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4,701 | Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of n-th roots. |
4,702 | If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define A1/2 = B. In general matrices may have multiple square roots or even an infinitude of them. For example, the 2 × 2 identity matrix has an infinity of square roots, though o... |
4,703 | Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2 − v2 = is proved using the commutativity of multiplication. If u and v are square roots of the same element, then u2 − v2 = 0. Because there are no zero divisors this implies u = v or u + v = 0, where the latt... |
4,704 | In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that −u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either... |
4,705 | Given an odd prime number p, let q = pe for some positive integer e. A non-zero element of the field Fq with q elements is a quadratic residue if it has a square root in Fq. Otherwise, it is a quadratic non-residue. There are /2 quadratic residues and /2 quadratic non-residues; zero is not counted in either class. The ... |
4,706 | The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid. |
4,707 | The MOSFET is a type of insulated-gate field-effect transistor that is fabricated by the controlled oxidation of a semiconductor, typically silicon. The voltage of the covered gate determines the electrical conductivity of the device; this ability to change conductivity with the amount of applied voltage can be used ... |
4,708 | The MOSFET is the basic building block of most modern electronics, and the most frequently manufactured device in history, with an estimated total of 13 sextillion MOSFETs manufactured between 1960 and 2018. It is the most common semiconductor device in digital and analog circuits, and the most common power device. I... |
4,709 | MOSFETs in integrated circuits are the primary elements of computer processors, semiconductor memory, image sensors, and most other types of integrated circuits. Discrete MOSFET devices are widely used in applications such as switch mode power supplies, variable-frequency drives, and other power electronics application... |
4,710 | The MOSFET is the most widely used type of transistor and the most critical device component in integrated circuit chips. Planar process, developed by Jean Hoerni at Fairchild Semiconductor in early 1959, was critical to the invention of the monolithic integrated circuit chip by Robert Noyce later in 1959. The same y... |
4,711 | Atalla realised that the main advantage of a MOS transistor was its ease of fabrication, particularly suiting it for use in the recently invented integrated circuits. In contrast to bipolar transistors which required a number of steps for the p–n junction isolation of transistors on a chip, MOSFETs required no such ste... |
4,712 | The earliest experimental MOS IC to be demonstrated was a 16-transistor chip built by Fred Heiman and Steven Hofstein at RCA in 1962. General Microelectronics later introduced the first commercial MOS integrated circuits in 1964, consisting of 120 p-channel transistors. It was a 20-bit shift register, developed by Robe... |
4,713 | There are various different types of MOS IC chips, which include the following. |
4,714 | With its high scalability, and much lower power consumption and higher density than bipolar junction transistors, the MOSFET made it possible to build high-density IC chips. By 1964, MOS chips had reached higher transistor density and lower manufacturing costs than bipolar chips. MOS chips further increased in complexi... |
4,715 | The MOSFET is the basis of every microprocessor, and was responsible for the invention of the microprocessor. The origins of both the microprocessor and the microcontroller can be traced back to the invention and development of MOS technology. The application of MOS LSI chips to computing was the basis for the first mi... |
4,716 | The earliest microprocessors were all MOS chips, built with MOS LSI circuits. The first multi-chip microprocessors, the Four-Phase Systems AL1 in 1969 and the Garrett AiResearch MP944 in 1970, were developed with multiple MOS LSI chips. The first commercial single-chip microprocessor, the Intel 4004, was developed by F... |
4,717 | Complementary metal–oxide–semiconductor logic was developed by Chih-Tang Sah and Frank Wanlass at Fairchild Semiconductor in 1963. CMOS had lower power consumption, but was initially slower than NMOS, which was more widely used for computers in the 1970s. In 1978, Hitachi introduced the twin-well CMOS process, which a... |
4,718 | The growth of digital technologies like the microprocessor has provided the motivation to advance MOSFET technology faster than any other type of silicon-based transistor. A big advantage of MOSFETs for digital switching is that the oxide layer between the gate and the channel prevents DC current from flowing through t... |
4,719 | The MOSFET's advantages in digital circuits do not translate into supremacy in all analog circuits. The two types of circuit draw upon different features of transistor behavior. Digital circuits switch, spending most of their time either fully on or fully off. The transition from one to the other is only of concern wit... |
4,720 | Nevertheless, MOSFETs are widely used in many types of analog circuits because of their own advantages . The characteristics and performance of many analog circuits can be scaled up or down by changing the sizes of the MOSFETs used. By comparison, in bipolar transistors the size of the device does not significantly af... |
4,721 | Some ICs combine analog and digital MOSFET circuitry on a single mixed-signal integrated circuit, making the needed board space even smaller. This creates a need to isolate the analog circuits from the digital circuits on a chip level, leading to the use of isolation rings and silicon on insulator . Since MOSFETs requi... |
4,722 | In the late 1980s, Asad Abidi pioneered RF CMOS technology, which uses MOS VLSI circuits, while working at UCLA. This changed the way in which RF circuits were designed, away from discrete bipolar transistors and towards CMOS integrated circuits. As of 2008, the radio transceivers in all wireless networking devices and... |
4,723 | MOSFET analog switches use the MOSFET to pass analog signals when on, and as a high impedance when off. Signals flow in both directions across a MOSFET switch. In this application, the drain and source of a MOSFET exchange places depending on the relative voltages of the source/drain electrodes. The source is the more ... |
4,724 | This analog switch uses a four-terminal simple MOSFET of either P or N type. |
4,725 | In the case of an n-type switch, the body is connected to the most negative supply and the gate is used as the switch control. Whenever the gate voltage exceeds the source voltage by at least a threshold voltage, the MOSFET conducts. The higher the voltage, the more the MOSFET can conduct. An N-MOS switch passes all v... |
4,726 | In the case of a P-MOS, the body is connected to the most positive voltage, and the gate is brought to a lower potential to turn the switch on. The P-MOS switch passes all voltages higher than Vgate − Vtp . |
4,727 | This "complementary" or CMOS type of switch uses one P-MOS and one N-MOS FET to counteract the limitations of the single-type switch. The FETs have their drains and sources connected in parallel, the body of the P-MOS is connected to the high potential and the body of the N-MOS is connected to the low potential . To t... |
4,728 | The voltage limits for this switch are the gate–source, gate–drain and source–drain voltage limits for both FETs. Also, the P-MOS is typically two to three times wider than the N-MOS, so the switch will be balanced for speed in the two directions. |
4,729 | Tri-state circuitry sometimes incorporates a CMOS MOSFET switch on its output to provide for a low-ohmic, full-range output when on, and a high-ohmic, mid-level signal when off. |
4,730 | The advent of the MOSFET enabled the practical use of MOS transistors as memory cell storage elements, a function previously served by magnetic cores in computer memory. The first modern computer memory was introduced in 1965, when John Schmidt at Fairchild Semiconductor designed the first MOS semiconductor memory, a 6... |
4,731 | MOS technology is the basis for DRAM . In 1966, Dr. Robert H. Dennard at the IBM Thomas J. Watson Research Center was working on MOS memory. While examining the characteristics of MOS technology, he found it was capable of building capacitors, and that storing a charge or no charge on the MOS capacitor could represent ... |
4,732 | Frank Wanlass, while studying MOSFET structures in 1963, noted the movement of charge through oxide onto a gate. While he did not pursue it, this idea would later become the basis for EPROM technology. In 1967, Dawon Kahng and Simon Sze proposed that floating-gate memory cells, consisting of floating-gate MOSFETs , co... |
4,733 | There are various different types of MOS memory. The following list includes various different MOS memory types. |
4,734 | A number of MOSFET sensors have been developed, for measuring physical, chemical, biological and environmental parameters. The earliest MOSFET sensors include the open-gate FET introduced by Johannessen in 1970, the ion-sensitive field-effect transistor invented by Piet Bergveld in 1970, the adsorption FET patented ... |
4,735 | By the mid-1980s, numerous other MOSFET sensors had been developed, including the gas sensor FET , surface accessible FET , charge flow transistor , pressure sensor FET , chemical field-effect transistor , reference ISFET , biosensor FET , enzyme-modified FET and immunologically modified FET . By the early 2000s, BioF... |
4,736 | The two main types of image sensors used in digital imaging technology are the charge-coupled device and the active-pixel sensor . Both CCD and CMOS sensors are based on MOS technology, with the CCD based on MOS capacitors and the CMOS sensor based on MOS transistors. |
4,737 | MOS technology is the basis for modern image sensors, including the charge-coupled device and the CMOS active-pixel sensor , used in digital imaging and digital cameras. Willard Boyle and George E. Smith developed the CCD in 1969. While researching the MOS process, they realized that an electric charge was the analogy... |
4,738 | The MOS active-pixel sensor was developed by Tsutomu Nakamura at Olympus in 1985. The CMOS active-pixel sensor was later developed by Eric Fossum and his team at NASA's Jet Propulsion Laboratory in the early 1990s. |
4,739 | MOS image sensors are widely used in optical mouse technology. The first optical mouse, invented by Richard F. Lyon at Xerox in 1980, used a 5 µm NMOS sensor chip. Since the first commercial optical mouse, the IntelliMouse introduced in 1999, most optical mouse devices use CMOS sensors. |
4,740 | MOS sensors, also known as MOSFET sensors, are widely used to measure physical, chemical, biological and environmental parameters. The ion-sensitive field-effect transistor , for example, is widely used in biomedical applications. |
4,741 | MOSFETs are also widely used in microelectromechanical systems , as silicon MOSFETs could interact and communicate with the surroundings and process things such as chemicals, motions and light. An early example of a MEMS device is the resonant-gate transistor, an adaptation of the MOSFET, developed by Harvey C. Nathans... |
4,742 | Common applications of other MOS sensors include the following. |
4,743 | The power MOSFET, which is commonly used in power electronics, was developed in the early 1970s. The power MOSFET enables low gate drive power, fast switching speed, and advanced paralleling capability. |
4,744 | The power MOSFET is the most widely used power device in the world. Advantages over bipolar junction transistors in power electronics include MOSFETs not requiring a continuous flow of drive current to remain in the ON state, offering higher switching speeds, lower switching power losses, lower on-resistances, and redu... |
4,745 | Switching power supplies are the most common applications for power MOSFETs. They are also widely used for MOS RF power amplifiers, which enabled the transition of mobile networks from analog to digital in the 1990s. This led to the wide proliferation of wireless mobile networks, which revolutionised telecommunication ... |
4,746 | LDMOS, a power MOSFET with lateral structure, is commonly used in high-end audio amplifiers and high-power PA systems. Their advantage is a better behaviour in the saturated region than the vertical MOSFETs. Vertical MOSFETs are designed for switching applications. |
4,747 | Power MOSFETs, including DMOS, LDMOS and VMOS devices, are commonly used for a wide range of other applications, which include the following. |
4,748 | RF DMOS, also known as RF power MOSFET, is a type of DMOS power transistor designed for radio-frequency applications. It is used in various radio and RF applications, which include the following. |
4,749 | MOSFETs are fundamental to the consumer electronics industry. According to Colinge, numerous consumer electronics would not exist without the MOSFET, such as digital wristwatches, pocket calculators, and video games, for example. |
4,750 | MOSFETs are commonly used for a wide range of consumer electronics, which include the following devices listed. Computers or telecommunication devices are not included here, but are listed separately in the Information and communications technology section below. |
4,751 | One of the earliest influential consumer electronic products enabled by MOS LSI circuits was the electronic pocket calculator, as MOS LSI technology enabled large amounts of computational capability in small packages. In 1965, the Victor 3900 desktop calculator was the first MOS LSI calculator, with 29 MOS LSI chips. I... |
4,752 | MOSFETs are commonly used for a wide range of audio-visual media technologies, which include the following list of applications. |
4,753 | Power MOSFETs are commonly used for a wide range of consumer electronics. Power MOSFETs are widely used in the following consumer applications. |
4,754 | MOSFETs are fundamental to information and communications technology , including modern computers, modern computing, telecommunications, the communications infrastructure, the Internet, digital telephony, wireless telecommunications, and mobile networks. According to Colinge, the modern computer industry and digital te... |
4,755 | MOSFETs are commonly used in a wide range of computers and computing applications, which include the following. |
4,756 | MOSFETs are commonly used in a wide range of telecommunications, which include the following applications. |
4,757 | The insulated-gate bipolar transistor is a power transistor with characteristics of both a MOSFET and bipolar junction transistor . As of 2010, the IGBT is the second most widely used power transistor, after the power MOSFET. The IGBT accounts for 27% of the power transistor market, second only to the power MOSFET , a... |
4,758 | The IGBT is widely used in the following applications. |
4,759 | In quantum physics and quantum mechanics, the MOSFET is the basis for two-dimensional electron gas and the quantum Hall effect. The MOSFET enables physicists to study electron behavior in a two-dimensional gas, called a two-dimensional electron gas. In a MOSFET, conduction electrons travel in a thin surface layer, and... |
4,760 | In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji observed the Hall effect in experiments carried out on the inversion layer of MOSFETs. In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper... |
4,761 | The MOSFET is used in quantum technology. A quantum field-effect transistor or quantum well field-effect transistor is a type of MOSFET that takes advantage of quantum tunneling to greatly increase the speed of transistor operation. |
4,762 | MOSFETs are widely used in transportation. For example, they are commonly used for automotive electronics in the automotive industry. MOS technology is commonly used for a wide range of vehicles and transportation, which include the following applications. |
4,763 | MOSFETs are widely used in the automotive industry, particularly for automotive electronics in motor vehicles. Automotive applications include the following. |
4,764 | Power MOSFETs are widely used in transportation technology, which includes the following vehicles. |
4,765 | In the automotive industry, power MOSFETs are widely used in automotive electronics, which include the following. |
4,766 | The insulated-gate bipolar transistor is a power transistor with characteristics of both a MOSFET and bipolar junction transistor . IGBTs are widely used in the following transportation applications. |
4,767 | In the space industry, MOSFET devices were adopted by NASA for space research in 1964, for its Interplanetary Monitoring Platform program and Explorers space exploration program. The use of MOSFETs was a major step forward in the electronics design of spacecraft and satellites. The IMP D , launched in 1966, was the fi... |
4,768 | The Cassini–Huygens to Saturn in 1997 had spacecraft power distribution accomplished 192 solid-state power switch devices, which also functioned as circuit breakers in the event of an overload condition. The switches were developed from a combination of two semiconductor devices with switching capabilities: the MOSFET... |
4,769 | MOSFETs are commonly used for a wide range of other applications, which include the following. |
4,770 | Housed in special high-speed memory, microcode translates machine instructions, state machine data, or other input into sequences of detailed circuit-level operations. It separates the machine instructions from the underlying electronics, thereby enabling greater flexibility in designing and altering instructions. More... |
4,771 | Through extensive microprogramming, microarchitectures of smaller scale and simplicity can emulate more robust architectures with wider word lengths, additional execution units, and so forth. This approach provides a relatively straightforward method of ensuring software compatibility between different products within ... |
4,772 | Some hardware vendors, notably IBM/Lenovo, use the term microcode interchangeably with firmware. In this context, all code within a device is termed microcode, whether it is microcode or machine code. For instance, updates to a hard disk drive's microcode often encompass updates to both its microcode and firmware. |
4,773 | At the hardware level, processors contain a number of separate areas of circuity, or "units", that perform different tasks. Commonly found units include the arithmetic logic unit which performs instructions such as addition or comparing two numbers, circuits for reading and writing data to external memory, and small a... |
4,774 | To properly perform an instruction, the various circuits have to be activated in order. For instance, it is not possible to add two numbers if they have not yet been loaded from memory. In RISC designs, the proper ordering of these instructions is largely up to the programmer, or at least to the compiler of the program... |
4,775 | As the sequence of instructions needed to complete this higher-level concept, "add these two numbers in memory", may require multiple instructions, this can represent a performance bottleneck if those instructions are stored in main memory. Reading those instructions one by one is taking up time that could be used to r... |
4,776 | Using the variation of the instruction, or "opcode", that most closely matches the ultimate operation can reduce the number of instructions to one, saving memory used by the program code and improving performance by leaving the data bus open for other operations. Internally, however, these instructions are not separate... |
4,777 | The basic idea behind microcode is to replace the custom logic implementing the instruction sequencing with a series of simple instructions run in a "microcode engine" in the processor. Whereas a custom logic system might have a series of diodes and gates that output a series of voltages on various control lines, the m... |
4,778 | The distinction between custom logic and microcode may seem small, one uses a pattern of diodes and gates to decode the instruction and produce a sequence of signals, whereas the other encodes the signals as microinstructions that are read in sequence to produce the same results. The critical difference is that in a cu... |
4,779 | The lowest layer in a computer's software stack is traditionally raw machine code instructions for the processor. In microcoded processors, fetching and decoding those instructions, and executing them, may be done by microcode. To avoid confusion, each microprogram-related element is differentiated by the micro prefix:... |
4,780 | Complex digital processors may also employ more than one control unit in order to delegate sub-tasks that must be performed essentially asynchronously in parallel. For example, the VAX 9000 has an IBox unit to fetch and decode instructions, which it hands to a microcoded EBox unit to be executed, and the VAX 8800 has... |
4,781 | A high-level programmer, or even an assembly language programmer, does not normally see or change microcode. Unlike machine code, which often retains some backward compatibility among different processors in a family, microcode only runs on the exact electronic circuitry for which it is designed, as it constitutes an i... |
4,782 | Engineers normally write the microcode during the design phase of a processor, storing it in a read-only memory or programmable logic array structure, or in a combination of both. However, machines also exist that have some or all microcode stored in static random-access memory or flash memory. This is traditionally... |
4,783 | Microprograms consist of series of microinstructions, which control the CPU at a very fundamental level of hardware circuitry. For example, a single typical horizontal microinstruction might specify the following operations: |
4,784 | To simultaneously control all processor's features in one cycle, the microinstruction is often wider than 50 bits; e.g., 128 bits on a 360/85 with an emulator feature. Microprograms are carefully designed and optimized for the fastest possible execution, as a slow microprogram would result in a slow machine instruction... |
4,785 | Microcode was originally developed as a simpler method of developing the control logic for a computer. Initially, CPU instruction sets were hardwired. Each step needed to fetch, decode, and execute the machine instructions was controlled directly by combinational logic and rather minimal sequential state machine circu... |
4,786 | Microcode simplified the job by allowing much of the processor's behaviour and programming model to be defined via microprogram routines rather than by dedicated circuitry. Even late in the design process, microcode could easily be changed, whereas hard-wired CPU designs were very cumbersome to change. Thus, this great... |
4,787 | From the 1940s to the late 1970s, a large portion of programming was done in assembly language; higher-level instructions mean greater programmer productivity, so an important advantage of microcode was the relative ease by which powerful machine instructions can be defined. The ultimate extension of this are "Directly... |
4,788 | Architectures with instruction sets implemented by complex microprograms included the IBM System/360 and Digital Equipment Corporation VAX. The approach of increasingly complex microcode-implemented instruction sets was later called complex instruction set computer . An alternate approach, used in many microprocessors,... |
4,789 | Microprogramming is still used in modern CPU designs. In some cases, after the microcode is debugged in simulation, logic functions are substituted for the control store. Logic functions are often faster and less expensive than the equivalent microprogram memory. |
4,790 | A processor's microprograms operate on a more primitive, totally different, and much more hardware-oriented architecture than the assembly instructions visible to normal programmers. In coordination with the hardware, the microcode implements the programmer-visible architecture. The underlying hardware need not have a ... |
4,791 | The IBM System/360 has a 32-bit architecture with 16 general-purpose registers, but most of the System/360 implementations use hardware that implements a much simpler underlying microarchitecture; for example, the System/360 Model 30 has 8-bit data paths to the arithmetic logic unit and main memory and implemented the... |
4,792 | A similar approach was used by Digital Equipment Corporation in their VAX family of computers. As a result, different VAX processors use different microarchitectures, yet the programmer-visible architecture does not change. |
4,793 | Microprogramming also reduces the cost of field changes to correct defects in the processor; a bug can often be fixed by replacing a portion of the microprogram rather than by changes being made to hardware logic and wiring. |
4,794 | In 1947, the design of the MIT Whirlwind introduced the concept of a control store as a way to simplify computer design and move beyond ad hoc methods. The control store is a diode matrix: a two-dimensional lattice, where one dimension accepts "control time pulses" from the CPU's internal clock, and the other connects ... |
4,795 | In 1951, Maurice Wilkes enhanced this concept by adding conditional execution, a concept akin to a conditional in computer software. His initial implementation consisted of a pair of matrices: the first one generated signals in the manner of the Whirlwind control store, while the second matrix selected which row of sig... |
4,796 | Microcode remained relatively rare in computer design as the cost of the ROM needed to store the code was not significantly different than using a custom control store. This changed through the early 1960s with the introduction of mass-produced core memory and core rope, which was far less expensive that dedicated logi... |
4,797 | But the real value in the 360 line was that one could build a series of machines that were completely different internally, yet run the same ISA. For a low-end machine, one might use an 8-bit ALU that requires multiple cycles to complete a single 32-bit addition, while a higher end machine might have a full 32-bit ALU ... |
4,798 | The outcome of this design was that customers could use a low-end model of the family to develop their software, knowing that if more performance was ever needed, they could move to a faster version and nothing else would change. This lowered the barrier to entry and the 360 was a runaway success. By the end of the dec... |
4,799 | Early minicomputers were far too simple to require microcode, and were more similar to earlier mainframes in terms of their instruction sets and the way they were decoded. But it was not long before their designers began using more powerful integrated circuits that allowed for more complex ISAs. By the mid-1970s, most ... |
4,800 | The same basic evolution occurred with microprocessors as well. Early designs were extremely simple, and even the more powerful 8-bit designs of the mid-1970s like the Zilog Z80 had instruction sets that were simple enough to be implemented in dedicated logic. By this time, the control logic could be patterned into the... |
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