Download Magnetostatics Analysis, Design, and Construction

Magnetostatics Analysis, Design, and
Construction of a Loudspeaker
Calin Galeriu, Becker College, Worcester, MA
M
aking a loudspeaker is a very rewarding hands-on
activity that can be used to teach about electromagnetism and sound waves. Several loudspeaker
designs have been described in this magazine.1-4 The simplest
loudspeaker4 has only a magnet, a coil, and three plastic cups.
The simpler devices3,4 require a powerful amplified output,
e.g., from a boom box. The more complex devices1,2 can
operate using the smaller electric current from a CD player
earphone output. Unfortunately, the procedure to make a
more efficient loudspeaker is lengthy and less recommended
to some high school students, involving a hot glue gun, a
safety razor, five-minute epoxy, etc. Our loudspeaker, a variation of Heller’s,2 is both simple in construction and efficient
in operation. An analysis of the magnetic field distribution
helped us in the design of this loudspeaker.
In its simplest form a loudspeaker is just a plastic (or paper) cup with a coil of wire glued (or taped) to its bottom and
with a permanent magnet inside or near the coil. The sound
is produced by the vibrations of the bottom of the cup, which
functions as a loudspeaker diaphragm. These vibrations are
the result of the electromagnetic interaction between the permanent magnet and the electric current flowing through the
coil. What matters in the operation of the loudspeaker is the
force perpendicular to the bottom of the cup, causing the vibrations. As a cross product, the electromagnetic force is perpendicular to both the magnetic field and the electric current.
Maximizing the electromagnetic force perpendicular to the
bottom of the cup is key to improving the performance of the
loudspeaker. This happens when, at the position of the coil, we
have a strong magnetic field with a radial direction (in a cylindrical reference system), as shown in Fig. 1.
Fig. 1. In the plane of the loop of wire
the electric current element has a
tangential direction. A magnetic field
with a radial direction will produce an
electromagnetic force perpendicular
to the plane of the loop.
Sometimes a second cup, of the same size, is used to hold
inside the permanent magnet and the cup with the coil. This
second cup will also work as a loudspeaker enclosure and help
dampen the out-of-phase sound waves produced at the rear of
the speaker. When the loudspeaker diaphragm moves forward
to create a layer of compressed air in front of the speaker, it
also creates a layer of rarefied air in the back. In the absence of
a loudspeaker enclosure, the 180oout-of-phase sound waves
will interfere in the room with the in-phase sound waves,
resulting in reduced loudspeaker efficiency, especially at low
frequencies.
Magnetostatics analysis
To calculate, visualize, and analyze the magnetic field lines
for different magnet configurations, we have used the Maxwell SV 9.0 computer program. The software and the getting
started guides can be downloaded from Ansoft’s website.5
Two tutorials are provided, one for an electrostatic problem
and one for a magnetostatic problem.
The Maxwell SV program can solve two-dimensional
problems, which means that the analyzed device must have
translational or rotational symmetry. The electromagnetic
field calculated is a cross section of the structure. The loudspeakers that we built use disk magnets and have cylindrical
symmetry.
Five steps are required to calculate the magnetic field distribution for any of the loudspeaker designs investigated. In
the first step a drawing is made in which the dimensions of
the solid parts (magnet, steel core, etc.) are specified. We have
used a grid of 0.03125 in and a 2-in by 2-in spatial domain. In
the second step the solid parts are assigned material proper-
Fig. 2. (a) A magnet. (b) Two magnets. (c) A magnet and
a steel core. (d) A magnet and a steel core inside a steel
cup. (e) A magnet on a metal bottle cap. Magnets are
shown in blue and steel parts are shown in gray. The
dotted line is the axis of rotational symmetry.
DOI: 10.1119/1.3502508
Fig. 3. Magnetic field lines for a
cylindrical magnet. The figure
shows the right half of a vertical
cross section. The rotation symmetry axis is at the left side of the
figure.
The Physics Teacher ◆ Vol. 48, November 2010
537
ties (NdFeB, steel). In the third step, free boundary conditions
are defined. In the fourth step, the computer solves the Maxwell equations using the finite element method. In the fifth
step, the calculated magnetic field lines are plotted. What we
get is the right half of a vertical cross section, with the rotation
symmetry axis at the left side of the figure.
Five magnet configurations have been investigated, all
shown in Fig. 2.
The simplest loudspeaker, with just a magnet inside a coil,
is not the most efficient because some of the magnetic field
lines travel far from the coil and don’t contribute to the electromagnetic force, as shown in Fig. 3.
If two magnets are used, then the magnetic field is stronger, and the sound produced is louder. A disadvantage of this
setup is the cost. But there is also an added bonus. The two
magnets can be placed on each side of the bottom of a second
cup, where they stick together. The cup with the coil is then
placed inside the cup with the magnets, as shown in Fig. 4.
This makes aligning the coil over the inner magnet a lot easier.
Furthermore, in this setup all the magnetic field lines reaching
the coil have the same radial direction (outward or inward).
Of course, on the other side of the bottom of the second cup
the radial direction of the field lines is reversed, as seen in
Fig. 5. In the absence of the second cup, if the coil extends
over both regions then the total electromagnetic force on the
Fig. 4. A loudspeaker with two cylindrical magnets and two plastic cups.
Magnets are shown in blue and the coil
is shown in red.
Fig. 5. Magnetic field lines for two magnets. In the upper half the radial component of the magnetic field is pointing
one way, while in the lower half it is
pointing the other way.
Fig. 7. Magnetic field lines for a magnet
and a steel core inside a steel cup. The
coil will be placed in the gap, between
the two steel parts, where the magnetic
field is strong.
538
coil will be reduced, since forces from some turns will oppose
forces from other turns.
The same discussion applies to the case of just one magnet.
In this situation the sound will be loudest when the lowest
turn of the coil is at the center of the magnet. Heller2 makes a
similar observation.
If a steel core is used inside the coil, then again all the magnetic field lines reaching the coil have the same radial direction, as seen in Fig. 6.
A great improvement in the performance of the loudspeaker is obtained when the magnet and the steel core are placed
inside a steel cup. This configuration confines almost all of
the magnetic field inside the metal and through the coil, as
seen in Fig. 7, resulting in a large electromagnetic force. The
magnetic field lines stay inside the metal because in this way
the magnetic field energy is minimized. Although this loudspeaker produced a really loud sound with just one magnet,
we do not recommend it for classroom use for several reasons.
Manufacturing the steel parts is not trivial. The magnet placed
inside the steel cup has a tendency to stick to the wall, and one
needs great skill and a couple of toothpicks to move it to the
center. Once the steel core is placed over the magnet, the three
parts become inseparable, hiding the details of the construction from future students, and thus reducing the educational
value of the experiment.
Fig. 6. Magnetic field lines for a magnet
(on the bottom) and a steel core (on
top). If the coil is placed around the
steel core, then all the magnetic field
lines intersecting the coil have the same
radial direction.
Fig. 8. Magnetic field lines for a magnet
on a metal bottle cap. This is an effective way to keep the magnetic field concentrated near the coil.
The Physics Teacher ◆ Vol. 48, November 2010
We have found that
a compromise between
simplicity and efficiency can be achieved
if a metal bottle cap
is used instead of the
two steel parts. The
magnetic field, while
not as confined as in
the previous case, is
still concentrated near
the coil, as seen in Fig.
8. Because this simple
design uses only one
rare-earth magnet
but still produces a
nice loud sound, this
loudspeaker is our
Fig. 9. An intermediate step during the
“winner.” Next we give construction of the loudspeaker.
a detailed step-by-step
procedure on how to make such a loudspeaker.
Materials needed
For each student:
• Two plastic cups (7 oz, flexible)
• One cylindrical rare-earth magnet6 (1/2-in diameter,
3/16-in height)
• Approximately 2 m of thin enamel-coated magnet wire
(30 gauge)
• One metal bottle cap
For students to share:
• 2-in wide Scotch packaging tape
• Scissors
• Sandpaper
• AA batteries
• Toothpicks
• An audio cable from a sacrificed set of earphones, with
exposed ends
Construction of the loudspeaker
Cut a square piece of Scotch tape (2 in x 2 in) and roll it on
a AA battery with the sticky side out in such a way that one
end gets attached to the other end, making a cylindrical tube
2 in long. Hold the battery from the “+” end, to prevent damaging the adhesive at the “–” end. Slide the battery through the
Scotch tape tube until about half an inch of tape extends over
the “–” end.
Sand out about half an inch of the enamel insulation and
leave about 10 in of free wire at both ends. Wind the coil,
about 20 turns and about a quarter-inch long, half an inch
from the end of the Scotch tape, as shown in Fig. 9. Twist the
Fig 10. A finished loudspeaker. A second cup will be
used to hold the metal bottle cap with the magnet and
the first cup with the coil, as shown in Fig. 4 for a related
design.
wire ends a couple of times to make sure that the coil will not
unwind, but not too tight because that could prevent the battery from sliding out.
Make four evenly spaced cuts in the exposed section of the
Scotch tape, as shown in Fig. 9 by the dotted lines. With the
help of two toothpicks bend the Scotch tape squares inside,
one by one, starting with two that are facing each other. Make
all four Scotch tape squares lie one on top of the other on the
flat bottom of the battery.
The Scotch tape square on top will have the sticky side exposed, and this side will be used to tape the coil to the center
of the bottom of a plastic cup, as shown in Fig. 10. Press on the
battery and from the inside on the bottom of the cup to make
sure that the bond is strong. Remove the battery. Very carefully cut the excess Scotch tape, slowly approaching the coil in
a spiral move. Make sure you don’t cut the wire too!
Place the magnet in the center of the metal bottle cap. Place
the bottle cap on the bottom of the second plastic cup. Slowly
slide the first cup inside the second cup, until the magnet is
inside the coil.
Connect the coil ends to the audio cable and turn the CD
player on. Gently adjust the position of the coil relative to the
magnet until the loudest sound is heard.
Conclusion
We have used Maxwell SV 9.0, a free and powerful computer program, to analyze the magnetic field distribution for
some very simple loudspeakers. Through conceptual and experimental investigations we have designed a simple, safe-tomake, low-cost, but nonetheless powerful loudspeaker. Every
student we asked was able to successfully build one.
Acknowledgments
The author wishes to thank the anonymous referees for
The Physics Teacher ◆ Vol. 48, November 2010
539
helpful comments, and Mr. Tate Ostiguy for manufacturing
the two steel parts in the Machine Shop at Bay Path High
School.
References
Frames of
REFERENCE
1. Allen Keeney and Brant Hershey, “Making your own dynamic
loudspeaker,” Phys. Teach. 35, 297–299 (May 1997).
2. Peter Heller, “Drinking-cup loudspeaker – A surprise demo,”
Phys. Teach. 35, 334 (Sept. 1997).
3. Molly Johnson and Virginia Stonick, “Sound science – A simple
and robust hands-on loudspeaker activity,” Phys. Teach. 37,
350–351 (Sept. 1999).
4. Rhett Herman, “As simple as possible,” Phys. Teach. 40, 182–183
(March 2002).
5. The Maxwell SV software and the getting started guides can be
downloaded from www.ansoft.com/maxwellsv.
6. Neodymium magnets can be ordered from K&J Magnetics,
www.kjmagnetics.com. This site also has information about the
safe handling of the magnets.
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Calin Galeriu is teaching physics and math at Becker College and math
at Bay Path Regional Vocational Technical High School. He has received a
BS degree in physics from the University of Bucharest, an MA degree from
Clark University, and a PhD degree from Worcester Polytechnic Institute.
Becker College, 61 Sever St., Worcester, MA, 01609; Calin.Galeriu@
becker.edu
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The Physics Teacher ◆ Vol. 48, November 2010