1.
Transmission line
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This article covers two-conductor transmission line such as parallel line, coaxial cable, stripline, and microstrip. Ordinary electrical cables suffice to carry low frequency alternating current, such as power, which reverses direction 100 to 120 times per second. However, they cannot be used to carry currents in the frequency range, above about 30 kHz, because the energy tends to radiate off the cable as radio waves. Radio frequency currents tend to reflect from discontinuities in the cable such as connectors and joints. These reflections act as bottlenecks, preventing the signal power from reaching the destination, Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. Types of transmission line include parallel line, coaxial cable, and planar transmission lines such as stripline, the higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the length of the cable is longer than a significant fraction of the transmitted frequencys wavelength. At microwave frequencies and above, power losses in transmission lines become excessive, and waveguides are used instead, some sources define waveguides as a type of transmission line, however, this article will not include them. At even higher frequencies, in the terahertz, infrared and light range, waveguides in turn become lossy, mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin and Oliver Heaviside. In 1855 Lord Kelvin formulated a model of the current in a submarine cable. The model correctly predicted the performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885 Heaviside published the first papers that described his analysis of propagation in cables, in many electric circuits, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a time can be assumed to be the same at all points. Stated another way, the length of the wire is important when the signal frequency components with corresponding wavelengths comparable to or less than the length of the wire. A common rule of thumb is that the cable or wire should be treated as a line if the length is greater than 1/10 of the wavelength. If the transmission line is uniform along its length, then its behaviour is described by a single parameter called the characteristic impedance. This is the ratio of the voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a cable, about 100 ohms for a twisted pair of wires
2.
Phasor
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In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude, angular frequency, and initial phase are time-invariant. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, a common situation in electrical networks is the existence of multiple sinusoids all with the same frequency, but different amplitudes and phases. The only difference in their analytic representations is the complex amplitude, a linear combination of such functions can be factored into the product of a linear combination of phasors and the time/frequency dependent factor that they all have in common. The origin of the term phasor rightfully suggests that a somewhat similar to that possible for vectors is possible for phasors as well. The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century, however, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required. The function A ⋅ e i is the representation of A ⋅ cos . Figure 2 depicts it as a vector in a complex plane. It is sometimes convenient to refer to the function as a phasor. But the term usually implies just the static vector, A e i θ. An even more compact representation of a phasor is the angle notation, multiplication of the phasor A e i θ e i ω t by a complex constant, B e i ϕ, produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid, Re = Re = A B cos In electronics, B e i ϕ would represent an impedance, in particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage, but the product of two phasors would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid, the time derivative or integral of a phasor produces another phasor. For example, Re = Re = Re = Re = ω A ⋅ cos Therefore, in phasor representation, similarly, integrating a phasor corresponds to multiplication by 1 i ω = e − i π /2 ω. The time-dependent factor, e i ω t, is unaffected, when we solve a linear differential equation with phasor arithmetic, we are merely factoring e i ω t out of all terms of the equation, and reinserting it into the answer. In polar coordinate form, it is,11 +2 ⋅ e − i ϕ, Therefore, v C =11 +2 ⋅ V P cos The sum of multiple phasors produces another phasor. A key point is that A3 and θ3 do not depend on ω or t, the time and frequency dependence can be suppressed and re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In angle notation, the operation shown above is written, A1 ∠ θ1 + A2 ∠ θ2 = A3 ∠ θ3, another way to view addition is that two vectors with coordinates and are added vectorially to produce a resultant vector with coordinates
3.
Voltage
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Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential energy between two points per unit electric charge. The voltage between two points is equal to the work done per unit of charge against an electric field to move the test charge between two points. This is measured in units of volts, voltage can be caused by static electric fields, by electric current through a magnetic field, by time-varying magnetic fields, or some combination of these three. A voltmeter can be used to measure the voltage between two points in a system, often a reference potential such as the ground of the system is used as one of the points. A voltage may represent either a source of energy or lost, used, given two points in space, x A and x B, voltage is the difference in electric potential between those two points. Electric potential must be distinguished from electric energy by noting that the potential is a per-unit-charge quantity. Like mechanical potential energy, the zero of electric potential can be chosen at any point, so the difference in potential, i. e. the voltage, is the quantity which is physically meaningful. The voltage between point A to point B is equal to the work which would have to be done, per unit charge, against or by the electric field to move the charge from A to B. The voltage between the two ends of a path is the energy required to move a small electric charge along that path. Mathematically this is expressed as the integral of the electric field. In the general case, both an electric field and a dynamic electromagnetic field must be included in determining the voltage between two points. Historically this quantity has also called tension and pressure. Pressure is now obsolete but tension is used, for example within the phrase high tension which is commonly used in thermionic valve based electronics. Voltage is defined so that negatively charged objects are pulled towards higher voltages, therefore, the conventional current in a wire or resistor always flows from higher voltage to lower voltage. Current can flow from lower voltage to higher voltage, but only when a source of energy is present to push it against the electric field. This is the case within any electric power source, for example, inside a battery, chemical reactions provide the energy needed for ion current to flow from the negative to the positive terminal. The electric field is not the only factor determining charge flow in a material, the electric potential of a material is not even a well defined quantity, since it varies on the subatomic scale. A more convenient definition of voltage can be found instead in the concept of Fermi level, in this case the voltage between two bodies is the thermodynamic work required to move a unit of charge between them
4.
Electrical network
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An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, a resistive circuit is a circuit containing only resistors and ideal current and voltage sources. Analysis of resistive circuits is less complicated than analysis of circuits containing capacitors and inductors, if the sources are constant sources, the result is a DC circuit. A network that contains active components is known as an electronic circuit. Such networks are generally nonlinear and require more complex design and analysis tools, an active network is a network that contains an active source – either a voltage source or current source. A passive network is a network that does not contain an active source, a network is linear if its signals obey the principle of superposition, otherwise it is non-linear. Sources can be classified as independent sources and dependent sources Ideal Independent Source maintains same voltage or current regardless of the elements present in the circuit. Its value is either constant or sinusoidal, the strength of voltage or current is not changed by any variation in connected network. Dependent Sources depend upon a particular element of the circuit for delivering the power or voltage or current depending upon the type of source it is, a number of electrical laws apply to all electrical networks. These include, Kirchhoffs current law, The sum of all currents entering a node is equal to the sum of all currents leaving the node, Kirchhoffs voltage law, The directed sum of the electrical potential differences around a loop must be zero. Ohms law, The voltage across a resistor is equal to the product of the resistance, nortons theorem, Any network of voltage or current sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor. Thévenins theorem, Any network of voltage or current sources and resistors is electrically equivalent to a voltage source in series with a single resistor. Other more complex laws may be needed if the network contains nonlinear or reactive components, non-linear self-regenerative heterodyning systems can be approximated. Applying these laws results in a set of equations that can be solved either algebraically or numerically. To design any electrical circuit, either analog or digital, electrical engineers need to be able to predict the voltages, simple linear circuits can be analyzed by hand using complex number theory. In more complex cases the circuit may be analyzed with specialized programs or estimation techniques such as the piecewise-linear model. More complex circuits can be analyzed numerically with software such as SPICE or GNUCAP, once the steady state solution is found, the operating points of each element in the circuit are known
5.
Electrical load
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An electrical load is an electrical component or portion of a circuit that consumes electric power. This is opposed to a source, such as a battery or generator. In electric power circuits examples of loads are appliances and lights, the term may also refer to the power consumed by a circuit. The term is used broadly in electronics for a device connected to a signal source. If an electric circuit has a port, a pair of terminals that produces an electrical signal. For example, if a CD player is connected to an amplifier, the CD player is the source, load affects the performance of circuits with respect to output voltages or currents, such as in sensors, voltage sources, and amplifiers. Mains power outlets provide an example, they supply power at constant voltage. When a high-power appliance switches on, it reduces the load impedance. If the load impedance is not very much higher than the power supply impedance, in a domestic environment, switching on a heating appliance may cause incandescent lights to dim noticeably. When discussing the effect of load on a circuit, it is helpful to disregard the circuits actual design and consider only the Thévenin equivalent. The Thévenin equivalent of a circuit looks like this, With no load, all of V S falls across the output, the voltage is V S. However. We would like to ignore the details of the circuit, as we did for the power supply. This current places a voltage drop across R S, so the voltage at the terminal is no longer V S. This illustration uses simple resistances, but similar discussion can be applied in alternating current circuits using resistive, capacitive and inductive elements
6.
Reflections of signals on conducting lines
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This can happen, for instance, if two lengths of dissimilar transmission lines are joined together. This article is about signal reflections on electrically conducting lines, such lines are loosely referred to as copper lines, and indeed, in telecommunications are generally made from copper, but other metals are used, notably aluminium in power lines. Reflections cause several undesirable effects, including modifying frequency responses, causing overload power in transmitters, however, the reflection phenomenon can also be made use of in such devices as stubs and impedance transformers. The special cases of open circuit and short circuit lines are of relevance to stubs. Reflections cause standing waves to be set up on the line, conversely, standing waves are an indication that reflections are present. There is a relationship between the measures of reflection coefficient and standing wave ratio, there are several approaches to understanding reflections, but the relationship of reflections to the conservation laws is particularly enlightening. A simple example is a voltage, V u, applied to one end of a lossless line. It cannot have any effect until the step actually reaches that point, the signal cannot have any foreknowledge of what is at the end of the line and is only affected by the local characteristics of the line. However, if the line is of length l the step will arrive at the circuit at time t = l / κ. Essentially, this is Kirchhoffs current law in operation and this equal and opposite current is the reflected current, i r, and since i r = v r Z0 there must also be a reflected voltage, v r, to drive the reflected current down the line. This reflected voltage must exist by reason of conservation of energy, the source is supplying energy to the line at a rate of v i i i. None of this energy is dissipated in the line or its termination, the only available direction is back up the line. As the reflection proceeds back up the line the reflected voltage continues to add to the incident voltage, if the generator is matched to the line with an impedance of Z0 the step transient will be absorbed in the generator internal impedance and there will be no further reflections. This counter-intuitive doubling of voltage may become clearer if the voltages are considered when the line is so short that it can be ignored for the purposes of analysis. The equivalent circuit of a generator matched to a load Z0 to which it is delivering a voltage V can be represented as in figure 2. That is, the generator can be represented as an ideal voltage generator of twice the voltage it is to deliver, however, if the generator is left open circuit, a voltage of 2 V appears at the generator output terminals as in figure 3. The same situation pertains if a short transmission line is inserted between the generator and the open circuit. But after an interval, a reflected transient will return from the end of the line with the information on what the line is terminated with
7.
Input impedance
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The input impedance of an electrical network is the measure of the opposition to current flow, both static and dynamic, into the load network being connected that is external to the electrical source. The input admittance is a measure of the propensity to draw current. The source network is the portion of the network that transmits power, and so the voltage across and current through the input terminals would be identical to the original load network. The Thévenins equivalent circuit of the network uses the concept of input impedance to determine the impedance of the equivalent circuit. To calculate the impedance, short the input terminals together. In this case, Z i n ≫ Z o u t The input impedance of the stage is much larger than the output impedance of the source. In AC circuits carrying power, the due to the reactive component of the impedance can be significant. These losses manifest themselves in a phenomenon called phase imbalance, where the current is out of phase with the voltage, therefore, the product of the current and the voltage is less than what it would be if the current and voltage were in phase. With DC sources, reactive circuits have no impact, therefore power factor correction is not necessary. For a circuit modeled with a source, an output impedance, and an input impedance. In this scenario the reactive component of the input impedance cancels the reactive component of the impedance of the load. From the ideal sources perspective the circuit between the output and input impedance is purely resistive in nature, and there are no losses due to imbalance in the source or the load. Z i n = X − j Im The maximum power will be transferred when the resistance of the source is equal to the resistance of the load, when this occurs the circuit is said to be complex conjugate matched to the signals impedance. Note this only maximizes the transfer, not the efficiency of the circuit. When the power transfer is optimized the circuit runs at 50% efficiency. Also note that equation does not work in reverse, the optimal output impedance of the source is 0 regardless of the input impedance of the load. The formula for complex conjugate matched is Z i n = Z o u t ∗ = | Z o u t | e − Θ o u t j = Re − Im j. When there is no reactive component this equation simplifies to Z i n = Z o u t as the part of Z o u t is zero
8.
International System of Units
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The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version