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To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again: d 2 d R L 2 ( R S 2 / R L + 2 R S + R L ) = 2 R S 2 / R L 3 . {\displaystyle {\frac {d^{2}}{dR_{\mathrm {L} }^{2}}}\left({R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }}\right)={2R_... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 336 | 867 | null |
When the source resistance can be varied, power transferred to the load can be increased by reducing R S {\displaystyle R_{\textrm {S}}} . For example, a 100 Volt source with an R S {\displaystyle R_{\textrm {S}}} of 10 Ω {\displaystyle 10\,\Omega } will deliver 250 watts of power to a 10 Ω {\displaystyle 10\,\Omega } ... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 170 | 616 | null |
Section: In reactive circuits. The power transfer theorem also applies when the source and/or load are not purely resistive. A refinement of the maximum power theorem says that any reactive components of source and load should be of equal magnitude but opposite sign. (See below for a derivation.) This means that the so... | Wikipedia - Maximum power transfer theorem - In reactive circuits | 320 | 1,468 | null |
Section: In reactive circuits > Proof. In this diagram, AC power is being transferred from the source, with phasor magnitude of voltage | V S | {\displaystyle |V_{\text{S}}|} (positive peak voltage) and fixed source impedance Z S {\displaystyle Z_{\text{S}}} (S for source), to a load with impedance Z L {\displaystyle Z... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 188 | 632 | null |
{\displaystyle |I|={|V_{\text{S}}| \over |Z_{\text{S}}+Z_{\text{L}}|}.} The average power P L {\displaystyle P_{\text{L}}} dissipated in the load is the square of the current multiplied by the resistive portion (the real part) R L {\displaystyle R_{\text{L}}} of the load impedance Z L {\displaystyle Z_{\text{L}}} : P L... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 324 | 656 | null |
2}{|V_{\text{S}}|^{2}R_{\text{L}} \over (R_{\text{S}}+R_{\text{L}})^{2}+(X_{\text{S}}+X_{\text{L}})^{2}},\end{aligned}}} where R S {\displaystyle R_{\text{S}}} and R L {\displaystyle R_{\text{L}}} denote the resistances, that is the real parts, and X S {\displaystyle X_{\text{S}}} and X L {\displaystyle X_{\text{L}}} d... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 222 | 489 | null |
2}{|V_{\text{S}}|^{2}R_{\text{L}} \over (R_{\text{S}}+R_{\text{L}})^{2}+(X_{\text{S}}+X_{\text{L}})^{2}},\end{aligned}}} where R S {\displaystyle R_{\text{S}}} and R L {\displaystyle R_{\text{L}}} denote the resistances, that is the real parts, and X S {\displaystyle X_{\text{S}}} and X L {\displaystyle X_{\text{L}}} d... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 439 | 1,052 | null |
To determine, for a given source, the voltage V S {\displaystyle V_{\text{S}}} and the impedance Z S , {\displaystyle Z_{\text{S}},} the value of the load impedance Z L , {\displaystyle Z_{\text{L}},} for which this expression for the power yields a maximum, one first finds, for each fixed positive value of R L {\displ... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 242 | 662 | null |
Since reactances can be negative, this is achieved by adapting the load reactance to: X L = − X S . {\displaystyle X_{\text{L}}=-X_{\text{S}}.} This reduces the above equation to: P L = 1 2 | V S | 2 R L ( R S + R L ) 2 {\displaystyle P_{\text{L}}={\frac {1}{2}}{\frac {|V_{\text{S}}|^{2}R_{\text{L}}}{(R_{\text{S}}+R_{\... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 219 | 556 | null |
This problem has the same form as in the purely resistive case, and the maximizing condition therefore is R L = R S . {\displaystyle R_{\text{L}}=R_{\text{S}}.} The two maximizing conditions: R L = R S {\displaystyle R_{\text{L}}=R_{\text{S}}} X L = − X S {\displaystyle X_{\text{L}}=-X_{\text{S}}} describe the complex ... | Wikipedia - Maximum power transfer theorem - In reactive circuits > Proof | 192 | 493 | null |
Article: Mechanical, electrical, and plumbing. Mechanical, Electrical, and Plumbing (MEP) refers to the installation of services which provide a functional and comfortable space for the building occupants. In residential and commercial buildings, these elements are often designed by specialized MEP engineers. MEP's des... | Wikipedia - Mechanical, electrical, and plumbing - Summary | 206 | 1,114 | null |
Section: Components of MEP > Mechanical. The mechanical component of MEP is an important superset of HVAC services. Thus, it incorporates the control of environmental factors (psychrometrics), either for human comfort or for the operation of machines. Heating, cooling, ventilation and exhaustion are all key areas to co... | Wikipedia - Mechanical, electrical, and plumbing - Components of MEP > Mechanical | 343 | 1,775 | null |
Section: Components of MEP > Electrical > Alternating current. Virtually all modern buildings integrate some form of AC mains electricity for powering domestic and everyday appliances. Such systems typically run between 100 and 500 volts, however their classifications and specifications vary greatly by geographical are... | Wikipedia - Mechanical, electrical, and plumbing - Components of MEP > Electrical > Alternating current | 243 | 1,333 | null |
Section: Components of MEP > Electrical > Information technology. Advances in technology and the advent of computer networking have led to the emergence of a new facet of electrical systems incorporating data and telecommunications wiring. As of 2019, several derivative acronyms have been suggested for this area, inclu... | Wikipedia - Mechanical, electrical, and plumbing - Components of MEP > Electrical > Information technology | 240 | 1,274 | null |
Section: Components of MEP > Plumbing. Competent design of plumbing systems is necessary to prevent conflicts with other trades, and to avoid expensive rework or surplus supplies. The scope of standard residential plumbing usually covers mains pressure potable water, heated water (in conjunction with mechanical and/or ... | Wikipedia - Mechanical, electrical, and plumbing - Components of MEP > Plumbing | 189 | 1,009 | null |
Article: Mesh analysis. Mesh analysis (or the mesh current method) is a circuit analysis method for planar circuits; planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop c... | Wikipedia - Mesh analysis - Summary | 155 | 735 | null |
Section: Mesh currents and essential meshes. Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three. A mesh... | Wikipedia - Mesh analysis - Mesh currents and essential meshes | 233 | 1,179 | null |
Section: Setting up the equations. Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current. For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the electronic component multiplied by the m... | Wikipedia - Mesh analysis - Setting up the equations | 189 | 975 | null |
The following is the same circuit from above with the equations needed to solve for all the currents in the circuit. { Mesh 1: I 1 = I s Mesh 2: − V s + R 1 ( I 2 − I 1 ) + 1 s C ( I 2 − I 3 ) = 0 Mesh 3: 1 s C ( I 3 − I 2 ) + R 2 ( I 3 − I 1 ) + s L I 3 = 0 {\displaystyle {\begin{cases}{\text{Mesh 1: }}I_{1}=I_{s}\\{\... | Wikipedia - Mesh analysis - Setting up the equations | 280 | 609 | null |
Section: Special cases > Supermesh. A supermesh occurs when a current source is contained between two essential meshes. The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the... | Wikipedia - Mesh analysis - Special cases > Supermesh | 227 | 744 | null |
Article: Metastability (electronics). In electronics, metastability is the ability of a digital electronic system to persist for an unbounded time in an unstable equilibrium or metastable state. In digital logic circuits, a digital signal is required to be within certain voltage or current limits to represent a '0' or ... | Wikipedia - Metastability (electronics) - Summary | 318 | 1,445 | null |
Section: Example. A simple example of metastability can be found in an SR NOR latch, when both Set and Reset inputs are true (R=1 and S=1) and then both transition to false (R=0 and S=0) at about the same time. Both outputs Q and Q are initially held at 0 by the simultaneous Set and Reset inputs. After both Set and Res... | Wikipedia - Metastability (electronics) - Example | 173 | 722 | null |
Section: Synchronous circuits. Synchronous circuit design techniques make digital circuits that are resistant to the failure modes that can be caused by metastability. A clock domain is defined as a group of flip-flops with a common clock. Such architectures can form a circuit guaranteed free of metastability (below a ... | Wikipedia - Metastability (electronics) - Synchronous circuits | 275 | 1,272 | null |
Section: Failure modes. Although metastability is well understood and architectural techniques to control it are known, it persists as a failure mode in equipment. Serious computer and digital hardware bugs caused by metastability have a fascinating social history. Many engineers have refused to believe that a bistable... | Wikipedia - Metastability (electronics) - Failure modes | 315 | 1,764 | null |
Article: MEMS. MEMS (micro-electromechanical systems) is the technology of microscopic devices incorporating both electronic and moving parts. MEMS are made up of components between 1 and 100 micrometres in size (i.e., 0.001 to 0.1 mm), and MEMS devices generally range in size from 20 micrometres to a millimetre (i.e.,... | Wikipedia - MEMS - Summary | 310 | 1,427 | null |
Section: History. An early example of a MEMS device is the resonant-gate transistor, an adaptation of the MOSFET, developed by Robert A. Wickstrom for Harvey C. Nathanson in 1965. Another early example is the resonistor, an electromechanical monolithic resonator patented by Raymond J. Wilfinger between 1966 and 1971. D... | Wikipedia - MEMS - History | 335 | 1,248 | null |
Section: Materials. The fabrication of MEMS evolved from the process technology in semiconductor device fabrication, i.e. the basic techniques are deposition of material layers, patterning by photolithography and etching to produce the required shapes. Silicon Silicon is the material used to create most integrated circ... | Wikipedia - MEMS - Materials | 349 | 1,894 | null |
MEMS devices can be made from polymers by processes such as injection molding, embossing or stereolithography and are especially well suited to microfluidic applications such as disposable blood testing cartridges. Metals Metals can also be used to create MEMS elements. While metals do not have some of the advantages d... | Wikipedia - MEMS - Materials | 329 | 1,598 | null |
Section: Basic processes > Lithography. Lithography in a MEMS context is typically the transfer of a pattern into a photosensitive material by selective exposure to a radiation source such as light. A photosensitive material is a material that experiences a change in its physical properties when exposed to a radiation ... | Wikipedia - MEMS - Basic processes > Lithography | 333 | 1,620 | null |
It was developed for manufacturing integrated circuits, and is also used for creating nanotechnology architectures. The primary advantage of electron beam lithography is that it is one of the ways to beat the diffraction limit of light and make features in the nanometer range. This form of maskless lithography has foun... | Wikipedia - MEMS - Basic processes > Lithography | 344 | 1,735 | null |
Random pattern, single-ion track structures and an aimed pattern consisting of individual single tracks can be generated. X-ray lithography is a process used in the electronic industry to selectively remove parts of a thin film. It uses X-rays to transfer a geometric pattern from a mask to a light-sensitive chemical ph... | Wikipedia - MEMS - Basic processes > Lithography | 159 | 789 | null |
Section: Basic processes > Etching processes > Wet etching. Wet chemical etching consists of the selective removal of material by dipping a substrate into a solution that dissolves it. The chemical nature of this etching process provides good selectivity, which means the etching rate of the target material is considera... | Wikipedia - MEMS - Basic processes > Etching processes > Wet etching | 297 | 1,462 | null |
Some single crystal materials, such as silicon, will have different etching rates depending on the crystallographic orientation of the substrate. This is known as anisotropic etching and one of the most common examples is the etching of silicon in KOH (potassium hydroxide), where Si <111> planes etch approximately 100 ... | Wikipedia - MEMS - Basic processes > Etching processes > Wet etching | 334 | 1,320 | null |
Section: Basic processes > Etching processes > Dry etching. Xenon difluoride (XeF2) is a dry vapor phase isotropic etch for silicon originally applied for MEMS in 1995 at University of California, Los Angeles. Primarily used for releasing metal and dielectric structures by undercutting silicon, XeF2 has the advantage o... | Wikipedia - MEMS - Basic processes > Etching processes > Dry etching | 336 | 1,509 | null |
Plasma etching can be isotropic, i.e., exhibiting a lateral undercut rate on a patterned surface approximately the same as its downward etch rate, or can be anisotropic, i.e., exhibiting a smaller lateral undercut rate than its downward etch rate. Such anisotropy is maximized in deep reactive ion etching. The use of th... | Wikipedia - MEMS - Basic processes > Etching processes > Dry etching | 342 | 1,425 | null |
Deep reactive-ion etching (DRIE) modifies the RIE technique to produce deep, narrow features. In reactive-ion etching (RIE), the substrate is placed inside a reactor, and several gases are introduced. A plasma is struck in the gas mixture using an RF power source, which breaks the gas molecules into ions. The ions acce... | Wikipedia - MEMS - Basic processes > Etching processes > Dry etching | 346 | 1,662 | null |
The first variation consists of three distinct steps (the original Bosch process) while the second variation only consists of two steps. In the first variation, the etch cycle is as follows: (i) SF6 isotropic etch; (ii) C4F8 passivation; (iii) SF6 anisotropic etch for floor cleaning. In the 2nd variation, steps (i) and... | Wikipedia - MEMS - Basic processes > Etching processes > Dry etching | 313 | 1,378 | null |
Section: Manufacturing technologies. Bulk micromachining is the oldest paradigm of silicon-based MEMS. The whole thickness of a silicon wafer is used for building the micro-mechanical structures. Silicon is machined using various etching processes. Bulk micromachining has been essential in enabling high performance pre... | Wikipedia - MEMS - Manufacturing technologies | 312 | 1,574 | null |
Wafer bonding involves joining two or more substrates (usually having the same diameter) to one another to form a composite structure. There are several types of wafer bonding processes that are used in microsystems fabrication including: direct or fusion wafer bonding, wherein two or more wafers are bonded together th... | Wikipedia - MEMS - Manufacturing technologies | 335 | 1,665 | null |
A new etching technology, deep reactive-ion etching, has made it possible to combine good performance typical of bulk micromachining with comb structures and in-plane operation typical of surface micromachining. While it is common in surface micromachining to have structural layer thickness in the range of 2 μm, in HAR... | Wikipedia - MEMS - Manufacturing technologies | 185 | 813 | null |
Section: Applications. Some common commercial applications of MEMS include: Inkjet printers, which use piezoelectrics or thermal bubble ejection to deposit ink on paper. Accelerometers in modern cars for a large number of purposes including airbag deployment and electronic stability control. Inertial measurement units ... | Wikipedia - MEMS - Applications | 341 | 1,556 | null |
Precision temperature-compensated resonators in real-time clocks. Silicon pressure sensors e.g., car tire pressure sensors, and disposable blood pressure sensors. Displays e.g., the digital micromirror device (DMD) chip in a projector based on DLP technology, which has a surface with several hundred thousand micromirro... | Wikipedia - MEMS - Applications | 315 | 1,470 | null |
Section: Industry structure. The global market for micro-electromechanical systems, which includes products such as automobile airbag systems, display systems and inkjet cartridges totaled $40 billion in 2006 according to Global MEMS/Microsystems Markets and Opportunities, a research report from SEMI and Yole Developme... | Wikipedia - MEMS - Industry structure | 235 | 1,273 | null |
Article: Microwave engineering. Microwave engineering pertains to the study and design of microwave circuits, components, and systems. Fundamental principles are applied to analysis, design and measurement techniques in this field. The short wavelengths involved distinguish this discipline from electronic engineering. ... | Wikipedia - Microwave engineering - Summary | 192 | 1,075 | null |
Section: The microwave domain. Microwave is a term used to identify electromagnetic waves above 103 megahertz (1 Gigahertz) up to 300 Gigahertz because of the short physical wavelengths of these frequencies. Short wavelength energy offers distinct advantages in many applications. For instance, sufficient directivity ca... | Wikipedia - Microwave engineering - The microwave domain | 321 | 1,750 | null |
Section: Relevance > Education. Many colleges and universities offer microwave engineering. A few examples follow. The University of Massachusetts Amherst provides research and educational programs in microwave remote sensing, antenna design and communications systems. Courses and project work are offered leading towar... | Wikipedia - Microwave engineering - Relevance > Education | 208 | 1,317 | null |
Article: Miller effect. In electronics, the Miller effect (named after its discoverer John Milton Miller) accounts for the increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of the effect of capacitance between the amplifier's input and output terminals, and is given by ... | Wikipedia - Miller effect - Summary | 247 | 1,023 | null |
Section: Derivation. Consider a circuit of an ideal inverting voltage amplifier of gain − A v {\displaystyle -A_{v}} with an impedance Z {\displaystyle Z} connected between its input and output nodes. The output voltage is therefore V o = − A v V i {\displaystyle V_{o}=-A_{v}V_{i}} . Assuming that the amplifier input d... | Wikipedia - Miller effect - Derivation | 263 | 673 | null |
The input impedance of the circuit is Z i n = V i I i = Z 1 + A v {\displaystyle Z_{in}={\frac {V_{i}}{I_{i}}}={\frac {Z}{1+A_{v}}}} . In the Laplace domain (where s {\displaystyle s} represents complex frequency), if Z {\displaystyle Z} consists of just a capacitor forming a complex impedance Z = 1 s C {\displaystyle ... | Wikipedia - Miller effect - Derivation | 332 | 807 | null |
Section: Effects. As most amplifiers are inverting ( A v {\displaystyle A_{v}} as defined above is positive), the effective capacitance at their inputs is increased due to the Miller effect. This can reduce the bandwidth of the amplifier, restricting its range of operation to lower frequencies. The tiny junction and st... | Wikipedia - Miller effect - Effects | 349 | 1,560 | null |
Section: Effects > Mitigation. The Miller effect may be undesired in many cases, and approaches may be sought to lower its impact. Several such techniques are used in the design of amplifiers. A current buffer stage may be added at the output to lower the gain A v {\displaystyle A_{v}} between the input and output term... | Wikipedia - Miller effect - Effects > Mitigation | 346 | 1,674 | null |
Section: Impact on frequency response. Figure 2A shows an example of Figure 1 where the impedance coupling the input to the output is the coupling capacitor C C {\displaystyle C_{C}} . Thévenin voltage source V A {\displaystyle V_{A}} drives the circuit with Thévenin resistance R A {\displaystyle R_{A}} . The output im... | Wikipedia - Miller effect - Impact on frequency response | 311 | 1,131 | null |
Therefore, the driver sees exactly the same loading in both circuits. On the output side, the same current from the output as drawn from the coupling capacitor in Figure 2A is instead drawn from a capacitor C M o {\displaystyle C_{Mo}} equal to: C M o = ( 1 + 1 A v ) C C . {\displaystyle C_{Mo}=(1+{\frac {1}{A_{v}}})C_... | Wikipedia - Miller effect - Impact on frequency response | 277 | 809 | null |
In this example, this transformation is equivalent to setting the currents equal, that is j ω C C ( V i − V O ) = j ω C M V i , {\displaystyle \ j\omega C_{C}(V_{i}-V_{O})=j\omega C_{M}V_{i},} or, rearranging this equation C M = C C ( 1 − V o V i ) = C C ( 1 + A v ) . {\displaystyle C_{M}=C_{C}\left(1-{\frac {V_{o}}{V_... | Wikipedia - Miller effect - Impact on frequency response | 345 | 999 | null |
However, when C C {\displaystyle C_{C}} is not zero, Figure 2B shows the large Miller capacitance appears at the input of the circuit. The voltage output of the circuit now becomes V o = − A v V i = − A v V A 1 + j ω C M R A , {\displaystyle V_{o}=-A_{v}V_{i}=-A_{v}{\frac {V_{A}}{1+j\omega C_{M}R_{A}}},} and rolls off ... | Wikipedia - Miller effect - Impact on frequency response | 350 | 1,403 | null |
Section: Impact on frequency response > Miller approximation. This example also assumes Av is frequency independent, but more generally there is frequency dependence of the amplifier contained implicitly in Av. Such frequency dependence of Av also makes the Miller capacitance frequency dependent, so interpretation of C... | Wikipedia - Miller effect - Impact on frequency response > Miller approximation | 163 | 859 | null |
Section: Explanation. Let e k {\displaystyle e_{k}} be the generators' voltages. Let R k {\displaystyle R_{k}} be the resistances on the branches with voltage generators e k {\displaystyle e_{k}} . Then Millman states that the voltage at the ends of the circuit is given by: v = ∑ e k R k ∑ 1 R k . {\displaystyle v={\fr... | Wikipedia - Millman's theorem - Explanation | 268 | 953 | null |
Section: Equivalent circuits. A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing physical components with other notional components that have the same effect. A particular technique might directly reduce the number of components, for ins... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits | 330 | 1,566 | null |
Section: Equivalent circuits > Impedances in series and in parallel. Some two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impedances in series or impedances in parallel. Impedances in series: Z e q = Z 1 + Z 2 + ⋯ + Z n . {\displaystyle Z_{\mathrm {eq} }=... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > Impedances in series and in parallel | 293 | 685 | null |
Section: Equivalent circuits > Delta-wye transformation. A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An n-terminal network can, at best, be reduced to n impedances (at worst ( n 2 ) {\displaystyle {\tbinom {n}{2}}} ). For a three terminal network, the... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > Delta-wye transformation | 234 | 1,064 | null |
Section: Equivalent circuits > Delta-wye transformation > Delta-to-star transformation equations. R a = R a c R a b R a c + R a b + R b c R b = R a b R b c R a c + R a b + R b c R c = R b c R a c R a c + R a b + R b c {\displaystyle {\begin{aligned}R_{a}&={\frac {R_{\mathrm {ac} }R_{\mathrm {ab} }}{R_{\mathrm {ac} }+R_... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > Delta-wye transformation > Delta-to-star transformation equations | 304 | 586 | null |
Section: Equivalent circuits > Delta-wye transformation > Star-to-delta transformation equations. R a c = R a R b + R b R c + R c R a R b R a b = R a R b + R b R c + R c R a R c R b c = R a R b + R b R c + R c R a R a {\displaystyle {\begin{aligned}R_{\mathrm {ac} }&={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{b}}}\\R... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > Delta-wye transformation > Star-to-delta transformation equations | 274 | 472 | null |
The resistance between any two nodes x, y is given by: R x y = R x R y ∑ i = 1 N 1 R i {\displaystyle R_{\mathrm {xy} }=R_{x}R_{y}\sum _{i=1}^{N}{\frac {1}{R_{i}}}} For a star-to-delta (N = 3) this reduces to: R a b = R a R b ( 1 R a + 1 R b + 1 R c ) = R a R b ( R a R b + R a R c + R b R c ) R a R b R c = R a R b + R ... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > General form of network node elimination | 326 | 551 | null |
{R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}}\end{aligned}}} For a series reduction (N = 2) this reduces to: R a b = R a R b ( 1 R a + 1 R b ) = R a R b ( R a + R b ) R a R b = R a + R b {\displaystyle R_{\mathrm {ab} }=R_{a}R_{b}\left({\frac {1}{R}}_{a}+{\frac {1}{R}}_{b}\right)={\frac {R_{a}R_{b}(R_{a}+R_{b})}{R_{a}R_{b... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > General form of network node elimination | 265 | 471 | null |
Section: Equivalent circuits > Source transformation. A generator with an internal impedance (i.e. non-ideal generator) can be represented as either an ideal voltage generator or an ideal current generator plus the impedance. These two forms are equivalent and the transformations are given below. If the two networks ar... | Wikipedia - Network analysis (electrical circuits) - Equivalent circuits > Source transformation | 214 | 828 | null |
Section: Simple networks > Current division of parallel components. Consider n admittances that are connected in parallel. The current I i {\displaystyle I_{i}} through any admittance Y i {\displaystyle Y_{i}} is I i = Y i V = ( Y i Y 1 + Y 2 + ⋯ + Y n ) I {\displaystyle I_{i}=Y_{i}V=\left({\frac {Y_{i}}{Y_{1}+Y_{2}+\c... | Wikipedia - Network analysis (electrical circuits) - Simple networks > Current division of parallel components | 165 | 399 | null |
Section: Nodal analysis. Nodal analysis uses the concept of a node voltage and considers the node voltages to be the unknown variables.: 2-8 - 2-9 For all nodes, except a chosen reference node, the node voltage is defined as the voltage drop from the node to the reference node. Therefore, there are N-1 node voltages fo... | Wikipedia - Network analysis (electrical circuits) - Nodal analysis | 288 | 1,313 | null |
Section: Superposition. In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed and either short-circuited in the case of voltage generators or open-circuited in the case of current generators. The total current through or the total voltage ... | Wikipedia - Network analysis (electrical circuits) - Superposition | 199 | 1,038 | null |
Section: Choice of method. Choice of method: 112–113 is to some extent a matter of taste. If the network is particularly simple or only a specific current or voltage is required then ad-hoc application of some simple equivalent circuits may yield the answer without recourse to the more systematic methods. Nodal analysi... | Wikipedia - Network analysis (electrical circuits) - Choice of method | 258 | 1,364 | null |
Section: Transfer function. A transfer function expresses the relationship between an input and an output of a network. For resistive networks, this will always be a simple real number or an expression which boils down to a real number. Resistive networks are represented by a system of simultaneous algebraic equations.... | Wikipedia - Network analysis (electrical circuits) - Transfer function | 215 | 1,142 | null |
Section: Transfer function > Two port network transfer function. Transfer functions, in general, in control theory are given the symbol H(s). Most commonly in electronics, transfer function is defined as the ratio of output voltage to input voltage and given the symbol A(s), or more commonly (because analysis is invari... | Wikipedia - Network analysis (electrical circuits) - Transfer function > Two port network transfer function | 261 | 975 | null |
Section: Transfer function > Two port network transfer function > Two port parameters. The concept of a two-port network can be useful in network analysis as a black box approach to analysis. The behaviour of the two-port network in a larger network can be entirely characterised without necessarily stating anything abo... | Wikipedia - Network analysis (electrical circuits) - Transfer function > Two port network transfer function > Two port parameters | 176 | 872 | null |
There are many others (see the main article for a full listing), one of these expresses all four parameters as impedances. It is usual to express the four parameters as a matrix; [ V 1 V 0 ] = [ z ( j ω ) 11 z ( j ω ) 12 z ( j ω ) 21 z ( j ω ) 22 ] [ I 1 I 0 ] {\displaystyle {\begin{bmatrix}V_{1}\\V_{0}\end{bmatrix}}={... | Wikipedia - Network analysis (electrical circuits) - Transfer function > Two port network transfer function > Two port parameters | 319 | 968 | null |
Section: Transfer function > Two port network transfer function > Distributed components. Where a network is composed of discrete components, analysis using two-port networks is a matter of choice, not essential. The network can always alternatively be analysed in terms of its individual component transfer functions. H... | Wikipedia - Network analysis (electrical circuits) - Transfer function > Two port network transfer function > Distributed components | 170 | 893 | null |
Section: Time-based network analysis with simulation. Most analysis methods calculate the voltage and current values for static networks, which are circuits consisting of memoryless components only but have difficulties with complex dynamic networks. In general, the equations that describe the behaviour of a dynamic ci... | Wikipedia - Network analysis (electrical circuits) - Time-based network analysis with simulation | 347 | 1,616 | null |
: 206-207 Since finding numerical results for the infinite number of time points from t0 to tf is not possible, this time period is discretized into discrete time instances, and the numerical solution is found for every instance. The time between the time instances is called the time step and can be fixed throughout th... | Wikipedia - Network analysis (electrical circuits) - Time-based network analysis with simulation | 256 | 1,050 | null |
Section: Non-linear networks. Most electronic designs are, in reality, non-linear. There are very few that do not include some semiconductor devices. These are invariably non-linear, the transfer function of an ideal semiconductor p-n junction is given by the very non-linear relationship; where; i and v are the instant... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks | 175 | 887 | null |
Section: Non-linear networks > Constitutive equations. The diode equation above is an example of an element constitutive equation of the general form, f ( v , i ) = 0 {\displaystyle f(v,i)=0} This can be thought of as a non-linear resistor. The corresponding constitutive equations for non-linear inductors and capacitor... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Constitutive equations | 153 | 521 | null |
Section: Non-linear networks > Existence, uniqueness and stability. An important consideration in non-linear analysis is the question of uniqueness. For a network composed of linear components there will always be one, and only one, unique solution for a given set of boundary conditions. This is not always the case in ... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Existence, uniqueness and stability | 225 | 1,118 | null |
Section: Non-linear networks > Methods > Boolean analysis of switching networks. A switching device is one where the non-linearity is utilised to produce two opposite states. CMOS devices in digital circuits, for instance, have their output connected to either the positive or the negative supply rail and are never foun... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Methods > Boolean analysis of switching networks | 336 | 1,547 | null |
Section: Non-linear networks > Methods > Graphical method of dc analysis. In a great many circuit designs, the dc bias is fed to a non-linear component via a resistor (or possibly a network of resistors). Since resistors are linear components, it is particularly easy to determine the quiescent operating point of the no... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Methods > Graphical method of dc analysis | 335 | 1,696 | null |
Section: Non-linear networks > Methods > Small signal equivalent circuit. This method can be used where the deviation of the input and output signals in a network stay within a substantially linear portion of the non-linear devices transfer function, or else are so small that the curve of the transfer function can be c... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Methods > Small signal equivalent circuit | 349 | 1,732 | null |
The most important parameter for transistors is usually the forward current gain, h21, in the common emitter configuration. This is designated hfe on data sheets. The small signal equivalent circuit in terms of two-port parameters leads to the concept of dependent generators. That is, the value of a voltage or current ... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Methods > Small signal equivalent circuit | 150 | 764 | null |
Section: Non-linear networks > Methods > Piecewise linear method. In this method, the transfer function of the non-linear device is broken up into regions. Each of these regions is approximated by a straight line. Thus, the transfer function will be linear up to a particular point where there will be a discontinuity. P... | Wikipedia - Network analysis (electrical circuits) - Non-linear networks > Methods > Piecewise linear method | 328 | 1,563 | null |
Article: Network analyzer (AC power). From 1929 to the late 1960s, large alternating current power systems were modelled and studied on AC network analyzers (also called alternating current network calculators or AC calculating boards) or transient network analyzers. These special-purpose analog computers were an outgr... | Wikipedia - Network analyzer (AC power) - Summary | 199 | 1,084 | null |
Section: Calculating methods. As AC power systems became larger at the start of the 20th century, with more interconnected devices, the problem of calculating the expected behavior of the systems became more difficult. Manual methods were only practical for systems of a few sources and nodes. The complexity of practica... | Wikipedia - Network analyzer (AC power) - Calculating methods | 340 | 1,819 | null |
Laboratory investigations of the stability of multiple-machine systems were constrained by the use of direct-operated indicating instruments (voltmeters, ammeters, and wattmeters). To ensure that the instruments negligibly loaded the model system, the machine power level used was substantial. Some workers in the 1920s ... | Wikipedia - Network analyzer (AC power) - Calculating methods | 178 | 900 | null |
Section: Scale model. A network analyzer system was essentially a scale model of the electrical properties of a specific power system. Generators, transmission lines, and loads were represented by miniature electrical components with scale values in proportion to the modeled system. Model components were interconnected... | Wikipedia - Network analyzer (AC power) - Scale model | 342 | 1,801 | null |
The model base quantities varied by manufacturer and date of design; as amplified indicating instruments became more common, lower base quantities were feasible. Model voltages and currents started off around 200 volts and 0.5 amperes in the MIT analyzer, which still allowed directly driven (but especially sensitive) i... | Wikipedia - Network analyzer (AC power) - Scale model | 327 | 1,684 | null |
The use of network analyzers allowed quick solutions to difficult calculation problems, and allowed problems to be analyzed that would otherwise be uneconomic to compute using manual calculations. Although expensive to build and operate, network analyzers often repaid their costs in reduced calculation time and expedit... | Wikipedia - Network analyzer (AC power) - Scale model | 277 | 1,447 | null |
Section: The MIT network analyzer. The network analyzer installed at Massachusetts Institute of Technology (MIT) grew out of a 1924 thesis project by Hugh H. Spencer and Harold Locke Hazen, investigating a power system modelling concept proposed by Vannevar Bush. Instead of miniature rotating machines, each generator w... | Wikipedia - Network analyzer (AC power) - The MIT network analyzer | 347 | 1,785 | null |
American Gas and Electric Company, the Tennessee Valley Authority, and many other organizations studied problems on the MIT analyzer in its first decade of operation. In 1940 the system was moved and expanded to handle more complex systems. By 1953 the MIT analyzer was beginning to fall behind the state of the art. Dig... | Wikipedia - Network analyzer (AC power) - The MIT network analyzer | 155 | 772 | null |
Section: Commercial manufacturers. By 1947, fourteen network analyzers had been built at a total cost of about two million US dollars. General Electric built two full-scale network analyzers for its own work and for services to its clients. Westinghouse built systems for their internal use and provided more than 20 ana... | Wikipedia - Network analyzer (AC power) - Commercial manufacturers | 342 | 1,710 | null |
Section: Other applications > Anacom. The Westinghouse Anacom was an AC-energized electrical analog computer system used extensively for problems in mechanical design, structural elements, lubrication oil flow, and various transient problems including those due to lightning surges in electric power transmission systems... | Wikipedia - Network analyzer (AC power) - Other applications > Anacom | 179 | 927 | null |
Section: Decline and obsolescence. Even during the Depression and the Second World War, many network analyzers were constructed because of their great value in solving calculations related to electric power transmission. By the mid 1950s, about thirty analyzers were available in the United States, representing an overs... | Wikipedia - Network analyzer (AC power) - Decline and obsolescence | 345 | 1,930 | null |
Section: Overview. Network analyzers are used mostly at high frequencies; operating frequencies can range from 1 Hz to 1.5 THz. Special types of network analyzers can also cover lower frequency ranges down to 1 Hz. These network analyzers can be used, for example, for the stability analysis of open loops or for the mea... | Wikipedia - Network analyzer (electrical) - Overview | 318 | 1,503 | null |
Section: Architecture > Test set. The test set takes the signal generator output and routes it to the device under test, and it routes the signal to be measured to the receivers. It often splits off a reference channel for the incident wave. In a SNA, the reference channel may go to a diode detector (receiver) whose ou... | Wikipedia - Network analyzer (electrical) - Architecture > Test set | 163 | 764 | null |
Section: Architecture > Receiver. The receivers make the measurements. A network analyzer will have one or more receivers connected to its test ports. The reference test port is usually labeled R, and the primary test ports are A, B, C, ... Some analyzers will dedicate a separate receiver to each test port, but others ... | Wikipedia - Network analyzer (electrical) - Architecture > Receiver | 313 | 1,549 | null |
Section: S-parameter measurement with vector network analyzer. A VNA is a test system that enables the RF performance of radio frequency and microwave devices to be characterised in terms of network scattering parameters, or S parameters. The diagram shows the essential parts of a typical 2-port vector network analyzer... | Wikipedia - Network analyzer (electrical) - S-parameter measurement with vector network analyzer | 343 | 1,439 | null |
The third port of DC1 couples off the power reflected from P1 via A1 and PC1, then feeding it to test receiver 1 (RX TEST1). Similarly, signals leaving P2 pass via A2, PC2 and DC2 to RX TEST2. RX REF1, RX TEST1, RX REF2 and RXTEST2 are known as coherent receivers as they share the same reference oscillator, and they ar... | Wikipedia - Network analyzer (electrical) - S-parameter measurement with vector network analyzer | 277 | 1,135 | null |
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