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Section: Noise of ideal resistors for moderate frequencies. Johnson's experiment (Figure 1) found that the thermal noise from a resistance R {\displaystyle R} at kelvin temperature T {\displaystyle T} and bandlimited to a frequency band of bandwidth Δ f {\displaystyle \Delta f} (Figure 3) has a mean square voltage of: ... | Wikipedia - Johnson–Nyquist noise - Noise of ideal resistors for moderate frequencies | 228 | 857 | null |
Section: Noise of ideal resistors for moderate frequencies > RMS noise voltage. The square root of the mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth Δ f {\displaystyle \Delta f} : V rms = V n 2 ¯ = 4 k B T R Δ f . {\displaystyle V_{\text{rms}}={\sqrt {\overline {V_{n}^{2}}}}=... | Wikipedia - Johnson–Nyquist noise - Noise of ideal resistors for moderate frequencies > RMS noise voltage | 226 | 775 | null |
Section: Thermal noise on capacitors. Ideal capacitors, as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (an RC circuit, a common low-pass filter) has what is called kTC noise. The equivalent noise bandwidth of an RC circuit is Δ f = 1 4 R C . {\displaystyle \Delta ... | Wikipedia - Johnson–Nyquist noise - Thermal noise on capacitors | 261 | 861 | null |
The mean-square and RMS noise voltage generated in such a filter are: V n 2 ¯ = 4 k B T R 4 R C = k B T C {\displaystyle {\overline {V_{n}^{2}}}={4k_{\text{B}}TR \over 4RC}={k_{\text{B}}T \over C}} V rms = 4 k B T R 4 R C = k B T C . {\displaystyle V_{\text{rms}}={\sqrt {4k_{\text{B}}TR \over 4RC}}={\sqrt {k_{\text{B}}... | Wikipedia - Johnson–Nyquist noise - Thermal noise on capacitors | 380 | 755 | null |
{\displaystyle V_{\text{rms}}={\sqrt {4k_{\text{B}}TR \over 4RC}}={\sqrt {k_{\text{B}}T \over C}}.} The noise charge Q n {\displaystyle Q_{n}} is the capacitance times the voltage: Q n = C V n = C k B T C = k B T C {\displaystyle Q_{n}=C\,V_{n}=C{\sqrt {k_{\text{B}}T \over C}}={\sqrt {k_{\text{B}}TC}}} Q n 2 ¯ = C 2 V ... | Wikipedia - Johnson–Nyquist noise - Thermal noise on capacitors | 349 | 849 | null |
Section: Thermal noise on capacitors > Reset noise. An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage must be interpreted not as a time average, ... | Wikipedia - Johnson–Nyquist noise - Thermal noise on capacitors > Reset noise | 312 | 1,376 | null |
Section: Thermometry. The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry". For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm. It accomplished this by using Jose... | Wikipedia - Johnson–Nyquist noise - Thermometry | 225 | 959 | null |
Section: Maximum transfer of noise power. The noise generated at a resistor R S {\displaystyle R_{\text{S}}} can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance R L {\displaystyle R_{\rm {L}}} of the remaining circuit matches R S {\displaystyle R_{\text{S}}}... | Wikipedia - Johnson–Nyquist noise - Maximum transfer of noise power | 197 | 719 | null |
Section: Maximum transfer of noise power > Available noise power in decibel-milliwatts. Signal power is often measured in dBm (decibels relative to 1 milliwatt). Available noise power would thus be 10 log 10 ( k B T Δ f 1 mW ) {\displaystyle 10\ \log _{10}({\tfrac {k_{\text{B}}T\Delta f}{\text{1 mW}}})} in dBm. At ro... | Wikipedia - Johnson–Nyquist noise - Maximum transfer of noise power > Available noise power in decibel-milliwatts | 184 | 575 | null |
Section: Nyquist's derivation of ideal resistor noise. Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell to explain Johnson's experimental result. Nyquist's thought experiment summed... | Wikipedia - Johnson–Nyquist noise - Nyquist's derivation of ideal resistor noise | 187 | 732 | null |
According to the conclusion of Figure 5, the total average power transferred over bandwidth Δ f {\displaystyle \Delta f} from R 1 {\displaystyle R_{1}} and absorbed by R 2 {\displaystyle R_{2}} was determined to be: P 1 ¯ = k B T Δ f . {\displaystyle {\overline {P_{1}}}=k_{\rm {B}}T\,\Delta f\,.} Simple application of ... | Wikipedia - Johnson–Nyquist noise - Nyquist's derivation of ideal resistor noise | 323 | 831 | null |
{\displaystyle {\overline {P_{1}}}=k_{\rm {B}}T\,\Delta f\,.} Simple application of Ohm's law says the current from V 1 {\displaystyle V_{1}} (the thermal voltage noise of only R 1 {\displaystyle R_{1}} ) through the combined resistance is I 1 = V 1 R 1 + R 2 = V 1 2 R 1 {\textstyle I_{1}{=}{\tfrac {V_{1}}{R_{1}+R_{2}}... | Wikipedia - Johnson–Nyquist noise - Nyquist's derivation of ideal resistor noise | 449 | 971 | null |
{\displaystyle P_{\text{1}}=I_{1}^{2}R_{2}=I_{1}^{2}R_{1}=\left({\frac {V_{1}}{2R_{1}}}\right)^{2}R_{1}={\frac {V_{1}^{2}}{4R_{1}}}\,.} Setting this P 1 {\textstyle P_{\text{1}}} equal to the earlier average power expression P 1 ¯ {\textstyle {\overline {P_{1}}}} allows solving for the average of V 1 2 {\textstyle V_{1... | Wikipedia - Johnson–Nyquist noise - Nyquist's derivation of ideal resistor noise | 329 | 844 | null |
Section: Generalized forms > Complex impedances. Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. Such a component can be described by a frequency-dependent complex electrical impedance Z ( f ) {\displaystyle... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Complex impedances | 247 | 847 | null |
The real part of impedance, Re [ Z ( f ) ] {\displaystyle \operatorname {Re} [Z(f)]} , is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies f 1 {\displaystyle f_{1}} to f 2 {\displaystyle f_{2}} can be found by taking the square root o... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Complex impedances | 343 | 894 | null |
{\displaystyle S_{i_{n}i_{n}}(f)=4k_{\text{B}}T\eta (f)\operatorname {Re} [Y(f)].} where Y ( f ) = 1 Z ( f ) {\displaystyle Y(f){=}{\tfrac {1}{Z(f)}}} is the electrical admittance; note that Re [ Y ( f ) ] = Re [ Z ( f ) ] | Z ( f ) | 2 . {\displaystyle \operatorname {Re} [Y(f)]{=}{\tfrac {\operatorname {Re} [Z(f)]... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Complex impedances | 176 | 338 | null |
Section: Generalized forms > Quantum effects at high frequencies or low temperatures. With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor η ( f ) {\displaystyle \eta (f)} mentioned earlier is in general given by:... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Quantum effects at high frequencies or low temperatures | 318 | 929 | null |
Section: Generalized forms > Quantum effects at high frequencies or low temperatures > Relation to Planck's law. Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of Planck's law of blackbody ... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Quantum effects at high frequencies or low temperatures > Relation to Planck's law | 188 | 866 | null |
Section: Generalized forms > Multiport electrical networks. Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators. Thermal noise appears at every port, and can be described as random series voltage sources in series wit... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Multiport electrical networks | 257 | 910 | null |
Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by S i m i n ( f ) = 2 k B T η ( f ) ( Y m n ( f ) + Y n m ( f ) ∗ ) {\displaystyle S_{i_{m}i_{n}}(f)=2k_{\text{B}}T\eta (f)(Y_{mn}(f)+Y_{nm}(f)^{*})} where Y = Z − ... | Wikipedia - Johnson–Nyquist noise - Generalized forms > Multiport electrical networks | 159 | 394 | null |
Section: Background. Resistors above about 1 ohm in value can be measured using a variety of techniques, such as an ohmmeter or by using a Wheatstone bridge. In such resistors, the resistance of the connecting wires or terminals is negligible compared to the resistance value. For resistors of less than an ohm, the resi... | Wikipedia - Kelvin bridge - Background | 246 | 1,256 | null |
Section: Principle of operation. The operation of the Kelvin bridge is very similar to the Wheatstone bridge, but uses two additional resistors. Resistors R1 and R2 are connected to the outside potential terminals of the four terminal known or standard resistor Rs and the unknown resistor Rx (identified as P1 and P′1 i... | Wikipedia - Kelvin bridge - Principle of operation | 251 | 1,088 | null |
The detector D is connected between the junction of R1 and R2 and the junction of R′1 and R′2. The balance equation of this bridge is given by the equation R x R s = R 2 R 1 + R par R s ⋅ R 1 ′ R 1 ′ + R 2 ′ + R par ⋅ ( R 2 R 1 − R 2 ′ R 1 ′ ) {\displaystyle {\frac {R_{x}}{R_{s}}}={\frac {R_{2}}{R_{1}}}+{\frac {R_{\tex... | Wikipedia - Kelvin bridge - Principle of operation | 290 | 608 | null |
The balance equation of this bridge is given by the equation R x R s = R 2 R 1 + R par R s ⋅ R 1 ′ R 1 ′ + R 2 ′ + R par ⋅ ( R 2 R 1 − R 2 ′ R 1 ′ ) {\displaystyle {\frac {R_{x}}{R_{s}}}={\frac {R_{2}}{R_{1}}}+{\frac {R_{\text{par}}}{R_{s}}}\cdot {\frac {R'_{1}}{R'_{1}+R'_{2}+R_{\text{par}}}}\cdot \left({\frac {R_{2}}{... | Wikipedia - Kelvin bridge - Principle of operation | 414 | 932 | null |
As a result, the last term of the above equation becomes zero and the balance equation becomes R x R s = R 2 R 1 {\displaystyle {\frac {R_{x}}{R_{s}}}={\frac {R_{2}}{R_{1}}}} Rearranging to make Rx the subject R x = R 2 ⋅ R s R 1 {\displaystyle R_{x}=R_{2}\cdot {\frac {R_{s}}{R_{1}}}} The parasitic resistance Rpar has ... | Wikipedia - Kelvin bridge - Principle of operation | 270 | 1,086 | null |
Section: Accuracy. The accuracy of measurements made using this bridge are dependent on a number of factors. The accuracy of the standard resistor (Rs) is of prime importance. Also of importance is how close the ratio of R1 to R2 is to the ratio of R′1 to R′2. As shown above, if the ratio is exactly the same, the error... | Wikipedia - Kelvin bridge - Accuracy | 345 | 1,664 | null |
Laboratory bridges are usually constructed with high accuracy variable resistors in the two potential arms of the bridge and achieve accuracies suitable for calibrating standard resistors. In such an application, the 'standard' resistor (Rs) will in reality be a sub-standard type (that is a resistor having an accuracy ... | Wikipedia - Kelvin bridge - Accuracy | 244 | 1,149 | null |
Article: Leading and lagging current. Leading and lagging current are phenomena that occur as a result of alternating current. In a circuit with alternating current, the value of voltage and current vary sinusoidally. In this type of circuit, the terms lead, lag, and in phase are used to describe current with reference... | Wikipedia - Leading and lagging current - Summary | 160 | 796 | null |
Section: Angle notation. Angle notation can easily describe leading and lagging current: A ∠ θ . {\displaystyle A\angle \theta .} In this equation, the value of theta is the important factor for leading and lagging current. As mentioned in the introduction above, leading or lagging current represents a time shift betwe... | Wikipedia - Leading and lagging current - Angle notation | 248 | 1,039 | null |
Section: Lagging current. A ∠ θ = A ∠ δ − ( β ) {\displaystyle A\angle \theta =A\angle \delta -(\beta )} Lagging current can be formally defined with respect to “an alternating current that reaches its maximum value up to 90 degrees later than the voltage that produces it.” This means that current lags the voltage when... | Wikipedia - Leading and lagging current - Lagging current | 286 | 1,219 | null |
Section: Leading current. A ∠ θ = A ∠ δ + ( β ) {\displaystyle A\angle \theta =A\angle \delta +(\beta )} Leading current can be formally defined as “an alternating current that reaches its maximum value up to 90 degrees ahead of the voltage that it produces.” This means that the current leads the voltage when β {\displ... | Wikipedia - Leading and lagging current - Leading current | 330 | 1,518 | null |
Article: Changzhi Li. Changzhi Li is a full professor and Whitacre Endowed Chair in Electrical & Computer Engineering, at Texas Tech University. He is also head of Biomedical Integrated Devices and Systems (BIDS). Professor Li specializes in portable radar sensor technologies that significantly advance healthcare, smar... | Wikipedia - Changzhi Li - Summary | 188 | 911 | null |
Section: Academic career. Currently, Li holds the position of Full Professor and Whitacre Endowed Chair in Electrical and Computer Engineering at Texas Tech University and is the head of the research group Biomedical Integrated Devices and Systems (BIDS). The Texas Tech Biomedical Integrated Devices and Systems (BIDS) ... | Wikipedia - Changzhi Li - Academic career | 154 | 920 | null |
Article: Linear timecode. Linear (or Longitudinal) Timecode (LTC) is an encoding of SMPTE timecode data in an audio signal, as defined in SMPTE 12M specification. The audio signal is commonly recorded on a VTR track or other storage media. The bits are encoded using the biphase mark code (also known as FM): a 0 bit has... | Wikipedia - Linear timecode - Summary | 208 | 900 | null |
Section: Generation and Distribution. In broadcast video situations, the LTC generator should be tied into house black burst, as should all devices using timecode, to ensure correct color framing and correct synchronization of all digital clocks. When synchronizing multiple clock-dependent digital devices together with... | Wikipedia - Linear timecode - Generation and Distribution | 320 | 1,501 | null |
It can also be distributed via 75 ohm video cable and video distribution amplifiers, although the voltage attenuation caused by using a 75 ohm system may cause the signal to drop to a level that can not be read by some equipment. Care has to be taken with analog audio to avoid audible 'breakthrough' (aka "crosstalk") f... | Wikipedia - Linear timecode - Generation and Distribution | 166 | 762 | null |
Section: Longitudinal timecode data format. The basic format is an 80-bit code that gives the time of day to the second, and the frame number within the second. Values are stored in binary-coded decimal, least significant bit first. There are thirty-two bits of user data, usually used for a reel number and date. Bit 10... | Wikipedia - Linear timecode - Longitudinal timecode data format | 350 | 1,434 | null |
Since the sync code includes an odd number of 1 bits, it is an odd parity bit over the data.) This keeps the phase of each frame consistent, so it always starts with a rising edge at the beginning of bit 0. This allows seamless splicing of different time codes, and lets it be more easily read with an oscilloscope. "Bin... | Wikipedia - Linear timecode - Longitudinal timecode data format | 287 | 1,236 | null |
Article: Linear transformation in rotating electrical machines. Transformation of three phase electrical quantities to two phase quantities is a usual practice to simplify analysis of three phase electrical circuits. Polyphase a.c machines can be represented by an equivalent two phase model provided the rotating polyph... | Wikipedia - Linear transformation in rotating electrical machines - Summary | 246 | 1,406 | null |
Section: Commonly used reference frames. Based on speed of reference frame there are four major type of reference frame. Arbitrary reference frame: Reference frame speed is unspecifie(ω), variables denoted by fdqos or fds, fqs and fos, transformation matrix denoted by Ks. Stationary reference frame: Reference frame spe... | Wikipedia - Linear transformation in rotating electrical machines - Commonly used reference frames | 325 | 1,464 | null |
Article: Comparison of EDA software. This page is a comparison of electronic design automation (EDA) software which is used today to design the near totality of electronic devices. Modern electronic devices are too complex to be designed without the help of a computer. Electronic devices may consist of integrated circu... | Wikipedia - Comparison of EDA software - Summary | 334 | 1,628 | null |
In the case of integrated circuits (ICs) for example, a single chip may contain today more than 20 billion transistors and, as a general rule, every single transistor in a chip must work as intended. Since a single VLSI mask set can cost up to 10-100 millions, trial and error approaches are not economically viable. To ... | Wikipedia - Comparison of EDA software - Summary | 165 | 747 | null |
Section: Comparison of proprietary EDA software > Mainstream EDA software bundles for ICs design. The world of electronic design automation (EDA) software for integrated circuit (IC) design is dominated by the three vendors Synopsys, Cadence Design Systems and Siemens EDA (Formerly Mentor Graphics, acquired in 2017 by ... | Wikipedia - Comparison of EDA software - Comparison of proprietary EDA software > Mainstream EDA software bundles for ICs design | 266 | 1,346 | null |
Article: Load line (electronics). In graphical analysis of nonlinear electronic circuits, a load line is a line drawn on the current–voltage characteristic graph for a nonlinear device like a diode or transistor. It represents the constraint put on the voltage and current in the nonlinear device by the external circuit... | Wikipedia - Load line (electronics) - Summary | 337 | 1,576 | null |
The load line (diagonal line), representing the relationship between current and voltage due to Kirchhoff's voltage law applied to the resistor and voltage source, is V D = V D D − I R {\displaystyle V_{D}=V_{DD}-IR\,} Since the same current flows through each of the three elements in series, and the voltage produced b... | Wikipedia - Load line (electronics) - Summary | 194 | 845 | null |
Section: Load lines for common configurations > Transistor load line. The load line diagram at right is for a resistive load in a common emitter circuit. The load line shows how the collector load resistor (RL) constrains the circuit voltage and current. The diagram also plots the transistor's collector current IC vers... | Wikipedia - Load line (electronics) - Load lines for common configurations > Transistor load line | 317 | 1,583 | null |
Section: DC and AC load lines. Semiconductor circuits typically have both DC and AC currents in them, with a source of DC current to bias the nonlinear semiconductor to the correct operating point, and the AC signal superimposed on the DC. Load lines can be used separately for both DC and AC analysis. The DC load line ... | Wikipedia - Load line (electronics) - DC and AC load lines | 310 | 1,476 | null |
Section: Loss-of-load-based reliability indices. Multiple reliability indices for the electrical generation are based on the loss of load being observed/calculated over a long interval (one or multiple years) in relatively small increments (an hour or a day). The total number of increments inside the long interval is d... | Wikipedia - Loss of load - Loss-of-load-based reliability indices | 333 | 1,295 | null |
Since LOLE uses the daily peak value for the whole day, LOLH (that uses different peak values for each hour) cannot be obtained by simply multiplying LOLE by 24; although in practice the relationship is close to linear, the coefficients vary from network to network; Loss of load events (LOLEV) a.k.a. loss of load frequ... | Wikipedia - Loss of load - Loss-of-load-based reliability indices | 158 | 600 | null |
Section: One-day-in-ten-years criterion. A typically accepted design goal for L O L E {\displaystyle LOLE} is 0.1 day per year ("one-day-in-ten-years criterion" a.k.a. "1 in 10"), corresponding to L O L P = 1 10 ⋅ 365 ≈ 0.000274 {\displaystyle {LOLP}={\frac {1}{10\cdot 365}}\approx 0.000274} . In the US, the threshold ... | Wikipedia - Loss of load - One-day-in-ten-years criterion | 164 | 547 | null |
Section: United States. In electrical power distribution, the US National Electrical Code (NEC), NFPA 70, article 725 (2005), defines low distribution system voltage (LDSV) as up to 49 V. The NFPA standard 79 article 6.4.1.1 defines distribution protected extra-low voltage (PELV) as nominal voltage of 30 Vrms or 60 V D... | Wikipedia - Low voltage - United States | 201 | 804 | null |
Article: Linear time-invariant system. In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined in the overview below. These properti... | Wikipedia - Linear time-invariant system - Summary | 297 | 1,365 | null |
Section: Overview. The defining properties of any LTI system are linearity and time invariance. Linearity means that the relationship between the input x ( t ) {\displaystyle x(t)} and the output y ( t ) {\displaystyle y(t)} , both being regarded as functions, is a linear mapping: If a {\displaystyle a} is a constant t... | Wikipedia - Linear time-invariant system - Overview | 307 | 1,005 | null |
Time invariance means that whether we apply an input to the system now or T seconds from now, the output will be identical except for a time delay of T seconds. That is, if the output due to input x ( t ) {\displaystyle x(t)} is y ( t ) {\displaystyle y(t)} , then the output due to input x ( t − T ) {\displaystyle x(t-... | Wikipedia - Linear time-invariant system - Overview | 259 | 974 | null |
This is called a continuous time system. Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in discrete time: y i = x i ∗ h i {\displaystyle y_{i}=x_{i}*h_{i}} where y, x, and h are sequences and the convolution, in discrete time, uses a discrete ... | Wikipedia - Linear time-invariant system - Overview | 345 | 1,369 | null |
As a result, if the input to a system is the complex waveform A s e s t {\displaystyle A_{s}e^{st}} for some complex amplitude A s {\displaystyle A_{s}} and complex frequency s {\displaystyle s} , the output will be some complex constant times the input, say B s e s t {\displaystyle B_{s}e^{st}} for some new complex am... | Wikipedia - Linear time-invariant system - Overview | 346 | 1,414 | null |
Section: Continuous-time systems > Impulse response and convolution. The behavior of a linear, continuous-time, time-invariant system with input signal x(t) and output signal y(t) is described by the convolution integral: where h ( t ) {\textstyle h(t)} is the system's response to an impulse: x ( τ ) = δ ( τ ) {\textst... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 273 | 907 | null |
When h ( τ ) {\textstyle h(\tau )} is zero for all negative τ {\textstyle \tau } , y ( t ) {\textstyle y(t)} depends only on values of x {\textstyle x} prior to time t {\textstyle t} , and the system is said to be causal. To understand why the convolution produces the output of an LTI system, let the notation { x ( u −... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 271 | 818 | null |
And in general, every value of the output can depend on every value of the input. This concept is represented by: y ( t ) = def O t { x } , {\displaystyle y(t)\mathrel {\stackrel {\text{def}}{=}} O_{t}\{x\},} where O t {\textstyle O_{t}} is the transformation operator for time t {\textstyle t} . In a typical system, y ... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 232 | 760 | null |
For a linear system, O {\textstyle O} must satisfy Eq.1: And the time-invariance requirement is: In this notation, we can write the impulse response as h ( t ) = def O t { δ ( u ) ; u } . {\textstyle h(t)\mathrel {\stackrel {\text{def}}{=}} O_{t}\{\delta (u);\ u\}.} Similarly: Substituting this result into the convolut... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 356 | 795 | null |
{\textstyle h(t)\mathrel {\stackrel {\text{def}}{=}} O_{t}\{\delta (u);\ u\}.} Similarly: Substituting this result into the convolution integral: ( x ∗ h ) ( t ) = ∫ − ∞ ∞ x ( τ ) ⋅ h ( t − τ ) d τ = ∫ − ∞ ∞ x ( τ ) ⋅ O t { δ ( u − τ ) ; u } d τ , {\displaystyle {\begin{aligned}(x*h)(t)&=\int _{-\infty }^{\infty }x(\ta... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 382 | 789 | null |
{\textstyle x_{\tau }(u)=\delta (u-\tau ).} Eq.2 then allows this continuation: ( x ∗ h ) ( t ) = O t { ∫ − ∞ ∞ x ( τ ) ⋅ δ ( u − τ ) d τ ; u } = O t { x ( u ) ; u } = def y ( t ) . {\displaystyle {\begin{aligned}(x*h)(t)&=O_{t}\left\{\int _{-\infty }^{\infty }x(\tau )\cdot \delta (u-\tau )\,\mathrm {d} \tau ;\ u\right... | Wikipedia - Linear time-invariant system - Continuous-time systems > Impulse response and convolution | 335 | 920 | null |
Section: Continuous-time systems > Exponentials as eigenfunctions. An eigenfunction is a function for which the output of the operator is a scaled version of the same function. That is, H f = λ f , {\displaystyle {\mathcal {H}}f=\lambda f,} where f is the eigenfunction and λ {\displaystyle \lambda } is the eigenvalue, ... | Wikipedia - Linear time-invariant system - Continuous-time systems > Exponentials as eigenfunctions | 187 | 613 | null |
The output of the system with impulse response h ( t ) {\displaystyle h(t)} is then ∫ − ∞ ∞ h ( t − τ ) A e s τ d τ {\displaystyle \int _{-\infty }^{\infty }h(t-\tau )Ae^{s\tau }\,\mathrm {d} \tau } which, by the commutative property of convolution, is equivalent to ∫ − ∞ ∞ h ( τ ) A e s ( t − τ ) d τ ⏞ H f = ∫ − ∞ ∞ h... | Wikipedia - Linear time-invariant system - Continuous-time systems > Exponentials as eigenfunctions | 346 | 699 | null |
}\,\mathrm {d} \tau \\[4pt]&=\overbrace {\underbrace {Ae^{st}} _{\text{Input}}} ^{f}\,\overbrace {\underbrace {H(s)} _{\text{Scalar}}} ^{\lambda }\,,\\\end{aligned}}} where the scalar H ( s ) = def ∫ − ∞ ∞ h ( t ) e − s t d t {\displaystyle H(s)\mathrel {\stackrel {\text{def}}{=}} \int _{-\infty }^{\infty }h(t)e^{-st}\... | Wikipedia - Linear time-invariant system - Continuous-time systems > Exponentials as eigenfunctions | 318 | 773 | null |
Section: Continuous-time systems > Exponentials as eigenfunctions > Direct proof. It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems. Let's set v ( t ) = e i ω t {\displaystyle v(t)=e^{i\omega t}} some complex exponential and v a ( t ) = e i ω ( t + a ) {\displaystyle v_{a}(t)=... | Wikipedia - Linear time-invariant system - Continuous-time systems > Exponentials as eigenfunctions > Direct proof | 312 | 751 | null |
So H [ v ] ( t + a ) = e i ω a H [ v ] ( t ) {\displaystyle H[v](t+a)=e^{i\omega a}H[v](t)} . Setting t = 0 {\displaystyle t=0} and renaming we get: H [ v ] ( τ ) = e i ω τ H [ v ] ( 0 ) {\displaystyle H[v](\tau )=e^{i\omega \tau }H[v](0)} i.e. that a complex exponential e i ω τ {\displaystyle e^{i\omega \tau }} as inp... | Wikipedia - Linear time-invariant system - Continuous-time systems > Exponentials as eigenfunctions > Direct proof | 158 | 383 | null |
Section: Continuous-time systems > Fourier and Laplace transforms. The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided Laplace transform H ( s ) = def L { h ( t ) } = def ∫ 0 ∞ h ( t ) e − s t d t {\displaystyle H(s)\mathrel {\stackrel {\text{def}}{=}}... | Wikipedia - Linear time-invariant system - Continuous-time systems > Fourier and Laplace transforms | 332 | 903 | null |
The Fourier transform H ( j ω ) = F { h ( t ) } {\displaystyle H(j\omega )={\mathcal {F}}\{h(t)\}} gives the eigenvalues for pure complex sinusoids. Both of H ( s ) {\displaystyle H(s)} and H ( j ω ) {\displaystyle H(j\omega )} are called the system function, system response, or transfer function. The Laplace transform... | Wikipedia - Linear time-invariant system - Continuous-time systems > Fourier and Laplace transforms | 299 | 1,285 | null |
The Fourier transform is often applied to spectra of infinite signals via the Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist. Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the t... | Wikipedia - Linear time-invariant system - Continuous-time systems > Fourier and Laplace transforms | 338 | 1,120 | null |
Section: Continuous-time systems > Important system properties > Stability. A system is bounded-input, bounded-output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying ‖ x ( t ) ‖ ∞ < ∞ {\displaystyle \ \|x(t)\|_{\infty }<\infty } leads to an output satisf... | Wikipedia - Linear time-invariant system - Continuous-time systems > Important system properties > Stability | 320 | 946 | null |
{\displaystyle \|h(t)\|_{1}=\int _{-\infty }^{\infty }|h(t)|\,\mathrm {d} t<\infty .} In the frequency domain, the region of convergence must contain the imaginary axis s = j ω {\displaystyle s=j\omega } . As an example, the ideal low-pass filter with impulse response equal to a sinc function is not BIBO stable, becaus... | Wikipedia - Linear time-invariant system - Continuous-time systems > Important system properties > Stability | 208 | 681 | null |
Section: Discrete-time systems > Discrete-time systems from continuous-time systems. In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back a... | Wikipedia - Linear time-invariant system - Discrete-time systems > Discrete-time systems from continuous-time systems | 298 | 1,248 | null |
Section: Discrete-time systems > Impulse response and convolution. Let { x [ m − k ] ; m } {\displaystyle \{x[m-k];\ m\}} represent the sequence { x [ m − k ] ; for all integer values of m } . {\displaystyle \{x[m-k];{\text{ for all integer values of }}m\}.} And let the shorter notation { x } {\displaystyle \{x\}} repr... | Wikipedia - Linear time-invariant system - Discrete-time systems > Impulse response and convolution | 320 | 1,006 | null |
(Thus the subscript, n.) In a typical system, y[n] depends most heavily on the elements of x whose indices are near n. For the special case of the Kronecker delta function, x [ m ] = δ [ m ] , {\displaystyle x[m]=\delta [m],} the output sequence is the impulse response: h [ n ] = def O n { δ [ m ] ; m } . {\displaystyl... | Wikipedia - Linear time-invariant system - Discrete-time systems > Impulse response and convolution | 324 | 973 | null |
To see how that is done, consider the identity: x [ m ] ≡ ∑ k = − ∞ ∞ x [ k ] ⋅ δ [ m − k ] , {\displaystyle x[m]\equiv \sum _{k=-\infty }^{\infty }x[k]\cdot \delta [m-k],} which expresses { x } {\displaystyle \{x\}} in terms of a sum of weighted delta functions. Therefore: y [ n ] = O n { x } = O n { ∑ k = − ∞ ∞ x [ k... | Wikipedia - Linear time-invariant system - Discrete-time systems > Impulse response and convolution | 370 | 749 | null |
Therefore: y [ n ] = O n { x } = O n { ∑ k = − ∞ ∞ x [ k ] ⋅ δ [ m − k ] ; m } = ∑ k = − ∞ ∞ x [ k ] ⋅ O n { δ [ m − k ] ; m } , {\displaystyle {\begin{aligned}y[n]=O_{n}\{x\}&=O_{n}\left\{\sum _{k=-\infty }^{\infty }x[k]\cdot \delta [m-k];\ m\right\}\\&=\sum _{k=-\infty }^{\infty }x[k]\cdot O_{n}\{\delta [m-k];\ m\},\... | Wikipedia - Linear time-invariant system - Discrete-time systems > Impulse response and convolution | 310 | 590 | null |
And because of Eq.5, we may write: O n { δ [ m − k ] ; m } = O n − k { δ [ m ] ; m } = def h [ n − k ] . {\displaystyle {\begin{aligned}O_{n}\{\delta [m-k];\ m\}&\mathrel {\stackrel {\quad }{=}} O_{n-k}\{\delta [m];\ m\}\\&\mathrel {\stackrel {\text{def}}{=}} h[n-k].\end{aligned}}} Therefore: which is the familiar disc... | Wikipedia - Linear time-invariant system - Discrete-time systems > Impulse response and convolution | 274 | 812 | null |
Section: Discrete-time systems > Exponentials as eigenfunctions. An eigenfunction is a function for which the output of the operator is the same function, scaled by some constant. In symbols, H f = λ f , {\displaystyle {\mathcal {H}}f=\lambda f,} where f is the eigenfunction and λ {\displaystyle \lambda } is the eigenv... | Wikipedia - Linear time-invariant system - Discrete-time systems > Exponentials as eigenfunctions | 252 | 760 | null |
Suppose the input is x [ n ] = z n {\displaystyle x[n]=z^{n}} . The output of the system with impulse response h [ n ] {\displaystyle h[n]} is then ∑ m = − ∞ ∞ h [ n − m ] z m {\displaystyle \sum _{m=-\infty }^{\infty }h[n-m]\,z^{m}} which is equivalent to the following by the commutative property of convolution ∑ m = ... | Wikipedia - Linear time-invariant system - Discrete-time systems > Exponentials as eigenfunctions | 327 | 687 | null |
The output of the system with impulse response h [ n ] {\displaystyle h[n]} is then ∑ m = − ∞ ∞ h [ n − m ] z m {\displaystyle \sum _{m=-\infty }^{\infty }h[n-m]\,z^{m}} which is equivalent to the following by the commutative property of convolution ∑ m = − ∞ ∞ h [ m ] z ( n − m ) = z n ∑ m = − ∞ ∞ h [ m ] z − m = z n ... | Wikipedia - Linear time-invariant system - Discrete-time systems > Exponentials as eigenfunctions | 351 | 792 | null |
Section: Discrete-time systems > Z and discrete-time Fourier transforms. The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform H ( z ) = Z { h [ n ] } = ∑ n = − ∞ ∞ h [ n ] z − n {\displaystyle H(z)={\mathcal {Z}}\{h[n]\}=\sum _{n=-\infty }^{\infty }h[... | Wikipedia - Linear time-invariant system - Discrete-time systems > Z and discrete-time Fourier transforms | 301 | 835 | null |
The discrete-time Fourier transform (DTFT) H ( e j ω ) = F { h [ n ] } {\displaystyle H(e^{j\omega })={\mathcal {F}}\{h[n]\}} gives the eigenvalues of pure sinusoids. Both of H ( z ) {\displaystyle H(z)} and H ( e j ω ) {\displaystyle H(e^{j\omega })} are called the system function, system response, or transfer functio... | Wikipedia - Linear time-invariant system - Discrete-time systems > Z and discrete-time Fourier transforms | 256 | 836 | null |
That is, y [ n ] = ( h ∗ x ) [ n ] = ∑ m = − ∞ ∞ h [ n − m ] x [ m ] = Z − 1 { H ( z ) X ( z ) } . {\displaystyle y[n]=(h*x)[n]=\sum _{m=-\infty }^{\infty }h[n-m]x[m]={\mathcal {Z}}^{-1}\{H(z)X(z)\}.} Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easie... | Wikipedia - Linear time-invariant system - Discrete-time systems > Z and discrete-time Fourier transforms | 155 | 378 | null |
Section: Discrete-time systems > Important system properties > Stability. A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if ‖ x [ n ] ‖ ∞ < ∞ {\displaystyle \|x[n]\|_{\infty }<\infty } implies that ‖ y [ n ] ‖ ∞ < ∞ {\displaystyle \|y[n]... | Wikipedia - Linear time-invariant system - Discrete-time systems > Important system properties > Stability | 331 | 966 | null |
Section: Purpose. Magnetically-controlled shunt reactors are intended for automatic control over reactive power and stabilization of voltage levels; these ensure the following: Elimination of daily and seasonal voltage variations in the power network; Improvement of electric power quality; Optimization and automation o... | Wikipedia - Magnetically controlled shunt reactor - Purpose | 218 | 1,290 | null |
Section: Field of application. On the assumption of tasks to be solved by MCSRs, as well as with consideration of existing experience of their operation, application field of controlled reactors covers (but not limited) the following areas of the power networks: networks with abrupt-changed load curves; networks with w... | Wikipedia - Magnetically controlled shunt reactor - Field of application | 339 | 1,825 | null |
Section: Operating principle. A magnetically-controlled shunt reactor is a transformer-type device which additionally provides functions of semiconducting key apparatus; this is ensured by means of reactor magnetic system operation in the domain of deep saturation. The basing principle allowed optimal employment of exi... | Wikipedia - Magnetically controlled shunt reactor - Operating principle | 330 | 1,691 | null |
In case of energy input to or output from the control circuit, the transient process of increase or decrease of network current and, respectively, of reactive power consumed by reactor is ensured. Reactor power winding current is regulated according to proportional control mode, when control angle of rectified current ... | Wikipedia - Magnetically controlled shunt reactor - Operating principle | 331 | 1,701 | null |
Article: Manley–Rowe relations. The Manley–Rowe relations are mathematical expressions developed originally for electrical engineers to predict the amount of energy in a wave that has multiple frequencies. They have since been found to describe systems in non-linear optics, fluid mechanics and the theory of non-linear ... | Wikipedia - Manley–Rowe relations - Summary | 205 | 950 | null |
Section: History. The original papers, written by two researchers at Bell Labs, J. M. Manley and H. E. Rowe between 1956 and 1960 was for an electrical circuit containing nonlinear capacitors and inductors. One or more oscillators, operating at specified frequencies, are connected to the input of this circuit. The Manl... | Wikipedia - Manley–Rowe relations - History | 347 | 1,762 | null |
Article: Maximum power transfer theorem. In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi pub... | Wikipedia - Maximum power transfer theorem - Summary | 341 | 1,801 | null |
Section: Maximizing power transfer versus power efficiency. The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient, since the power dissipated as heat in the battery would always be equal to the power de... | Wikipedia - Maximum power transfer theorem - Maximizing power transfer versus power efficiency | 253 | 1,147 | null |
The efficiency η is the ratio of the power dissipated by the load resistance RL to the total power dissipated by the circuit (which includes the voltage source's resistance of RS as well as RL): η = P L P T o t a l = I 2 ⋅ R L I 2 ⋅ ( R L + R S ) = R L R L + R S = 1 1 + R S / R L . {\displaystyle \eta ={\frac {P_{\math... | Wikipedia - Maximum power transfer theorem - Maximizing power transfer versus power efficiency | 318 | 726 | null |
{\displaystyle \eta ={\frac {P_{\mathrm {L} }}{P_{\mathrm {Total} }}}={\frac {I^{2}\cdot R_{\mathrm {L} }}{I^{2}\cdot (R_{\mathrm {L} }+R_{\mathrm {S} })}}={\frac {R_{\mathrm {L} }}{R_{\mathrm {L} }+R_{\mathrm {S} }}}={\frac {1}{1+R_{\mathrm {S} }/R_{\mathrm {L} }}}\,.} Consider three particular cases (note that voltag... | Wikipedia - Maximum power transfer theorem - Maximizing power transfer versus power efficiency | 319 | 721 | null |
If R L / R S = 1 {\displaystyle R_{\mathrm {L} }/R_{\mathrm {S} }=1} , then η = 1 2 . {\displaystyle \eta ={\tfrac {1}{2}}.} Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer). If R L / R S → ∞ {\displaystyle R_{\mathrm {L} }/R_{\mathrm {S} }\to... | Wikipedia - Maximum power transfer theorem - Maximizing power transfer versus power efficiency | 185 | 586 | null |
Section: Calculus-based proof for purely resistive circuits. In the simplified model of powering a load with resistance RL by a source with voltage V and source resistance RS, then by Ohm's law the resulting current I is simply the source voltage divided by the total circuit resistance: I = V R S + R L . {\displaystyle... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 161 | 538 | null |
{\displaystyle I={\frac {V}{R_{\mathrm {S} }+R_{\mathrm {L} }}}.} The power PL dissipated in the load is the square of the current multiplied by the resistance: P L = I 2 R L = ( V R S + R L ) 2 R L = V 2 R S 2 / R L + 2 R S + R L . {\displaystyle P_{\mathrm {L} }=I^{2}R_{\mathrm {L} }=\left({\frac {V}{R_{\mathrm {S} }... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 348 | 758 | null |
{\displaystyle P_{\mathrm {L} }=I^{2}R_{\mathrm {L} }=\left({\frac {V}{R_{\mathrm {S} }+R_{\mathrm {L} }}}\right)^{2}R_{\mathrm {L} }={\frac {V^{2}}{R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }}}.} The value of RL for which this expression is a maximum could be calculated by differentiating ... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 308 | 682 | null |
Differentiating the denominator with respect to RL: d d R L ( R S 2 / R L + 2 R S + R L ) = − R S 2 / R L 2 + 1. {\displaystyle {\frac {d}{dR_{\mathrm {L} }}}\left(R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }\right)=-R_{\mathrm {S} }^{2}/R_{\mathrm {L} }^{2}+1.} For a maximum or minimum, the... | Wikipedia - Maximum power transfer theorem - Calculus-based proof for purely resistive circuits | 330 | 795 | null |
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