so we want to somehow approximate a horizontal line with the superposition of the
magnitude of sinusoidals. Given n electromagnets, it can be shown that the optimal phase
shift is pin
radians between adjacent electromagnets. Since we add torque curves, this phase
shift will not change regardless of the number of magnets used.
There is one problem with this approach - it neglects the force of attraction between the
embedded magnets and the iron cores of our electromagnets. Because of this, we need to
find by trial and error an optimal distance.
3 Design
Generally, we designed the motor to be adaptable, rather than optimized for a specific trait.
Interchangeable parts and adjustable mechanics allow for precise calibration and optimiza-
17
tion of every specification.
3.1 Design Goals
1. RPM
2. Torque
(a) Startup Torque
(b) Running Torque
3. Power
4. Efficiency
5. RPM vs. Voltage
3.1.1 RPM
Our primary goal was to maximize our motor’s RPM. To achieve maximum angular ve-
locity, we planned to minimize rotational inertia while maximizing torque and eliminating
unnecessary power losses (due to vibrations, for example).
3.1.2 Torque
The frame of our motor has been designed to accommodate a switch between a low-torque
high-RPM axle and a high-torque slightly-lower-RPM axle very easily. We made two axles;
one designed for high RPM and the other for torque (large rotational inertia). The magnets
of the axle designed for torque are located farther from the axle’s rotational axis. Of course,
we will have to adjust the position of our electromagnets to accommodate the larger axle,
but that is easily accomplished by changing the nut position on the electromagnets’ shafts
which fasten them to the frame.
3.1.3 Efficiency
To maximize efficiency, we focus on minimizing energy losses by using electromagnets with
little depth but large diameters, therefore reducing resistance in the wire while still main-
taining a powerful field. B = µni implies that it is turn density that matters (turns per unit
18
length), not number of turns. Therefore, having a long electromagnet is wasteful. Any addi-
tion of extra wire that fails to increase turn density only raises the resistance and ultimately
hurts the electromagnet’s efficiency. For example, a 10cm electromagnet with 1000 turns has
the same turn density (and thus strength) as a 1cm electromagnet with 100 turns. However,
the latter electromagnet has ten times less resistance. Additionally, our electromagnets are
completely adjustable, allowing us to place them at the ideal distance for motor power.
Electrical power losses are important, but more important are mechanical power losses
(largely from a wobbling frame and from rotational friction of the axle’s shaft on the frame).
To counteract this, we secure the frame with 6 L brackets on a large, stable base. We also
employ ceramic ball bearings to minimize frictional losses.
3.1.4 Power
Power is the derivative of energy with respect to time. Therefore, optimization of power will
come not from the axle, but from the setup of the electromagnets and reduction of energy
losses. We implemented all the choices detailed in previous subsections, especially those
in Section 3.1.3. We also choose to use wires and electrical components rated for higher
currents. After all, if we simply pump more current though our circuit (increasing input
power), we could output more power as well.
3.1.5 RPM vs. Voltage
We attempt to maximize RPM at a given voltage by lowering the overall resistance of our
circuits. Of course, we also implement all of our optimizations for a rotational speed (skinny
shaft, etc.) and efficiency. At a set voltage, lowering resistance increases current, which
in turn increases the magnetic field strength, torque, and thus RPM. The most relevant
increase in speed (compared to other groups) will be from our low-resistance electromag-
nets. Additional depth only adds resistance to the circuit, but doesn’t boost magnetic field
strength.
3.2 Component Specs
3.2.1 Wire
We used 26 gauge (AWG) unstranded copper wire. This wire has a diameter of 0.404 mm,
a resistance per unit length of 133.9 mΩm
, and a maximum continuous current rating of 2.2A
[5].
19
3.2.2 Resistor
We used 100Ω, 1 kΩ, and 10 kΩ resistors in our circuits.
3.2.3 Hall Chip
Our Hall chip was a latching SS461A type, meaning that it toggles output states (and locks
therein) with the presence of any external magnetic field. The Hall chip can handle a supply
voltage between 3.8 and 30V, and can accept at most 10mA of current to the supply pin. It
outputs up to 20 mA at up to 0.40 V. It can switch states in approximately 0.1µs. (0.05µs
rise, 0.15µs fall) [6].
3.2.4 Transistor
We decided to deal with only NPN transistors, rather than PNP transistors. Specifically,
we used a TIP120G NPN BJT power transistor, rated for 5A continuous collector-emitter
current, 8A peak collector-emitter current, a collector-emitter and collector-base voltage
difference of 60V, and an emitter-base voltage of 5V. The NPN is rated for maximum current
to the base of 120 mA [7].
3.2.5 Relay
We used a DS2E-M-DC5V relay, with a switching capacity of 60W or 125V and an enormous
breakdown voltage of either 1000V (between contacts) or 1500V (between contacts and coil).
Excluding contact bounce time, the relay operates, releases, sets, and resets in 3ms each.
The nominal voltage drop is 5V across the internal inductor. The relay does not latch. The
maximum voltage allowed (at 50° C) is 7.5V [8].
3.2.6 Magnets
We used cylindrical Neodymium Iron Borate (NeFeB) magnets (the strongest possible) with
radius 12.70 mm and 12.70 mm depth.
3.3 Circuitry
We designed four circuits, two of which made it to final testing. We spent a lot of time
working on an H bridge circuit (see Appendix B) for unlimited push-pull capabilities, but
20
in the end we were unable to make it work. We additional designed a power circuit (see
Appendix C), but it was quickly clear that that was unreasonable.
The two circuits we used for testing are described below.
3.3.1 Hall Chip, Transistor and Relay
The most effective circuit utilized a combination of a hall chip, a transistor, and a relay. The
circuit diagram is shown in Figure 2.
Current from the output of the Hall chip powers an NPN transistor’s base. If there is
current to the base, current from the transistor will activate the relay’s coil, switching the
direction of current in the electromagnet. If the electromagnet is off, the far switches are
closed, and current will flow one direction through the electromagnets; on the other hand, if
the transistor’s base has power, the relay’s coil is active, so the close switch position will be
enabled, allowing current to flow the opposite direction through the electromagnets.
The Hall chip is placed at such a position that the electromagnets always apply positive
torque to the axle.
This circuit allows for a push-pull electromagnet setup. However, since it relies on a
mechanical switch, it is in theory limited by the switching speed of the relay. The relay
is also somewhat inefficient (we hear rapid mechanical clicks), so we lose some power by
incorporating a relay.
3.3.2 Hall Chip and Transistor
We also built a simple, efficient circuit using just a Hall chip and transistor. The circuit
diagram can be seen in Figure 3. The Hall chip controls whether or not current flows to the
base of the transistor. The incoming current will be at a very low voltage (having passed
the pull up resistor), so any current at all (even very little) will allow lots of current to flow
from the collector to the emitter and out to the electromagnets before returning to ground.
This circuit is good because it is simple and efficient, without unnecessary components
adding complexity or reducing efficiency (such as a relay’s reliance on motion). However,
it is bad because it does not create ’push-pull’ electromagnets; rather, current through the
electromagnets is either on - in one particular direction - or off. Therefore, we can only ever
push, and are losing some potential for high RPMs. We rely on the moment of inertia of the
axle to maintain angular velocity until the electromagnets can push again.
As before, we position the Hall chip so that the electromagnets apply positive torque to
the axle. The innate attraction of the magnets to the iron core provides a pull.
21
Pull-up Resistor
1 kΩ
Hall Chip
GND
Out
Vcc Power Relay
R1
1 kΩ
NPN
Electromagnet
Electromagnet
Figure 2: Current from the Hall chip’s output powers the base of an NPN transistor. Theemitter of the NPN transistor goes to the relay’s coil. Clever wiring from the relay’s out pinsallows the transistor’s signal to control the direction of current flow in the electromagnets.The relay is grounded through a resistor for safety.
22
Power
20 V
Pull Up
1 kΩ
NPN
Hall Chip
Out
GND
Vcc
Electromagnet Electromagnet
Figure 3: Simpler is better. The Hall chip will toggle (a small amount of) current to thebase of the transistor. The NPN will allow current to flow to the electromagnets if and onlyif there is current to its base.
23
3.4 Frame
The frame was designed to support up to four electromagnets. While the third and fourth
electromagnets were never added due to time constraints, they easily could have been added,
given a later deadline. The two solenoids, which rest in the center of the frame, are supported
by a unique design which allows for the magnets to be scrolled in and out in order to
accommodate different sized axles. By sliding the backside of the solenoid’s iron core, an
iron bolt, though a drill pressed hole in the center of the frame, nuts can attached on
both sides of the frame. When these nuts are loosened or tightened together they slide the
electromagnet towards or away from the axle. The upright pieces of wood, to which the
electromagnets are fastened, remain very rigid and stable through the use of L brackets.
One L bracket was attached on the inside of each upright and two on the outside. We did
not use L brackets to secure the cap of the frame, since it needed to be removable in order to
change between the different axles. To make this possible, small brass pins were inserted into
holes both in the top of the uprights as well as the bottom of the cap. The holes were snug
and still retained the structural integrity of the frame while allowing for quick axle swaps.
When attaching the ball bearings, which house the axle and are located in the center of the
cap and bottom piece, we were afraid to use epoxy to secure them in place in the fear that
the epoxy would damage the bearings. Instead we decided to use electrical tape to shrink
the diameter of the hole just enough to create a snug fit. This worked exceedingly well since
the ball bearings spin perfectly with little inhibition, avoiding a problem many other groups
faced.
3.4.1 Poles and Phases
In the end, two electromagnets were placed diametrically opposite each other, and two
magnets were embedded into the axle. The axial magnets were arranged in a N-S N-S
manner for one main reason. We needed the magnets to hold themselves together, since
we wanted to avoid epoxy and couldn’t possibly have the magnets repel. In the magnet’s
rotating reference frame, the centrifugal pseudoforce from rotation was countered by this
magnetic attraction, keeping the magnets firmly in their sockets.
We chose 2 electromagnets so that their ideal phase shift is π for a push-pull motor. We
wired the electromagnet leads such that when one was N, the other was S. In theory, the
electromagnet’s direction flip should occur when the magnet passes the center line between
the two iron cores, and in practice we found this to be true.
24
3.5 Axles
We designed three axles, but only two made it to final testing. Our two final axles were one
for RPM and one for torque. To see more about our failed axle, see Appendix A.
3.5.1 Small Radius Axle
The small radius axle (see Figure 4) sports a thin shaft with sunken holes, both aimed at
decreasing the moment of inertia of the system. The main purpose of this axle is to achieve
the highest possible rpm’s and all aspects are designed according to this goal. The shaft was
printed hallow in order to cut down on the rotors weight; however, the 3D printer laid down
structural wax which we were never able to remove. Sadly the plan to hollow out the shaft
backfired since it severally weakened the shafts structure, losing the much needed rigidity
demanded for high speed rotations. A large amount of energy was lost to vibrations due
to the axle bending at high speed. The magnets were then placed in the sunken holes in a
north south orientation which attracted each other and held them both in place without the
need for glue. See Figure 4 for detailed measurements.
3.5.2 Large Radius Axle
The large radius axle (see Figure 5) was in actuality was very poorly designed. Given limited
time to refine it before printing, we made the crucial mistake of only adding only two magnet
holes instead of four. With its large radius, when the permanent magnets are halfway through
their cycle the drop off in magnetic field results in little force on the magnets. Additional
holes would have allowed magnets to be closer to the solenoids at all times. However, the
large radius axle was not a complete failure. Since our frame design allowed for quick axle
swaps, when it came time to measure torque, we achieved a much higher result than we
would have with its smaller and more RPM oriented counterpart. This result, however, was
not repeatable.
25
19.
05
88.
10
98.
40
7.95
12
.70
7.95
19.05
9.53
19.05
98.
40
88.
10
15
2
MotorAxleSmallMagnetsWEIGHT:
A4
SHEET 1 OF 1SCALE:1:2
DWG NO.
TITLE:
REVISIONDO NOT SCALE DRAWING
MATERIAL:
DATESIGNATURENAME
DEBUR AND BREAK SHARP EDGES
FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:
Q.A
MFG
APPV'D
CHK'D
DRAWN
Figure 4: This axle maximizes efficiency and RPMs. It has a very narrow shaft (to minimizemoment of inertia), sunken magnet holes (to maximize potential for close electromagnets andmagnets), and a narrow flagpole on top, on which to attach a flag for RPM measurements.
26
80
9
12
.70
80
19.
05
80
MotorAxleTorqueWEIGHT:
A4
SHEET 1 OF 1SCALE:1:2
DWG NO.
TITLE:
REVISIONDO NOT SCALE DRAWING
MATERIAL:
DATESIGNATURENAME
DEBUR AND BREAK SHARP EDGES
FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:
Q.A
MFG
APPV'D
CHK'D
DRAWN
Figure 5: This axle is a simple 3D-printed cylinder (with a cylindrical, coaxial hole) andsunken magnet holes in the side. We inserted an straight wooden dowel as our driveshaft.
27
4 Results
Table 2: Motor Specifications Summary
Spec Top Average St. Dev. Trials
RPM 4545.6 - - 23
Startup Torque (N·m) 0.0625 0.0564 0.005 3
Running Torque (N·m) 0.0054 - - 3
Efficiency (%) 1.94 1.84 0.09 5
Power (W) 0.976 - - 14
RPMs/Volt 394 - - 3
4.1 RPM
Over 23 trials, we measured an top RPM of 4545.6. We don’t report a mean or standard
deviation, because all these RPM measurements were taken with various circuit parameters.
Additionally, the 4545.6 RPM is hard to repeat - it is easy to reach over 3900 RPM, but
eking out the last 600 RPMs takes significant fiddling.
We measured our motor’s RPM by attaching a makeshift flag to the protruding top of our
axle. To ensure structural integrity, we taped a popsicle stick to the flag. As the axle rotates,
so does the flag. We used a break-beam light sensor (which detects changes in voltage from
light intensity as a proxy for whether or not the laser is broken) to determine the frequency
of our motor’s rotation (in Hz). From there, we extrapolated to RPM by multiplying by 60.
An example RPM readout is shown below in Figure 6.
28
Figure 6: The output, in voltage, of the break-beam oscilloscope. Each period corresponds to
one revolution of our motor. By measuring the time between adjacent peaks, we can measure
the frequency of our motor, and thus the RPMs. Note that we are actually underestimating
the frequency, as the left cursor is too far left, and should be made closer, increasing RPM
further.
4.2 Torque
4.2.1 Startup Torque
Over 3 trials, we measured an average startup torque of 0.0564 N·m, with a maximum of
0.0625 N·m and a standard deviation of 0.005 N·m.
Interestingly, these torque numbers are coming from our skinny axle, rather than our
torque axle. Since the distance from the electromagnets to the magnets in a large axle can
be so great, the torque curve for that axle is very nonlinear. Therefore, it was incredibly
29
difficult to find a reliable torque measurement for our torque axle. The torque depends very
much on the angle of rotation, which we can’t control with our setup. Nevertheless, we had
one very clean measurement, of 1.958N at 0.08m from the axis of rotation, giving 0.157 N·mof startup torque. However, this measurement was unrepeatable, so we cannot in good faith
report that measurement as our true startup torque.
Initially, we were going to measure startup torque using friction bands and spring gauges
(see Appendix E). However, that failed, so we resorted to using a string and a spring gauge.
We tied the string to both our axle and the spring gauge. When the motor spins up, it quickly
rotates until the string is taut (force spike), at which point the force levels off somewhat.
However, there are still oscillations, as the axle is bobbing back and force. The sensor reading
gives force at a radial distance from the axis of rotation, so we calculate τ = F · r.An example startup torque readout is shown below in Figure 7
Figure 7: The output, in Newtons, of the force sensor. The initial spike corresponds to the
tightening and ’bounce’ of the string connecting the gauge and the axle. The continuous
portion represents our startup torque. The oscillations at the end are from us untangling
the string from the axle.
30
4.2.2 Running Torque
Over 3 trials, we measured a top running torque of 0.0054 N·m. Since running torque is a
function only of the mass (when we ignore the marginal acceleration), the concept of average
and standard deviation aren’t as relevant - they vary with every change in the motor.
We demonstrate how to calculate running torque in Section 4.5.
An example running torque readout is shown below in Figure 8. Note that this graph
isn’t perfect, as the velocity never is truly constant. As such, we’re underestimating our
power. Nevertheless, other trials had a more constant velocity.
Figure 8: The output, in m (top) and ms
(bottom), of the distance sensor. The highlighted
region indicates the approximately constant velocity of the mass rising. The spike before the
constant period is the initial rapid acceleration, and the drop off at the end corresponds to
when the mass fell off the pulley.
4.3 Power
Over 14 trials, we measured an maximum power of 0.976W.
We demonstrate how to calculate power in Section 4.5.
31
4.4 Efficiency
Over 5 trials, we calculated an average efficiency of 1.84% and a top efficiency of 1.94% with
a standard deviation of 0.09%.
Efficiency is defined as
ηmotor =PoutPin
.
To calculate Pin accurately without an ammeter, we need to know the resistance of our
circuit. We put a 100Ω resistor in series with the entire circuit, and measured the voltage
drop across both the resistor and the entire circuit. From the resistor’s known resistance
and measured voltage drop, we can calculate the current that enters the circuit. With this
current, and a measured voltage drop across the rest of the circuit, we can calculate the
equivalent resistance of each of our circuits.
Rt =Vt
VresistRresist
For our simple circuit, we measured an equivalent resistance of 23.8Ω. For our relay
circuit, we measured an equivalent resistance of 7.125Ω. Both were tested at 12V total
power input.
We can calculate Pout as shown in Section 4.5.
Note: These efficiency calculations are inaccurate.
4.5 A Note on Calculating Running Torque, Power, and Efficiency
In our setup, a thin, low-mass kite string passes over a low-mass pulley and is attached to a
mass. There is a planar surface (light cardboard sheet) beneath. On the floor is a distance
sensor. When the motor spins, it pulls the string, raising the mass. The distance sensor
gives us position and velocity of the mass. As explained above, we will always calculate
input power by Pin = V 2
Rt. There are two main cases to consider:
Call the mass of the weight and the cardboard sheet m, and neglect the mass of the
string and the pulley.
4.5.1 Constant Velocity
If, at some point in the trial, the mass hits constant velocity, we can easily calculate running
torque, output power, and efficiency. Constant velocity means zero acceleration, so the
upward force of tension T (ultimately from the motor) is equal to the weight of the mass
32
mg. Since our axle is cylindrical, the lever arm distance is simply the radius of the cylinder.
Therefore, τrunning = mgr
Additionally, Pout = τω, and ω = vr, so Pout = mgr v
r= mgv. Notice that if we instead
view power as the derivative of energy with respect to time. Since velocity is constant,
the rotational energy of the axle isn’t changing, and the kinetic energy of the mass isn’t
changing, so the only change in energy is from the increase in gravitational potential energy
of the mass. Pout = dughdt
= mg dhdt
= mgv, which is the same, which is good.
As before, efficiency is just the ratio of output power to input power. ηmotor = Pout
Pin=
mgvV 2
R
= mgvRV 2
In summary,
τrunning = mgr
Pout = mgv
ηmotor =mgvR
V 2
4.5.2 Changing Velocity, Constant Acceleration
Let acceleration be a constant a. When acceleration is constant, the math becomes harder,
since now both rotational and kinetic energies increase too. We are sure to define the positive
y direction as away from the center of earth, and the negative y direction as into the earth.
We can still calculate running torque, since ΣF = T + mg = ma =⇒ T = ma −mg =⇒τ = (ma−mg)r
Calculating output power is difficult. In a short time interval dt, height of the mass,
velocity of the mass, and angular velocity of the axle will change.
v′ = v0 + adt
h′ = h0 + v0dt+1
2a(dt)2
ω′ =v′
r=v0 + adt
r= ω0 +
a
rdt
33
The change in total energy is given by
∆Energy =
(1
2m(v′)2 +mgh′ +
1
2I(ω′)2
)−(
1
2mv2
0 +mgh0 +1
2Iω2
0
)=
(1
2m(v0 + adt)2 +mg(h0 + v0dt+
1
2a(dt)2) +
1
2I(ω0 +
a
rdt)2
)−(
1
2mv2
0 +mgh0 +1
2Iω2
0
)=
(mv0adt+
1
2ma2(dt)2 +mgv0dt+
1
2mga(dt)2 + Iω0
a
rdt+
1
2Ia2
r2(dt)2
)= dt
(mv0a+mgv0 + I
a
r
)+ (dt)2
(1
2ma2 +
1
2mga+
1
2Ia2
r2
)Power is the derivative of energy, so
P =dEnergy
dt
=dt(mv0a+mgv0 + I a
r
)+ (dt)2
(12ma2 + 1
2mga+ 1
2I a
2
r2
)dt
=(mv0a+mgv0 + I
a
r
)+ dt
(1
2ma2 +
1
2mga+
1
2Ia2
r2
)limdt→0
P = mv0a+mgv0 + Ia
r
Notice that with a = 0, this equation reduces to the simple case we established in
Section 4.5.1. Note, however, that this implies that power is not constant when the mass is
accelerating, since v0 is monotonically increasing, which makes sense intuitively.
We would calculate efficiency in the same way we have been doing.
4.5.3 Practical Choice of Mass
Luckily, we can vary the mass until it hits constant velocity at some point in the trial.
Therefore, we don’t ever need to worry about the dubious mathematical manipulation of
differentials in Section 4.5.2.
4.6 RPM vs. Voltage
We ran a linear regression on three data points and found a statistically significant correlation
between voltage and RPM. Our model gives ˆRPM = 393.84·V . With one degree of freedom,
34
y = 393.84x R² = 0.97601
n = 3
3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000
7 8 9 10 11
RPM
Voltage (V)
RPM vs. Voltage
RPM
Linear(RPM)
Figure 9: RPM appears to increase linearly with voltage over the ranges of voltages weexamined.
the T test doesn’t give much information. A graph of RPM vs. voltage is shown in Figure
9.
We forced 0 RPM at 0V, but if we had not, the regression predicts 73RPM at 0V, which is
reassuring. While it’s likely the RPM v. voltage curve is not linear, our linear approximation
seems to be rather accurate, at least around voltages we tested.
However, there are some potential errors with this approach. First, we have only three
data points. With just three data points, anything looks linear. It was difficult to obtain more
than three data points, because for voltages less than 8V, the attractive force between the
magnets and the iron core was too much for the repulsive force between the electromagnets
and the magnets to overcome. For voltages greater than 12V, the RPM measurement was
too inconsistent, as vibrations began to significantly effect output.
35
Figure 10: With just three data points, it’s hard to determine the type of relationshipbetween power and input voltage. The data appear positively correlated, with a weak lineartrend.
4.7 Additional Results
4.7.1 Circuit Comparison
While our choice of two axles was essentially narrowed to only one functioning motor, both
of our circuits functioned as we hoped.
As expected, the relay circuit was able to put out both more running torque (0.0054
N·m compared to 0.0031 N·m) and more power (0.976W compared to 0.814W). We can’t do
statistical significance tests, since we have only one sample for the relay circuit.
4.7.2 Power vs. RPM
The following diagram (Figure 10) is for our skinny axle with the simple circuit. There are
standard error bars, but the standard error is so small that they don’t show up.
36
5 Conclusion
5.1 Overview
The majority of the project was spent on the initial steps of designing, building and optimiz-
ing our brushless DC motor. From the start of the project the group had one single goal in
mind: placing our motor in the first spot of the RPM hall of fame. Sadly, this did not hap-
pen. Our motor did produce speeds capable entering the hall of fame in RPM category, just
not winning it. To begin, we focused on exploring different circuit elements. After creating
a simple push-only circuit, we decided to try to wire up an H Bridge. The H Bridge would
allow for both push and pull from our solenoids without mechanical limitations. Perfecting
this circuit would be have been ideal for our high RPM goals. Unfortunately, we were never
able to get our H Bridge to work, and lost several class periods to troubleshooting that never
panned out. After admitting temporary defeat with our H Bridge, we constructed the frame
for the motor. We reached speeds of around 3,000 RPM on the simple push circuit with
magnets taped to the axle. After we 3D-printed the real axle with sunken holes, we switched
to using relays in our circuitry in order to provide both push and pull. By moving around
the Hall chip, we were able to reach our top rotational speed with the relay circuit.
During the ASR motor project, we not only designed, built and optimized our motors,
but also benchmarked them by running several tests. The first test measured the fastest
rpm that our motor could sustain. By altering the Hall chip position as well as the voltage,
we were able to achieve a maximum speed of 4,500 RPM. We measured this figure using
an oscilloscope and a break beam sensor. Next, we moved on to measuring the startup
torque, utilizing a LabQuest wired up to a force sensor. By tying a taut rope around the
axle, we were able to measure a maximum startup torque of 0.0625 N·m. After 3 trials we
had an average startup torque of 0.0564 N·m. Each of the startup torque readings were
measured with our small diameter torque axle. Our next test measured running torque by
lifting a small weight using a pulley system while monitoring its velocity with a distance
sensor. Using the velocity data gathered by the LabQuest, we were able to calculate running
torque as 0.0054 N·m. We also calculated our efficiency to be 1.94% by using our running
torque to calculate output power and comparing it to our input power. We derived our
input power by making the (incorrect) assumption that any given circuit has an equivalent
resistance independent of frequency, making our efficiency improperly calculated and most
likely inaccurate. Our final test was to find the relationship between rotational speed and
voltage. We did so in a similar manner as we did for our maximum RPM; however, we varied
37
the voltage for each test.
5.2 Sources of Error
There were a few clear potential sources of error in our measurements.
5.2.1 Flag
We taped a flag to the top of our motor so that it could interrupt the oscilloscope’s beam.
However, in early iterations it became clear that the flag would not always keep pace with the
axle. It is possible, therefore that our RPM measurements are limited less by the capacity
of the motor and more by the capacity of the tape.
5.2.2 Simplifying Assumptions
In various physics calculations, we made many simplifying assumptions, which hold to vary-
ing degrees. For example, we assumed V = IR, even in the presence of a varying external
magnetic field. We chose to ignore the back emf induced in our electromagnets or the re-
lay’s coil, as well as the mass of the pulley and string for torque, efficiency, and power
measurements. We assumed that our electromagnets have zero depth, and we ignore the
magnetization of the iron core after repeated exposure to a magnetic field. While most of
these shortcuts have little impact on the final conclusions, its possible that some error derives
from our intentional trade of accuracy for ease of calculation.
5.2.3 Equivalent Resistance
In all of our input power calculations, we assumed that our circuits have an equivalent
resistance. This is inaccurate. The current is alternating, not to mention that inductors have
a back emf that will change resistance. Frequency affects resistance. The only computation
that incorporated input power was efficiency. . . so it would be wise to consider alternative
explanations for low efficiency. In the future, we could read the voltage drop across a known
resistor to calculate the current as a function of time. The current could be used to accurately
calculate input power - which could then be combined with our current data to obtain a more
accurate efficiency measurement.
38
5.3 Future Improvements
There are many improvements we would make if we were to do this project again.
5.3.1 Non-Wobbling Axle
During the 3D printing process, all axles become slightly warped due to the fact that they
are placed in a very hot oven to melt the extraneous printing wax. Additionally, the axle was
further damaged by the attraction of iron core of the solenoid which was positioned slightly
closer on one side than the other for roughly a week. The magnets pull, compounded over a
few days, was enough to add a noticeable additional amount of warp. In the future, all of our
axles would have a cylindrical coaxial hole, rather than a 3D-printed shaft. We would have
the driveshaft be a straight cylindrical wooden dowel (as we did with our torque motor).
This change would increase power output (and all related specs - efficiency, torque, RPM).
5.3.2 Shorter Driveshaft
Our driveshafts were 3D printed to the size of our frame - 7 inches tall, with some extra
room to slot into ball bearings and attach the flag. We chose 7 inches because it gave us
enough room to work on a daily basis. In retrospect, we should have used a much shorter
driveshaft, which would have lowered the axle’s moment of inertia and increased its RPM.
5.3.3 H Bridge
A very significant portion of our time was spent on the H Bridge circuit, and it’s sad that
we never got it to work. With more time, we would build and perfect the H Bridge for use
in our motor. As explained in Appendix B, a functioning H Bridge circuit would allow us
to send current through our electromagnets in opposite directions, without the mechanical
limitations of a relay.
5.3.4 Vibrations
A big loss of power for our motor was due to vibrations. While we did secure the walls of
the frame to the base, the base itself was much too big for our purposes, and often hindered
our measurements. The unnecessary base area also vibrated constantly, diminishing power
to our motor.
39
While our incorporation of pegs for the top portion of the frame was clever, we had some
errors with the drill press, and only had three functioning peg holes. Drilling more accurate,
symmetric holes would also have reduced energy losses due to vibrations.
5.4 Big Picture
Motors are everywhere. They have a strong presence throughout history, and are ubiquitous
in modern life. Building an electric motor was useful not only because it gave us the chance
to practice our own engineering skills (and refine our physics knowledge), but also because
it gave us an appreciation for the detail, both mechanical and electrical, that goes into the
construction of a simple motor. Imagine how much thought, design, and ingenious ideas must
go into every car, or toaster, or refrigerator, or television. Computers, for one, are praised
for their complexity - but who praises the complexity of a digital camera? In the scheme
of things, this motor project is just one of many steps in a lifelong journey of engineering,
creation, discovery, and invention.
6 Acknowledgements
We would like to thank Dr. Dann and Mr. Del Carlo for their perpetual ability to trou-
bleshoot any electrical and mechanical engineering problems we had and their willingness to
stay after school and on weekends so that we could optimize the motor.
We would also like to thank Jordan and Eric for collaborating on circuitry design and
selling us extremely smooth ball bearings.
40
References
[1] “Joseph Henry - Life of an Early Electrical Pioneer.” Joseph Henry - Life of an Early
Electrical Pioneer. Edison Tech Center, n.d. Web. 16 Oct. 2013.
[2] ”History of Brushless DC Motors.” Brushless DC Motors A. Minebea, n.d. Web. 15
Oct. 2013.
[3] Akita, H.; Nakahara, Y.; Miyake, N.; Oikawa, T., “New core structure and
manufacturing method for high efficiency of permanent magnet motors,” Industry
Applications Conference, 2003. 38th IAS Annual Meeting. Conference Record of the,
vol.1, no., pp.367,372 vol.1, 12-16 Oct. 2003
[4] Milestones:Maxwell’s Equations, 1860-1871. IEEE Global History Network.
http://www.ieeeghn.org/wiki/index.php/Milestones:Maxwell’s Equations, 1860-1871
[5] Wire Gauge and Current Limits Including Skin Depth and Strength. Power Stream.
http://www.powerstream.com/Wire Size.htm
[6] Solid State Sensors. Honeywell Sensing and Control.
http://images.ihscontent.net/vipimages/VipMasterIC/IC/HONY/HONYS00556/
HONYS00556-1.pdf
[7] TIP120G - Plastic Medium-Power Complementary Silicon Transistors - ON
Semiconductor. All Data Sheets.
http://pdf1.alldatasheet.com/datasheet-pdf/view/175430/ONSEMI/TIP120G.html
[8] DS2E-M-DC5V - HIGHLY SENSITIVE 1500 V FCC SURGE WITHSTAND-
ING MINIATURE RELAY - Nais(Matsushita Electric Works). All Data Sheets.
http://pdf1.alldatasheet.com/datasheet-pdf/view/80538/NAIS/DS2E-M-DC5V.html
41
A Appendix A - Arc Magnets
We thought that curved magnets (arcs of a cylinder) would be ideal for a DC motor, since
they do not protrude at all, making a nice, symmetrical cylinder that (we hoped) would allow
us to get extremely close. The 90° magnets slotted perfectly around our axle in a N-S-N-S
configuration. However, in practice we down that our axle spun significantly slower with the
arc magnets than with the cylindrical magnets. We believe this is because the transition
between north and south faces of the axle is too gradual, so the axle fights itself as it tries
to both pull and push the magnets.
B Appendix B - H Bridge
B.1 Goals
Our goal was to construct a circuit that allowed for a push-pull electromagnet without the
restriction of a relay’s top switching speed. The circuit diagram is shown in Figure 11.
B.2 Theory
In theory, an H Bridge should allow current to flow both directions through an electromagnet
(or pair of electromagnets). An external magnetic field controls whether or not current
is present in the output wire of the Hall chip, as discussed previously. This current is
split. One path passes through a not gate. One of the resulting two wires will always
have current, but never both. If the unmodified wire is on, current will flow to the base
of the upper left transistor, and through a resistor (to lower voltage) to the lower right
transistor. Current from the main power line must flow through the upper left transistor,
through the electromagnets from left to right, and out the bottom right transistor to ground.
If, conversely, the negated wire has current, the bases of the upper right and lower left
transistors will have power (the lower base again having passed through a resistor). In this
case, current can only flow from the main power line through the upper right transistor,
through the electromagnet from right to left and out the lower left transistor.
In practice, however, this circuit did not behave as expected. We hypothesized that
current was forcing its way through a closed transistor, short-circuiting the motor.
42
Pull Up Resistor
1 kΩ
Hall Chip
GND
Out
Vcc Power
NOT
NPN
NPN NPN
NPN
Electromagnet
1 kΩ
1 kΩ
Electromagnet
Figure 11: Circuit Diagram for the H Bridge Circuit. Current from the output of a hall chipgets split. One half is negated and powers the base of two diagonally opposite transistors.The other half powers the base of another pair of diagonally opposite transistors. The wiresthat go to the lower transistors pass first through a resistor to ensure that base voltage isless than collector voltage.
43
C Appendix C - Power Circuit
Our transistors are not rated for much more than 5A (pulsed) or 3A (continuous), and one
power supply can only supply around 6A. So, to maximize current, we hooked up three power
supplies (in parallel) to three transistors to maximize current through our electromagnets.
In contrast to the H Bridge, a circuit in which we invested a lot of time and energy, the
power circuit was mostly a product of one afternoon in the lab. In essence, we pumped an
obscene amount of current through our circuit. The circuit diagram is shown in Figure 12.
As you can see, there is potential for a ridiculous amount of current to pass through our
electromagnets, all the while staying within reasonable bounds for the transistors and the
Hall chip.
As expected, when we ran this circuit, something broke. In particular, the middle transis-
tor effectively melted (fizzled and smoked), while the other two remained cool to the touch.
We are still unsure why one transistor took the majority of the current, but it’s clear that
we won’t be trying a circuit like this again any time soon.
D Appendix D - Extra Physics
D.1 Magnetic Force
To understand the magnetic force between two magnets, we first must consider the origins
of magnetic fields in general.
Magnetic fields are generated from the motion of electrons in orbitals around atoms.
According to Ampere, a moving charge (a.k.a. an electron) generates an magnetic field. A
charge moving in a circle acts as a magnetic dipole. Thus, the origins of magnetic fields are
really orbiting electrons. We tend to approach problems relating to magnetic interactions
through one of two models - the Amperian model, where we consider the circular motion of
electrons, or the Gilbert model, where we imagine magnetic monopoles smeared at two ends
of a magnet. The Gilbert model is easier to understand and work with, but is marginally
less accurate in some cases. Similar to the Coulomb force (derivable from Gauss’s Law), the
force between two magnetic ’monopoles’ is
F =µqm1qm2
4πr2
This suggests that (as common sense indicates), similar poles repel (positive repulsive
44
Pull Up
1 kΩ
Hall Chip
Out
GND
Vcc
Electromagnet Electromagnet
NPN
NPN
NPN
Figure 12: As with the simple circuit, here the bases of transistors are controlled by theoutput of a Hall chip. The only difference - we have three transistors, all operating at closeto maximum ratings, and three power supplies (voltages add in parallel). Current recombinesafter the transistors and passes through the electromagnets.
45
force), and opposite poles attract (negative attractive force).
E Appendix E - Measuring Torque
Initially, we planned to measure startup torque using a friction band and spring gauges.
Much like our final method of measuring torque, our idea would utilize the resistive spring
force to bring our motor to a halt. However, we were going to use a friction band, rather
than a string, to connect our motor to the sensor. In theory, the band should grip the axle,
and should be pulled as the axle rotates. On one side of the axle, the band builds up slack,
and in the other, tension. The tension measured gives force at a radial distance from the
center of rotation.
However, in practice we did not have a friction belt, and instead tried to use a rubber
band. It was necessary to poke a hole in the rubber band to attach it to our spring gauge,
and under high force, the hole expanded and the rubber band snapped in two, rendering
data collection impossible.
F Appendix F - Python Code
We wrote a quick Python program to convert our raw readings to useful measurements of
running torque and power.
rad = 0.004 #Radius o f a x l e i s 4mm
#Takes in a ( v e l o c i t y , mass , v o l t a g e ) t u p l e
#Return a ( running torque , power ) t u p l e
def c a l c ( tup ) :
velo , mass , v o l t = tup
return ( mass ∗ 9 .81 ∗ rad , mass ∗ 9 .81 ∗ velo , v o l t )
#Simple C i r cu i t
s imple1 = [ ( v , 0 . 056 , 13 . 9 ) for v in [ 0 . 8 9 0 , 0 . 965 , 0 . 954 , 0 . 894 , 0 . 8 4 2 ] ]
s imple2 = [ ( v , 0 . 056 , 17 . 92 ) for v in [ 1 . 0 8 4 , 1 . 065 , 1 . 0 6 8 ] ]
s imple3 = [ ( v , 0 . 080 , 25 . 0 ) for v in [ 1 . 0 3 0 , 1 . 012 , 1 . 040 , 1 . 043 , 1 . 0 5 9 ] ]
46
s imple = [ simple1 , s imple2 , s imple3 ]
#Relay C i r cu i t
r e l ay1 = [ ( v , 0 . 138 , 12 . 83 ) for v in [ 0 . 7 2 1 ] ]
r e l a y = [ r e l ay1 ]
print ”Running Torque , Power Output , Volts f o r Simple C i r c u i t ”
for s in s imple :
for x in s :
print c a l c ( x )
print ”∗∗∗∗∗”
print ”Running Torque , Power Output , Volts f o r Relay C i r c u i t ”
for r in r e l a y :
for x in r :
print c a l c ( x )
print ”∗∗∗∗∗”
47
