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Design News Gadget Freak Case #164 Page 1 THE GADGET FREAK FILES CASE #164 Pet owners who want to spend a day hiking, at the beach, shopping at the mall, or even a weekend far from home can build an automatic pet feeder to dispense water and dry food at a preset time. Tom Thompson, Julie Redmond, Curtis Siebenaller, and Nathan Woodworth at Colorado State University explain how to combine electronic and mechanical devices to do just that. Their design would satisfy even Morris, the world's most finicky cat. The automatic feeder uses four Microchip Technology PIC microcontrollers (MCUs) to deliver food once per day via a motor-driven feeder wheel. Water delivery depends on the amount of water left in the pet's bowl. The feeder uses a menu system and a keypad that let a pet owner: Set the current date and time. Select the amount of food to dispense (in 0.5 cup, or 120 cm 3 Set the time to dispense food. increments). Calibrate the water sensors. Adjust the contrast on the LCD displays. Turn the water or food dispensers on or off individually. System photos and additional images appear at the end of this document. Download the PICBASIC sourceJATcode from: www.gfreak.com/GF163/GF163_Feeder.zip. The feeder uses an integrated real-time clock to determine when to dispenses food once a day at the time set by the pet owner who also can set the amount dispensed. The feeder system includes the capability to operate a water valve or pump and to monitor the water levels in a built-in water dish and reservoir tank. Tom and his team fabricated capacitive sensors that the water-control MCU uses to determine the water level around each sensor. Watch the video for this project for a tour of its operations. NOTE: The as-built project included an optional spring scale that would indicate to the Feeder MCU that the dry food supply had decreased to a preset point and it needed a refill. The following description does not discuss the optional scale, although it appears in several photographs. The schematic diagram shows a normally-open pushbutton labeled, "Scale Trip Switch" connected to pin 10 on the Feeder MCU. Substitute a wire jumper for this pushbutton in the circuit. Note that this Gadget Freak design is not a cut-and-assembly project. It requires some engineering and ingenuity from the builder. But, builders are free to adapt the design as they see fit. Some metal-working experience will help, as will hands-on experience with small motors, gears, and pulley drives. Please read the entire Gadget Freak article before you start this project. BUILD INSTRUCTIONS Dispenser Motor

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Page 1: THE GADGET FREAK FILES CASE #164 - …downloads.deusm.com/designnews/25233-Gadget_Freak_Case_164_L… · THE GADGET FREAK FILES CASE #164 . Pet owners who want to spend a day hiking,

Design News Gadget Freak Case #164 Page 1

THE GADGET FREAK FILES CASE #164 Pet owners who want to spend a day hiking, at the beach, shopping at the mall, or even a weekend far from home can build an automatic pet feeder to dispense water and dry food at a preset time. Tom Thompson, Julie Redmond, Curtis Siebenaller, and Nathan Woodworth at Colorado State University explain how to combine electronic and mechanical devices to do just that. Their design would satisfy even Morris, the world's most finicky cat. The automatic feeder uses four Microchip Technology PIC microcontrollers (MCUs) to deliver food once per day via a motor-driven feeder wheel. Water delivery depends on the amount of water left in the pet's bowl. The feeder uses a menu system and a keypad that let a pet owner:

� Set the current date and time. � Select the amount of food to dispense (in 0.5 cup, or 120 cm3

� Set the time to dispense food. increments).

� Calibrate the water sensors. � Adjust the contrast on the LCD displays. � Turn the water or food dispensers on or off individually.

System photos and additional images appear at the end of this document. Download the PICBASIC sourceJATcode from: www.gfreak.com/GF163/GF163_Feeder.zip. The feeder uses an integrated real-time clock to determine when to dispenses food once a day at the time set by the pet owner who also can set the amount dispensed. The feeder system includes the capability to operate a water valve or pump and to monitor the water levels in a built-in water dish and reservoir tank. Tom and his team fabricated capacitive sensors that the water-control MCU uses to determine the water level around each sensor. Watch the video for this project for a tour of its operations.

NOTE: The as-built project included an optional spring scale that would indicate to the Feeder MCU that the dry food supply had decreased to a preset point and it needed a refill. The following description does not discuss the optional scale, although it appears in several photographs. The schematic diagram shows a normally-open pushbutton labeled, "Scale Trip Switch" connected to pin 10 on the Feeder MCU. Substitute a wire jumper for this pushbutton in the circuit.

Note that this Gadget Freak design is not a cut-and-assembly project. It requires some engineering and ingenuity from the builder. But, builders are free to adapt the design as they see fit. Some metal-working experience will help, as will hands-on experience with small motors, gears, and pulley drives. Please read the entire Gadget Freak article before you start this project. BUILD INSTRUCTIONS Dispenser Motor

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Design News Gadget Freak Case #163 Page 2

In the original design, Tom's group used the motor and gearing from a Black and Decker battery-powered drill and controlled it with a pulse-width modulation (PWM) output from the Feeder PIC16F88 microcontroller shown in the main schematic diagram (Figure 1.1). Schematic diagrams appear on full pages at the end of this document. In the final design, the team "geared down" the motor with a set of pulleys (which added parts) to reduce the output speed yet retain torque. The PWM-drive transistors can control a motor that draws as much as 3 amps at running speed. (The information in Figure 1.2 provides additional schematic-diagram details.) In practice, a builder could locate a more suitable motor with an attached gearbox. The Anaheim Automation permanent-magnet brushless-DC motor, model number BDPG-38-86-12V-3000-R139, can provide a peak torque of 50 in-lb, which should suffice to prevent dry food from jamming when the wheel is adjusted correctly. (For information, see: Design and Assembly of the Feeder Wheel, below.) Builders could choose to include a series of pulleys and belts, or even a chain drive, to reduce motor output speed. The belt-and-pulley arrangement would let the motor "slip" if someone inadvertently puts a finger in the inlet to the dispenser wheel. During testing, powering the motor at the wrong time could cause a painful experience for a person adjusting the dispensing wheel. The pulley set connected to the drill motor used a round vacuum-cleaner belt and two pulleys that were turned on a lathe. The PWM signal that controls the drill motor let the team use the drill without any "gearing down" to achieve the desired output speed. The control loop changes the PWM duty cycle based on the wheel speed (see: Encoder Setup, below). An output speed of 20 rpm or less is best for the wheel. Selecting a motor geared to operate the feeder wheel at 10 rpm could let you exclude the PWM circuitry and code and use an on-off relay, instead of the PWM-drive transistors, to turn the motor on or off. Builders can buy pulleys, drive belts, gears, bearings, bushings, and other mechanical components from Stock Drive products/Sterling Instruments: www.sdp-si.com, and from Small Parts: www.smallparts.com. Select a Valve or Pump This type of pet feeder can rely on gravity-fed water stored in a tank or on pressurized water from a home's water system. If you use a gravity-fed tank, a small 5V or 12V pump would suffice. Simply wire it to the proper voltage from the power supply via the normally-open relay, shown as Relay Switch in Fig. 1-1. (The schematic diagram shows the connections for a 120V AC normally-closed water valve.) As an alternative, you can connect to a pressurized-water supply and use a sprinkler valve as long as the water connections keep the flow at a steady rate into the water dish. DripWorks (www.dripworksusa.com) sells a variety of 24V DC valves used in drip-irrigation systems. The VBAC34 valve, for example, takes 3/4-inch plastic fittings, Builders can use plastic tubing on the outlet and pressure-compensating emitters to

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Design News Gadget Freak Case #164 Page 3

reduce the flow to 1/2, 1, 2, or 6 gallons/hour. Many home-improvement and building-supply stores sell similar valves, fittings, and reducers. Making the Capacitive Sensors Each of the two water-level sensors comprises two thin waterproofed aluminum plates, 2-inches (5 cm) wide by 10-inches (25 cm) long, attached to a lead wire and separated with a nonconductive spacer, as shown in Figure 2. The lead wire to each plate must make a good electrical contact with the plate. Make two small holes in a plate near a short edge and weave the wire into the holes. Then fold over the short edge onto the wire once or twice and crimp the connection with pliers. You will need four of these plate-wire assemblies.

Figure 2. Water-level sensor plates. Next, wrap each plate--with wire attached--with a nontoxic dielectric material to seal the plates from the water. The original plates use several layers of electrical tape. The amount of tape used to seal the plates made them much thicker than the original metal. Any sealant should also cover about the first 1/2-inch (12 mm) or so of the wire as it leaves the aluminum plate. [Editor's note: Consider using a material such as Plasti Dip in which you can dip each electrode to create a integral molded coating (www.plastidip.com). Plasti Dip is nontoxic when fully cured. Or, try Starbrite Liquid Electrical Tape (www.starbrite.com), available

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Design News Gadget Freak Case #163 Page 4

at hardware and building-supply stores. Also, some copying stores have laminators that could encapsulate the electrodes and their lead wire.] After the coating material has dried, place a sealed electrode in a container of tap water, but keep the unconnected end of the wire lead out of the water. Let the sealed plate sit in the water for several hours. Then attach a fixed resistor with a value between 10 kohms and 100 kohms to the wire and apply between 5V and 12V DC to the unconnected end of the resistor. A 9V battery will work well. Place the power supply's, or battery's, ground wire into the water. If water has penetrated the sealed electrode, you will measure a small voltage across the resistor. A reading of 0V indicates a good seal around the electrode. Note: If you use a high resistance resistor, a voltmeter may show a small voltage across the resistor even with no current flow. If you suspect this is the case, use a lower resistance, say, 10 kohms. If you think a defect exists in a plate's seal, dry it and attempt to seal the leak. Or, create a new electrode. After you have completely insulated four plates, glue six (6) 1/4-inch (6.5 mm) thick plastic spacers between each pair of plates at regular intervals. [Editor's note: plastic spacers, such as Allied part #839-1097 or 839-1114 would work well.] The space between the plates lets water flow freely. Use a nontoxic water-insoluble glue such as a silicone-based sealant. As water fills the space between the fixed pair of electrodes, the capacitance across the connections increases. You can create the sensor plates in any 2D shape. The best results will occur with plates of the same surface area and spacing, as noted above. Sensors with slightly larger capacitance (larger surface area) should work, too. Water Storage The feeder requires a lined or sealed water dish that will prevent water from soaking into the wooden frame. Likewise, the water reservoir requires a similar lined or sealed container. You could use a removable water container and dish, which would make for easier cleaning and construction. Use a non-toxic paint on all parts of the feeder. Main Box Assembly For most of the prototype we used 1/2-inch-thick medium-density fiberboard (MDF), however 1/2-inch plywood might be a better choice if you plan to create a built-in water-storage section rather than use a removable reservoir and dish. The pieces are joined with screws and 1x2 and 2x2 (inch) dimensioned lumber made from furring strips. Wood glue provides the best bond. Again, use a non-toxic paint to protect the wood from moisture. For CAD drawings of the box assembly, see Figures 3 through 6. Your main box might vary, depending on how you decide to construct it.

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Figure 3.

Figure 4.

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Figure 5. Front view of the feeder without front cover. The water container is on the left, the food container on the right.

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Figure 6. Front view of the feeder, less the front piece that completes the water bowl and food holder. Design and Assembly of the Feeder Wheel The feeder uses a "paddle wheel" with four paddles, or dividers, to dispense dry food for a pet as the wheel rotates. The height of the four dividers, relative to the sides of the wheel (the height difference shown in Figure 7), depends on the type of food you intend to use. For large pieces of dry food, increase this dimension. During the wheel's operation, the dry food should not get cut into pieces as the wheel rotates, and the food should not cause the wheel to jam, as the divider passes the edge of the outer shield near the top. Figure 8 shows a 3D view of an assembled wheel.

Figure 7. This front view of the reeder wheel shows the position of the dividers and cross braces, the encoder disc and the infrared LEDs and photo-transistors. Note the height difference between the diameter of the wheel and the top of the divider.

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Figure 8. Food-dispensing wheel with ramp and outer shields in place. Food enters through an opening above the wheel to the upper left. Choose an axle and bearings that match. Create a center hub for the wheel to match your shaft diameter and cut it to 2.1 inches, or 5.3 cm, in length. Take care when you press the bearings into the wood (you want them to fit tightly into the housing to hold the wheel in place) so that the wood that makes up the housing is not damaged. The nice thing about this type of setup is that you can align and adjust the bearings and prevent their axial movement. A wheel with four dividers that measure about 2.1-inches (5.3 cm) wide and 2-inches (5 cm) deep provides about 1/2-cup (120 cm3

) of dry food for each 90-degree turn of the shaft. Builders can change the dimensions to account for larger or smaller volumes, but they must properly adjust the rest of the design to accommodate their new feeder-wheel volume.

Tom's team fabricated the wheel shown from 20-gauge aluminum sheet. The wheel requires two circular pieces for the wheel sides, four rectangular pieces for individual dividers, and 16 narrow rectangular pieces for braces, as shown in Figure 9. As an easier alternative, you cut out four dividers with integral tabs (Figure 10) and use them instead.

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Figure 9. Partially assembled feeder wheel. This view shows the use of individual dividers and braces.

Figure 10. Instead of using individual dividers and braces, a feeder wheel could use all-in-one pieces of aluminum with integral tabs. Roughen the ends of the braces, or the tabs, and the roughen the area around the slots on one side of each wheel side-piece. During assembly, ensure the roughened side of the wheel side-pieces faces out, away from the spacers.

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Whether you use the tabbed spacers or individual braces and spacers, assembly proceeds in basically the same way. Insert the tabs or braces in the slots in one wheel side-piece and bend them into position. If you use separate dividers, slide them into place between pairs of dividers. Next, place the second wheel side-piece over the tabs or braces and bend them into position. Make the slots in the wheel side-pieces slightly wider than the thickness of the material you use for the dividers to ensure a tight assembly, but that still leaves the wheel flexible enough so you can aligned it after assembly. Use an epoxy or other high-strength glue to bond the bent tabs on the outside of the wheel side-pieces to the side-pieces. If you used separate braces and spacers, glue the tabs the same way and also use some glue to hold the spacers in place against the inside of the wheel side-pieces, the spacers, and the wheel 's center hub. Now that the parts are assembled and glued, let the glue set. Then, clamp down the wheel and use threaded alignment spacers for the bearings or, if you matched the axle and bearings, then use nuts and washers. You must establish solid contact between the sides of the wheel and the wheel spacer and the pieces that are pressing the sides because the assembly must transmit torque to the wheel from the axle. [Editor's note: You drill the hollow feeder-wheel hub or spindle and tap it for set screws. Then machine (or file) a flat section on the center shaft, slide the wheel on the shaft and tighten the set screws. A drop of Loctite on the set-screw threads will hold the tightened screws in place.] Making the Wheel Housing/Motor Mounts A wheel/ramp housing is shown in Figures 11 and 12, and earlier in Figure 8. This housing doesn't include space for a motor mount, but builders can extend the space in the housing under the wheel for a motor, depending on whether or not they use a pulley or chain drive for the feeder wheel. A design could leave space under the water tank so the motor could directly drive the feeder wheel. Small braces (Fig. 8) hold the ramp/outer shields in place. Tom's team used small wood blocks attached to the large frame on the outside, and glue from a hot-glue gun stabilized and sealed the ramp edges where they met the side of the housing.

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Figure 11. Feeder wheel and encoder assembly. Food enters the wheel through the top opening in the shield and goes down the incline into the feeder dish.

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Figure 12. Feeder wheel assembly, front view. The ramp in the foreground delivers dry food to the feed container on the front of the feeder system. Encoder Setup A disc of plastic or metal creates an optical encoder wheel that the control circuit uses to determine wheel position and speed. The encoder comprises a thin disc of opaque material with two concentric rings of small holes that align individually with an infrared photo-transistor on one side of the wheel and an IR-emitting LED on the other. As shown in Figure 13, the outer ring has 20 evenly spaced holes (18 degrees apart) that the feeder MCU uses to determine the wheel's speed. The inner ring has only four

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equally spaced holes (90 degrees apart) that let the MCU measuring the number of quarter rotations the wheel has made. Figure 8 illustrates the placement of the encoder disc on the wheel shaft. The aluminum encoder measures 3-inches (7.6 cm) in diameter and the holes measure 0.1-inches (2.5 mm) in diameter. You can place the holes radially as you choose for your mounting configuration. Tom recommends at least a 0.4-inch (1 cm) space between holes.

Figure 13. The feeder MCU uses the speed data to adjust the PWM duty cycle for the motor and it uses the quarter turn data to measure the food. The program for the feeder MCU lets it recognize a complete stall that would occur if food bits prevent the wheel from turning. When a stall occurs, the feeder MCU adds to the PWM output cycles to boost power to the motor and clear the jam.

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Figure 14. This view shows the horizontal water sensor in the water bowl and the inset image on the right shows the sensor in the water-reservoir section. If you use discrete infrared (IR) sensors and emitters, pay careful attention to their placement. You should put both IR Sensor LEDs on the same side of the encoder wheel and the IR phototransistors on the opposite side (Fig. 8). Place the IR photo transistors and LEDs so they align with their respective holes. Keeping the LED/photo-transistor pairs far from each other helps reduce any interference, or "splash," from the IR radiation produced by one LED from influencing the photo-transistor from the other pair. You will need sufficient space between the holes on the encoder wheel so the infrared sensors do not produce an output greater than 1.0V when blocked by the wheel. Some spurious IR radiation from the IR LEDs and from ambient conditions could cause the IR photo-transistor's output to exceed the 1V threshold and signal that condition to the MCU. The IR photo-transistor's output should only exceed 1V when the encoder wheel positions a hole between them. You could use a small-diameter paper tube around each LED and photo-transistor to provide a shield that reduces interfering light. If your design does not need the 20 holes in the encoder wheel for speed control, you could cut four slots in the outer edge of the encoder wheel and use a slotted optical

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switch, or optical interrupter, such as the H21A (Allied part #263-0581) in place of the discrete IR LED and photo-transistor. The H21A combines an IR LED coupled with a phototransistor in a single plastic package with about a 3-mm slot between them. This would require changing the resistances involved with the IR devices to ensure that the current limitations of the IR emitter and detector are not exceeded. The 30-ohm shown in figure 1.1 for the emitters allows about 100mA of current which exceeds the limits of the H21A emitter. To check the alignment of discrete LED/phototransistor pairs, power the circuit and measure the voltage at pins 8 and 9 on the Feeder PIC16F88 MCU. This voltage should measure below 0.75V when you have the opaque encoder wheel between the LED and its corresponding phototransistor. The voltage should measure between 1.8 and 5V when you rotate the encoder wheel to position a hole between the LED and the phototransistor. The Feeder MCU should detect a logic one when a phototransistor's output voltage rises above 1.15 to 1.4V. If the voltage output by a photo-transistor remains too high when you block the light path with an opaque portion of the encoder wheel, you might need to create an encoder wheel with a larger diameter so you can increase the spacing between the holes. If that does not solve the problem, lower the resistance between the input pin on the Feeder MCU and ground. If the voltage doesn't rise high enough during the best line-of-sight placement, make sure the emitter and detector are lined up properly and if they are then increase the resistance from the pin on the controller to ground. A good setup produces a voltage at the respective Feeder MCU below 0.6V when the wheel blocks the IR light path and blocked and a voltage above 2.5V when a hole permits IR light to reach the corresponding photo-transistor. Sensor Setup You must place the two capacitive water-level sensors in the dish and tank so that water can rise and fall freely in the space between the plates in each sensor. Tom placed the reservoir-tank sensor vertically and the dish sensor horizontally (Figure 14). If a sensor gets disconnected the MCU should produce a sensor "reading" at or near 0. If the dish sensor gets disconnected, the MCU defaults to the valve-relay-off state. If the tank sensor indicates a disconnected condition, the MCU will ignore the tank reading. If your design will not use a tank or reservoir to store water, use capacitors as substitutes for the tank sensor and follow the procedure below to calibrate the parameters the MCU uses. Replaced the tank sensor with one 22-pF capacitor to perform the tank-empty calibration and then place a second 22-pF capacitor in parallel with the other 22-pF capacitor to perform the tank-full calibration. Leave the two capacitors in place so the MCU will determine that the water tank is always full. The calibration of each sensor is very important for the accurate measurement of the water levels. During calibration, a pet owner uses the LCD menu and the keypad. Start with dry sensors fixed in their permanent mounting locations. After calibrating the tank-

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empty condition, fill the dish and tank to the desired water line and perform the full calibration. Menus on the LCD guide you through the process. Contrast Controller Connect the three signal lines between the Master MCU (18F2525, pins 15, 16, and 17, shown in Fig. 1.1) and the Contrast MCU to ground through 1000-ohm to 10-kohm resistors, one resistor per line. This arrangement should prevent the contrast MCU from receiving false logic-high signals if the 18F2525 is physically disconnected from these three lines during power-on. If the Master 18F2525 MCU does not maintain 0V on the LCD contrast controller's enable line (18F2525, pin 17), the contrast MCU could detect false logic-high signals on this input. A false logic high will cause the contrast to change drastically, most often to a lighter-contrast condition that makes the display appear blank and nonfunctional. A builder could remove the contrast controller MCU and connect the contrast lines of the LCD to a 10 kohm potentiometer for each LCD to provide contrast that can only be adjusted by gaining access to the pots themselves (this is a standard way to provide contrast adjustment to an LCD display and is shown in the data sheets for the LCD displays). The programs will work without the contrast-controller MCU because no other MCUs depend on signals from the contrast-controller MCU. If you include the contrast-controller MCU and the displays flicker while using the contract controller following the instructions at the end of the code for the contrast-controller MCU to adjust the period of the PWM signals for the displays. Troubleshooting the Electronics Good connections are essential! If you suspect problems with a serial signal, use an oscilloscope to probe the input side of each signal. The protocols for serial communication between the MCUs in this project use three lines between each MCU. One line provides analog signaling (handshaking) from the sending MCU to the receiving MCU to indicate information is ready to be sent, when the receiving MCU registers this high signal, it readies its portion of code related to serial comm. Then the receiver sends a high signal along a second handshake line to the sender, indicating that it is ready to receive data. When this high signal reaches the sending MCU the serial comm. is sent to the receiving MCU. To obtain a representation of the serial signal use an oscilloscope probe connected directly to the sending MCU’s data pin. The data rate is specified by PICBASIC PRO with the number 16416, which establishes the serial comm. driven true, with no parity, at a baud rate of 19200 bits/sec, and data can be sent in either direction on the same data line. PICBASIC PRO is available from microEngineering Labs: microengineeringlabs.com. Builders can reduce the baud rate to 9600 bits/sec by changing the value 16416 to 16468 in all of the serout2/serin2 commands in all of the programs. You must make this change in the code for both receivers and transmitters.

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GF #163, Pet Feeder Bill of Materials Amt Part Description Allied Part #

1 Keypad, 12 key 948-0013 1 PIC18F2525 MCU 383-1727 3 PIC16F88 MCU 383-0496 1 Piezoelectric Buzzer 854-0084

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1 NTE392 Transistor 935-8009 1 TIP120 Transistor 568-1140 3 PN2222A Transistor 568-1125 2 DPDT Relay 788-1089 1 24x2 Character LCD 355-0020 1 16x2 Character LCD 355-0010 4 0.1 μF Ceramic Capacitors 541-0340 1 0.1 μF Polyester Capacitor 862-2142 13 100-ohm Resistor, 1/4W 296-6638 8 1000Ω Resistor, 1/4W 296-4741 1 10kΩ Resistor, 1/4W 296-4743 2 1MΩ Resistor, 1/4W 296-6331 4 3MΩ Resistor, 1/4W 296-6574 1 1500Ω Resistor, 1/4W 895-0540 1 3000Ω Resistor, 1/4W 895-3159 3 6000Ω Resistor, 1/4W 895-0605 4 1N5059 Diode, 2A 431-0632 3 12Ω Resistor, 2W 296-2323 1 Heatsink for NTE392 619-0087 1 Heatsink for TIP120 296-6966 3 18-Pin DIP Socket 374-5536 1 28-Pin DIP Socket 374-5548 2 Infrared LED 387-0014 2 Photo-transistors 387-1799

Other Parts

1 Water valve or small pump, 5V DC or 12 VDC, must operate via relay.

1 Geared motor, 5V to 12V with high torque and less than or equal to 20 rpm.

1 Computer Power Supply, RAIDMAX RX-380K 380W ATX12V. www.newegg.com

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Fig A. The MCU circuit board.

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Art B. The front panel provides a keypad for data entry and two displays. The bottom view shows the optional scale platform removed from its mount and placed on its side. The unpainted wood piece "floats" up and down on springs and fits against the Food Dispenser opening in the green painted area. A sensor pushbutton indicates when the springs can lift the platform.

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Fig. C. The front view of the Pet Feeder.

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Fig. D. This top view of the open feeder shows the water-storage tank and the food dispenser wheel below the food-scale platform.

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Fig. E. The power transistors require a heat sink, as shown here.

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Fig. F. This view of the feeder wheel shows the arrangement of dividers and braces. Using an all-in-one divider that includes tabs eliminates some of the work involved with discrete dividers and braces.

Fig G.

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Fig. F.

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