Mech - User contributions [en]

ME 224 Experimental Engineering

2006-11-28T03:24:32Z

67.128.15.151:

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Rotary Encoder

2006-11-28T02:31:41Z

67.128.15.151:

[[Category:Sensors]]
__NOTOC__

[[image:encoder disk.jpg|200px]]

A digital optical encoder is a device that converts
motion into a sequence of digital pulses. By counting a single
bit or by decoding a set of bits, the pulses can be converted to
relative or absolute position measurements. Encoders have both
linear and rotary configurations, but the most common type is
rotary. Rotary encoders are manufactured in two basic forms: 1) the
absolute encoder where a unique digital word corresponds to each
rotational position of the shaft, and 2) the incremental encoder,
which produces digital pulses as the shaft rotates, allowing
measurement of relative position of shaft. Most rotary encoders
are composed of a glass or plastic slotted disk. As
radial lines in each track interrupt the beam between a
photoemitter-detector pair (or [[Optointerrupter]]), digital pulses are produced.

 

[[image:encoder diagram.png|center]]

<h4>Absolute encoder</h4>

The optical disk of the absolute encoder is designed to
produce a digital word that distinguishes N distinct positions of
the shaft. For example, if there are 8 tracks, the encoder is
capable of producing 256 distinct positions or an angular
resolution of 1.406 (360/256) degrees. The most common types of
numerical encoding used in the absolute encoder are gray and
binary codes. To illustrate the acion of an absolute encoder, the
gray code and natural binary code dsisk track patterns for a
simple 4-track (4-bit) encoder are illustrated in Fig 2 and 3.
The linear patterns and associated timing diagrams are what the
photodetectors sense as the code disk circular tracks rotate with
the shaft. The output bit codes for both coding schemes are
listed in Table 1.

{|
|+
|[[image:gray.jpg|300px]]||[[image:binary.jpg|300px]]
|}

<table border="1" width="37%">
<tr>
<td width="19%">
Decimal code
</td>
<td width="37%">
Rotation range (deg.)
</td>
<td width="22%">
Binary code
</td>
<td width="22%">
Gray code
</td>
</tr>
<tr>
<td width="19%">
0
</td>
<td width="37%">
0-22.5
</td>
<td width="22%">
0000
</td>
<td width="22%">
0000
</td>
</tr>
<tr>
<td width="19%">
1
</td>
<td width="37%">
22.5-45
</td>
<td width="22%">
0001
</td>
<td width="22%">
0001
</td>
</tr>
<tr>
<td width="19%">
2
</td>
<td width="37%">
45-67.5
</td>
<td width="22%">
0010
</td>
<td width="22%">
0011
</td>
</tr>
<tr>
<td width="19%">
3
</td>
<td width="37%">
67.5-90
</td>
<td width="22%">
0011
</td>
<td width="22%">
0010
</td>
</tr>
<tr>
<td width="19%">
4
</td>
<td width="37%">
90-112.5
</td>
<td width="22%">
0100
</td>
<td width="22%">
0110
</td>
</tr>
<tr>
<td width="19%">
5
</td>
<td width="37%">
112.5-135
</td>
<td width="22%">
0101
</td>
<td width="22%">
0111
</td>
</tr>
<tr>
<td width="19%">
6
</td>
<td width="37%">
135-157.5
</td>
<td width="22%">
0110
</td>
<td width="22%">
0101
</td>
</tr>
<tr>
<td width="19%">
7
</td>
<td width="37%">
15.75-180
</td>
<td width="22%">
0111
</td>
<td width="22%">
0100
</td>
</tr>
<tr>
<td width="19%">
8
</td>
<td width="37%">
180-202.5
</td>
<td width="22%">
1000
</td>
<td width="22%">
1100
</td>
</tr>
<tr>
<td width="19%">
9
</td>
<td width="37%">
202.5-225
</td>
<td width="22%">
1001
</td>
<td width="22%">
1101
</td>
</tr>
<tr>
<td width="19%">
10
</td>
<td width="37%">
225-247.5
</td>
<td width="22%">
1010
</td>
<td width="22%">
1111
</td>
</tr>
<tr>
<td width="19%">
11
</td>
<td width="37%">
247.5-270
</td>
<td width="22%">
1011
</td>
<td width="22%">
1110
</td>
</tr>
<tr>
<td width="19%">
12
</td>
<td width="37%">
270-292.5
</td>
<td width="22%">
1100
</td>
<td width="22%">
1010
</td>
</tr>
<tr>
<td width="19%">
13
</td>
<td width="37%">
292.5-315
</td>
<td width="22%">
1101
</td>
<td width="22%">
1011
</td>
</tr>
<tr>
<td width="19%">
14
</td>
<td width="37%">
315-337.5
</td>
<td width="22%">
1110
</td>
<td width="22%">
1001
</td>
</tr>
<tr>
<td width="19%">
15
</td>
<td width="37%">
337.5-360
</td>
<td width="22%">
1111
</td>
<td width="22%">
1000
</td>
</tr>
</table> 
Table 1. 4-Bit gray and natural binary codes
 

[[image:conv.jpg|395px|right|frame|Fig 4. Gray code to binary code conversion]]

The gray code is designed so that only one track (one bit)
will change state for each count transition, unlike the binary
code where multiple tracks (bits) change at certain count
transitions. This effect can be seen clearly in Table 1. For the
gray code, the uncertainty during a transition is only one count,
unlike with the binary code, where the uncertainty could be
multiple counts.

Since the gray code provides data with the least uncertainty
but the natural binary code is the preferred choice for direct
interface to computers and other digital devices, a circuit to
convert from gray to binary code is desirable. Figure 4 shows a
simple circuit that utilizes exclusive OR gates (XOR) to perform
this function.For a gray code to binary code conversion of any
number of bits N, the most signficant bits (MSB) of the binary
and gray code are always identical, and for each other bit, the
binary bit is the exlcusive OR (XOR) combination of adjacent gray
code bits.
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<h4>Incremental encoder</h4>

[[image:increment.jpg|352px|right]]

The incremental encoder, sometimes called a relative
encoder, is simpler in design than the absolute encoder. It
consists of two tracks and two sensors whose outputs are called
channels A and B. As the shaft rotates, pulse trains occur on
these channels at a frequency proportional to the shaft speed,
and the phase relationship between the signals yields the
direction of rotation. The code disk pattern and output signals A
and B are illustrated in Figure 5. By counting the number of
pulses and knowing the resolution of the disk, the angular motion
can be measured. The A and B channels are used to determine the
direction of rotation by assessing which channels "leads" the
other. The signals from the two channels are a 1/4 cycle out of
phase with each other and are known as quadrature signals. Often
a third output channel, called INDEX, yields one pulse per
revolution, which is useful in counting full revolutions. It is
also useful as a reference to define a home base or zero
position.

Figure 5 illustrates two separate tracks for the A and B
channels, but a more common configuration uses a single track
with the A and B sensors offset a 1/4 cycle on the track to yield
the same signal pattern. A single-track code disk is simpler and
cheaper to manufacture.

The quadrature signals A and B can be decoded to yield the
direction of rotation as hown in Figure 6. Decoding transitions
of A and B by using sequential logic circuits in different ways
can provide three different resolutions of the output pulses: 1X,
2X, 4X. 1X resolution only provides a single pulse for each cycle
in one of the signals A or B, 4X resolution provides a pulse at
every edge transition in the two signals A and B providing four
times the 1X resolution. The direction of rotation(clockwise or
counter-clockwise) is determined by the level of one signal
during an edge transition of the second signal. For example, in
the 1X mode, A=<math>\downarrow</math> with B =1 implies a clockwise pulse, and B=<math>\downarrow</math> with A=1 implies a
counter-clockwise pulse. If we only had a single output channel A
or B, it would be impossible to determine the direction of
rotation. Furthermore, shaft jitter around an edge transition in
the single signal woudl result in erroneous pulses..

[[image:quadrature.jpg|500px]]

[[image:encoder parts.png|center]]

<h4>Connecting an Encoder to the PC/104 Stack</h4>

To connect an encoder to the PC/104 stack, you will have to create a suitable ribbon cable connector. The connector that came with the encoder will most likely not match the correct pinout to attach. Consult the [[PC104 I/O]] page to see how to connect the encoder.

==Datasheets==
Datasheet for the Maxon Tacho 103935 encoder: [[Media:maxon_tacho_103935_encoder.pdf]]

==References==
Introduction to Mechatronics and Measurement Systems, Histand & Alciatore, 1999 McGraw Hill

Brushed DC Motor Theory

2006-11-28T01:36:32Z

67.128.15.151: /* Introduction */

[[Category:Actuators]][[Category:Motors]]

==Introduction==

The specific type of motor we are addressing is the permanent magnet brushed DC motor (PMDC). These motors have two terminals. Applying a voltage across the terminals results in a proportional speed of the output shaft.

There are two pieces to the motor: 1) ''stator'' and 2) ''rotor''. The ''stator'' includes the housing, ''windings'' and ''brushes''. The ''rotor'' consists of the output shaft, ''windings'' and ''commutators''. The image below shows a cut-away view of a [http://www.maxon.com Maxon] motor. ''Note this picture has a gearbox and encoder attached to the motor''.

[[image:motor cutaway.png|center]]
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==Motor Physics==

The forces inside a motor that cause the rotor to rotate are called ''Lorentz Forces''. If an electron is moving through an electric field, it experiences a force that is perpendicular to both the magnetic field and the direction it's moving. If we have a wire instead of a single electron, the wire experiences a force equal to

{| align="center"
|<math>\vec F = \vec I \times \vec B\,</math>
|}

You can easily find the direction of this force using the ''Right Hand Rule''. The ''Right Hand Rule'' states that if you point your right hand fingers along the direction of current, <math>I</math>, and curl them towards the direction of the magnetic flux, <math>B</math>, the direction of force is along the thumb. See the picture below.

[[image:Lorentz Force.png|center]]

Now, imagine this single wire is replaced with a loop of wire. Between the magnets poles, this looks like two wires with current flowing in opposite directions. The forces on the wires cause the loop to rotate with forces as shown.

[[image:motor coils.png|center]]

This coil is attached to the rotor and rotates. As it does so, the magnitude and direction of the force on the wires remains constant. However, the resultant torque varies with the angle. Look at the picture below. When the coil starts, there is maximum torque. As the coil moves, the moment arm is reduced and the torque decreases. Finally, when the coil is vertical (in the picture), there is no torque.

[[image:rotating coils.png|center]]

To keep a (almost) constant torque on the rotor, there are two things that happen. First, the current through the coil is reversed every half turn. So instead of an alternating torque like the one in the first figure below, the torque is always in the same direction. Also, additional coils can be used. When these coils are offset at different angles around the motor, the resultant torque becomes the sum of the colored torque curves in the figure below. The resultant torque is alway greater than zero, but is not constant. This variation is called ''torque ripple''.

[[image:torque graphs.png|center]]

The process of switching current direction is called ''commutation''. To switch the direction of curent, brushed DC motors use brushes and commutators. Commutation can also be done electronically (see [[Brushless DC Motors]]). The following diagram shows how brushes and commutators work.

[[image:Motor Commutators.jpg|center|400px]]

==Equations==

[[image:dc motor power.png|center]]

We start by writing the equation for conservation of energy in the motor. The power is input as electrical power, <math>P_{elec}\,</math> and the motor converts that to mechanical power, <math>P_{mech}\,</math>. However, some of the power is lost as heat, <math>P_{heat}\,</math>.

<center><math>P_{elec}=P_{heat}+P_{mech}\,</math></center>

We can re-write this in terms of electrical and mechanical quantities as

<center><math>iv=i^2R+\tau\omega\,</math></center>

The wire coils have both a resistance, <math>R</math>, and an inductance, <math>L</math>. When the motor is turning, the current is switching, causing a voltage,

<center><math>\begin{matrix}v_{emf} = L \frac{di}{dt}\end{matrix}</math></center>

This voltage is called the ''back-emf'' ('''e'''lectro'''m'''otive '''f'''orce). The motor constants relate various quantities in a motor. They are a property of the design and material of each individual motor. The speed constant relates the back-emf to the output speed of the shaft,

<center><math>n=k_nv_{emf}\,</math></center>

where <math>n</math> is the rotation in '''revolutions-per-minute''' (rpm). The torque constant relates the input current (in the windings) to the torque of the output,

<center><math>\tau = k_Mi\,</math></center>

These two constants are actually the same number with different units. The relationship between the two is

<center><math>k_n \cdot k_M = \begin{matrix} \frac{30,000}{\pi} \end{matrix}\,</math></center>

Now look at the electrical side. If we sum the voltages, we get

<center><math>v-v_{emf}-iR=0\,</math></center>

This back-emf voltage is working against the voltage we apply across the terminals. Using the constants from above, we get this equation,

<center><math>v=\begin{matrix}\frac{n}{k_n}\end{matrix}+\begin{matrix}\frac{\tau}{k_M}\end{matrix}R\,</math></center>

The maximum torque, or ''stall torque'', is the torque at which <math>n = 0</math> or

<center><math>\tau_{stall} = \begin{matrix}\frac{vk_M}{R}\end{matrix}\,</math></center>

From this we can find the stall current, or ''starting current'',

<center><math>i_0 = \begin{matrix}\frac{V}{R}\end{matrix}\,</math></center>

The ''no-load speed'' is the maximum speed the motor can turn. To find its value, we set the torque equal to zero and we get

<center><math>n_0 = k_nv\,</math></center>

This relationship means that, given a constant voltage, the motor will settle at a constant speed. This linear relationship produces the ''constant voltage speed-torque line''

<center><math>n = n_0 - \begin{matrix}\frac{n_0}{\tau_{stall}}\end{matrix}\tau\,</math></center>

==Reading Datasheets==

The datasheet below is for the Maxon motor available in the Mechatronics Lab, which also has an integrated 6:1 gearhead and a 100 line encoder.

[[image:maxon-characteristics-small.jpg|center]]

'''1.''' Nominal voltage: The recommended maximum voltage across the motor terminals, and the voltage for which the speed-torque curve (below) is plotted. The motor can be powered with less or more voltage, but higher voltage should be used with care, to prevent the motor coils from overheating due to ohmic (resistive) heating.

'''2.''' No load speed, <math>n_0\,</math>: The speed of the motor powered by the nominal voltage when the motor provides zero torque.

'''3.''' No load current: The current required to spin the motor at the no load condition (i.e., the current needed to provide the torque necessary to overcome friction).

'''4.''' Nominal speed: The speed of the motor at the maximum continuous torque.

'''5.''' Nominal torque (max continuous torque): The maximum torque the motor can provide continuously.

'''6.''' Nominal current (max continuous current): The current that yields the maximum continuous torque. This maximum is determined by thermal characteristics of the motor. The power dissipated by the motor as heat is the product of the voltage across the motor and the current through it. Larger currents are acceptable intermittently, but larger continuous currents may cause the motor to overheat.

'''7.''' Stall torque, <math>\tau_{stall}\,</math>: The maximum torque achievable by the motor at the nominal voltage. This torque is achieved at zero velocity (stall).

'''8.''' Starting current, <math>i_0\,</math>: The current through the motor at zero velocity, equal to the nominal voltage divided by the terminal resistance. Also called the stall current.

'''9.''' Max efficiency: The maximum efficiency of the motor in converting electrical power to mechanical power. This maximum efficiency typically occurs at high speed and low torque; the efficiency is zero at zero speed and zero torque.

'''10.''' Terminal resistance <math>R\,</math>: The resistance of the motor windings.

'''11.''' Terminal inductance <math>L\,</math>: The inductance of the motor windings.

'''12.''' Torque constant, <math>k_M\,</math>: The constant of proportionality relating current to torque. In SI units (Newton-meters per amp), the torque constant is equivalent to the speed constant in SI units (radians per volt-second).

'''13.''' Speed constant, <math>k_n\,</math>: The constant of proportionality relating speed to voltage. Equivalent to the torque constant when expressed in SI units.

'''14.''' Speed/torque gradient: A representation of the slope of the speed-torque curve (see graph below), approximately equal to the no load speed divided by the stall torque.

'''15.''' Mechanical time constant: The time it takes the unloaded motor to reach 63% of its no load speed under a constant voltage, starting from rest. Proportional to the inertia of the rotor and inversely proportional to the square of the the torque constant.

'''16.''' Rotor inertia: The inertia of the rotating element (the rotor) about the axis of rotation.

Much of the data sheet can be expressed in the speed-torque curve, plotted for the constant nominal voltage (below). The mechanical power output is the product of the torque and the speed, and is maximized at half the maximum speed and torque.

[[image:new-speed-torque-small.gif|center]]



==References==
* Carl R. Nave, "DC Motors," http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motdc.html
* Maxon Motor Guide ([[media:Maxon Motor Guide.pdf|pdf]])

Rotary Encoder

2006-11-27T23:39:21Z

67.128.15.151:

[[Category:Sensors]]
__NOTOC__

[[image:encoder disk.jpg|200px]]

A digital optical encoder is a device that converts
motion into a sequence of digital pulses. By counting a single
bit or by decoding a set of bits, the pulses can be converted to
relative or absolute position measurements. Encoders have both
linear and rotary configurations, but the most common type is
rotary. Rotary encoders are manufactured in two basic forms: 1) the
absolute encoder where a unique digital word corresponds to each
rotational position of the shaft, and 2) the incremental encoder,
which produces digital pulses as the shaft rotates, allowing
measurement of relative position of shaft. Most rotary encoders
are composed of a glass or plastic slotted disk. As
radial lines in each track interrupt the beam between a
photoemitter-detector pair (or [[Optointerrupter]]), digital pulses are produced.

 

[[image:encoder diagram.png|center]]

<h4>Absolute encoder</h4>

The optical disk of the absolute encoder is designed to
produce a digital word that distinguishes N distinct positions of
the shaft. For example, if there are 8 tracks, the encoder is
capable of producing 256 distinct positions or an angular
resolution of 1.406 (360/256) degrees. The most common types of
numerical encoding used in the absolute encoder are gray and
binary codes. To illustrate the acion of an absolute encoder, the
gray code and natural binary code dsisk track patterns for a
simple 4-track (4-bit) encoder are illustrated in Fig 2 and 3.
The linear patterns and associated timing diagrams are what the
photodetectors sense as the code disk circular tracks rotate with
the shaft. The output bit codes for both coding schemes are
listed in Table 1.

{|
|+
|[[image:gray.jpg|300px]]||[[image:binary.jpg|300px]]
|}

<table border="1" width="37%">
<tr>
<td width="19%">
Decimal code
</td>
<td width="37%">
Rotation range (deg.)
</td>
<td width="22%">
Binary code
</td>
<td width="22%">
Gray code
</td>
</tr>
<tr>
<td width="19%">
0
</td>
<td width="37%">
0-22.5
</td>
<td width="22%">
0000
</td>
<td width="22%">
0000
</td>
</tr>
<tr>
<td width="19%">
1
</td>
<td width="37%">
22.5-45
</td>
<td width="22%">
0001
</td>
<td width="22%">
0001
</td>
</tr>
<tr>
<td width="19%">
2
</td>
<td width="37%">
45-67.5
</td>
<td width="22%">
0010
</td>
<td width="22%">
0011
</td>
</tr>
<tr>
<td width="19%">
3
</td>
<td width="37%">
67.5-90
</td>
<td width="22%">
0011
</td>
<td width="22%">
0010
</td>
</tr>
<tr>
<td width="19%">
4
</td>
<td width="37%">
90-112.5
</td>
<td width="22%">
0100
</td>
<td width="22%">
0110
</td>
</tr>
<tr>
<td width="19%">
5
</td>
<td width="37%">
112.5-135
</td>
<td width="22%">
0101
</td>
<td width="22%">
0111
</td>
</tr>
<tr>
<td width="19%">
6
</td>
<td width="37%">
135-157.5
</td>
<td width="22%">
0110
</td>
<td width="22%">
0101
</td>
</tr>
<tr>
<td width="19%">
7
</td>
<td width="37%">
15.75-180
</td>
<td width="22%">
0111
</td>
<td width="22%">
0100
</td>
</tr>
<tr>
<td width="19%">
8
</td>
<td width="37%">
180-202.5
</td>
<td width="22%">
1000
</td>
<td width="22%">
1100
</td>
</tr>
<tr>
<td width="19%">
9
</td>
<td width="37%">
202.5-225
</td>
<td width="22%">
1001
</td>
<td width="22%">
1101
</td>
</tr>
<tr>
<td width="19%">
10
</td>
<td width="37%">
225-247.5
</td>
<td width="22%">
1010
</td>
<td width="22%">
1111
</td>
</tr>
<tr>
<td width="19%">
11
</td>
<td width="37%">
247.5-270
</td>
<td width="22%">
1011
</td>
<td width="22%">
1110
</td>
</tr>
<tr>
<td width="19%">
12
</td>
<td width="37%">
270-292.5
</td>
<td width="22%">
1100
</td>
<td width="22%">
1010
</td>
</tr>
<tr>
<td width="19%">
13
</td>
<td width="37%">
292.5-315
</td>
<td width="22%">
1101
</td>
<td width="22%">
1011
</td>
</tr>
<tr>
<td width="19%">
14
</td>
<td width="37%">
315-337.5
</td>
<td width="22%">
1110
</td>
<td width="22%">
1001
</td>
</tr>
<tr>
<td width="19%">
15
</td>
<td width="37%">
337.5-360
</td>
<td width="22%">
1111
</td>
<td width="22%">
1000
</td>
</tr>
</table> 
Table 1. 4-Bit gray and natural binary codes
 

[[image:conv.jpg|395px|right|frame|Fig 4. Gray code to binary code conversion]]

The gray code is designed so that only one track (one bit)
will change state for each count transition, unlike the binary
code where multiple tracks (bits) change at certain count
transitions. This effect can be seen clearly in Table 1. For the
gray code, the uncertainty during a transition is only one count,
unlike with the binary code, where the uncertainty could be
multiple counts.

Since the gray code provides data with the least uncertainty
but the natural binary code is the preferred choice for direct
interface to computers and other digital devices, a circuit to
convert from gray to binary code is desirable. Figure 4 shows a
simple circuit that utilizes exclusive OR gates (XOR) to perform
this function.For a gray code to binary code conversion of any
number of bits N, the most signficant bits (MSB) of the binary
and gray code are always identical, and for each other bit, the
binary bit is the exlcusive OR (XOR) combination of adjacent gray
code bits.
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<h4>Incremental encoder</h4>

[[image:increment.jpg|352px|right]]

The incremental encoder, sometimes called a relative
encoder, is simpler in design than the absolute encoder. It
consists of two tracks and two sensors whose outputs are called
channels A and B. As the shaft rotates, pulse trains occur on
these channels at a frequency proportional to the shaft speed,
and the phase relationship between the signals yields the
direction of rotation. The code disk pattern and output signals A
and B are illustrated in Figure 5. By counting the number of
pulses and knowing the resolution of the disk, the angular motion
can be measured. The A and B channels are used to determine the
direction of rotation by assessing which channels "leads" the
other. The signals from the two channels are a 1/4 cycle out of
phase with each other and are known as quadrature signals. Often
a third output channel, called INDEX, yields one pulse per
revolution, which is useful in counting full revolutions. It is
also useful as a reference to define a home base or zero
position.

Figure 5 illustrates two separate tracks for the A and B
channels, but a more common configuration uses a single track
with the A and B sensors offset a 1/4 cycle on the track to yield
the same signal pattern. A single-track code disk is simpler and
cheaper to manufacture.

The quadrature signals A and B can be decoded to yield the
direction of rotation as hown in Figure 6. Decoding transitions
of A and B by using sequential logic circuits in different ways
can provide three different resolutions of the output pulses: 1X,
2X, 4X. 1X resolution only provides a single pulse for each cycle
in one of the signals A or B, 4X resolution provides a pulse at
every edge transition in the two signals A and B providing four
times the 1X resolution. The direction of rotation(clockwise or
counter-clockwise) is determined by the level of one signal
during an edge transition of the second signal. For example, in
the 1X mode, A=<math>\downarrow</math> with B =1 implies a clockwise pulse, and B=<math>\downarrow</math> with A=1 implies a
counter-clockwise pulse. If we only had a single output channel A
or B, it would be impossible to determine the direction of
rotation. Furthermore, shaft jitter around an edge transition in
the single signal woudl result in erroneous pulses..

[[image:quadrature.jpg|500px]]

[[image:encoder parts.png|center]]

<h4>Connecting an Encoder to the PC/104 Stack</h4>

To connect an encoder to the PC/104 stack, you will have to create a suitable ribbon cable connector. The connector that came with the encoder will most likely not match the correct pinout to attach. Consult the [[PC104 I/O]] page to see how to connect the encoder.

==Datasheets==
Datasheet for the Maxon Tacho 103935 encoder: [[Media:maxon_tacho_103935_encoder.pdf]]

==References==
Introduction to Mechatronics and Measurement Systems, Histand & Alciatore, 1999 McGraw Hill

ME 224 Experimental Engineering

2006-11-27T23:35:48Z

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