ME2402-Mechatronics-Lecture-notes.pdf

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Unit 1

Introduction to Mechatronics Systems

Mechatronics is the synergistic combination of Mechanical engineering, Electronic

engineering, Computer engineering, Control engineering, and Systems Design engineering in order

to design, and manufacture useful products. The term mechatronics is defined as a multidisciplinary

engineering system design, that is to say it rejects splitting engineering into separate disciplines.

A mechatronics engineer unites the principles of mechanics, electronics, and computing to

generate a simpler, more economical and reliable system. Mechatronics is centered on mechanics,

electronics, computing, control engineering, molecular engineering (from nanochemistry and

biology), and optical engineering, which, combined, make possible the generation of simpler, more

economical, reliable and versatile systems. The portmanteau "mechatronics" was coined by Tetsuro

Mori, the senior engineer of the Japanese company Yaskawa in 1969. An industrial robot is a prime

example of a mechatronics system; it includes aspects of electronics, mechanics, and computing to

do its day-to-day jobs.

The development of mechatronics has gone through three stages: The first stage corresponds

to the years around the introduction of word mechatronics.

During this stage, technologies used in mechatronics systems developed rather independently of

each other and individually.With start of eighties a synergic integration of different technologies

started taking place.A notable example is opto-electronics, an integration of optics and electronics.

The concept of hardware/software co-design also started in this year.

The third stage, which is considered as start of ‗Mechatronics Age‘, starts with the early

nineties. The most notable aspect of this stage are more and more integration of different engineering

disciplinesand increased use of computational intelligence in the mechatronics products and

systems.Another important development in the third stage is the concept of ‗micromechatronis‘, i.e.,

start of miniaturization the components such as microactuators and microsensors.Design of such

products and processes, therefore, has to be the outcome of a multi-disciplinary activity rather than

an interdisciplinary one.

Hence mechatronics challenges the traditional engineering thinking, because the way it is operating,

is crossing the boundaries between the traditional engineering disciplines.

SENSORS

A sensor is a device which receives and responds to a signal. A sensor's sensitivity indicates

how much the sensor's output changes when the measured quantity changes. For instance, if the

mercury in a thermometer moves 1 cm when the temperature changes by 1 °C, the sensitivity is

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1 cm/°C (it is basically the slope Dy/Dx assuming a linear characteristic). Sensors that measure very

small changes must have very high sensitivities.

A good sensor obeys the following rules:

Is sensitive to the measured property

Is insensitive to any other property likely to be encountered in its application

Does not influence the measured property

Characteristics of sensor

The sensitivity may in practice differ from the value specified. This is called a sensitivity

error, but the sensor is still linear.

Since the range of the output signal is always limited, the output signal will eventually reach

a minimum or maximum when the measured property exceeds the limits. The full scale range

defines the maximum and minimum values of the measured property.

If the output signal is not zero when the measured property is zero, the sensor has an offset or

bias. This is defined as the output of the sensor at zero input.

If the sensitivity is not constant over the range of the sensor, this is called nonlinearity.

Usually this is defined by the amount the output differs from ideal behavior over the full

range of the sensor, often noted as a percentage of the full range.

If the deviation is caused by a rapid change of the measured property over time, there is a

dynamic error. Often, this behaviour is described with a bode plot showing sensitivity error

and phase shift as function of the frequency of a periodic input signal.

If the output signal slowly changes independent of the measured property, this is defined as

drift (telecommunication).

Long term drift usually indicates a slow degradation of sensor properties over a long period

of time.

Noise is a random deviation of the signal that varies in time.

Hysteresis is an error caused by when the measured property reverses direction, but there is

some finite lag in time for the sensor to respond, creating a different offset error in one

direction than in the other.

If the sensor has a digital output, the output is essentially an approximation of the measured

property. The approximation error is also called digitization error.

If the signal is monitored digitally, limitation of the sampling frequency also can cause a

dynamic error, or if the variable or added noise noise changes periodically at a frequency

near a multiple of the sampling rate may induce aliasing errors.

The sensor may to some extent be sensitive to properties other than the property being

measured. For example, most sensors are influenced by the temperature of their environment.

DISPLACEMENT AND POSITION SENSORS

Displacement Measurement

Measurement of displacement is the basis of measuring:

Position

Velocity

Acceleration

Stress

Force

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Pressure

Proximity

Thickness

Displacement Sensors types

• Potentiometers displacement sensors

• Inductive displacement sensors

• Capacitive displacement sensors

• Eddy current displacement sensors

• Piezoelectric displacement sensors

• Ultrasonic displacement sensors

• Magnetostrictive displacement sensors

• Optical encoder displacement sensors

• Strain Gages displacement sensors

Resistive displacement sensors: An electrically conductive wiper that slides against a fixed

resistive element. To measure displacement, a potentiometer is typically wired in a voltage divider

configuration.

A known voltage is applied to the resistor ends. The contact is attached to the moving

object of interest The output voltage at the contact is proportional to the displacement.

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Inductive displacement sensors

The coil acts as a source of magnetomotive force that drives the flux through the magnetic

circuit and the air gap. The presence of the air gap causes a large increase in circuit reluctance and a

corresponding decrease in the flux. Hence, a small variation in the air gap results in a

measurable change in inductance.

Linear Variable Differential Transformer (LVDT)

Motion of a magnetic core changes the mutual inductance of two secondary coils relative to a

primary coil Primary coil voltage: VSsin(wt)

Secondary coil induced emf:

V1=k1sin(wt) and V2=k2sin(wt)

k1 and k2 depend on the amount of coupling between the primary and the secondary coils,

which is proportional to the position of the coil.

When the coil is in the central position,

k1=k2; VOUT=V1-V2=0

When the coil is is displaced x units,

k1 not equal to k2 ;

VOUT=(k1-k2)sin(wt)

Positive or negative displacements are determined from the phase

of VOUT.

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The linear variable differential transformer (LVDT) is a type of electrical transformer used for

measuring linear displacement. The transformer has three solenoidal coils placed end-to-end around

a tube. The center coil is the primary, and the two outer coils are the secondaries. A cylindrical

ferromagnetic core, attached to the object whose position is to be measured, slides along the axis of

the tube.

An alternating current is driven through the primary, causing a voltage to be induced in each

secondary proportional to its mutual inductance with the primary. The frequency is usually in the

range 1 to 10 kHz.

As the core moves, these mutual inductances change, causing the voltages induced in the secondaries

to change. The coils are connected in reverse series, so that the output voltage is the difference

(hence "differential") between the two secondary voltages. When the core is in its central position,

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equidistant between the two secondaries, equal but opposite voltages are induced in these two coils,

so the output voltage is zero.

When the core is displaced in one direction, the voltage in one coil increases as the other decreases,

causing the output voltage to increase from zero to a maximum. This voltage is in phase with the

primary voltage. When the core moves in the other direction, the output voltage also increases from

zero to a maximum, but its phase is opposite to that of the primary. The magnitude of the output

voltage is proportional to the distance moved by the core (up to its limit of travel), which is why the

device is described as "linear". The phase of the voltage indicates the direction of the displacement.

Because the sliding core does not touch the inside of the tube, it can move without friction, making

the LVDT a highly reliable device. The absence of any sliding or rotating contacts allows the LVDT

to be completely sealed against the environment.

LVDTs are commonly used for position feedback in servomechanisms, and for automated

measurement in machine tools and many ot her industrial and scientific applications.

A proximity sensor is a sensor able to detect the presence of nearby objects without any physical

contact. A proximity sensor often emits an electromagnetic or electrostatic field, or a beam of

electromagnetic radiation (infrared, for instance), and looks for changes in the field or return signal.

The object being sensed is often referred to as the proximity sensor's target. Different proximity

sensor targets demand different sensors. For example, a capacitive or photoelectric sensor might be

suitable for a plastic target; an inductive proximity sensor requires a metal target.

The maximum distance that this sensor can detect is defined "nominal range". Some sensors have

adjustments of the nominal range or means to report a graduated detection distance.

Proximity sensors can have a high reliability and long functional life because of the absence of

mechanical parts and lack of physical contact between sensor and the sensed object.

Proximity sensors are also used in machine vibration monitoring to measure the variation in distance

between a shaft and its support bearing. This is common in large steam turbines, compressors, and

motors that use sleeve-type bearings.

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A thermocouple is a junction between two different metals that produces a voltage related to a

temperature difference. Thermocouples are a widely used type of temperature sensor for

measurement and control[1]

and can also be used to convert heat into electric power. They are

inexpensive[2]

and interchangeable, are supplied fitted with standard connectors, and can measure a

wide range of temperatures. The main limitation is accuracy: system errors of less than one degree

Celsius (C) can be difficult to achieve.[3]

Any junction of dissimilar metals will produce an electric potential related to temperature.

Thermocouples for practical measurement of temperature are junctions of specific alloys which have

a predictable and repeatable relationship between temperature and voltage. Different alloys are used

for different temperature ranges. Properties such as resistance to corrosion may also be important

when choosing a type of thermocouple. Where the measurement point is far from the measuring

instrument, the intermediate connection can be made by extension wires which are less costly than

the materials used to make the sensor. Thermocouples are usually standardized against a reference

temperature of 0 degrees Celsius; practical instruments use electronic methods of cold-junction

compensation to adjust for varying temperature at the instrument terminals. Electronic instruments

can also compensate for the varying characteristics of the thermocouple, and so improve the

precision and accuracy of measurements.

Thermocouples are widely used in science and industry; applications include temperature

measurement for kilns, gas turbine exhaust, diesel engines, and other industrial processes.

Resistance thermometers, also called resistance temperature detectors or resistive thermal

devices (RTDs), are temperature sensors that exploit the predictable change in electrical resistance

of some materials with changing temperature. As they are almost invariably made of platinum, they

are often called platinum resistance thermometers (PRTs). They are slowly replacing the use of

thermocouples in many industrial applications below 600 °C, due to higher accuracy and

repeatability.[1]

There are many categories; carbon resistors, film, and wire-wound types are the most widely used.

Carbon resistors are widely available and are very inexpensive. They have very reproducible

results at low temperatures. They are the most reliable form at extremely low temperatures.

They generally do not suffer from significant hysteresis or strain gauge effects. Carbon

resistors have been used for many years because of their advantages.

Film thermometers have a layer of platinum on a substrate; the layer may be extremely thin,

perhaps one micrometer. Advantages of this type are relatively low cost (the high cost of

platinum being offset by the tiny amount required) and fast response. Such devices have

improved performance although the different expansion rates of the substrate and platinum

give "strain gauge" effects and stability problems.

Wire-wound thermometers can have greater accuracy, especially for wide temperature ranges. The

coil diameter provides a compromise between mechanical stability and allowing expansion of the

wire to minimize strain and consequential drift.

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ME1402 MECHATRONICS

(Common to Mechanical and Production- VI Semester)

OBJECTIVE

•To understand the interdisciplinary applications of Electronics, Electrical, Mechanical and

Computer Systems for the Control of Mechanical and Electronic Systems.

1.MECHATRONICS, SENSORS AND TRANSDUCERS 9

Introduction to Mechatronics Systems – Measurement Systems – Control Systems – Microprocessor

based Controllers. Sensors and Transducers – Performance Terminology – Sensors for

Displacement, Position and Proximity; Velocity, Motion, Force, Fluid Pressure, Liquid Flow, Liquid

Level, Temperature, Light Sensors – Selection of Sensors

2.ACTUATION SYSTEMS 9

Pneumatic and Hydraulic Systems – Directional Control Valves – Rotary Actuators. Mechanical

Actuation Systems – Cams – Gear Trains – Ratchet and pawl – Belt and Chain Drives – Bearings.

Electrical Actuation Systems – Mechanical Switches – Solid State Switches – Solenoids – D.C

Motors – A.C Motors – Stepper Motors.

3.SYSTEM MODELS AND CONTROLLERS 9

Building blocks of Mechanical, Electrical, Fluid and Thermal Systems, Rotational – Transnational

Systems, Electromechanical Systems – Hydraulic – Mechanical Systems. Continuous and discrete

process Controllers – Control Mode – Two – Step mode – Proportional Mode – Derivative Mode –

Integral Mode – PID Controllers – Digital Controllers – Velocity Control – Adaptive Control –

Digital Logic Control – Micro Processors Control.

4. PROGRAMMING LOGIC CONTROLLERS 9

Programmable Logic Controllers – Basic Structure – Input / Output Processing – Programming –

Mnemonics – Timers, Internal relays and counters – Shift Registers – Master and Jump Controls –

Data Handling – Analogs Input / Output – Selection of a PLC Problem.

5.DESIGN OF MECHATRONICS SYSTEM 9

Stages in designing Mechatronics Systems – Traditional and Mechatronic Design - Possible Design

Solutions Case Studies of Mechatronics Systems, Pick and place robot – automatic Car Park

Systems – Engine Management Systems.

TOTAL : 45

TEXT BOOKS

W. Bolton, ―Mechatronics‖, Pearson Education, Second Edition, 1999.

REFERENCES

Michael B. Histand and David G. Alciatore, ― Introduction to Mechatronics and Measurement

Systems‖, McGraw-Hill International Editions, 2000.

Bradley D. A., Dawson D., Buru N.C. and. Loader A.J, ―Mechatronics‖, Chapman and Hall, 1993.

Dan Necsulesu, ―Mechatronics‖, Pearson Education Asia, 2002 (Indian Reprint).

Lawrence J. Kamm, ―Understanding Electro – Mechanical Engineering‖, An Introduction to

Mechatronics, Prentice – Hall of India Pvt., Ltd., 2000.

Nitaigour Premchand Mahadik, ―Mechatronics‖, Tata McGraw-Hill publishing Company Ltd, 2003

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Unit 1

Unit 2 ACTUATION SYSTEMS

Pneumatic and Hydraulic Systems – Directional Control Valves – Rotary Actuators. Mechanical

Actuation Systems – Cams – Gear Trains – Ratchet and pawl – Belt and Chain Drives – Bearings.

Electrical Actuation Systems – Mechanical Switches – Solid State Switches – Solenoids – D.C

Motors – A.C Motors – Stepper Motors.

Pneumatic systems

Pneumatics is a branch of technology, which deals with the study and application of use of

pressurized gas to affect mechanical motion.

Pneumatic systems are extensively used in industry, where factories are commonly plumbed with

compressed air or other compressed inert gases. This is because a centrally-located and electrically-

powered compressor that powers cylinders and other pneumatic devices through solenoid valves is

often able to provide motive power in a cheaper, safer, more flexible, and more reliable way than a

large number of electric motors and actuators.

Pneumatics also has applications in dentistry, construction, mining, and other areas.

Gases used in pneumatic systems

Pneumatic systems in fixed installations such as factories use compressed air because a sustainable

supply can be made by compressing atmospheric air. The air usually has moisture removed and a

small quantity of oil added at the compressor, to avoid corrosion of mechanical components and to

lubricate them.

Factory-plumbed, pneumatic-power users need not worry about poisonous leakages as the gas is

commonly just air. Smaller or stand-alone systems can use other compressed gases which are an

asphyxiation hazard, such as nitrogen - often referred to as OFN (oxygen-free nitrogen), when

supplied in cylinders.

Any compressed gas other than air is an asphyxiation hazard - including nitrogen, which makes up

approximately 80% of air. Compressed oxygen (approx. 20% of air) would not asphyxiate, but it

would be an extreme fire hazard, so is never used in pneumatically powered devices.

Portable pneumatic tools and small vehicles such as Robot Wars machines and other hobbyist

applications are often powered by compressed carbon dioxide because containers designed to hold it

such as soda stream canisters and fire extinguishers are readily available, and the phase change

between liquid and gas makes it possible to obtain a larger volume of compressed gas from a lighter

container than compressed air would allow. Carbon dioxide is an asphyxiant and can also be a

freezing hazard when vented inappropriately.

Advantages of pneumatics

Simplicity of Design And Control

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o Machines are easily designed using standard cylinders & other components. Control

is as easy as it is simple ON - OFF type control.

Reliability o Pneumatic systems tend to have long operating lives and require very little

maintenance.

o Because gas is compressible, the equipment is less likely to be damaged by shock.

The gas in pneumatics absorbs excessive force, whereas the fluid of hydraulics

directly transfers force.

Storage o Compressed Gas can be stored, allowing the use of machines when electrical power is

lost.

Safety

o Very low chance of fire (compared to hydraulic oil).

o Machines can be designed to be overload safe.

Hydraulic systems

Hydraulic machinery are machines and tools which use fluid power to do simple work. Heavy

equipment is a common example.

In this type of machine, high-pressure liquid — called hydraulic fluid — is transmitted throughout

the machine to various hydraulic motors and hydraulic cylinders. The fluid is controlled directly or

automatically by control valves and distributed through hoses and tubes.

The popularity of hydraulic machinery is due to the very large amount of power that can be

transferred through small tubes and flexible hoses, and the high power density and wide array of

actuators that can make use of this power.

Hydraulic machinery is operated by the use of hydraulics, where a liquid is the powering medium

Advantages of hydraulics

Liquid (as a gas is also a 'fluid') does not absorb any of the supplied energy.

Capable of moving much higher loads and providing much higher forces due to the

incompressibility.

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The hydraulic working fluid is basically incompressible, leading to a minimum of spring

action. When hydraulic fluid flow is stopped, the slightest motion of the load releases the

pressure on the load; there is no need to "bleed off" pressurized air to release the pressure on

the load.

Directional control valves are one of the most fundamental parts in hydraulic machinery. They

allow fluid flow into different paths from one or more sources. They usually consist of a piston

inside a cylinder which is electrically controlled. The movement of the cylinder restricts or permits

the flow, thus it controls the fluid flow.

Directional control valves are mainly two types:

Hydraulic and

Pneumatic.

Hydraulic directional control valves are for a liquid working fluid (e.g. water, hydraulic oil) and

pneumatic directional control valves are for a gaseous (usually air) working fluid.

Control valves

Directional control valves route the fluid to the desired actuator. They usually consist of a spool

inside a cast iron or steel housing. The spool slides to different positions in the housing, intersecting

grooves and channels route the fluid based on the spool's position.

The spool has a central (neutral) position maintained with springs; in this position the supply fluid is

blocked, or returned to tank. Sliding the spool to one side routes the hydraulic fluid to an actuator

and provides a return path from the actuator to tank. When the spool is moved to the opposite

direction the supply and return paths are switched. When the spool is allowed to return to neutral

(center) position the actuator fluid paths are blocked, locking it in position.

Directional control valves are usually designed to be stackable, with one valve for each hydraulic

cylinder, and one fluid input supplying all the valves in the stack.

Tolerances are very tight in order to handle the high pressure and avoid leaking, spools typically

have a clearance with the housing of less than a thousandth of an inch (25 µm). The valve block will

be mounted to the machine's frame with a three point pattern to avoid distorting the valve block and

jamming the valve's sensitive components.

The spool position may be actuated by mechanical levers, hydraulic pilot pressure, or solenoids

which push the spool left or right. A seal allows part of the spool to protrude outside the housing,

where it is accessible to the actuator.

The main valve block is usually a stack of off the shelf directional control valves chosen by flow

capacity and performance. Some valves are designed to be proportional (flow rate proportional to

valve position), while others may be simply on-off. The control valve is one of the most expensive

and sensitive parts of a hydraulic circuit.

Pressure relief valves are used in several places in hydraulic machinery; on the return circuit

to maintain a small amount of pressure for brakes, pilot lines, etc... On hydraulic cylinders, to

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prevent overloading and hydraulic line/seal rupture. On the hydraulic reservoir, to maintain a

small positive pressure which excludes moisture and contamination.

Pressure regulators reduce the supply pressure of hydraulic fluids as needed for various

circuits.

Sequence valves control the sequence of hydraulic circuits; to ensure that one hydraulic

cylinder is fully extended before another starts its stroke, for example.

Shuttle valves provide a logical or function.

Check valves are one-way valves, allowing an accumulator to charge and maintain its

pressure after the machine is turned off, for example.

Pilot controlled Check valves are one-way valve that can be opened (for both directions) by

a foreign pressure signal. For instance if the load should not be hold by the check valve

anymore. Often the foreign pressure comes from the other pipe that is connected to the motor

or cylinder.

Counterbalance valves are in fact a special type of pilot controlled check valve. Whereas

the check valve is open or closed, the counterbalance valve acts a bit like a pilot controlled

flow control.

Cartridge valves are in fact the inner part of a check valve; they are off the shelf components

with a standardized envelope, making them easy to populate a proprietary valve block. They

are available in many configurations; on/off, proportional, pressure relief, etc. They generally

screw into a valve block and are electrically controlled to provide logic and automated

functions.

Hydraulic fuses are in-line safety devices designed to automatically seal off a hydraulic line

if pressure becomes too low, or safely vent fluid if pressure becomes too high.

Auxiliary valves in complex hydraulic systems may have auxiliary valve blocks to handle

various duties unseen to the operator, such as accumulator charging, cooling fan operation,

air conditioning power, etc. They are usually custom valves designed for the particular

machine, and may consist of a metal block with ports and channels drilled. Cartridge valves

are threaded into the ports and may be electrically controlled by switches or a microprocessor

to route fluid power as needed.

Hydraulic rotary actuators

The hydraulic rotary actuator is a device which transform hydraulic power (pressure and flow) in

rotational mechanical power (torque and speed).

It is used for alternative movements with a limited rotation angle (max 280°).The simplicity of

construction allows to obtain very high mechanical efficiency values, close to 95%.

Mechanical actuation systems

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Mechanical linear actuators operate by conversion of rotary motion into linear motion. Conversion is

commonly made via a few simple types of mechanism:

Screw: Screw jack, ball screw and roller screw actuators all operate on the principle of the

simple machine known as the screw. By rotating the actuator's nut, the screw shaft moves in

a line.

Wheel and axle: Hoist, winch, rack and pinion, chain drive, belt drive, rigid chain and rigid

belt actuators operate on the principle of the wheel and axle. By rotating a wheel/axle (e.g.

drum, gear, pulley or shaft) a linear member (e.g. cable, rack, chain or belt) moves.[1]

Cam: Cam actuators function on a principle similar to that of the wedge, but provide

relatively limited travel. As a wheel-like cam rotates, its eccentric shape provides thrust at the

base of a shaft.

Some mechanical linear actuators only pull (e.g. hoist, chain drive and belt drive) and others only

push (e.g. cam actuator).

Cams A linear actuator is an actuator that, when driven by a non-linear motion, creates linear motion (as

opposed to rotary motion, e.g. of an electric motor). Mechanical and hydraulic actuation are the most

common methods of achieving the linear motion.

A cam is a rotating or sliding piece in a mechanical linkage used especially in transforming rotary

motion into linear motion or vice-versa.[1][2]

It is often a part of a rotating wheel (e.g. an eccentric

wheel) or shaft (e.g. a cylinder with an irregular shape) that strikes a lever at one or more points on

its circular path. The cam can be a simple tooth, as is used to deliver pulses of power to a steam

hammer, for example, or an eccentric disc or other shape that produces a smooth reciprocating (back

and forth) motion in the follower, which is a lever making contact with the cam.

Gear train

A gear train is a set or system of gears arranged to transfer rotational torque from one part of a

mechanical system to another.

Gear trains may consist of:

Driving gears - attached to the input shaft

Driven gears/Motor gears - attached to the output shaft

Idler gears - interposed between the driving and driven gear in order to maintain the direction

of the output shaft the same as the input shaft or to increase the distance between the drive

and driven gears. A compound gear train refers to two or more gears used to transmit motion.

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Types of gear trains include:

Simple gear train

Compound gear train

Epicyclic gear train

Reverted gear train

A gear train is two or more gear working together by meshing their teeth and turning each other in a

system to generate power and speed. It reduces speed and increases torque. To create large gear

ratio, gears are connected together to form gear trains. They often consist of multiple gears in the

train. The smaller gears are one-fifth of the size of the larger gear. Electric motors are used with the

gear systems to reduce the speed and increase the torque. Electric motor is connected to the driving

end of each train and is mounted on the test platform. The output end output end of the gear train is

connected to a large magnetic particle brake that is used to measure the output torque.

Types of Gear Trains

Simple Gear Train - The most common of the gear train is the gear pair connecting parallel

shafts. The teeth of this type can be spur, helical or herringbone. The angular velocity is

simply the reverse of the tooth ratio. The main limitation of a simple gear train is that the

maximum speed change ratio is 10:1. For larger ratio, large size of gear trains are required.

The sprockets and chain in the bicycle is an example of simple gear train. When the paddle is

pushed, the front gear is turned and that meshes with the links in the chain. The chain moves

and meshes with the links in the rear gear that is attached to the rear wheel. This enables the

bicycle to move.

Compound Gear Train - For large velocities, compound arrangement is preferred. Two

keys are keyed to a single shaft. A double reduction train can be arranged to have its input

and output shafts in a line, by choosing equal center distance for gears and pinions.

Epicyclic Gear Train - It is the system of epicyclic gears in which at least one wheel axis

itself revolves around another fixed axis.

Planetary Gear Train - It is made of few components, a small gear at the center called the

sun, several medium sized gears called the planets and a large external gear called the ring

gear. The planet gear rolls and revolves about the sun gear and the ring gear rolls on the

planet gear. Planetary gear trains have several advantages. They have higher gear ratios.

They are popular for automatic transmissions in automobiles. They are also used in bicycles

for controlling power of pedaling automatically or manually. They are also used for power

train between internal combustion engine and an electric motor.

Applications Gear trains are used in representing the phases of moon on a watch or clock dial. It is also

used for driving a conventional two-disk lunar phase display off the day-of-the-week shaft of

the calendar.

Ratchet & pawl

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A ratchet is a device that allows continuous linear or rotary motion in only one direction while

preventing motion in the opposite direction. Because most socket wrenches today use ratcheting

handles, the term "ratchet" alone is often used to refer to a ratcheting wrench, and the terms "ratchet"

and "socket" are closely associated in many users' minds.

A ratchet consists of a round gear (see Figure 1) or linear rack with teeth, and a pivoting,

springloaded finger called a pawl (or click[1]

) that engages the teeth. The teeth are uniform but

asymmetrical, with each tooth having a moderate slope on one edge and a much steeper slope on the

other edge.

When the teeth are moving in the unrestricted (i.e., forward) direction (see Figure 2), the pawl easily

slides up and over the gently sloped edges of the teeth, with a spring forcing it (often with an audible

'click') into the depression between the teeth as it passes the tip of each tooth. When the teeth move

in the opposite (backward) direction, however, the pawl will catch against the steeply sloped edge of

the first tooth it encounters, thereby locking it against the tooth and preventing any further motion in

that direction.

Backlash

Because the ratchet can only stop backward motion at discrete points (i.e., at tooth boundaries), a

ratchet does allow a limited amount of backward motion. This backward motion—which is limited

to a maximum distance equal to the spacing between the teeth—is called backlash. In cases where

backlash must be minimized, a smooth, toothless ratchet with a high friction surface such as rubber

is sometimes used. The pawl bears against the surface at an angle so that any backward motion will

cause the pawl to jam against the surface and thus prevent any further backward motion. Since the

backward travel distance is primarily a function of the compressibility of the high friction surface,

this mechanism can result in significantly reduced backlash.

Belt drives

A belt is a loop of flexible material used to link two or more rotating shafts mechanically. Belts may

be used as a source of motion, to transmit power efficiently, or to track relative movement. Belts are

looped over pulleys. In a two pulley system, the belt can either drive the pulleys in the same

direction, or the belt may be crossed, so that the direction of the shafts is opposite. As a source of

motion, a conveyor belt is one application where the belt is adapted to continually carry a load

between two points.

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Belts are the cheapest utility for power transmission between shafts that may not be axially aligned.

Power transmission is achieved by specially designed belts and pulleys. The demands on a belt drive

transmission system are large and this has led to many variations on the theme. They run smoothly

and with little noise, and cushion motor and bearings against load changes, albeit with less strength

than gears or chains. However, improvements in belt engineering allow use of belts in systems that

only formerly allowed chains or gears.

Pros and cons

Belt drive, moreover, is simple, inexpensive, and does not require axially aligned shafts. It helps

protect the machinery from overload and jam, and damps and isolates noise and vibration. Load

fluctuations are shock-absorbed (cushioned). They need no lubrication and minimal maintenance.

They have high efficiency (90-98%, usually 95%), high tolerance for misalignment, and are

inexpensive if the shafts are far apart. Clutch action is activated by releasing belt tension. Different

speeds can be obtained by step or tapered pulleys.

The angular-velocity ratio may not be constant or equal to that of the pulley diameters, due to slip

and stretch. However, this problem has been largely solved by the use of toothed belts. Temperatures

ranges from −31 °F (−35 °C) to 185 °F (85 °C). Adjustment of center distance or addition of an idler

pulley is crucial to compensate for wear and stretch.

Flat belts

The drive belt: used to transfer power from the engine's flywheel. Here shown driving a threshing

machine.

Flat belts were used early in line shafting to transmit power in factories.[1]

It is a simple system of

power transmission that was well suited for its day. It delivered high power for high speeds (500 hp

for 10,000 ft/min), in cases of wide belts and large pulleys. These drives are bulky, requiring high

tension leading to high loads, so vee belts have mainly replaced the flat-belts except when high

speed is needed over power. The Industrial Revolution soon demanded more from the system, and

flat belt pulleys needed to be carefully aligned to prevent the belt from slipping off. Because flat

belts tend to climb towards the higher side of the pulley, pulleys were made with a slightly convex or

"crowned" surface (rather than flat) to keep the belts centered. Flat belts also tend to slip on the

pulley face when heavy loads are applied and many proprietary dressings were available that could

be applied to the belts to increase friction, and so power transmission. Grip was better if the belt was

assembled with the hair (i.e. outer) side of the leather against the pulley although belts were also

often given a half-twist before joining the ends (forming a Möbius strip), so that wear was evenly

distributed on both sides of the belt (DB). Belts were joined by lacing the ends together with leather

thonging,[2][3]

or later by steel comb fasteners.[4]

A good modern use for a flat belt is with smaller

pulleys and large central distances. They can connect inside and outside pulleys, and can come in

both endless and jointed construction.

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Round belts

Round belts are a circular cross section belt designed to run in a pulley with a circular (or near

circular) groove. They are for use in low torque situations and may be purchased in various lengths

or cut to length and joined, either by a staple, gluing or welding (in the case of polyurethane). Early

sewing machines utilized a leather belt, joined either by a metal staple or glued, to a great effect.

Vee belts

Belts on a Yanmar 2GM20 marine diesel engine.

A multiple-V-belt drive on an air compressor.

Vee belts (also known as V-belt or wedge rope) solved the slippage and alignment problem. It is

now the basic belt for power transmission. They provide the best combination of traction, speed of

movement, load of the bearings, and long service life. The V-belt was developed in 1917 by John

Gates of the Gates Rubber Company. They are generally endless, and their general cross-section

shape is trapezoidal. The "V" shape of the belt tracks in a mating groove in the pulley (or sheave),

with the result that the belt cannot slip off. The belt also tends to wedge into the groove as the load

increases — the greater the load, the greater the wedging action — improving torque transmission

and making the V-belt an effective solution, needing less width and tension than flat belts. V-belts

trump flat belts with their small center distances and high reduction ratios. The preferred center

distance is larger than the largest pulley diameter, but less than three times the sum of both pulleys.

Optimal speed range is 1000–7000 ft/min. V-belts need larger pulleys for their larger thickness than

flat belts. They can be supplied at various fixed lengths or as a segmented section, where the

segments are linked (spliced) to form a belt of the required length. For high-power requirements, two

or more vee belts can be joined side-by-side in an arrangement called a multi-V, running on

matching multi-groove sheaves. The strength of these belts is obtained by reinforcements with fibers

like steel, polyester or aramid (e.g. Twaron or Kevlar). This is known as a multiple-V-belt drive (or

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sometimes a "classical V-belt drive"). When an endless belt does not fit the need, jointed and link V-

belts may be employed. However they are weaker and only usable at speeds up to 4000 ft/min. A

link v-belt is a number of rubberized fabric links held together by metal fasteners. They are length

adjustable by disassembling and removing links when needed.

Multi-groove belts

A multi-groove or polygroove belt[5]

is made up of usually 5 or 6 "V" shapes along side each other.

This gives a thinner belt for the same drive surface, thus is more flexible, although often wider. The

added flexibility offers an improved efficiency, as less energy is wasted in the internal friction of

continually bending the belt. In practice this gain of efficiency is overshadowed by the reduced

heating effect on the belt, as a cooler-running belt lasts longer in service.

A further advantage of the polygroove belt, and the reason they have become so popular, stems from

the ability to be run over pulleys on the ungrooved back of the belt. Although this is sometimes done

with vee belts and a single idler pulley for tensioning, a polygroove belt may be wrapped around a

pulley on its back tightly enough to change its direction, or even to provide a light driving force.[6]

Any vee belt's ability to drive pulleys depends on wrapping the belt around a sufficient angle of the

pulley to provide grip. Where a single-vee belt is limited to a simple convex shape, it can adequately

wrap at most three or possibly four pulleys, so can drive at most three accessories. Where more must

be driven, such as for modern cars with power steering and air conditioning, multiple belts are

required. As the polygroove belt can be bent into concave paths by external idlers, it can wrap any

number of driven pulleys, limited only by the power capacity of the belt.[6]

This ability to bend the belt at the designer's whim allows it to take a complex or "serpentine" path.

This can assist the design of a compact engine layout, where the accessories are mounted more

closely to the engine block and without the need to provide movable tensioning adjustments. The

entire belt may be tensioned by a single idler pulley.

Ribbed belt

A ribbed belt is a power transmission belt featuring lengthwise grooves. It operates from contact

between the ribs of the belt and the grooves in the pulley. Its single-piece structure it reported to

offer an even distribution of tension across the width of the pulley where the belt is in contact, a

power range up to 600 kW, a high speed ratio, serpentine drives (possibility to drive off the back of

the belt), long life, stability and homogeneity of the drive tension, and reduced vibration. The ribbed

belt may be fitted on various applications : compressors, fitness bikes, agricultural machinery, food

mixers, washing machines, lawn mowers, etc.[7]

Film belts

Though often grouped with flat belts, they are actually a different kind. They consist of a very thin

belt (0.5-15 millimeters or 100-4000 micrometres) strip of plastic and occasionally rubber. They are

generally intended for low-power (10 hp or 7 kW), high-speed uses, allowing high efficiency (up to

98%) and long life. These are seen in business machines, printers, tape recorders, and other light-

duty operations.

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Timing belts

Timing belt

Belt-drive cog on a belt-driven bicycle

Timing belts, (also known as Toothed, Notch, Cog, or Synchronous belts) are a positive transfer

belt and can track relative movement. These belts have teeth that fit into a matching toothed pulley.

When correctly tensioned, they have no slippage, run at constant speed, and are often used to

transfer direct motion for indexing or timing purposes (hence their name). They are often used in

lieu of chains or gears, so there is less noise and a lubrication bath is not necessary. Camshafts of

automobiles, miniature timing systems, and stepper motors often utilize these belts. Timing belts

need the least tension of all belts, and are among the most efficient. They can bear up to 200 hp

(150 kW) at speeds of 16,000 ft/min.

Timing belts with a helical offset tooth design are available. The helical offset tooth design forms a

chevron pattern and causes the teeth to engage progressively. The chevron pattern design is self-

aligning. The chevron pattern design does not make the noise that some timing belts make at

idiosyncratic speeds, and is more efficient at transferring power (up to 98%).

Disadvantages include a relatively high purchase cost, the need for specially fabricated toothed

pulleys, less protection from overloading and jamming, and the lack of clutch action.

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Specialty belts

Belts normally transmit power on the tension side of the loop. However, designs for continuously

variable transmissions exist that use belts that are a series of solid metal blocks, linked together as in

a chain, transmitting power on the compression side of the loop.

Rolling roads

Belts used for rolling roads for wind tunnels can be capable of 250 km/h.[8]

Flying rope

For transmission of mechanical power over distance without electrical energy, a flying rope can be

used[9]

. A wire or manila rope can be used to transmit mechanical energy from a steam engine or

water wheel to a factory or pump which is located a considerable distance (10 to 100s of meters or

more) from the power source. A flying rope way could be supported on poles and pulleys similar to

the cable on a chair lift or aerial tramway. Transmission efficiency is generally high.

Chain drives

Chain drive is a way of transmitting mechanical power from one place to another. It is often used to

convey power to the wheels of a vehicle, particularly bicycles and motorcycles. It is also used in a

wide variety of machines besides vehicles.

Most often, the power is conveyed by a roller chain, known as the drive chain or transmission

chain,[1]

passing over a sprocket gear, with the teeth of the gear meshing with the holes in the links

of the chain. The gear is turned, and this pulls the chain putting mechanical force into the system.

Sometimes the power is output by simply rotating the chain, which can be used to lift or drag

objects. In other situations, a second gear is placed and the power is recovered by attaching shafts or

hubs to this gear. Though drive chains are often simple oval loops, they can also go around corners

by placing more than two gears along the chain; gears that do not put power into the system or

transmit it out are generally known as idler-wheels. By varying the diameter of the input and output

gears with respect to each other, the gear ratio can be altered, so that, for example, the pedals of a

bicycle can spin all the way around more than once for every rotation of the gear that drives the

wheels.

Chains versus belts

Drive chains are most often made of metal, while belts are often rubber, plastic, or other substances.

Although well-made chains may prove stronger than belts, their greater mass increases drive train

inertia.

Drive belts can often slip (unless they have teeth) which means that the output side may not rotate at

a precise speed, and some work gets lost to the friction of the belt against its rollers. Teeth on

toothed drive belts generally wear faster than links on chains, but wear on rubber or plastic belts and

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their teeth is often easier to observe; you can often tell a belt is wearing out and about to break more

easily than a chain.

Chains are often narrower than belts, and this can make it easier to shift them to larger or smaller

gears in order to vary the gear ratio. Multi-speed bicycles with derailleurs make use of this. Also, the

more positive meshing of a chain can make it easier to build gears that can increase or shrink in

diameter, again altering the gear ratio.

Both can be used to move objects by attaching pockets, buckets, or frames to them; chains are often

used to move things vertically by holding them in frames, as in industrial toasters, while belts are

good at moving things horizontally in the form of conveyor belts. It is not unusual for the systems to

be used in combination; for example the rollers that drive conveyor belts are themselves often driven

by drive chains.

Drive shafts are another common method used to move mechanical power around that is sometimes

evaluated in comparison to chain drive; in particular shaft drive versus chain drive is a key design

decision for most motorcycles. Drive shafts te

Bearing

A bearing is a device to allow constrained relative motion between two or more parts, typically

rotation or linear movement. Bearings may be classified broadly according to the motions they allow

and according to their principle of operation as well as by the directions of applied loads they can

handle.

Plain bearings use surfaces in rubbing contact, often with a lubricant such as oil or graphite. A plain

bearing may or may not be a discrete device. It may be nothing more than the bearing surface of a

hole with a shaft passing through it, or of a planar surface that bears another (in these cases, not a

discrete device); or it may be a layer of bearing metal either fused to the substrate (semi-discrete) or

in the form of a separable sleeve (discrete). With suitable lubrication, plain bearings often give

entirely acceptable accuracy, life, and friction at minimal cost. Therefore, they are very widely used.

However, there are many applications where a more suitable bearing can improve efficiency,

accuracy, service intervals, reliability, speed of operation, size, weight, and costs of purchasing and

operating machinery.

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Thus, there are many types of bearings, with varying shape, material, lubrication, principal of

operation, and so on. For example, rolling-element bearings use spheres or drums rolling between

the parts to reduce friction; reduced friction allows tighter tolerances and thus higher precision than a

plain bearing, and reduced wear extends the time over which the machine stays accurate. Plain

bearings are commonly made of varying types of metal or plastic depending on the load, how

corrosive or dirty the environment is, and so on. In addition, bearing friction and life may be altered

dramatically by the type and application of lubricants. For example, a lubricant may improve bearing

friction and life, but for food processing a bearing may be lubricated by an inferior food-safe

lubricant to avoid food contamination; in other situations a bearing may be run without lubricant

because continuous lubrication is not feasible, and lubricants attract dirt that damages the bearings.

Principles of operation

There are at least six common principles of operation:

plain bearing, also known by the specific styles: bushings, journal bearings, sleeve bearings,

rifle bearings

rolling-element bearings such as ball bearings and roller bearings

jewel bearings, in which the load is carried by rolling the axle slightly off-center

fluid bearings, in which the load is carried by a gas or liquid

magnetic bearings, in which the load is carried by a magnetic field

flexure bearings, in which the motion is supported by a load element which bends.

angular contact bearing

Motions

Common motions permitted by bearings are:

Axial rotation e.g. shaft rotation

Linear motion e.g. drawer

spherical rotation e.g. ball and socket joint

hinge motion e.g. door, elbow, knee

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electrical actuators

Relay

A solid state relay (SSR) is an electronic switching device in which a small control signal controls a

larger load current or voltage. It comprises a voltage or current sensor which responds to an

appropriate input (control signal), a solid-state electronic switching device of some kind which

switches power to the load circuitry either on or off, and some coupling mechanism to enable the

control signal to activate this switch without mechanical parts. The relay may be designed to switch

either AC or DC to the load. It serves the same function as an electromechanical relay, but has no

moving parts.

Operation

An SSR based on a single MOSFET, or multiple MOSFETs in a paralleled array, works well for DC

loads.

There is an inherent substrate diode in all MOSFETs that conducts in the reverse direction. This

means that a single MOSFET cannot block current in both directions. For AC (bi-directional)

operation two MOSFETs are arranged back to back with their source pins tied together. Their drain

pins are connected to either side of the output. The substrate diodes then are alternately reverse

biased in order to block current when the relay is off. When the relay is on, the common source is

always riding on the instantaneous signal level and both gates are biased positive relative to the

source by the photo-diode.

It is common to provide access to the common source so that multiple MOSFETs can be wired in

parallel if switching a DC load. There is also commonly some circuitry to discharge the gate when

the LED is turned off, speeding the relay's turn-off.

Advantages over mechanical relays

Most of the relative advantages of solid state and electromechanical relays are common to all solid-

state as against electromechanical devices.

SSRs are faster than electromechanical relays; their switching time is dependent on the time

needed to power the LED on and off, of the order of microseconds to milliseconds

Lower (if any) minimum output current (latching current) required

Increased lifetime, particularly if activated many times, as there are no moving parts to wear

o Output resistance remains constant regardless of amount of use

Clean, bounceless operation

Decreased electrical noise when switching

No sparking, allowing use in explosive environments where it is critical that no spark is

generated during switching

Totally silent operation

Inherently smaller than a mechanical relay of similar specification (if desired may have the

same "casing" form factor for interchangeability).

Much less sensitive to storage and operating environment. For example much less sensitive

to mechanical shock and vibration, humidity.

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Solenoid

solenoid[nb 1]

is a coil wound into a tightly packed helix. In physics, the term solenoid refers

to a long, thin loop of wire, often wrapped around a metallic core, which produces a

magnetic field when an electric current is passed through it. Solenoids are important because

they can create controlled magnetic fields and can be used as electromagnets. The term

solenoid refers specifically to a magnet designed to produce a uniform magnetic field in a

volume of space (where some experiment might be carried out).

In engineering, the term solenoid may also refer to a variety of transducer devices

that convert energy into linear motion. The term is also often used to refer to a solenoid

valve, which is an integrated device containing an electromechanical solenoid which actuates

either a pneumatic or hydraulic valve, or a solenoid switch, which is a specific type of relay

that internally uses an electromechanical solenoid to operate an electrical switch; for

example, an automobile starter solenoid, or a linear solenoid, which is an electromechanical

solenoid.

AC motor

An AC motor is an electric motor that is driven by an alternating current. It consists of two

basic parts, an outside stationary stator having coils supplied with alternating current to

produce a rotating magnetic field, and an inside rotor attached to the output shaft that is given

a torque by the rotating field. (There are examples, such as some Papst motors, which have the

stator on the inside and the rotor on the outside to increase the inertia and cooling. They were

common on high quality tape and film machines where speed stability is important.)

There are two types of AC motors, depending on the type of rotor used (not including eddy

current motors also AC/DC mechanically commutated machines in which speed is dependent

on voltage and winding connection). The first is the synchronous motor, which rotates

exactly at the supply frequency or a submultiple of the supply frequency. The magnetic field

on the rotor is either generated by current delivered through slip rings or by a permanent

magnet. The second type is the induction motor, which runs slightly slower than the supply

frequency. The magnetic field on the rotor of this motor is created by an induced current.

Squirrel-cage rotors

Most common AC motors use the squirrel cage rotor, which will be found in virtually all domestic

and light industrial alternating current motors. The squirrel cage refers to the rotating exercise cage

for pet animals. The motor takes its name from the shape of its rotor "windings"- a ring at either end

of the rotor, with bars connecting the rings running the length of the rotor. It is typically cast

aluminum or copper poured between the iron laminates of the rotor, and usually only the end rings

will be visible. The vast majority of the rotor currents will flow through the bars rather than the

higher-resistance and usually varnished laminates. Very low voltages at very high currents are

typical in the bars and end rings; high efficiency motors will often use cast copper to reduce the

resistance in the rotor.

In operation, the squirrel cage motor may be viewed as a transformer with a rotating secondary.

When the rotor is not rotating in sync with the magnetic field, large rotor currents are induced; the

large rotor currents magnetize the rotor and interact with the stator's magnetic fields to bring the

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rotor almost into synchronization with the stator's field. An unloaded squirrel cage motor at rated no-

load speed will consume electrical power only to maintain rotor speed against friction and resistance

losses. As the mechanical load increases, so will the electrical load - the electrical load is inherently

related to the mechanical load. This is similar to a transformer, where the primary's electrical load is

related to the secondary's electrical load.

This is why a squirrel cage blower motor may cause household lights to dim upon starting, but

doesn't dim the lights on startup when its fan belt (and therefore mechanical load) is removed.

Furthermore, a stalled squirrel cage motor (overloaded or with a jammed shaft) will consume current

limited only by circuit resistance as it attempts to start. Unless something else limits the current (or

cuts it off completely) overheating and destruction of the winding insulation is the likely outcome.

To prevent the currents induced in the squirrel cage from superimposing itself back onto the supply,

the squirrel cage is generally constructed with a prime number of bars, or at least a small multiple of

a prime number (rarely more than 2). There is an optimum number of bars in any design, and

increasing the number of bars beyond that point merely serves to increase the losses of the motor

particularly when starting.

Virtually every washing machine, dishwasher, standalone fan, record player, etc. uses some variant

of a squirrel cage motor.

Calecon Effect

If the rotor of a squirrel runs at the true synchronous speed, the flux in the rotor at any given place on

the rotor would not change, and no current would be created in the squirrel cage. For this reason,

ordinary squirrel-cage motors run at some tens of rpm slower than synchronous speed, even at no

load. Because the rotating field (or equivalent pulsating field) actually or effectively rotates faster

than the rotor, it could be said to slip past the surface of the rotor. The difference between

synchronous speed and actual speed is called slip, and loading the motor increases the amount of slip

as the motor slows down slightly.

Two-phase AC servo motors

A typical two-phase AC servo-motor has a squirrel cage rotor and a field consisting of two windings:

1. a constant-voltage (AC) main winding.

2. a control-voltage (AC) winding in quadrature (i.e., 90 degrees phase shifted) with the main

winding so as to produce a rotating magnetic field. Reversing phase makes the motor reverse.

An AC servo amplifier, a linear power amplifier, feeds the control winding. The electrical resistance

of the rotor is made high intentionally so that the speed/torque curve is fairly linear. Two-phase

servo motors are inherently high-speed, low-torque devices, heavily geared down to drive the load.

Single-phase AC induction motors

Three-phase motors produce a rotating magnetic field. However, when only single-phase power is

available, the rotating magnetic field must be produced using other means. Several methods are

commonly used:

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Shaded-pole motor

A common single-phase motor is the shaded-pole motor and is used in devices requiring low starting

torque, such as electric fans or the drain pump of washing machines and dishwashers or in other

small household appliances. In this motor, small single-turn copper "shading coils" create the

moving magnetic field. Part of each pole is encircled by a copper coil or strap; the induced current in

the strap opposes the change of flux through the coil. This causes a time lag in the flux passing

through the shading coil, so that the maximum field intensity moves across the pole face on each

cycle. This produces a low level rotating magnetic field which is large enough to turn both the rotor

and its attached load. As the rotor picks up speed the torque builds up to its full level as the principal

magnetic field is rotating relative to the rotating rotor.

A reversible shaded-pole motor was made by Barber-Colman several decades ago. It had a single

field coil, and two principal poles, each split halfway to create two pairs of poles. Each of these four

"half-poles" carried a coil, and the coils of diagonally-opposite half-poles were connected to a pair of

terminals. One terminal of each pair was common, so only three terminals were needed in all.

The motor would not start with the terminals open; connecting the common to one other made the

motor run one way, and connecting common to the other made it run the other way. These motors

were used in industrial and scientific devices.

An unusual, adjustable-speed, low-torque shaded-pole motor could be found in traffic-light and

advertising-lighting controllers. The pole faces were parallel and relatively close to each other, with

the disc centred between them, something like the disc in a watthour meter. Each pole face was split,

and had a shading coil on one part; the shading coils were on the parts that faced each other. Both

shading coils were probably closer to the main coil; they could have both been farther away, without

affecting the operating principle, just the direction of rotation.

Applying AC to the coil created a field that progressed in the gap between the poles. The plane of

the stator core was approximately tangential to an imaginary circle on the disc, so the travelling

magnetic field dragged the disc and made it rotate.

The stator was mounted on a pivot so it could be positioned for the desired speed and then clamped

in position. Keeping in mind that the effective speed of the travelling magnetic field in the gap was

constant, placing the poles nearer to the centre of the disc made it run relatively faster, and toward

the edge, slower.

It is possible that these motors are still in use in some older installations.

Split-phase induction motor

Another common single-phase AC motor is the split-phase induction motor,[5]

commonly used in

major appliances such as washing machines and clothes dryers. Compared to the shaded pole motor,

these motors can generally provide much greater starting torque by using a special startup winding in

conjunction with a centrifugal switch.

In the split-phase motor, the startup winding is designed with a higher resistance than the running

winding. This creates an LR circuit which slightly shifts the phase of the current in the startup

winding. When the motor is starting, the startup winding is connected to the power source via a set

of spring-loaded contacts pressed upon by the stationary centrifugal switch. The starting winding is

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wound with fewer turns of smaller wire than the main winding, so it has a lower inductance (L) and

higher resistance (R). The lower L/R ratio creates a small phase shift, not more than about 30

degrees, between the flux due to the main winding and the flux of the starting winding. The starting

direction of rotation may be reversed simply by exchanging the connections of the startup winding

relative to the running winding.

The phase of the magnetic field in this startup winding is shifted from the phase of the mains power,

allowing the creation of a moving magnetic field which starts the motor. Once the motor reaches

near design operating speed, the centrifugal switch activates, opening the contacts and disconnecting

the startup winding from the power source. The motor then operates solely on the running winding.

The starting winding must be disconnected since it would increase the losses in the motor.

Capacitor start motor

Schematic of a capacitor start motor.

A capacitor start motor is a split-phase induction motor with a starting capacitor inserted in series

with the startup winding, creating an LC circuit which is capable of a much greater phase shift (and

so, a much greater starting torque). The capacitor naturally adds expense to such motors.

Resistance start motor

A resistance start motor is a split-phase induction motor with a starter inserted in series with the

startup winding, creating capacitance. This added starter provides assistance in the starting and

initial direction of rotation.

Permanent-split capacitor motor

Another variation is the permanent-split capacitor (PSC) motor (also known as a capacitor start and

run motor).[6]

This motor operates similarly to the capacitor-start motor described above, but there is

no centrifugal starting switch,[6]

and what correspond to the start windings (second windings) are

permanently connected to the power source (through a capacitor), along with the run windings.[6]

PSC motors are frequently used in air handlers, blowers, and fans (including ceiling fans) and other

cases where a variable speed is desired.

A capacitor ranging from 3 to 25 microfarads is connected in series with the "start" windings and

remains in the circuit during the run cycle.[6]

The "start" windings and run windings are identical in

this motor,[6]

and reverse motion can be achieved by reversing the wiring of the 2 windings,[6]

with

the capacitor connected to the other windings as "start" windings. By changing taps on the running

winding but keeping the load constant, the motor can be made to run at different speeds. Also,

provided all 6 winding connections are available separately, a 3 phase motor can be converted to a

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capacitor start and run motor by commoning two of the windings and connecting the third via a

capacitor to act as a start winding.

Wound rotors

An alternate design, called the wound rotor, is used when variable speed is required. In this case, the

rotor has the same number of poles as the stator and the windings are made of wire, connected to slip

rings on the shaft. Carbon brushes connect the slip rings to an external controller such as a variable

resistor that allows changing the motor's slip rate. In certain high-power variable speed wound-rotor

drives, the slip-frequency energy is captured, rectified and returned to the power supply through an

inverter. With bidirectionally controlled power, the wound-rotor becomes an active participant in the

energy conversion process with the wound-rotor doubly-fed configuration showing twice the power

density.

Compared to squirrel cage rotors and without considering brushless wound-rotor doubly-fed

technology, wound rotor motors are expensive and require maintenance of the slip rings and brushes,

but they were the standard form for variable speed control before the advent of compact power

electronic devices. Transistorized inverters with variable-frequency drive can now be used for speed

control, and wound rotor motors are becoming less common.

Several methods of starting a polyphase motor are used. Where the large inrush current and high

starting torque can be permitted, the motor can be started across the line, by applying full line

voltage to the terminals (direct-on-line, DOL). Where it is necessary to limit the starting inrush

current (where the motor is large compared with the short-circuit capacity of the supply), reduced

voltage starting using either series inductors, an autotransformer, thyristors, or other devices are

used. A technique sometimes used is (star-delta, YΔ) starting, where the motor coils are initially

connected in star for acceleration of the load, then switched to delta when the load is up to speed.

This technique is more common in Europe than in North America. Transistorized drives can directly

vary the applied voltage as required by the starting characteristics of the motor and load.

This type of motor is becoming more common in traction applications such as locomotives, where it

is known as the asynchronous traction motor.

The speed of the AC motor is determined primarily by the frequency of the AC supply and the

number of poles in the stator winding, according to the relation:

Ns = 120F / p

where

Ns = Synchronous speed, in revolutions per minute

F = AC power frequency

p = Number of poles per phase winding

Actual RPM for an induction motor will be less than this calculated synchronous speed by an

amount known as slip, that increases with the torque produced. With no load, the speed will be very

close to synchronous. When loaded, standard motors have between 2-3% slip, special motors may

have up to 7% slip, and a class of motors known as torque motors are rated to operate at 100% slip

(0 RPM/full stall).

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The slip of the AC motor is calculated by:

S = (Ns − Nr) / Ns

where

Nr = Rotational speed, in revolutions per minute.

S = Normalised Slip, 0 to 1.

As an example, a typical four-pole motor running on 60 Hz might have a nameplate rating of 1725

RPM at full load, while its calculated speed is 1800 RPM.

The speed in this type of motor has traditionally been altered by having additional sets of coils or

poles in the motor that can be switched on and off to change the speed of magnetic field rotation.

However, developments in power electronics mean that the frequency of the power supply can also

now be varied to provide a smoother control of the motor speed.

Three-phase AC synchronous motors

If connections to the rotor coils of a three-phase motor are taken out on slip-rings and fed a separate

field current to create a continuous magnetic field (or if the rotor consists of a permanent magnet),

the result is called a synchronous motor because the rotor will rotate synchronously with the rotating

magnetic field produced by the polyphase electrical supply.

The synchronous motor can also be used as an alternator.

Nowadays, synchronous motors are frequently driven by transistorized variable-frequency drives.

This greatly eases the problem of starting the massive rotor of a large synchronous motor. They may

also be started as induction motors using a squirrel-cage winding that shares the common rotor: once

the motor reaches synchronous speed, no current is induced in the squirrel-cage winding so it has

little effect on the synchronous operation of the motor, aside from stabilizing the motor speed on

load changes.

Synchronous motors are occasionally used as traction motors; the TGV may be the best-known

example of such use.

One use for this type of motor is its use in a power factor correction scheme. They are referred to as

synchronous condensers. This exploits a feature of the machine where it consumes power at a

leading power factor when its rotor is over excited. It thus appears to the supply to be a capacitor,

and could thus be used to correct the lagging power factor that is usually presented to the electric

supply by inductive loads. The excitation is adjusted until a near unity power factor is obtained

(often automatically). Machines used for this purpose are easily identified as they have no shaft

extensions. Synchronous motors are valued in any case because their power factor is much better

than that of induction motors, making them preferred for very high power applications.

Some of the largest AC motors are pumped-storage hydroelectricity generators that are operated as

synchronous motors to pump water to a reservoir at a higher elevation for later use to generate

electricity using the same machinery. Six 350-megawatt generators are installed in the Bath County

Pumped Storage Station in Virginia, USA. When pumping, each unit can produce 563,400

horsepower (420 megawatts).[7]

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Universal motors and series wound motors

AC motors can also have brushes. The universal motor is widely used in small home appliances and

power tools.

Repulsion motor

Repulsion motors are wound-rotor single-phase AC motors that are similar to universal motors. In a

repulsion motor, the armature brushes are shorted together rather than connected in series with the

field. By transformer action ,the stator induces currents in the rotor, which create torque by repulsion

instead of attraction as in other motors.

DC MOTOR

A DC motor is an electric motor that runs on direct current (DC) electricity.

Stepper motor

A stepper motor (or step motor) is a brushless, synchronous electric motor that can divide a full

rotation into a large number of steps. The motor's position can be controlled precisely without any

feedback mechanism (see Open-loop controller), as long as the motor is carefully sized to the

application. Stepper motors are similar to switched reluctance motors (which are very large stepping

motors with a reduced pole count, and generally are closed-loop commutated.)

Fundamentals of operation

Stepper motors operate differently from DC brush motors, which rotate when voltage is applied to

their terminals. Stepper motors, on the other hand, effectively have multiple "toothed"

electromagnets arranged around a central gear-shaped piece of iron. The electromagnets are

energized by an external control circuit, such as a microcontroller. To make the motor shaft turn,

first one electromagnet is given power, which makes the gear's teeth magnetically attracted to the

electromagnet's teeth. When the gear's teeth are thus aligned to the first electromagnet, they are

slightly offset from the next electromagnet. So when the next electromagnet is turned on and the first

is turned off, the gear rotates slightly to align with the next one, and from there the process is

repeated. Each of those slight rotations is called a "step", with an integer number of steps making a

full rotation. In that way, the motor can be turned by a precise angle.

Unit 3.SYSTEM MODELS AND CONTROLLERS

Building blocks of Mechanical, Electrical, Fluid and Thermal Systems, Rotational –

Transnational Systems, Electromechanical Systems – Hydraulic – Mechanical Systems.

Continuous and discrete process Controllers – Control Mode – Two – Step mode –

Proportional Mode – Derivative Mode – Integral Mode – PID Controllers – Digital Controllers

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– Velocity Control – Adaptive Control – Digital Logic Control – Micro Processors Control.

Control loop basics

A familiar example of a control loop is the action taken when adjusting hot and cold faucet valves to

maintain the faucet water at the desired temperature. This typically involves the mixing of two

process streams, the hot and cold water. The person touches the water to sense or measure its

temperature. Based on this feedback they perform a control action to adjust the hot and cold water

valves until the process temperature stabilizes at the desired value.

Sensing water temperature is analogous to taking a measurement of the process value or process

variable (PV). The desired temperature is called the setpoint (SP). The input to the process (the water

valve position) is called the manipulated variable (MV). The difference between the temperature

measurement and the setpoint is the error (e) and quantifies whether the water is too hot or too cold

and by how much.

After measuring the temperature (PV), and then calculating the error, the controller decides when to

change the tap position (MV) and by how much. When the controller first turns the valve on, it may

turn the hot valve only slightly if warm water is desired, or it may open the valve all the way if very

hot water is desired. This is an example of a simple proportional control. In the event that hot water

does not arrive quickly, the controller may try to speed-up the process by opening up the hot water

valve more-and-more as time goes by. This is an example of an integral control.

Making a change that is too large when the error is small is equivalent to a high gain controller and

will lead to overshoot. If the controller were to repeatedly make changes that were too large and

repeatedly overshoot the target, the output would oscillate around the setpoint in either a constant,

growing, or decaying sinusoid. If the oscillations increase with time then the system is unstable,

whereas if they decrease the system is stable. If the oscillations remain at a constant magnitude the

system is marginally stable.

In the interest of achieving a gradual convergence at the desired temperature (SP), the controller may

wish to damp the anticipated future oscillations. So in order to compensate for this effect, the

controller may elect to temper their adjustments. This can be thought of as a derivative control

method.

If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller

will be in response to changes in other measured or unmeasured inputs to the process that impact on

the process, and hence on the PV. Variables that impact on the process other than the MV are known

as disturbances. Generally controllers are used to reject disturbances and/or implement setpoint

changes. Changes in feedwater temperature constitute a disturbance to the faucet temperature control

process.

In theory, a controller can be used to control any process which has a measurable output (PV), a

known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant

PV. Controllers are used in industry to regulate temperature, pressure, flow rate, chemical

composition, speed and practically every other variable for which a measurement exists.

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Proportional control

The fly-ball governor is a classic example of proportional control.

A proportional control system is a type of linear feedback control system. Two classic mechanical

examples are the toilet bowl float proportioning valve and the fly-ball governor.

The proportional control system is more complex than an on-off control system like a bi-metallic

domestic thermostat, but simpler than a proportional-integral-derivative (PID) control system used in

something like an automobile cruise control. On-off control will work where the overall system has a

relatively long response time, but will result in instability if the system being controlled has a rapid

response time. Proportional control overcomes this by modulating the output to the controlling

device, such as a continuously variable valve.

An analogy to on-off control is driving a car by applying either full power or no power and varying

the duty cycle, to control speed. The power would be on until the target speed is reached, and then

the power would be removed, so the car reduces speed. When the speed falls below the target, with a

certain hysteresis, full power would again be applied. It can be seen that this looks like pulse-width

modulation, but would obviously result in poor control and large variations in speed. The more

powerful the engine, the greater the instability; the heavier the car, the greater the stability. Stability

may be expressed as correlating to the power-to-weight ratio of the vehicle.

Proportional control is how most drivers control the speed of a car. If the car is at target speed and

the speed increases slightly, the power is reduced slightly, or in proportion to the error (the actual

versus target speed), so that the car reduces speed gradually and reaches the target point with very

little, if any, "overshoot", so the result is much smoother control than on-off control.

Further refinements like PID control would help compensate for additional variables like hills, where

the amount of power needed for a given speed change would vary, which would be accounted for by

the integral function of the PID control.

Proportional Control Theory

In the proportional control algorithm, the controller output is proportional to the error signal, which

is the difference between the set point and the process variable. In other words, the output of a

proportional controller is the multiplication product of the error signal and the proportional gain.

This can be mathematically expressed as

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where

Pout: Output of the proportional controller

Kp: Proportional gain

e(t): Instantaneous process error at time t. e(t) = SP − PV

SP: Set point

PV: Process variable

PID controller

A proportional–integral–derivative controller (PID controller) is a generic control loop feedback

mechanism (controller) widely used in industrial control systems – a PID is the most commonly used

feedback controller. A PID controller calculates an "error" value as the difference between a

measured process variable and a desired setpoint. The controller attempts to minimize the error by

adjusting the process control inputs.

The PID controller calculation (algorithm) involves three separate parameters, and is accordingly

sometimes called three-term control: the proportional, the integral and derivative values, denoted

P, I, and D. Heuristically, these values can be interpreted in terms of time: P depends on the present

error, I on the accumulation of past errors, and D is a prediction of future errors, based on current

rate of change.[1]

The weighted sum of these three actions is used to adjust the process via a control

element such as the position of a control valve or the power supply of a heating element.

In the absence of knowledge of the underlying process, a PID controller is the best controller.[2]

By

tuning the three constants in the PID controller algorithm, the controller can provide control action

designed for specific process requirements. The response of the controller can be described in terms

of the responsiveness of the controller to an error, the degree to which the controller overshoots the

setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does

not guarantee optimal control of the system or system stability.

Some applications may require using only one or two modes to provide the appropriate system

control. This is achieved by setting the gain of undesired control outputs to zero. A PID controller

will be called a PI, PD, P or I controller in the absence of the respective control actions. PI

controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the

absence of an integral value may prevent the system from reaching its target value due to the control

action.

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PID controller theory

The PID control scheme is named after its three correcting terms, whose sum constitutes the

manipulated variable (MV). Hence:

Where Pout, Iout, and Dout are the contributions to the output from the PID controller from each of the

three terms, as defined below.

The proportional term (sometimes called gain) makes a change to the output that is proportional to

the current error value. The proportional response can be adjusted by multiplying the error by a

constant Kp, called the proportional gain.

The proportional term is given by:

where

Pout: Proportional term of output

Kp: Proportional gain, a tuning parameter

SP: Setpoint, the desired value

PV: Process value (or process variable), the measured value

e: Error = SP − PV

t: Time or instantaneous time (the present)

A high proportional gain results in a large change in the output for a given change in the error. If the

proportional gain is too high, the system can become unstable (see the section on loop tuning). In

contrast, a small gain results in a small output response to a large input error, and a less responsive

(or sensitive) controller. If the proportional gain is too low, the control action may be too small when

responding to system disturbances.

Droop

A pure proportional controller will not always settle at its target value, but may retain a steady-state

error. Specifically, the process gain - drift in the absence of control, such as cooling of a furnace

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towards room temperature, biases a pure proportional controller. If the process gain is down, as in

cooling, then the bias will be below the set point, hence the term "droop".

Droop is proportional to process gain and inversely proportional to proportional gain. Specifically

the steady-state error is given by:

e = G / Kp

Droop is an inherent defect of purely proportional control. Droop may be mitigated by adding a

compensating bias term (setting the setpoint above the true desired value), or corrected by adding an

integration term (in a PI or PID controller), which effectively computes a bias adaptively.

Despite droop, both tuning theory and industrial practice indicate that it is the proportional term that

should contribute the bulk of the output change.

Integral term

The contribution from the integral term (sometimes called reset) is proportional to both the

magnitude of the error and the duration of the error. Summing the instantaneous error over time

(integrating the error) gives the accumulated offset that should have been corrected previously. The

accumulated error is then multiplied by the integral gain and added to the controller output. The

magnitude of the contribution of the integral term to the overall control action is determined by the

integral gain, Ki.

The integral term is given by:

where

Iout: Integral term of output

Ki: Integral gain, a tuning parameter

SP: Setpoint, the desired value

PV: Process value (or process variable), the measured value

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e: Error = SP − PV

t: Time or instantaneous time (the present)

τ: a dummy integration variable

The integral term (when added to the proportional term) accelerates the movement of the process

towards setpoint and eliminates the residual steady-state error that occurs with a proportional only

controller. However, since the integral term is responding to accumulated errors from the past, it can

cause the present value to overshoot the setpoint value (cross over the setpoint and then create a

deviation in the other direction). For further notes regarding integral gain tuning and controller

stability, see the section on loop tuning.

Derivative term

The rate of change of the process error is calculated by determining the slope of the error over time

(i.e., its first derivative with respect to time) and multiplying this rate of change by the derivative

gain Kd. The magnitude of the contribution of the derivative term (sometimes called rate) to the

overall control action is termed the derivative gain, Kd.

The derivative term is given by:

where

Dout: Derivative term of output

Kd: Derivative gain, a tuning parameter

SP: Setpoint, the desired value

PV: Process value (or process variable), the measured value

e: Error = SP − PV

t: Time or instantaneous time (the present)

The derivative term slows the rate of change of the controller output and this effect is most

noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude

of the overshoot produced by the integral component and improve the combined controller-process

stability. However, differentiation of a signal amplifies noise and thus this term in the controller is

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highly sensitive to noise in the error term, and can cause a process to become unstable if the noise

and the derivative gain are sufficiently large. Hence an approximation to a differentiator with a

limited bandwidth is more commonly used. Such a circuit is known as a Phase-Lead compensator.

Summary

The proportional, integral, and derivative terms are summed to calculate the output of the PID

controller. Defining u(t) as the controller output, the final form of the PID algorithm is:

where the tuning parameters are:

Proportional gain, Kp

Larger values typically mean faster response since the larger the error, the larger the

proportional term compensation. An excessively large proportional gain will lead to process

instability and oscillation.

Integral gain, Ki

Larger values imply steady state errors are eliminated more quickly. The trade-off is larger

overshoot: any negative error integrated during transient response must be integrated away

by positive error before reaching steady state.

Derivative gain, Kd

Larger values decrease overshoot, but slow down transient response and may lead to

instability due to signal noise amplification in the differentiation of the error.

Motion control

Motion control is a sub-field of automation, in which the position and/or velocity of machines are

controlled using some type of device such as a hydraulic pump, linear actuator, or an electric motor,

generally a servo. Motion control is an important part of robotics and CNC machine tools, however

it is more complex than in the use of specialized machines, where the kinematics are usually simpler.

The latter is often called General Motion Control (GMC). Motion control is widely used in the

packaging, printing, textile, semiconductor production, and assembly industries.

The basic architecture of a motion control system contains:

A motion controller to generate set points (the desired output or motion profile) and close a

position and/or velocity feedback loop.

A drive or amplifier to transform the control signal from the motion controller into a higher

power electrical current or voltage that is presented to the actuator. Newer "intelligent" drives

can close the position and velocity loops internally, resulting in much more accurate control.

An actuator such as a hydraulic pump, air cylinder, linear actuator, or electric motor for

output motion.

One or more feedback sensors such as optical encoders, resolvers or Hall effect devices to

return the position and/or velocity of the actuator to the motion controller in order to close

the position and/or velocity control loops.

Mechanical components to transform the motion of the actuator into the desired motion,

including: gears, shafting, ball screw, belts, linkages, and linear and rotational bearings.

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The interface between the motion controller and drives it controls is very critical when coordinated

motion is required, as it must provide tight synchronization. Historically the only open interface was

an analog signal, until open interfaces were developed that satisfied the requirements of coordinated

motion control, the first being SERCOS in 1991 which is now enhanced to SERCOS III. Later

interfaces capable of motion control include Profinet IRT, Ethernet Powerlink, and EtherCAT.

Common control functions include:

Velocity control.

Position (point-to-point) control: There are several methods for computing a motion

trajectory. These are often based on the velocity profiles of a move such as a triangular

profile, trapezoidal profile, or an S-curve profile.

Pressure or Force control.

Trans-mutational vector mapping.

Electronic gearing (or cam profiling): The position of a slave axis is mathematically linked to

the position of a master axis. A good example of this would be in a system where two

rotating drums turn at a given ratio to each other. A more advanced case of electronic gearing

is electronic camming. With electronic camming, a slave axis follows a profile that is a

function of the master position. This profile need not be salted, but it must be an animated

function.

Adaptive control

Adaptive control involves modifying the control law used by a controller to cope with the fact that

the parameters of the system being controlled are slowly time-varying or uncertain. For example, as

an aircraft flies, its mass will slowly decrease as a result of fuel consumption; we need a control law

that adapts itself to such changing conditions. Adaptive control is different from robust control in the

sense that it does not need a priori information about the bounds on these uncertain or time-varying

parameters; robust control guarantees that if the changes are within given bounds the control law

need not be changed, while adaptive control is precisely concerned with control law changes.

Classification of adaptive control techniques

In general one should distinguish between:

1. Feedforward Adaptive Control

2. Feedback Adaptive Control

Applications

When designing adaptive control systems, special consideration is necessary of convergence and

robustness issues.

Typical applications of adaptive control are (in general):

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Self-tuning of subsequently fixed linear controllers during the implementation phase for one

operating point;

Self-tuning of subsequently fixed robust controllers during the implementation phase for

whole range of operating points;

Self-tuning of fixed controllers on request if the process behaviour changes due to ageing,

drift, wear etc.;

Adaptive control of linear controllers for nonlinear or time-varying processes;

Adaptive control or self-tuning control of nonlinear controllers for nonlinear processes;

Adaptive control or self-tuning control of multivariable controllers for multivariable

processes (MIMO systems);

Usually these methods adapt the controllers to both the process statics and dynamics. In special cases

the adaptation can be limited to the static behavior alone, leading to adaptive control based on

characteristic curves for the steady-states or to extremum value control, optimizing the steady state.

Hence, there are several ways to apply adaptive control algorithms.

BOOLEAN VARIABLES & TRUTH TABLES

BOOLEAN ALGEBRA DIFFERS IN A MAJOR WAY FROM ORDINARY ALGEBRA IN

THAT BOOLEAN CONSTANTS AND VARIABLES ARE ALLOWED TO HAVE ONLY

TWO POSSIBLE VALUES, 0 OR 1.

BOOLEAN 0 AND 1 DO NOT REPRESENT ACTUAL NUMBERS BUT INSTEAD

REPRESENT THE STATE OF A VOLTAGE VARIABLE, OR WHAT IS CALLED ITS

LOGIC LEVEL.

SOME COMMON REPRESENTATION OF 0 AND 1 IS SHOWN IN THE FOLLOWING

DIAGRAM.

IN BOOLEAN ALGEBRA, THERE ARE THREE BASIC LOGIC OPERATIONS: AND

,OR, AND NOT.

THESE LOGIC GATES ARE DIGITAL CIRCUITS CONSTRUCTED FROM DIODES,

TRANSISTORS, AND RESISTORS CONNECTED IN SUCH A WAY THAT THE

CIRCUIT OUTPUT IS THE RESULT OF A BASIC LOGIC OPERATION (OR, AND,

NOT) PERFORMED ON THE INPUTS.

TRUTH TABLE

A TRUTH TABLE IS A MEANS FOR DESCRIBING HOW A LOGIC CIRCUIT'S

OUTPUT DEPENDS ON THE LOGIC LEVELS PRESENT AT THE CIRCUIT'S INPUTS.

IN THE FOLLOWING TWO-INPUT LOGIC CIRCUIT, THE TABLE LISTS ALL

POSSIBLE COMBINATIONS OF LOGIC LEVELS PRESENT AT INPUTS A AND B

ALONG WITH THE CORRESPONDING OUTPUT LEVEL X.

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WHEN EITHER INPUT A OR B IS 1, THE OUTPUT X IS 1. THEREFORE THE "?" IN

THE BOX IS AN OR GATE.

OR OPERATION

THE EXPRESSION X = A + B READS AS "X EQUALS A OR B". THE + SIGN STANDS

FOR THE OR OPERATION, NOT FOR ORDINARY ADDITION.

THE OR OPERATION PRODUCES A RESULT OF 1 WHEN ANY OF THE INPUT

VARIABLE IS 1.

THE OR OPERATION PRODUCES A RESULT OF 0 ONLY WHEN ALL THE INPUT

VARIABLES ARE 0.

An example of three input OR gate and its truth table is as follows:

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With the OR operation, 1 + 1 = 1, 1+ 1 + 1 = 1 and so on.

AND Operation

The expression X = A * B reads as "X equals A AND B".

The multiplication sign stands for the AND operation, same for ordinary multiplication of 1s

and 0s.The AND operation produces a result of 1 occurs only for the single case when all of

the input variables are 1.The output is 0 for any case where one or more inputs are 0

An example of three input AND gate and its truth table is as follows:

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With the AND operation, 1*1 = 1, 1*1*1 = 1 and so on.

NOT Operation

The NOT operation is unlike the OR and AND operations in that it can be performed on a

single input variable. For example, if the variable A is subjected to the NOT operation, the

result x can be expressed as x = A' where the prime (') represents the NOT operation. This

expression is read as:

x equals NOT A

x equals the inverse of A

x equals the complement of A

Each of these is in common usage and all indicate that the logic value of x = A' is o

pposite to the logic value of A. The truth table of the NOT operation is as follows:

1'=0 because NOT 1 is 0

0' = 1 because NOT 0 is 1

The NOT operation is also referred to as inversion or complementation, and these terms are

used interchangeably.

NOR Operation

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NOR and NAND gates are used extensively in digital circuitry. These gates combine the

basic operations AND, OR and NOT, which make it relatively easy to describe then using

Boolean algebra.

NOR gate symbol is the same as the OR gate symbol except that it has a small circle on the

output. This small circle represents the inversion operation. Therefore the output expression

of the two input NOR gate is:

X = (A + B)'

An example of three inputs OR gate can be constructed by a NOR gate plus a NOT gate:

NAND Operation

NAND gate symbol is the same as the AND gate symbol except that it has a small circle on

the output. This small circle represents the inversion operation. Therefore the output

expression of the two input NAND gate is:

X = (AB)'

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Describing Logic Circuits Algebraically

Any logic circuit, no matter how complex, may be completely described using the Boolean

operations, because the OR gate, AND gate, and NOT circuit are the basic building blocks of

digital systems.

This is an example of the circuit using Boolean expression:

If an expression contains both AND and OR operations, the AND operations are performed

first (X=AB+C: AB is performed first), unless there are parentheses in the expression, in

which case the operation inside the parentheses is to be performed first (X= (A+B) +C: A+B

is performed first).

Circuits containing Inverters

Whenever an INVERTER is present in a logic-circuit diagram, its output expression is

simply equal to the input expression with a prime (') over it.

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Evaluating Logic Circuit Outputs

Once the Boolean expression for a circuit output has been obtained, the output logic level can

be determined for any set of input levels.

These are two examples of the evaluating logic circuit output:

Let A=0, B=1, C=1, D=1

X = A'BC (A+D)'

= 0'*1*1* (0+1)'

= 1 *1*1* (1)'

= 1 *1*1* 0

= 0

Let A=0, B=0, C=1, D=1, E=1

X = [D+ ((A+B)C)'] * E

= [1 + ((0+0)1 )'] * 1

= [1 + (0*1)'] * 1

= [1+ 0'] *1

= [1+ 1 ] * 1

= 1

In general, the following rules must always be followed when evaluating a Boolean

expression:

1. First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0.

2. Then perform all operations within parentheses.

3. Perform an AND operation before an OR operation unless parentheses indicate otherwise.

4. If an expression has a bar over it, perform the operations of the expression first and then

invert the result.

Determining Output Level from a Diagram

The output logic level for given input levels can also be determined directly from the circuit

diagram without using the Boolean expression.

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Implementing Circuits from Boolean Expression

If the operation of a circuit is defined by a Boolean expression, a logic-circuit diagram can he

implemented directly from that expression.

Suppose that we wanted to construct a circuit whose output is y = AC+BC' + A'BC. This

Boolean expression contains three terms (AC, BC', A'BC), which are ORed together. This

tells us that a three-input OR gate is required with inputs that are equal to AC, BC', and

A'BC, respectively.

Each OR-gate input is an AND product term, which means that an AND gate with

appropriate inputs can be used to generate each of these terms. Note the use of INVERTERs

to produce the A' and C' terms required in the expression.

Boolean Theorems

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Investigating the various Boolean theorems (rules) can help us to simplify logic expressions

and logic circuits.

Multivariable Theorems

The theorems presented below involve more than one variable:

(9) x + y = y + x (commutative law)

(10) x * y = y * x (commutative law)

(11) x+ (y+z) = (x+y) +z = x+y+z (associative law)

(12) x (yz) = (xy) z = xyz (associative law)

(13a) x (y+z) = xy + xz

(13b) (w+x)(y+z) = wy + xy + wz + xz

(14) x + xy = x [proof see below]

(15) x + x'y = x + y

Proof of (14)

x + xy = x (1+y)

= x * 1 [using theorem (6)]

= x [using theorem (2)]

DeMorgan's Theorem

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DeMorgan's theorems are extremely useful in simplifying expressions in which a product or

sum of variables is inverted. The two theorems are:

(16) (x+y)' = x' * y'

Theorem (16) says that when the OR sum of two variables is inverted, this is

the same as inverting each variable individually and then ANDing these

inverted variables.

(17) (x*y)' = x' + y'

Theorem (17) says that when the AND product of two variables is inverted, this is the

same as inverting each variable individually and then ORing them.

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Example

X = [(A'+C) * (B+D')]'

= (A'+C)' + (B+D')' [by theorem (17)]

= (A''*C') + (B'+D'') [by theorem (16)]

= AC' + B'D

Three Variables DeMorgan's Theorem

(18) (x+y+z)' = x' * y' * z'

(19) (xyz)' = x' + y' + z'

Universality of NAND & NOR Gates

It is possible to implement any logic expression using only NAND gates and no other type of

gate. This is because NAND gates, in the proper combination, can be used to perform each of

the Boolean operations OR, AND, and INVERT.

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In a similar manner, it can be shown that NOR gates can be arranged to implement any of the

Boolean operations.

Alternate Logic Gate Representations

The left side of the illustration shows the standard symbol for each logic gate, and the right

side shows the alternate symbol. The alternate symbol for each gate is obtained from the

standard symbol by doing the following:

1. Invert each input and output of the standard symbol. This is done by adding

bubbles (small circles) on input and output lines that do not have bubbles, and by

removing bubbles that are already there.

2. Change the operation symbol from AND to OR, or from OR to AND. (In the

special case of the INVERTER, the operation symbol is not changed.)

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Several points should be stressed regarding the logic symbol equivalences:

1. The equivalences are valid for gates with any number of inputs.

2. None of the standard symbols have bubbles on their inputs, and all the alternate

symbols do.

3. The standard and alternate symbols for each gate represent the same physical

circuit: there is no difference in the circuits represented by the two symbols.

4. NAND and NOR gates are inverting gates, and so both the standard and alternate

symbols for each will have a bubble on either the input or the output. AND and OR

gates are noninverting gates, and so the alternate symbols for each will have bubbles

on both inputs and output.

Concept of Active Logic Levels:

When an input or output line on a logic circuit symbol has no bubble on it, that line is said to

be active-HIGH. When an input or output line does have a bubble on it, that line is said to be

active-LOW. The presence or absence of a bubble, then, determines the active-HIGH/active-

LOW status of a circuit's inputs and output, and is used to interpret the circuit operation.

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Boolean Function

A Boolean function is an algebraic expression consists of binary variables, the

constants 0 & 1, and the Boolean operators.For a set of given values of the variables,

the function is evaluated to either 0 or 1

e.g. f = x • y + x • z‘

The Boolean function f has 3 binaryvariables x, y and z

The function is 1 if x and y are both 1 or if x is 1 and z is 0. Otherwise, f = 0

Operator Precedence

The operator precedence is:

1. Parentheses

2. NOT

3. AND

4. OR

e.g. f = x • y + x • z‘

Precedence: z‘, x • y, x • z‘, x • y + x • z‘

e.g. f = (a +b) • (c+d‘)

Precedence: a+b, d‘, c+d‘, (a +b) • (c+d‘)

The parentheses precedence is the same as in normal algebra

Boolean Function Truth Table

Boolean function can be represented by truth table as well.If the function has n variables, its

truth table will have 2n rows

e.g. f = x • y + x • z‘

f has 3 variables so 23 combinations

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f is 1 when the expression is evaluated to 1 otherwise it is 0.

Minterm

In a Boolean function, a binary variable (x) may appear either in its normal form (x) or in its

complement form (x‘).Consider 2 binary variables x and y and an AND operation, there are 4

and only 4 possible combinations: x‘•y‘, x‘•y, x•y‘ & x•y

Each of the 4 product terms is called a MINTERM or STANDARD PRODUCT

By definition, a Minterm is a product which consists of all the variables in the normal form

or the complement form but NOT BOTH.

e.g. for a function with 2 variables x and y:

x•y‘ is a minterm but x‘ is NOT a minterm

e.g. for a function with 3 variables x, y andz:

x‘yz‘ is a minterm but xy‘ is NOT a minterm

Maxterm

Consider 2 binary variables x and y and an OR operation, there are 4 and only 4

possible combinations: x‘+y‘, x‘+y, x+y‘, x+y.Each of the 4 sum terms is called a

MAXTERM or STANDARD SUM.By definition, a Maxterm is a sum in which each

variable appears once and only once either in its normal form or its complement

form but NOT BOTH.

Minterms and Maxterms for 3 Variables

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Minterm Boolean Expression

Boolean functions can be expressed with minterms,

e.g.f1(x,y,z) = m1 + m4 + m6 = Σm(1, 4, 6)

f2(x,y,z) = m2 + m4 + m6+ m7

= Σm(2, 4, 6, 7)

Maxterm Boolean Expression

Boolean functions can also be expressed with maxterms,

e.g.f1‘ = x‘y‘z‘+x‘yz‘+x‘yz+xy‘z+xyz

f1 = (x‘y‘z‘+x‘yz‘+x‘yz+xy‘z+xyz)‘

= (x+y+z)(x+y‘+z)(x+y‘+z‘)(x‘+y+z‘)(x‘+y‘+z‘)

= M0•M2•M3•M5•M7

= Π M(0, 2, 3, 5, 7)

f2 = M0•M1•M3•M5

= Π M(0, 1, 3, 5)

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Literal

A Literal is a variable in a product or sum term

xy‘ is a 2-literal product term

x‘yz has 3 literals

x‘ + xy‘ + x‘yz is an expression of sum of products with 3 product terms.The 3

product terms have 1, 2 and 3 literals respectively

x‘(x+y‘)(x‘+y+z) is an expression of product of sums.The 3 sum terms have 1, 2 and

3 literals

Express Boolean Functions in Minterms

If product terms in a Boolean function are not minterms, they can be converted to minterms

e.g. f(a,b,c) = a‘ + bc‘ + ab‘c

Function f has 3 variables, therefore, each minterm must have 3 literals

Neither a‘ nor bc‘ are minterms.They can be converted to minterm.ab‘c is a minterm

Conversion to Minterms

e.g. f(a,b,c) = a‘ + bc‘ + ab‘c

To convert a‘ to a minterm, the 2 variables (b, c) must be added, without changing its

functionality .Since a‘=a‘•1 & 1 = b+b‘, a‘= a‘(b + b‘) = a‘b + a‘b‘

Similarly, a‘b = a‘b(c + c‘) = a‘bc + a‘bc‘ and a‘b‘ = a‘b‘(c+c‘) = a‘b‘c + a‘b‘c‘

bc‘ = bc‘(a+a‘) = abc‘ + a‘bc‘

f = a‘bc+a‘bc‘+a‘b‘c+a‘b‘c‘+abc‘+a‘bc‘+ab‘c

Express Boolean Functions in Maxterms

By using the Distribution Law: x+yz = (x+y)(x+z), a Boolean function can

be converted to an expression in product of maxterms

e.g. f(a,b,c) = a‘+bc‘

= (a‘+b)(a‘+c‘) {not maxterms}

= (a‘+b+cc‘)(a‘+c‘+bb‘) {cc‘=0}

= (a‘+b+c)(a‘+b+c‘)(a‘+c‘+b)(a‘+c‘+b‘)

= (a‘+b+c)(a‘+b+c‘)(a‘+c‘+b‘)

Boolean Function Manipulation

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Boolean functions can be manipulated with Boolean algebra.Manipulation can transform

logic expressions, but still keep the same logic functionality.Manipulation can reduce the

complexity, hence, easier to be implemented in hardware, i.e. fewer logic gates

Boolean Function Manipulation Example

f = xy‘ + xyz + x‘z

= x(y‘ + yz) + x‘z {common factor}

= x[(y‘+y)(y‘+z)] + x‘z {Distribution law}

= x(y‘+z) + x‘z {y‘ + y = 1}

= xy‘ + xz + x‘z {Distribution law}

= xy‘ + (x + x‘)z {common factor}

= xy‘ + z {x + x‘ = 1}

Simplify f1=abc+a‘b+abc‘ and f2=(a+b)‘(a‘+b‘) to the minimum literals

f1 = abc+a‘b+abc‘ = ab(c+c‘) + a‘b = ab + a‘b = (a+a‘)b = b

f2 =(a+b)‘(a‘+b‘) = a‘b‘(a‘+b‘) {DeMorgan}

= a‘b‘a‘+a‘b‘b‘

= a‘b‘ + a‘b‘ = a‘b‘

QUINE-McCLUSKEY MINIMIZATION

Quine-McCluskey minimization method uses the same theorem to produce the solution as the

K-map method, namely X(Y+Y')=X

Minimization Technique

The expression is represented in the canonical SOP form if not already in that form.

The function is converted into numeric notation.

The numbers are converted into binary form.

The minterms are arranged in a column divided into groups.

Begin with the minimization procedure.

Each minterm of one group is compared with each minterm in the group immediately below.

Each time a number is found in one group which is the same as a number in the group below

except for one digit, the numbers pair is ticked and a new composite is created.

This composite number has the same number of digits as the numbers in the pair except the

digit different which is replaced by an "x".

The above procedure is repeated on the second column to generate a third column.

The next step is to identify the essential prime implicants, which can be done using a prime

implicant chart.

Where a prime implicant covers a minterm, the intersection of the corresponding row and

column is marked with a cross.

Those columns with only one cross identify the essential prime implicants. -> These prime

implicants must be in the final answer.

The single crosses on a column are circled and all the crosses on the same row are also

circled, indicating that these crosses are covered by the prime implicants selected.

Once one cross on a column is circled, all the crosses on that column can be circled since the

minterm is now covered.

If any non-essential prime implicant has all its crosses circled, the prime implicant is

redundant and need not be considered further.

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Next, a selection must be made from the remaining nonessential prime implicants, by

considering how the non-circled crosses can be covered best.

One generally would take those prime implicants which cover the greatest number of crosses

on their row.

If all the crosses in one row also occur on another row which includes further crosses, then

the latter is said to dominate the former and can be selected.

The dominated prime implicant can then be deleted.

Example

Find the minimal sum of products for the Boolean expression,

f= (1,2,3,7,8,9,10,11,14,15), using Quine-McCluskey method.

Firstly these minterms are represented in the binary form as shown in the table below. The

above binary representations are grouped into a number of sections in terms of the number of

1's as shown in the table below.

Binary representation of minterms

Minterms U V W X

1 0 0 0 1

2 0 0 1 0

3 0 0 1 1

7 0 1 1 1

8 1 0 0 0

9 1 0 0 1

10 1 0 1 0

11 1 0 1 1

14 1 1 1 0

15 1 1 1 1

Group of minterms for different number of 1's

No of 1's Minterms U V W X

1 1 0 0 0 1

1 2 0 0 1 0

1 8 1 0 0 0

2 3 0 0 1 1

2 9 1 0 0 1

2 10 1 0 1 0

3 7 0 1 1 1

3 11 1 0 1 1

3 14 1 1 1 0

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4 15 1 1 1 1

Any two numbers in these groups which differ from each other by only one variable can be

chosen and combined, to get 2-cell combination, as shown in the table below.

2-Cell combinations

Combinations U V W X

(1,3) 0 0 - 1

(1,9) - 0 0 1

(2,3) 0 0 1 -

(2,10) - 0 1 0

(8,9) 1 0 0 -

(8,10) 1 0 - 0

(3,7) 0 - 1 1

(3,11) - 0 1 1

(9,11) 1 0 - 1

(10,11) 1 0 1 -

(10,14) 1 - 1 0

(7,15) - 1 1 1

(11,15) 1 - 1 1

(14,15) 1 1 1 -

From the 2-cell combinations, one variable and dash in the same position can be combined to

form 4-cell combinations as shown in the figure below.

Combinations U V W X

(1,3,9,11) - 0 - 1

(2,3,10,11) - 0 1 -

(8,9,10,11) 1 0 - -

(3,7,11,15) - - 1 1

(10,11,14,15) 1 - 1 -

The cells (1,3) and (9,11) form the same 4-cell combination as the cells (1,9) and (3,11). The

order in which the cells are placed in a combination does not have any effect. Thus the

(1,3,9,11) combination could be written as (1,9,3,11).

From above 4-cell combination table, the prime implicants table can be plotted as shown in

table below.

Prime Implicants Table

Prime 1 2 3 7 8 9 10 11 14 15

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Implicants

(1,3,9,11) X - X - - X - X - -

(2,3,10,11) - X X - - - X X - -

(8,9,10,11) - - - - X X X X - -

(3,7,11,15) - - - - - - X X X X

- X X - X X - - - X -

The columns having only one cross mark correspond to essential prime implicants. A yellow

cross is used against every essential prime implicant. The prime implicants sum gives the

function in its minimal SOP form.

Y = V'X + V'W + UV' + WX + UW

Logic

Combinational logic blocks have the outputs depending on the combinations of the current

inputs. Sequential logic blocks have the outputs depending on the current inputs as well as

any previous inputs.

Binary Adder

Binary Adder is for binary number addition

Logic Circuit to be discussed:

Half Adder

Full Adder

Ripple Adder

Carry Look Ahead Adder

Half Adder

o Half adder is for addition of 2 single bits

o It has two 1-bit inputs and two 1-bit outputs

o The inputs are the 2 bits to be added (a, b)

o The outputs are 1-bit sum (s) & 1-bit carry (c)

The logic is:

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Binary Addition

The half adder adds 2 single-bit inputs

It cannot complete a full addition

To complete a full addition, the adder needs to take in 3 inputs: a, b and the carry from the

previous bit.

Full Adder

To carry the addition, an adder with 3 inputs is required. A Full Adder takes in 3 inputs (a, b

and ci) and produces 2 outputs (s, co) a & b are the 2 bits to be added, ci is the carry input

(carry over from the previous bit) and co is the carry output (to the next bit)

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Logic for Full Adder Logic equations derived from the truth table:

s = a b ci

co = ab + bci + aci

Full Adder

The below implementation shows implementing the full adder with AND-OR gates, instead

of using XOR gates. The basis of the circuit below is from the above Kmap.

Circuit-SUM

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Circuit-CARRY

Full adder can be built from 2 half adders

s = a b ci

co = ab+bci+aci

= ab+(a‘bci+abci)+(abci+ab‘ci)

= ab + abci + ci (a‘b+ab‘) = ab + ci (a b)

n-bit Ripple Adder

To perform an addition of 2 n-bit numbers An-1…A1A0 & Bn-1…B1B0, where An-1 &

Bn-1 are theMSB & A0B0 are the LSB, we need a n-bit adder,which can be built from ‗n

‗fulladders

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Ripple Adder: Carry ripples through the chain

Carry-Look-Ahead Adder

The delay generated by an N-bit adder is proportional to the length N of the two numbers X

and Y that are added because the carry signals have to propagate from one full-adder to the

next. For large values of N, the delay becomes unacceptably large so that a special solution

needs to be adopted to accelerate the calculation of the carry bits. This solution involves a

"look-ahead carry generator" which is a block that simultaneously calculates all the carry bits

involved. Once these bits are available to the rest of the circuit, each individual three-bit

addition (Xi+Yi+carry-ini) is implemented by a simple 3-input XOR gate. The design of the

look-ahead carry generator involves two Boolean functions named Generate and Propagate.

For each input bits pair these functions are defined as:

Gi = Xi . Yi

Pi = Xi + Yi

The carry bit c-out(i) generated when adding two bits Xi and Yi is '1' if the corresponding

function Gi is '1' or if the c-out(i-1)='1' and the function Pi = '1' simultaneously. In the first

case, the carry bit is activated by the local conditions (the values of Xi and Yi). In the second,

the carry bit is received from the less significant elementary addition and is propagated

further to the more significant elementary addition. Therefore, the carry_out bit

corresponding to a pair of bits Xi and Yi is calculated according to the equation:

carry_out(i) = Gi + Pi.carry_in(i-1)

For a four-bit adder the carry-outs are calculated as follows

carry_out0 = G0 + P0 . carry_in0

carry_out1 = G1 + P1 . carry_out0 = G1 + P1G0 + P1P0 . carry_in0

carry_out2 = G2 + P2G1 + P2P1G0 + P2P1P0 . carry_in0

carry_out3 = G3 + P3G2 + P3P2G1 + P3P2P1G0 + P3P2P1 . carry_in0

The set of equations above are implemented by the circuit below and a complete adder with a

look-ahead carry generator is next. The input signals need to propagate through a maximum

of 4 logic gate in such an adder as opposed to 8 and 12 logic gates in its counterparts

illustrated earlier.

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Pi is called Carry Propagate

Gi is called Carry Generate

With Pi and Gi, we obtain the sum & carry for the full adder:

Ci+1= Gi + PiCi

C1 = G0 + P0C0

C2 = G1 + P1C1

= G1 + P1(G0 + P0C0)

= G1 + P1G0 + P1P0C0

C3 = G2 + P2C2

= G2 + P2(G1 + P1G0 + P1P0C0)

= G2 + P2G1 + P2P1G0 + P2P1P0C0

Carry no longer depend on its previous stage

Look-Ahead Carry Generator

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Speed: 2 gate delays for all carry

Cost: more gates

Sums can be calculated from the following equations, where carryout is taken from the carry

calculated in the above circuit.

sum_out0 = X 0 Y0 carry_out0

sum_out1 = X 1 Y1 carry_out1

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sum_out2 = X 2 Y2 carry_out2

sum_out3 = X 3 Y3 carry_out3

MSI Adder

Adders are available in Medium Scale Integration (MSI) devices

Both TTL and CMOS are available, e.g.

74183: TTL 1-bit Full Adder

7482: TTL 4-bit Carry-Look-Ahead Adder

4008: CMOS 4-bit Carry-Look-Ahead Adder

74182: 4-bit Look-Ahead Carry Generator

4-bit Addition

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To add 2 4-bit numbers: Z = X + Y

8-bit Addition

To add 2 8-bit numbers: Z = X + Y

Subtractor

Subtractor circuits take two binary numbers as input and subtract one binary number input

from the other binary number input. Similar to adders, it gives out two outputs, difference

and borrow (carry-in the case of Adder). There are two types of subtractors.

Half Subtractor

Full Subtractor

Half Subtractor

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The half-subtractor is a combinational circuit which is used to perform subtraction of two

bits. It has two inputs, X (minuend) and Y (subtrahend) and two outputs D (difference) and B

(borrow). The logic symbol and truth table are shown below.

Symbol

Truth Table

X Y D B

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

From the above table we can draw the Kmap as shown below for "difference" and "

borrow". The boolean expression for the difference and Borrow can be written.

From the equation we can draw the half-subtractor as shown in the figure below.

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Full Subtractor

A full subtractor is a combinational circuit that performs subtraction involving three bits,

namely minuend, subtrahend, and borrow-in. The logic symbol and truth table are shown

below.

Symbol

Truth Table

X Y Bin D Bout

0 0 0 0 0

0 0 1 1 1

0 1 0 1 1

0 1 1 0 1

1 0 0 1 0

1 0 1 0 0

1 1 0 0 0

1 1 1 1 1

From above table we can draw the Kmap as shown below for "difference" and "borrow".

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The boolean expression for difference and borrow can be written as

D = X'Y'Bin + X'YBin' + XY'Bin' + XYBin

= (X'Y' + XY)Bin + (X'Y + XY')Bin'

= (X Y)'Bin + (X Y)Bin'

= X Y Bin

Bout = X'.Y + X'.Bin + Y.Bin

From the equation we can draw the full-subtractor as shown in figure below.

Full-subtractor circuit is more or less same as a full-adder with slight modification.

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Parallel Binary Subtractor

Parallel binary subtractor can be implemented by cascading several full-subtractors.

Implementation and associated problems are those of a parallel binary adder, seen before in

parallel binary adder section.

Below is the block level representation of a 4-bit parallel binary subtractor, which subtracts

4-bit Y3Y2Y1Y0 from 4-bit X3X2X1X0. It has 4-bit difference output D3D2D1D0 with

borrow output Bout.

A serial subtractor can be obtained by converting the serial adder using the 2's complement

system. The subtrahend is stored in the Y register and must be 2's complemented before it is

added to the minuend stored in the X register.

The circuit for a 4-bit serial subtractor using full-adder is shown in the figure below.

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Comparator

Comparator compares binary numbers.

Logic comparing 2 bits: a and b

Magnitude Comparator

Comparator compares binary numbers

4-bit Magnitude Comparator:

Inputs: A3A2A1A0 & B3B2B1B0

Outputs: Y A>B, Y A<B, Y A=B

For each bit, let:

Si = AiBi + Ai‘Bi‘ = (AiBi‘ + Ai‘Bi)‘

Si is true when Ai = Bi

For A = B, we must have:

A3=B3 and A2=B2 and A1=B1 and A0=B0

Hence, Y A=B = S3•S2•S1•S0 136

Logic For A > B

For A > B, there are 4 cases:

1. A3B3 is 10 and A2A1A0 & B2B1B0 can be anything:

A=1xxx, B=0xxx

2. A3=B3 and A2B2 is 10 and A1A0 & B1B0 can be

anything: A=11xx, B=10xx or A=01xx, B=00xx

3. A3=B3 and A2=B2 and A1B1=10 and A0B0 is xx: e.g.

A=011x, B=010x

4. A3=B3 and A2=B2 and A1=B1 and A0B0 is 10: e.g.

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A=1011, B=1010

Y A>B=A3B3’+S3A2B2’+S3S2A1B1’+S3S2S1A0B0’

Logic For A < B

For A < B, there are also 4 cases:

1) A3B3 is 01 and A2A1A0 & B2B1B0 can be anything:

1. A=0xxx, B=1xxx

2) A3=B3 and A2B2 is 01 and A1A0 & B1B0 can be

1. anything: A=10xx, B=11xx or A=00xx, B=01xx

3) A3=B3 and A2=B2 and A1B1=01 and A0B0 is xx: e.g.

1. A=110x, B=111x

4) A3=B3 and A2=B2 and A1=B1 and A0B0 is 01: e.g.

1. A=1000, B=1001

Y A<B=A3 ’B3+S3A2 ’B2+S3S2A1 ’ B1+S3S2S1A0 ’ B0

4-bit Comparator Logic Circuit

MSI: 7485 4-bit Magnitude Comparator

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Comparison of 4-bit Numbers

Comparison of 8 - bit Numbers

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Decoder

A decoder is a multiple-input, multiple-output logic circuit that converts coded inputs into

coded outputs, where the input and output codes are different.Binary Decoder has n inputs

and 2noutputs also called as n-to-2

n decoder. Inputs have all the 2

n combinations and

the corresponding output will be activated for each input combinations.Decoding is necessary

in applications such as data multiplexing, 7 segment display and memory address decoding.

Enable inputs must be on for the decoder to function, otherwise its outputs assume a single

"disabled" output code word. Figure below shows the pseudo block of a decoder.

A binary decoder has n inputs and 2n outputs. Only one output is active at any one time,

corresponding to the input value. Figure below shows a representation of Binary n-to-2n

decoder

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e.g. 3-to-8 decoder has 3 inputs and 8 outputs

3-to-8 Decoder Function Table

3-to-8 Decoder Logic Circuit

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2-to-4 Decoder with Output Enable

Implement Logic Function with Decoder

Any n-variable logic function, in canonical sum-of-minterms form can be implemented using

a single n-to-2n decoder to generate the minterms, and an OR gate to form the sum.

The output lines of the decoder corresponding to the minterms of the function are used as

inputs to the or gate.

Any combinational circuit with n inputs and m outputs can be implemented with an n-to-2n

decoder with m OR gates.

Suitable when a circuit has many outputs, and each output function is expressed with few

minterms.

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(Ex) Full adder using decoder

S(x, y, z) = (1,2,4,7)

C(x, y, z) = (3,5,6,7)

Truth Table

From

the truth

table we

know

the

values

for

which

the sum

(s) is

active and also the carry (c) is active. Thus we have the equation as shown above and a

circuit can be drawn as shown below from the equation derived.

Use a 3-to-8 decoder to implement:

f = x‘y‘z + xy‘z + xyz

(m1 + m5 + m7)

X Y Z C S

0 0 0 0 0

0 0 1 0 1

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 1

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MSI Decoders

1. 2-to-4 Decoder

2. 3-to-8 Decoder

3. 4-to-16 Decoder

4. BCD-to-Decimal Decoder

5. BCD-to-Seven-Segment Decoder

e.g. Low Power Schottky TTL:

74LS138 3-to-8 Decoder where G1, G2A and G2B are enable pins

Logic Symbol

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74LS138 3-to-8 Decoder

Implement Logic Function with74LS138

Use a 3-to-8 decoder to implement:

f = x‘y‘z + xy‘z + xyz

(m1 + m5 + m7)

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4-to-16 Decoder

Use 2 3-to-8 decoders

Inputs: D, C, B, A

Outputs: Y0 – Y15

When D = 0, top decoder is enabled

When D = 1,bottom decoderis enabled

En‘ is enable

Binary Encoders

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An encoder is a combinational circuit that performs the inverse operation of a decoder. If a

device output code has fewer bits than the input code has, the device is usually called an

encoder. e.g. 2n-to-n, priority encoders.

The simplest encoder is a 2n-to-n binary encoder, where it has only one of 2

n inputs = 1 and

the output is the n-bit binary number corresponding to the active input. It can be built from

OR gates

e.g. 4-to-2 Encoder

Octal-to-Binary Encoder

Octal-to-Binary take 8 inputs and provides 3 outputs, thus doing the opposite of what the 3-

to-8 decoder does. At any one time, only one input line has a value of 1. The figure below

shows the truth table of an Octal-to-binary encoder.

Truth Table

I0 I1 I2 I3 I4 I5 I6 I7 Y2 Y1 Y0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 1

0 0 1 0 0 0 0 0 0 1 0

0 0 0 1 0 0 0 0 0 1 1

0 0 0 0 1 0 0 0 1 0 0

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0 0 0 0 0 1 0 0 1 0 1

0 0 0 0 0 0 1 0 1 1 0

0 0 0 0 0 0 0 1 1 1 1

For an 8-to-3 binary encoder with inputs I0-I7 the logic expressions of the outputs Y0-Y2

are:

Y0 = I1 + I3 + I5 + I7

Y1= I2 + I3 + I6 + I7

Y2 = I4 + I5 + I6 +I7

Based on the above equations, we can draw the circuit as shown below

Priority Encoder

If more then two inputs are active simultaneously, the output is unpredictable or rather it is

not what we expect it to be.This ambiguity is resolved if priority is established so that only

one input is encoded, no matter how many inputs are active at a given point of time. The

priority encoder includes a priority function. The operation of the priority encoder is such

that if two or more inputs are active at the same time, the input having the highest priority

will take precedence.

e.g. 4-to-2 PriorityEncoder

A3 has the highest priority

A0 has the lower priority

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74148 8-to-3 Priority Encoder

16-to-4 Priority Encoder

Cascade two 74148 8-to-3 priority encoders. The Input 15 has highest priority

Multiplexer

A multiplexer (MUX) is a digital switch which connects data from one of n sources to the

output. A number of select inputs determine which data source is connected to the output.

The block diagram of MUX with n data sources of b bits wide and s bits wide select line is

shown in below figure.

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MUX acts like a digitally controlled multi-position switch where the binary code applied to

the select inputs controls the input source that will be switched on to the output as shown in

the figure below. At any given point of time only one input gets selected and is connected to

output, based on the select input signal.

The operation of a multiplexer can be better explained using a mechanical switch as shown in

the figure below. This rotary switch can touch any of the inputs, which is connected to the

output. As you can see at any given point of time only one input gets transferred to output.

2x1 MUX

A 2 to 1 line multiplexer is shown in figure below, each 2 input lines A to B is applied to one

input of an AND gate. Selection lines S are decoded to select a particular AND gate. The

truth table for the 2:1 mux is given in the table below.

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Design of a 2:1 Mux

To derive the gate level implementation of 2:1 mux we need to have truth table as s

hown in figure. And once we have the truth table, we can draw the K-map as shown

in figure for all the cases when Y is equal to '1'.

Combining the two 1' as shown in figure, we can drive the output y as shown below

Y = A.S‘ + B.S

Truth Table

B A S Y

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 1

1 0 1

1 1 1 1

Kmap

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Circuit

MSI MUX

74150: 16-to-1

74153: Dual 4-to-1

74157: Quad 2-to-1

74151: 8-to-1

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16-to-1 MUX

Use two 74151

D = 0 enables top MUX

D = 1 enables bottom MUX

W = Y‘

= (Y1+Y2)‘

= (W1‘+W2‘)‘

= W1W2

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Larger Multiplexers

Larger multiplexers can be constructed from smaller ones. An 8-to-1 multiplexer can be

constructed from smaller multiplexers as shown below.

8-to-1 multiplexer from Smaller MUX

16-to-1 multiplexer from 4:1 mux

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Quadruple 2-to-1 MUX

It is 2-to-1 MUX with 4 bits for each input

There is 1 output of 4 bits

There is 1 select signal

When 1 input is selected, the whole group of 4 bits goes to the output

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Quad 2-to-1 MUX

Implementing Functions Multiplexers

Any n-variable logic function can be implemented using a smaller 2n-1

-to-1 multiplexer and a

single inverter (e.g 4-to-1 mux to implement 3 variable functions) as follows.

Express function in canonical sum-of-minterms form. Choose n-1 variables as inputs to mux

select lines. Construct the truth table for the function, but grouping inputs by selection line

values (i.e select lines as most significant inputs).

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Determine multiplexer input line i values by comparing the remaining input variable and the

function F for the corresponding selection lines value i.

We have four possible mux input line i values:

Connect to 0 if the function is 0 for both values of remaining variable.

Connect to 1 if the function is 1 for both values of remaining variable.

Connect to remaining variable if function is equal to the remaining variable.

Connect to the inverted remaining variable if the function is equal to the remaining

variable inverted

3-variable Function Using 8-to-1 mux

Implement the function F(X,Y,Z) = S(1,3,5,6) using an 8-to-1 mux. Connect the input

variables X, Y, Z to mux select lines. Mux data input lines 1, 3, 5, 6 that correspond to the

function minterms are connected to 1. The remaining mux data input lines 0, 2, 4, 7 are

connected to 0.

3-variable Function Using 4-to-1 mux

Implement the function F(X,Y,Z) = S(0,1,3,6) using a single 4-to-1 mux and an inverter. We

choose the two most significant inputs X, Y as mux select lines.

Truth Table

Select i X Y Z F Mux Input i

0 0 0 0 1 1

0 0 0 1 1 1

1 0 1 0 0 Z

1 0 1 1 1 Z

2 1 0 0 0 0

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2 1 0 1 0 0

3 1 1 0 1 Z'

3 1 1 1 0 Z'

We determine multiplexer input line i values by comparing the remaining input variable

Z and the function F for the corresponding selection lines value i

when XY=00 the function F is 1 (for both Z=0, Z=1) thus mux input0 = 1

when XY=01 the function F is Z thus mux input1 = Z

when XY=10 the function F is 0 (for both Z=0, Z=1) thus mux input2 = 0

when XY=11 the function F is Z' thus mux input3 = Z'

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Example for logic function implementation using MUX

De-multiplexers

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They are digital switches which connect data from one input source to one of n

outputs.Usually implemented by using n-to-2n binary decoders where the decoder enable line

is used for data input of the de-multiplexer.The figure below shows a de-multiplexer block

diagram which has got s-bits-wide select input, one b-bits-wide data input and n b-bits-wide

outputs.

The operation of a de-multiplexer can be better explained using a mechanical switch as

shown in the figure below. This rotary switch can touch any of the outputs, which is

connected to the input. As you can see at any given point of time only one output gets

connected to input.

1-to-4 De-multiplexer

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Truth Table

S1 S0 F0 F1 F2 F3

0 0 D 0 0 0

0 1 0 D 0 0

1 0 0 0 D 0

1 1 0 0 0 D

Unit 4. PROGRAMMING LOGIC CONTROLLERS

Programmable Logic Controllers – Basic Structure – Input / Output Processing – Programming –

Mnemonics – Timers, Internal relays and counters – Shift Registers – Master and Jump Controls –

Data Handling – Analogs Input / Output – Selection of a PLC Problem.

A programmable logic controller (PLC) or programmable controller is a digital computer used

for automation of electromechanical processes, such as control of machinery on factory assembly

lines, amusement rides, or lighting fixtures. PLCs are used in many industries and machines. Unlike

general-purpose computers, the PLC is designed for multiple inputs and output arrangements,

extended temperature ranges, immunity to electrical noise, and resistance to vibration and impact.

Programs to control machine operation are typically stored in battery-backed or non-volatile

memory. A PLC is an example of a hard real time system since output results must be produced in

response to input conditions within a bounded time, otherwise unintended operation will result.

Early PLCs were designed to replace relay logic systems. These PLCs were programmed in "ladder

logic", which strongly resembles a schematic diagram of relay logic. This program notation was

chosen to reduce training demands for the existing technicians. Other early PLCs used a form of

instruction list programming, based on a stack-based logic solver.

Modern PLCs can be programmed in a variety of ways, from ladder logic to more traditional

programming languages such as BASIC and C. Another method is State Logic, a very high-level

programming language designed to program PLCs based on state transition diagrams.

Many early PLCs did not have accompanying programming terminals that were capable of graphical

representation of the logic, and so the logic was instead represented as a series of logic expressions

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in some version of Boolean format, similar to Boolean algebra. As programming terminals evolved,

it became more common for ladder logic to be used, for the aforementioned reasons. Newer formats

such as State Logic and Function Block (which is similar to the way logic is depicted when using

digital integrated logic circuits) exist, but they are still not as popular as ladder logic. A primary

reason for this is that PLCs solve the logic in a predictable and repeating sequence, and ladder logic

allows the programmer (the person writing the logic) to see any issues with the timing of the logic

sequence more easily than would be possible in other formats.

Ladder logic is a programming language that represents a program by a graphical diagram based on

the circuit diagrams of relay-based logic hardware. It is primarily used to develop software for

Programmable Logic Controllers (PLCs) used in industrial control applications. The name is based

on the observation that programs in this language resemble ladders, with two vertical rails and a

series of horizontal rungs between them.

An argument that aided the initial adoption of ladder logic was that a wide variety of engineers and

technicians would be able to understand and use it without much additional training, because of the

resemblance to familiar hardware systems. This argument has become less relevant given that most

ladder logic programmers have a software background in more conventional programming

languages, and in practice implementations of ladder logic have characteristics - such as sequential

execution and support for control flow features—that make the analogy to hardware somewhat

inaccurate.

Ladder logic is widely used to program PLCs, where sequential control of a process or

manufacturing operation is required. Ladder logic is useful for simple but critical control systems or

for reworking old hardwired relay circuits. As programmable logic controllers became more

sophisticated it has also been used in very complex automation systems. Often the ladder logic

program is used in conjunction with an HMI program operating on a computer workstation.

Manufacturers of programmable logic controllers generally also provide associated ladder logic

programming systems. Typically the ladder logic languages from two manufacturers will not be

completely compatible; ladder logic is better thought of as a set of closely related programming

languages rather than one language. (The IEC 61131-3 standard has helped to reduce unnecessary

differences, but translating programs between systems still requires significant work.) Even different

models of programmable controllers within the same family may have different ladder notation such

that programs cannot be seamlessly interchanged between models.

Ladder logic can be thought of as a rule-based language rather than a procedural language. A "rung"

in the ladder represents a rule. When implemented with relays and other electromechanical devices,

the various rules "execute" simultaneously and immediately. When implemented in a programmable

logic controller, the rules are typically executed sequentially by software, in a continuous loop

(scan). By executing the loop fast enough, typically many times per second, the effect of

simultaneous and immediate execution is relatively achieved to within the tolerance of the time

required to execute every rung in the "loop" (the "scan time"). It is somewhat similar to other rule-

based languages, like spreadsheets or SQL. However, proper use of programmable controllers

requires understanding the limitations of the execution order of rungs.

Example of a simple ladder logic program

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The language itself can be seen as a set of connections between logical checkers (contacts) and

actuators (coils). If a path can be traced between the left side of the rung and the output, through

asserted (true or "closed") contacts, the rung is true and the output coil storage bit is asserted (1) or

true. If no path can be traced, then the output is false (0) and the "coil" by analogy to

electromechanical relays is considered "de-energized". The analogy between logical propositions

and relay contact status is due to Claude Shannon.

Ladder logic has contacts that make or break circuits to control coils. Each coil or contact

corresponds to the status of a single bit in the programmable controller's memory. Unlike

electromechanical relays, a ladder program can refer any number of times to the status of a single

bit, equivalent to a relay with an indefinitely large number of contacts.

So-called "contacts" may refer to physical ("hard") inputs to the programmable controller from

physical devices such as pushbuttons and limit switches via an integrated or external input module,

or may represent the status of internal storage bits which may be generated elsewhere in the

program.

Each rung of ladder language typically has one coil at the far right. Some manufacturers may allow

more than one output coil on a rung.

--( )-- a regular coil, energized whenever its rung is closed

--(\)-- a "not" coil, energized whenever its rung is open

--[ ]-- A regular contact, closed whenever its corresponding coil or an input which controls it is

energized.

--[\]-- A "not" contact, open whenever its corresponding coil or an input which controls it is

energized.

The "coil" (output of a rung) may represent a physical output which operates some device connected

to the programmable controller, or may represent an internal storage bit for use elsewhere in the

program.

Examples

Here is an example of what one rung in a ladder logic program might look like. In real life, there

may be hundreds or thousands of rungs.

For example:

1. ----[ ]---------|--[ ]--|------( )

X | Y | S

| |

|--[ ]--|

Z

The above realizes the function: S = X AND ( Y OR Z )

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Typically, complex ladder logic is 'read' left to right and top to bottom. As each of the lines (or

rungs) are evaluated the output coil of a rung may feed into the next stage of the ladder as an input.

In a complex system there will be many "rungs" on a ladder, which are numbered in order of

evaluation.

1. ----[ ]-----------|---[ ]---|----( )

X | Y | S

| |

|---[ ]---|

Z

2. ----[ ]----[ ]-------------------( )

S X T

2. T = S AND X where S is equivalent to #1. above

This represents a slightly more complex system for rung 2. After the first line has been evaluated,

the output coil (S) is fed into rung 2, which is then evaluated and the output coil T could be fed into

an output device (buzzer, light etc..) or into rung 3 on the ladder. (Note that the contact X on the

second rung serves no useful purpose, as X is already defined in the 'AND' function of S from the 1st

rung.)

This system allows very complex logic designs to be broken down and evaluated.

For more practical examples see below:

------[ ]--------------[ ]----------------( )

Key Switch 1 Key Switch 2 Door Motor

This circuit shows two key switches that security guards might use to activate an electric motor on a

bank vault door. When the normally open contacts of both switches close, electricity is able to flow

to the motor which opens the door. This is a logical AND.

+-------+

----------------------------+ +----

+-------+

Remote Receiver

--|-------[ ]-------+-----------------( )

| Remote Unlock | Lock Solenoid

| |

|-------[ ]-------|

Interior Unlock

This circuit shows the two things that can trigger a car's power door locks. The remote receiver is

always powered. The lock solenoid gets power when either set of contacts is closed. This is a logical

OR.

Often we have a little green "start" button to turn on a motor, and we want to turn it off with a big

red "Stop" button. The stop button itself is wired as a normally closed switch, the PLC input is read

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as normally open. When the stop button is pushed, the input will go false. Making the rung false,

stopping the "run" output.

--+----[ ]--+----[\]----( )

| start | stop run

| |

+----[ ]--+

run

-------[ ]--------------( )

run motor

This latch configuration is a common idiom in ladder logic. In ladder logic it is referred to as seal-in

logic.

Additional functionality

Additional functionality can be added to a ladder logic implementation by the PLC manufacturer as

a special block. When the special block is powered, it executes code on predetermined arguments.

These arguments may be displayed within the special block.

+-------+

-----[ ]--------------------+ A +----

Remote Unlock +-------+

Remote Counter

+-------+

-----[ ]--------------------+ B +----

Interior Unlock +-------+

Interior Counter

+--------+

--------------------+ A + B +-----------

+ into C +

+--------+

Adder

In this example, the system will count the number of times that the interior and remote unlock

buttons are pressed. This information will be stored in memory locations A and B. Memory location

C will hold the total number of times that the door has been unlocked electronically.

PLCs have many types of special blocks. They include timers, arithmetic operators and comparisons,

table lookups, text processing, PID control, and filtering functions. More powerful PLCs can operate

on a group of internal memory locations and execute an operation on a range of addresses, for

example,to simulate a physical sequential drum controller or a finite state machine. In some cases,

users can define their own special blocks, which effectively are subroutines or macros. The large

library of special blocks along with high speed execution has allowed use of PLCs to implement

very complex automation systems.

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Anatomy of a Ladder Program

Input instructions are entered on the left

Output instructions are entered on the right

The power rails simulate the power supply lines

L1 and L2 for AC circuits and +24 v and ground for DC circuits

Most PLCs allow more than one output per rung

The processor (or ―controller‖) scans ladder rungs from top-to-bottom and from left-to-right.

The basic sequence is altered whenever jump or subroutine instructions are executed.

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Logic Functions

PLC programming is a logical procedure

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In a PLC program, ―things‖ (inputs and rungs) are either TRUE or FALSE

If the proper input conditions are TRUE:The rung becomes TRUE and an output action

occurs (for example, a motor turns on)

If the proper input conditions are not TRUE:The rung becomes FALSE and an output action

does not occur

Ladder logic is based on the following logic functions:

AND, OR, Sometimes called ―inclusive OR‖, Exclusive OR.

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