A control system manages, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large Industrial control systems which are used for controlling processes or machines. For continuously modulated control, a feedback controller is used to automatically control a process or operation; the control system compares the value or status of the process variable being controlled with the desired value or setpoint, applies the difference as a control signal to bring the process variable output of the plant to the same value as the setpoint. For sequential and combinational logic, software logic, such as in a programmable logic controller, is used. There are two common classes of control action: closed loop. In an open-loop control system, the control action from the controller is independent of the process variable. An example of this is a central heating boiler controlled only by a timer.
The control action is the switching on or off of the boiler. The process variable is the building temperature; this controller operates the heating system for a constant time regardless of the temperature of the building. In a closed-loop control system, the control action from the controller is dependent on the desired and actual process variable. In the case of the boiler analogy, this would utilise a thermostat to monitor the building temperature, feed back a signal to ensure the controller output maintains the building temperature close to that set on the thermostat. A closed loop controller has a feedback loop which ensures the controller exerts a control action to control a process variable at the same value as the setpoint. For this reason, closed-loop controllers are called feedback controllers. In the case of linear feedback systems, a control loop including sensors, control algorithms, actuators is arranged in an attempt to regulate a variable at a setpoint. An everyday example is the cruise control on a road vehicle.
The PID algorithm in the controller restores the actual speed to the desired speed in the optimum way, with minimal delay or overshoot, by controlling the power output of the vehicle's engine. Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent. Open-loop control systems do not make use of feedback, run only in pre-arranged ways. Logic control systems for industrial and commercial machinery were implemented by interconnected electrical relays and cam timers using ladder logic. Today, most such systems are constructed with microcontrollers or more specialized programmable logic controllers; the notation of ladder logic is still in use as a programming method for PLCs. Logic controllers may respond to switches and sensors, can cause the machinery to start and stop various operations through the use of actuators. Logic controllers are used to sequence mechanical operations in many applications.
Examples include washing machines and other systems with interrelated operations. An automatic sequential control system may trigger a series of mechanical actuators in the correct sequence to perform a task. For example, various electric and pneumatic transducers may fold and glue a cardboard box, fill it with product and seal it in an automatic packaging machine. PLC software can be written in many different ways -- SFC or statement lists. On -- off control uses a feedback controller. A simple bi-metallic domestic thermostat can be described as an on-off controller; when the temperature in the room goes below the user setting, the heater is switched on. Another example is a pressure switch on an air compressor; when the pressure drops below the setpoint the compressor is powered. Refrigerators and vacuum pumps contain similar mechanisms. Simple on -- off control systems like these can be effective. Linear control systems use negative feedback to produce a control signal to maintain the controlled PV at the desired SP.
There are several types of linear control systems with different capabilities. Proportional control is a type of linear feedback control system in which a correction is applied to the controlled variable, proportional to the difference between the desired value and the measured value. 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, but simpler than a proportional-integral-derivative control system used, for instance, in an automobile cruise control. On–off control will work for systems that do not require high accuracy or responsiveness, but is not effective for rapid and timely corrections and responses. Proportional control overcomes this by modulating the manipulated variable, such as a control valve, at a gain level which avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional correction. A drawback of proportional control is that it cannot eliminate the residual SP–PV error, as it requires an error to generate a proportional output.
A PI controller can be used to overcome this. The PI controller uses a proportional term to remove the gross error, an integral term to eliminate the residual offset error by integrating the error over time. In some systems there are practical limits to the range of the MV. For example, a heater has a limit to how much heat it can produce
In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye. The SI unit of luminous intensity is an SI base unit. Photometry deals with the measurement of visible light; the human eye can only see light in the visible spectrum and has different sensitivities to light of different wavelengths within the spectrum. When adapted for bright conditions, the eye is most sensitive to greenish-yellow light at 555 nm. Light with the same radiant intensity at other wavelengths has a lower luminous intensity; the curve which measures the response of the human eye to light is a defined standard, known as the luminosity function. This curve, denoted V or y ¯, is based on an average of differing experimental data from scientists using different measurement techniques. For instance, the measured responses of the eye to violet light varied by a factor of ten.
Luminous intensity should not be confused with another photometric unit, luminous flux, the total perceived power emitted in all directions. Luminous intensity is the perceived power per unit solid angle. If a lamp has a 1 lumen bulb and the optics of the lamp are set up to focus the light evenly into a 1 steradian beam the beam would have a luminous intensity of 1 candela. If the optics were changed to concentrate the beam into 1/2 steradian the source would have a luminous intensity of 2 candela; the resulting beam is brighter, though its luminous flux remains unchanged. Luminous intensity is not the same as the radiant intensity, the corresponding objective physical quantity used in the measurement science of radiometry. Like other SI base units, the candela has an operational definition—it is defined by the description of a physical process that will produce one candela of luminous intensity. By definition, if one constructs a light source that emits monochromatic green light with a frequency of 540 THz, that has a radiant intensity of 1/683 watts per steradian in a given direction, that light source will emit one candela in the specified direction.
The frequency of light used in the definition corresponds to a wavelength in a vacuum of 555 nm, near the peak of the eye's response to light. If the source emitted uniformly in all directions, the total radiant flux would be about 18.40 mW, since there are 4π steradians in a sphere. A typical candle produces roughly one candela of luminous intensity. Prior to the definition of the candela, a variety of units for luminous intensity were used in various countries; these were based on the brightness of the flame from a "standard candle" of defined composition, or the brightness of an incandescent filament of specific design. One of the best-known of these standards was the English standard: candlepower. One candlepower was the light produced by a pure spermaceti candle weighing one sixth of a pound and burning at a rate of 120 grains per hour. Germany and Scandinavia used the Hefnerkerze, a unit based on the output of a Hefner lamp. In 1881, Jules Violle proposed the Violle as a unit of luminous intensity, it was notable as the first unit of light intensity that did not depend on the properties of a particular lamp.
All of these units were superseded by the definition of the candela. The luminous intensity for monochromatic light of a particular wavelength λ is given by I v = 683 ⋅ y ¯ ⋅ I e, where Iv is the luminous intensity in candelas, Ie is the radiant intensity in watts per steradian, y ¯ is the standard luminosity function. If more than one wavelength is present, one must sum or integrate over the spectrum of wavelengths present to get the luminous intensity: I v = 683 ∫ 0 ∞ y ¯ ⋅ d I e d λ d λ. Brightness International System of Quantities Radiance
A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks. They are used in engineering in hardware design, electronic design, software design, process flow diagrams. Block diagrams are used for higher level, less detailed descriptions that are intended to clarify overall concepts without concern for the details of implementation. Contrast this with the schematic diagrams and layout diagrams used in electrical engineering, which show the implementation details of electrical components and physical construction; as an example, a block diagram of a radio is not expected to show each and every connection and dial and switch, but the schematic diagram is. The schematic diagram of a radio does not show the width of each connection in the printed circuit board, but the layout does. To make an analogy to the map making world, a block diagram is similar to a highway map of an entire nation.
The major cities are listed but the minor county roads and city streets are not. When troubleshooting, this high level map is useful in narrowing down and isolating where a problem or fault is. Block diagrams rely on the principle of the black box where the contents are hidden from view either to avoid being distracted by the details or because the details are not known. We know what goes in, we know what goes out, but we can't see how the box does its work. In electrical engineering, a design will begin as a high level block diagram, becoming more and more detailed block diagrams as the design progresses ending in block diagrams detailed enough that each individual block can be implemented; this is known as top down design. Geometric shapes are used in the diagram to aid interpretation and clarify meaning of the process or model; the geometric shapes are connected by lines to indicate association and direction/order of traversal. Each engineering discipline has their own meaning for each shape.
Block diagrams are used in every discipline of engineering. They are a valuable source of concept building and educationally beneficial in non-engineering disciplines. In process control, block diagrams are a visual language for describing actions in a complex system in which blocks are black boxes that represent mathematical or logical operations that occur in sequence from left to right and top to bottom, but not the physical entities, such as processors or relays, that perform those operations, it is possible to create such block diagrams and implement their functionality with specialized programmable logic controller programming languages. In biology there is an increasing use of engineering principles, techniques of analysis and methods of diagramming. There is some similarity between the block diagram and what is called Systems Biology Graphical Notation; as it is there is use made in systems biology of the block diagram technique harnessed by control engineering where the latter itself is an application of control theory.
An example of this is the function block diagram, one of five programming languages defined in part 3 of the IEC 61131 standard, formalized, with strict rules for how diagrams are to be built. Directed lines are used to connect input variables to block inputs, block outputs to output variables and inputs of other blocks. Black box Bond graph Data flow diagram Functional flow block diagram One-line diagram Reliability block diagram Schematic diagram Signal-flow graph
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave, it is named after the function sine. It occurs in pure and applied mathematics, as well as physics, signal processing and many other fields, its most basic form as a function of time is: y = A sin = A sin where: A, the peak deviation of the function from zero. F, ordinary frequency, the number of oscillations that occur each second of time. Ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second φ, specifies where in its cycle the oscillation is at t = 0. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds. A negative value represents a delay, a positive value represents an advance; the sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform; this property makes it acoustically unique.
In general, the function may have: a spatial variable x that represents the position on the dimension on which the wave propagates, a characteristic parameter k called wave number, which represents the proportionality between the angular frequency ω and the linear speed ν. The wavenumber is related to the angular frequency by:. K = ω v = 2 π f v = 2 π λ where λ is the wavelength, f is the frequency, v is the linear speed; this equation gives a sine wave for a single dimension. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed; this wave pattern occurs in nature, including wind waves, sound waves, light waves. A cosine wave is said to be sinusoidal, because cos = sin , a sine wave with a phase-shift of π/2 radians.
Because of this head start, it is said that the cosine function leads the sine function or the sine lags the cosine. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics. To the human ear, a sound, made of more than one sine wave will have perceptible harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, the reason why the same musical note played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform, including square waves. Fourier used it as an analytical tool in the study of waves and heat flow, it is used in signal processing and the statistical analysis of time series.
Since sine waves propagate without changing form in distributed linear systems, they are used to analyze wave propagation. Sine waves traveling in two directions in space can be represented as u = A sin When two waves having the same amplitude and frequency, traveling in opposite directions, superpose each other a standing wave pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed end
The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is referred to as a maximally flat magnitude filter, it was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers". Butterworth had a reputation for solving "impossible" mathematical problems. At the time, filter design required a considerable amount of designer experience due to limitations of the theory in use; the filter was not in common use for over 30 years after its publication. Butterworth stated that: "An ideal electrical filter should not only reject the unwanted frequencies but should have uniform sensitivity for the wanted frequencies"; such an ideal filter cannot be achieved but Butterworth showed that successively closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated a substantial ripple in the passband, the choice of component values was interactive.
Butterworth showed that a low pass filter could be designed whose frequency response was G = 1 1 + ε 2 2 n, where ω is the angular frequency in radians per second,n is the number of poles in the filter—equal to the number of reactive elements in a passive filter, ε is the maximum passband gain, ωc is the cutoff frequency. If ωc= 1 ε =1, the amplitude response of this type of filter in the passband is 1/√2 ≈ 0.707, half power or −3 dB. Butterworth only dealt with filters with an number of poles in his paper, he may have been unaware. He built his higher order filters from 2-pole filters separated by vacuum tube amplifiers, his plot of the frequency response of 2, 4, 6, 8, 10 pole filters is shown as A, B, C, D, E in his original graph. Butterworth solved the equations for two- and four-pole filters, showing how the latter could be cascaded when separated by vacuum tube amplifiers and so enabling the construction of higher-order filters despite inductor losses. In 1930, low-loss core materials such as molypermalloy had not been discovered and air-cored audio inductors were rather lossy.
Butterworth discovered that it was possible to adjust the component values of the filter to compensate for the winding resistance of the inductors. He used coil forms of 3 ″ length with plug-in terminals. Associated capacitors and resistors were contained inside the wound coil form; the coil formed part of the plate load resistor. Two poles were used per vacuum tube and RC coupling was used to the grid of the following tube. Butterworth showed that the basic low-pass filter could be modified to give low-pass, high-pass, band-pass and band-stop functionality; the frequency response of the Butterworth filter is maximally flat in the passband and rolls off towards zero in the stopband. When viewed on a logarithmic Bode plot, the response slopes off linearly towards negative infinity. A first-order filter's response rolls off at −6 dB per octave. A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on. Butterworth filters have a monotonically changing magnitude function with ω, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.
Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic filters can achieve. A transfer function of a third-order low-pass Butterworth filter design shown in the figure on the right looks like this: V o V i = R s 3 + s 2 + s + R. A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with C2 = 4/3 F, R4 = 1 Ω, L1 = 3/2 H, L3 = 1/2 H. Taking the impedance of the capacitors C to be 1/ and the impedance of the inductors L to be Ls, where s = σ + jω is the complex frequency, the circuit equations yield the transfer function for this device: H = V o V i (
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group/phase delay, which preserves the wave shape of filtered signals in the passband. Bessel filters are used in audio crossover systems; the filter's name is a reference to German mathematician Friedrich Bessel, who developed the mathematical theory on which the filter is based. The filters are called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949; the Bessel filter is similar to the Gaussian filter, tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot, the Bessel filter has a small amount of overshoot, but still much less than common frequency domain filters. Compared to finite-order approximations of the Gaussian filter, the Bessel filter has better shaping factor, flatter phase delay, flatter group delay than a Gaussian of the same order, though the Gaussian has lower time delay and zero overshoot.
A Bessel low-pass filter is characterized by its transfer function: H = θ n θ n where θ n is a reverse Bessel polynomial from which the filter gets its name and ω 0 is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of 1 / ω 0. Since θ n is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that θ n = lim x → 0 θ n; the transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following: n = 1. 2 n − k k!! K = 0, 1, …, n; the transfer function for a third-order Bessel low-pass filter with ω 0 = 1 is H = 15 s 3 + 6 s 2 + 15 s + 15, where the numerator has been chosen to given unity gain at zero frequency. The roots of the denominator polynomial, the filter's poles, include a real pole at s = −2.3222, a complex-conjugate pair of poles at s = −1.8389 ± j1.7544, plotted above. The gain is G = | H | = 15 ω 6 + 6 ω 4 + 45 ω 2 + 225.
The phase is ϕ = − arg (
Frequency is the number of occurrences of a repeating event per unit of time. It is referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency; the period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals, radio waves, light. For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics and radio, frequency is denoted by a Latin letter f or by the Greek letter ν or ν; the relation between the frequency and the period T of a repeating event or oscillation is given by f = 1 T.
The SI derived unit of frequency is the hertz, named after the German physicist Heinrich Hertz. One hertz means. If a TV has a refresh rate of 1 hertz the TV's screen will change its picture once a second. A previous name for this unit was cycles per second; the SI unit for period is the second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz. As a matter of convenience and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are described by their frequency instead of period; these used conversions are listed below: Angular frequency denoted by the Greek letter ω, is defined as the rate of change of angular displacement, θ, or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument to the sine function: y = sin = sin = sin d θ d t = ω = 2 π f Angular frequency is measured in radians per second but, for discrete-time signals, can be expressed as radians per sampling interval, a dimensionless quantity.
Angular frequency is larger than regular frequency by a factor of 2π. Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.: y = sin = sin d θ d x = k Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has an inverse relationship to the wavelength, λ. In dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ. In the special case of electromagnetic waves moving through a vacuum v = c, where c is the speed of light in a vacuum, this expression becomes: f = c λ; when waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can done in the following ways, Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period dividing the count by the length of the time period.
For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz If the number of counts is not large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count; this is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T