雖然作者曾在

《時間序列︰生成函數《十二》》

文本中，以『變換』思維視野之重要性，簡略的介紹了

Laplace transform︰

生成函數『形式』何其多？

『變換』思維道其妙︰

Laplace transform

In mathematics the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/). It takes a function of a positive real variable t (often time) to a function of a complex variable s (frequency).

The Laplace transform is very similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. Laplace transforms are usually restricted to functions of t with t > 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Also techniques of complex variables can be used directly to study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.

The Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (time). Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.^[1] So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. It has many applications in the sciences and technology.

『對數』概念始其用︰

歐拉公式 開其門︰

歷代耕耘方法傳。

History

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called the z-transform) in his work on probability theory.^[2] The current widespread use of the transform (mainly in engineering) came about during and soon after World War II ^[3] although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich.

The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form

 

as solutions of differential equations but did not pursue the matter very far.^[4]

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

which some modern historians have interpreted within modern Laplace transform theory.^{[5][6][clarification needed]}

These types of integrals seem first to have attracted Laplace’s attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.^[7] However, in 1785, Laplace took the critical step forward when, rather than just looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form

 

akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.^[8]

Laplace also recognised that Joseph Fourier‘s method of Fourier series for solving the diffusion equation could only apply to a limited region of space because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.^[9]

莫說『形式』無『實質』︰

Formal definition

The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable. The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by

 

where s is a complex number frequency parameter

  , with real numbers σ and ω.

Other notations for the Laplace transform include L{f} , or alternatively L{f(t)} instead of F.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it to be a conditionally convergent improper integral at ∞. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral^[10]

An important special case is where μ is a probability measure, for example, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes

 

where the lower limit of 0⁻ is shorthand notation for

 

This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

總綱立論『心要』在︰

觀其『會通』理念純︰

『出入自然』自為功！

s-domain equivalent circuits and impedances

The Laplace transform is often used in circuit analysis, and simple conversions to the s-domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.

Here is a summary of equivalents:

Note that the resistor is exactly the same in the time domain and the s-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-domain account for that.

The equivalents for current and voltage sources are simply derived from the transformations in the table above.

 

不過踏腳石也。

欲求論理嚴謹運用從容者，最好能深耕呦︰

---

• From Continuous Fourier Transform to Laplace Transform
• Region of Convergence (ROC)
• Zeros and Poles of the Laplace Transform
• Properties of ROC
• Properties of Laplace Transform
• Laplace Transform of Typical Signals
• Representation of LTI Systems by Laplace Transform
• LTI Systems Characterized by LCCDEs
• Evaluation of Fourier Transform from Pole-Zero Plot
  • First order system
  • Second order system
  • Root locus of Second Order System
  • Frequency Response for Small
  • Higher Order Systems
• System Algebra and Block Diagram
• Unilateral Laplace Transform
• Initial and Final Value Theorems
• Solving LCCDEs by Unilateral Laplace Transform
• About this document …

---

Ruye Wang 2012-01-28

 

倘若還能善使工具，豈非如虎添翼耶☆

 

※ 參考

【電阻】

 

【電容】

並聯表現︰

串聯表現︰

 

【電感】

串聯表現︰

並聯表現︰

 

Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function.^[1] The concept was discovered in 1702 by both Johann Bernoulli and Gottfried Leibniz independently.^[2]

In symbols, one can use partial fraction expansion to change a rational fraction in the form

where f and g are polynomials, into an expression of the form

where g_j (x) are polynomials that are factors of g(x), and are in general of lower degree. Thus, the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of rational fractions, which produces a single rational fraction with a numerator and denominator usually of high degree. The full decomposition pushes the reduction as far as it can go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that fraction as a sum of a polynomial and one of several fractions, such that:

• the denominator of each fraction is a power of an irreducible (not factorable) polynomial and
• the numerator is a polynomial of smaller degree than this irreducible polynomial.

As factorization of polynomials may be difficult, a coarser decomposition is often preferred, which consists of replacing factorization by square-free factorization. This amounts to replace “irreducible” by “square-free” in the preceding description of the outcome.

 

 

 

 

 

 

 

《呂氏春秋‧慎行論》

察傳

夫得言不可以不察，數傳而白為黑，黑為白。故狗似玃，玃似母猴 ，母猴似人，人之與狗則遠矣。此愚者之所以大過也。聞而審則為福矣，聞而不審，不若無聞矣。

齊桓公聞管子於鮑叔，楚莊聞孫叔敖於沈尹筮，審之也，故國霸諸侯也。吳王聞越王句踐於太宰嚭，智伯聞趙襄子於張武，不審也，故國亡身死也。

凡聞言必熟論，其於人必驗之以理。魯哀公問於孔子曰：「樂正夔一足，信乎？」孔子曰：「昔者舜欲以樂傳教於天下，乃令重黎舉夔於草莽之中而進之，舜以為樂正。夔於是正六律，和五聲 ，以通八風，而天下大服。重黎又欲益求人，舜曰：『夫樂，天地之精也 ，得失之節也，故唯聖人為能和。樂之本也。夔能和之 ，以平天下 。若夔者一而足矣。』故曰夔一足，非一足也。」

宋之丁氏，家無井而出溉汲，常一人居外。及其家穿井，告人曰：「吾穿井得一人 。」有聞而傳之者曰：「丁氏穿井得一人 。」國人道之，聞之於宋君，宋君令人問之於丁氏，丁氏對曰：「得一人之使，非得一人於井中也。」求能之若此，不若無聞也。

子夏之晉，過衛，有讀史記者曰：「晉師三豕涉河。」子夏曰：「非也，是己亥也。夫『己』與『三』相近，『豕』與『亥』相似 。」至於晉而問之，則曰「晉師己亥涉河」也。辭多類非而是 ，多類是而非。是非之經，不可不分，此聖人之所慎也。然則何以慎？緣物之情及人之情以為所聞則得之矣。

古早的中國喜用『類比』來論事說理，難道這就是『科學』不興的原因嗎？李約瑟在其大著《中國的科學與文明》試圖解決這個今稱『李約瑟難題』之大哉問！終究還是百家爭鳴也？若是比喻的說︰一個孤立隔絕系統之演化，常因內部機制的折衝協調，周遭環境之影響相對的小很多。因此秦之『大一統』，歷代的『戰亂』頻起，能不達於『社會』之『平衡』的耶？？如此『主流價值』亦是已然確立成為『文化內涵』的吧！！所以『天不變』、『道不變』，人亦『不變』乎！！？？雖然李約瑟曾經明示『類比』── 關聯式 corelative thinking 思考 ── 難以建立完整的『邏輯體系』， 或是『科學』不興的理由耶？？！！如果『自然事物』之『邏輯推理』能形成系統『大樹』，那麼『類比關聯』將創造體系『森林』矣，豈可不『慎察』也。

類比（英語：Analogy，源自古希臘語：ἀναλογία，analogia，意為等比例的），或類推，是一種認知過程，將某個特定事物所附帶的訊息轉移到其他特定事物之上。類比通過比較兩件事情，清楚揭示二者之間的相似點，並將已知事物的特點，推衍到未知事物中，但兩者不一定有實質上的同源性，其類比也不見得「合理 」。在記憶、溝通與問題解決等過程中扮演重要角色；於不同學科中也有各自的定義。

舉例而言，原子中的原子核以及由電子組成的軌域，可類比成太陽系中行星環繞太陽的樣子。除此之外，修辭學中的譬喻法有時也是一種類比，例如將月亮比喻成銀幣。生物學中因趨同演化而形成的的同功或同型解剖構造，例如哺乳類、爬行類與鳥類的翅膀也是類似概念。

───

Analogy

Analogy (from Greek ἀναλογία, analogia, “proportion”^[1][2]) is a cognitive process of transferring information or meaning from a particular subject (the analogue or source) to another (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from one particular to another particular, as opposed to deduction, induction, and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a similarity, as in the biological notion of analogy.

Analogy plays a significant role in problem solving such as, decision making, perception, memory, creativity, emotion, explanation, and communication. It lies behind basic tasks such as the identification of places, objects and people, for example, in face perception and facial recognition systems. It has been argued that analogy is “the core of cognition”.^[3] Specific analogical language comprises exemplification, comparisons, metaphors, similes, allegories, and parables, but not metonymy. Phrases like and so on, and the like, as if, and the very word like also rely on an analogical understanding by the receiver of a message including them. Analogy is important not only in ordinary language and common sense (where proverbs and idioms give many examples of its application) but also in science, philosophy, and the humanities. The concepts of association, comparison, correspondence, mathematical and morphological homology, homomorphism, iconicity, isomorphism, metaphor, resemblance, and similarity are closely related to analogy. In cognitive linguistics, the notion of conceptual metaphor may be equivalent to that of analogy.

Analogy has been studied and discussed since classical antiquity by philosophers, scientists, and lawyers. The last few decades have shown a renewed interest in analogy, most notably in cognitive science.

Rutherford’s model of the atom (modified by Niels Bohr) made an analogy between the atom and the solar system.

………

 

植種大樹，走入森林，方知

縱使宇宙萬有同源，萬象表現實在是錯綜複雜耶！！方了

世間書籍雖然汗牛充棟，原創概念往往卻沒有幾個？？

─ 《W!O+ 的《小伶鼬工坊演義》︰神經網絡【轉折點】四中》

 

如何了解『電容器』之『充』、『放』電呢？

維基百科詞條如是說︰

DC circuits

See also: RC circuit

A simple resistor-capacitor circuit demonstrates charging of a capacitor.

A series circuit containing only a resistor, a capacitor, a switch and a constant DC source of voltage V₀ is known as a charging circuit.^[26] If the capacitor is initially uncharged while the switch is open, and the switch is closed at t₀, it follows from Kirchhoff’s voltage law that

Taking the derivative and multiplying by C, gives a first-order differential equation:

At t = 0, the voltage across the capacitor is zero and the voltage across the resistor is V₀. The initial current is then I(0) =V₀/R. With this assumption, solving the differential equation yields

where τ₀ = RC, the time constant of the system. As the capacitor reaches equilibrium with the source voltage, the voltages across the resistor and the current through the entire circuit decay exponentially. In the case of a discharging capacitor, the capacitor’s initial voltage (V_Ci) replaces V₀. The equations become

 

lcapy 軟體這麼講︰

Simple transient analysis

Let’s consider a series R-C network in series with a DC voltage source

>>> from lcapy import *
>>> n = Vstep(20) + R(5) + C(10, 0)
>>> n
Vstep(20) + R(5) + C(10, 0)
>>> Voc = n.Voc(s)
>>> Voc
20
──
s
>>> n.Isc(s)
   4
────────
s + 1/50
>>> isc = n.Isc.transient_response()
>>> isc
⎧   -t
⎪   ───
⎨    50
⎪4⋅ℯ     for t ≥ 0

 

Here n is network formed by the components in series, and n.Voc(s) is the open-circuit s-domain voltage across the network. Note, this is the same as the s-domain value of the voltage source. n.Isc(s) is the short-circuit s-domain voltage through the network. The method transient_response converts this to the time-domain. Note, since the capacitor has the initial value specified, this network is analysed as an initial value problem and thus the result is not known for  . If the initial capacitor voltage is not specified, the network cannot be analysed.

Of course, the previous example can be performed symbolically,

>>> from lcapy import *
>>> n = Vstep('V_1') + R('R_1') + C('C_1', 0)
>>> n
Vstep(V₁) + R(R₁) + C(C₁, 0)
>>> Voc = n.Voc(s)
>>> Voc
V₁
──
s
>>> n.Isc(s)
      V₁
──────────────
   ⎛      1  ⎞
R₁⋅⎜s + ─────⎟
   ⎝    C₁⋅R₁⎠
>>> isc = n.Isc.transient_response()
>>> isc
⎧     -t
⎪    ─────
⎪    C₁⋅R₁
⎨V₁⋅ℯ
⎪─────────  for t ≥ 0
⎪    R₁
⎩

 

※ 參考

 

內容十分一致。

若問那個 RC 電路是『直流電路』嗎？

lcapy 『電路分析』 

Circuit Analysis

>>> cct1 = (Vstep(10) + R(1)) | C(2)

>>> cct2 = Circuit()
>>> cct2.add('V 1 0 step 10')
>>> cct2.add('R 1 2 1')
>>> cct2.add('C 2 0 2')

The two approaches have many attributes and methods in common. For example,

>>> cct1.is_causal
True
>>> cct2.is_causal
True
>>> cct1.is_dc
False
>>> cct2.is_dc
False

However, there are subtle differences. For example,

>>> cct1.Voc.s
   5
──────
 2   s
s  + ─
     2

>>> cct2.Voc(2, 0).s
   5
──────
 2   s
s  + ─
     2

Notice, the second example requires specific nodes to determine the open-circuit voltage across.

……

Laplace analysis

The response due to a transient excitation from an independent source can be analysed using Laplace analysis. Since the unilateral transform is not unique (it ignores the circuit behaviour for  , the response can only be determined for  .

If the independent sources are known to be causal (a causal signal is zero for  analogous to a causal impulse response) and the initial conditions (i.e., the voltages across capacitors and currents through inductors) are zero, then the response is 0 for  . Thus in this case, the response can be specified for all  .

The response due to a general non-causal excitation is hard to determine using Laplace analysis. One strategy is to use circuit analysis techniques to determine the response for  , compute the pre-initial conditions, and then use Laplace analysis to determine the response for  . Note, the pre-initial conditions at  are required. These differ from the initial conditions at  whenever a Dirac delta (or its derivative) excitation is considered. Determining the initial conditions is not straightforward for arbitrary excitations and at the moment Lcapy expects you to do this!

The use of pre-initial conditions also allows switching circuits to be considered (see Switching analysis). In this case the independent sources are ignored for  and the result is only known for  .

Note if any of the pre-initial conditions are non-zero and the independent sources are causal then either we have an initial value problem or a mistake has been made. Lcapy assumes that if any of the inductors and capacitors have explicit initial conditions, then the circuit is to be analysed as an initial value problem with the independent sources ignored for  . In this case a DC source is not DC since it is considered to switch on at  .

………

Switching analysis

Whenever a circuit has a switch it is time variant. The opening or closing of switch changes the circuit and can produce transients. While a switch violates the LTI requirements for linear circuit analysis, the circuit prior to the switch changing can be analysed and Vnoiused to determine the initial conditions for the circuit after the switched changed. Lcapy requires that you do this! The independent sources are ignored for  and the result is only known for  .

───

 

範例

※ 註

28.3. pdb — The Python Debugger

Source code: Lib/pdb.py

---

The module pdb defines an interactive source code debugger for Python programs. It supports setting (conditional) breakpoints and single stepping at the source line level, inspection of stack frames, source code listing, and evaluation of arbitrary Python code in the context of any stack frame. It also supports post-mortem debugging and can be called under program control.

The debugger is extensible – it is actually defined as the class Pdb. This is currently undocumented but easily understood by reading the source. The extension interface uses the modules bdb and cmd.

The debugger’s prompt is (Pdb). Typical usage to run a program under control of the debugger is:

>>> import pdb
>>> import mymodule
>>> pdb.run('mymodule.test()')
> <string>(0)?()
(Pdb) continue
> <string>(1)?()
(Pdb) continue
NameError: 'spam'
> <string>(1)?()
(Pdb)

 

裡有答案︰不是！

 

但是讀過『直流』

Direct current

Direct Current (red line). The vertical axis shows current or voltage and the horizontal ‘t’ axis measures time and shows the zero value.

Direct current (DC) is the unidirectional flow of electric charge. A battery is a good example of a DC power supply. Direct current may flow in a conductor such as a wire, but can also flow through semiconductors,insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.^[1]

The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.^[2][3]

Direct current may be obtained from an alternating current supply by use of a rectifier, which contains electronic elements (usually) or electromechanical elements (historically) that allow current to flow only in one direction. Direct current may be converted into alternating current with an inverter or a motor-generator set.

Direct current is used to charge batteries and as power supply for electronic systems. Very large quantities of direct-current power are used in production of aluminum and other electrochemical processes. It is also used for some railways, especially in urban areas. High-voltage direct current is used to transmit large amounts of power from remote generation sites or to interconnect alternating current power grids.

……

Various definitions

Types of direct current

The term DC is used to refer to power systems that use only one polarity of voltage or current, and to refer to the constant, zero-frequency, or slowly varying local mean value of a voltage or current.^[9] For example, the voltage across a DC voltage source is constant as is the current through a DC current source. The DC solution of an electric circuit is the solution where all voltages and currents are constant. It can be shown that any stationary voltage or current waveform can be decomposed into a sum of a DC component and a zero-mean time-varying component; the DC component is defined to be the expected value, or the average value of the voltage or current over all time.

Although DC stands for “direct current”, DC often refers to “constant polarity”. Under this definition, DC voltages can vary in time, as seen in the raw output of a rectifier or the fluctuating voice signal on a telephone line.

Some forms of DC (such as that produced by a voltage regulator) have almost no variations in voltage, but may still have variations in output power and current.

Circuits

A direct current circuit is an electrical circuit that consists of any combination of constant voltage sources, constant current sources, and resistors. In this case, the circuit voltages and currents are independent of time. A particular circuit voltage or current does not depend on the past value of any circuit voltage or current. This implies that the system of equations that represent a DC circuit do not involve integrals or derivatives with respect to time.

If a capacitor or inductor is added to a DC circuit, the resulting circuit is not, strictly speaking, a DC circuit. However, most such circuits have a DC solution. This solution gives the circuit voltages and currents when the circuit is in DC steady state. Such a circuit is represented by a system of differential equations. The solution to these equations usually contain a time varying or transient part as well as constant or steady state part. It is this steady state part that is the DC solution. There are some circuits that do not have a DC solution. Two simple examples are a constant current source connected to a capacitor and a constant voltage source connected to an inductor.

In electronics, it is common to refer to a circuit that is powered by a DC voltage source such as a battery or the output of a DC power supply as a DC circuit even though what is meant is that the circuit is DC powered.

 

詞條的人，又會怎麼判斷呢？？

因此藉『類比』想象者，能不慎察『異同』也！！

Hydraulic analogy

The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) ^[1] is the most widely used analogy for “electron fluid” in a metal conductor. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. Electricity (as well as heat) was originally understood to be a kind of fluid, and the names of certain electric quantities (such as current) are derived from hydraulic equivalents. As with all analogies, it demands an intuitive and competent understanding of the baseline paradigms (electronics and hydraulics).

Analogy between a hydraulic circuit (left) and an electronic circuit (right).

Limits to the analogy

If taken too far, the water analogy can create misconceptions. For it to be useful, one must remain aware of the regions where electricity and water behave very differently.

Fields (Maxwell equations, Inductance): Electrons can push or pull other distant electrons via their fields, while water molecules experience forces only from direct contact with other molecules. For this reason, waves in water travel at the speed of sound, but waves in a sea of charge will travel much faster as the forces from one electron are applied to many distant electrons and not to only the neighbors in direct contact. In a hydraulic transmission line, the energy flows as mechanical waves through the water, but in an electric transmission line the energy flows as fields in the space surrounding the wires, and does not flow inside the metal. Also, an accelerating electron will drag its neighbors along while attracting them, both because of magnetic forces.

Charge: Unlike water, movable charge carriers can be positive or negative, and conductors can exhibit an overall positive or negative net charge. The mobile carriers in electric currents are usually electrons, but sometimes they are charged positively, such as the positive ions in an electrolyte, the H⁺ ions in proton conductors or holes in p-type semiconductors and some (very rare) conductors.

Leaking pipes: The electric charge of an electrical circuit and its elements is usually almost equal to zero, hence it is (almost) constant. This is formalized in Kirchhoff’s current law, which does not have an analogy to hydraulic systems, where amount of the liquid is not usually constant. Even with incompressible liquid the system may contain such elements as pistons and open pools, so the volume of liquid contained in a part of the system can change. For this reason, continuing electric currents require closed loops rather than hydraulics’ open source/sink resembling spigots and buckets.

Fluid velocity and resistance of metals: As with water hoses, the carrier drift velocity in conductors is directly proportional to current. However, water only experiences drag via the pipes’ inner surface, while charges are slowed at all points within a metal, as with water forced through a filter. Also, typical velocity of charge carriers within a conductor is less than centimeters per minute, and the “electrical friction” is extremely high. If charges ever flowed as fast as water can flow in pipes, the electric current would be immense, and the conductors would become incandescently hot and perhaps vaporize. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe. Resistance in most electrical conductors is a linear function: as current increases, voltage drop increases proportionally (Ohm’s Law). Liquid resistance in pipes is not linear with volume, varying as the square of volumetric flow (see Darcy–Weisbach equation).

Quantum Mechanics: Solid conductors and insulators contain charges at more than one discrete level of atomic orbit energy, while the water in one region of a pipe can only have a single value of pressure. For this reason there is no hydraulic explanation for such things as a battery‘s charge pumping ability, a diode‘s depletion layer and voltage drop, solar cell functions, Peltier effect, etc., however equivalent devices can be designed which exhibit similar responses, although some of the mechanisms would only serve to regulate the flow curves rather than to contribute to the component’s primary function.

In order for the model to be useful, the reader or student must have a substantial understanding of the model (hydraulic) system’s principles. It also requires that the principles can be transferred to the target (electrical) system. Hydraulic systems are deceptively simple: the phenomenon of pump cavitation is a known, complex problem that few people outside of the fluid power or irrigation industries would understand. For those who do, the hydraulic analogy is amusing, as no “cavitation” equivalent exists in electrical engineering. The hydraulic analogy can give a mistaken sense of understanding that will be exposed once a detailed description of electrical circuit theory is required.

One must also consider the difficulties in trying to make an analogy match reality completely. The above “electrical friction” example, where the hydraulic analog is a pipe filled with sponge material, illustrates the problem: the model must be increased in complexity beyond any realistic scenario.

 

 

 

 

 

 

 

 

大自然中與日常生活裡，有許多靜電現象的例子，像是塑膠袋與手之間的吸引、冬天脫毛衣的劈啪聲、彷彿是自發性的穀倉爆炸與用手接觸電子元件可能的損毀和影印機的運作原理等等。

─── 《【SONIC Π】電聲學導引《二》》

 

懷古之幽思，重新認識『電容器』！！

Capacitor

Type	Passive
Invented	Ewald Georg von Kleist
Electronic symbol

A capacitor is a passive two-terminal electrical component that stores potential energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator.^[1] The original name is still widely used in many languages, but not in English.

The physical form and construction of practical capacitors vary widely and many capacitor types are in common use. Most capacitors contain at least two electrical conductors often in the form of metallic plates or surfaces separated by a dielectric medium. A conductor may be a foil, thin film, sintered bead of metal, or an electrolyte. The nonconducting dielectric acts to increase the capacitor’s charge capacity. Materials commonly used as dielectrics include glass, ceramic, plastic film, paper, mica, and oxide layers. Capacitors are widely used as parts of electrical circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not dissipate energy.

When two conductors experience a potential difference, for example, when a capacitor is attached across a battery, an electric field develops across the dielectric, causing a net positive charge to collect on one plate and net negative charge to collect on the other plate. No current actually flows through the dielectric, however, there is a flow of charge through the source circuit. If the condition is maintained sufficiently long, the current through the source circuit ceases. However, if a time-varying voltage is applied across the leads of the capacitor, the source experiences an ongoing current due to the charging and discharging cycles of the capacitor.

Capacitance is defined as the ratio of the electric charge on each conductor to the potential difference between them. The unit of capacitance in the International System of Units (SI) is the farad (F), defined as one coulomb per volt (1 C/V). Capacitance values of typical capacitors for use in general electronics range from about 1 picofarad (pF) (10⁻¹² F) to about 1 millifarad (mF) (10⁻³ F).

The capacitance of a capacitor is proportional to the surface area of the plates (conductors) and inversely related to the gap between them. In practice, the dielectric between the plates passes a small amount of leakage current. It has an electric field strength limit, known as the breakdown voltage. The conductors and leads introduce an undesired inductance and resistance.

Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems, they stabilize voltage and power flow.^[2] The property of energy storage in capacitors was exploited as dynamic memory in early digital computers.^[3]

Theory of operation

Overview

A simple demonstration capacitor made of two parallel metal plates, using an air gap as the dielectric.

Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance.

A capacitor consists of two conductors separated by a non-conductive region.^[17] The non-conductive region can either be a vacuum or an electrical insulator material known as a dielectric. Examples of dielectric media are glass, air, paper, plastic, ceramic, and even a semiconductor depletion region chemically identical to the conductors. From Coulomb’s law a charge on one conductor will exert a force on the charge carriers within the other conductor, attracting opposite polarity charge and repelling like polarity charges, thus an opposite polarity charge will be induced on the surface of the other conductor. The conductors thus hold equal and opposite charges on their facing surfaces,^[18] and the dielectric develops an electric field.

An ideal capacitor is characterized by a constant capacitance C, in farads in the SI system of units, defined as the ratio of the positive or negative charge Q on each conductor to the voltage V between them:^[17]

A capacitance of one farad (F) means that one coulomb of charge on each conductor causes a voltage of one volt across the device.^[19] Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than when the conductors are separated, yielding a larger capacitance.

In practical devices, charge build-up sometimes affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes:

Hydraulic analogy

In the hydraulic analogy, a capacitor is analogous to a rubber membrane sealed inside a pipe— this animation illustrates a membrane being repeatedly stretched and un-stretched by the flow of water, which is analogous to a capacitor being repeatedly charged and discharged by the flow of charge

In the hydraulic analogy, charge carriers flowing through a wire are analogous to water flowing through a pipe. A capacitor is like a rubber membrane sealed inside a pipe. Water molecules cannot pass through the membrane, but some water can move by stretching the membrane. The analogy clarifies a few aspects of capacitors:

• The current alters the charge on a capacitor, just as the flow of water changes the position of the membrane. More specifically, the effect of an electric current is to increase the charge of one plate of the capacitor, and decrease the charge of the other plate by an equal amount. This is just as when water flow moves the rubber membrane, it increases the amount of water on one side of the membrane, and decreases the amount of water on the other side.
• The more a capacitor is charged, the larger its voltage drop; i.e., the more it “pushes back” against the charging current. This is analogous to the more a membrane is stretched, the more it pushes back on the water.
• Charge can flow “through” a capacitor even though no individual electron can get from one side to the other. This is analogous to water flowing through the pipe even though no water molecule can pass through the rubber membrane. The flow cannot continue in the same direction forever; the capacitor experiences dielectric breakdown, and analogously the membrane will eventually break.
• The capacitance describes how much charge can be stored on one plate of a capacitor for a given “push” (voltage drop). A very stretchy, flexible membrane corresponds to a higher capacitance than a stiff membrane.
• A charged-up capacitor is storing potential energy, analogously to a stretched membrane.

Parallel-plate model

Parallel plate capacitor model consists of two conducting plates, each of area A, separated by a gap of thickness d containing a dielectric.

The simplest model capacitor consists of two thin parallel conductive plates each with an area of  separated by a uniform gap of thickness  filled with a dielectric with permittivity  . It is assumed the gap  is much smaller than the dimensions of the plates. This model applies well to many practical capacitors which are constructed of metal sheets separated by a thin layer of insulating dielectric, since manufacturers try to keep the dielectric very uniform in thickness to avoid thin spots which can cause failure of the capacitor.

Since the separation between the plates is uniform over the plate area, the electric field between the plates  is constant, and directed perpendicularly to the plate surface, except for an area near the edges of the plates where the field decreases because the electric field lines “bulge” out of the sides of the capacitor. This “fringing field” area is approximately the same width as the plate separation,  , and assuming  is small compared to the plate dimensions, it is small enough to be ignored. Therefore, if a charge of  is placed on one plate and  on the other plate, the charge on each plate will be spread evenly in a surface charge layer of constant charge density  coulombs per square meter, on the inside surface of each plate. From Gauss’s law the magnitude of the electric field between the plates is  . The voltage  between the plates is defined as the line integralof the electric field over a line from one plate to another

The capacitance is defined as  . Substituting  above into this equation

Therefore, in a capacitor the highest capacitance is achieved with a high permittivity dielectric material, large plate area, and small separation between the plates.

Since the area  of the plates increases with the square of the linear dimensions and the separation  increases linearly, the capacitance scales with the linear dimension of a capacitor (), or as the cube root of the volume.

A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. The capacitor’s dielectric material has a dielectric strength U_d which sets the capacitor’s breakdown voltage at V = V_bd = U_dd. The maximum energy that the capacitor can store is therefore

The maximum energy is a function of dielectric volume, permittivity, and dielectric strength. Changing the plate area and the separation between the plates while maintaining the same volume causes no change of the maximum amount of energy that the capacitor can store, so long as the distance between plates remains much smaller than both the length and width of the plates. In addition, these equations assume that the electric field is entirely concentrated in the dielectric between the plates. In reality there are fringing fields outside the dielectric, for example between the sides of the capacitor plates, which increase the effective capacitance of the capacitor. This is sometimes called parasitic capacitance. For some simple capacitor geometries this additional capacitance term can be calculated analytically.^[20] It becomes negligibly small when the ratios of plate width to separation and length to separation are large.

Energy stored in a capacitor

To increase the charge and voltage on a capacitor, work must be done by an external power source to move charge from the negative to the positive plate against the opposing force of the electric field.^[21][22] If the voltage on the capacitor is  , the work  required to move a small increment of charge  from the negative to the positive plate is  . The energy is stored in the increased electric field between the plates. The total energy stored in a capacitor is equal to the total work done in establishing the electric field from an uncharged state.^[23][22][21]

where  is the charge stored in the capacitor,  is the voltage across the capacitor, and  is the capacitance. This potential energy will remain in the capacitor until the charge is removed. If charge is allowed to move back from the positive to the negative plate, for example by connecting a circuit with resistance between the plates, the charge moving under the influence of the electric field will do work on the external circuit.

If the gap between the capacitor plates  is constant, as in the parallel plate model above, the electric field between the plates will be uniform (neglecting fringing fields) and will have a constant value  . In this case the stored energy can be calculated from the electric field strength

The last formula above is equal to the energy density per unit volume in the electric field multiplied by the volume of field between the plates, confirming that the energy in the capacitor is stored in its electric field.

Current–voltage relation

The current I(t) through any component in an electric circuit is defined as the rate of flow of a charge Q(t) passing through it, but actual charges—electrons—cannot pass through the dielectric layer of a capacitor. Rather, one electron accumulates on the negative plate for each one that leaves the positive plate, resulting in an electron depletion and consequent positive charge on one electrode that is equal and opposite to the accumulated negative charge on the other. Thus the charge on the electrodes is equal to the integralof the current as well as proportional to the voltage, as discussed above. As with any antiderivative, a constant of integration is added to represent the initial voltage V(t₀). This is the integral form of the capacitor equation:^[24]

Taking the derivative of this and multiplying by C yields the derivative form:^[25]

The dual of the capacitor is the inductor, which stores energy in a magnetic field rather than an electric field. Its current-voltage relation is obtained by exchanging current and voltage in the capacitor equations and replacing C with the inductance L.

 

 

 

 

 

 

 

什麼是『人工智慧』AI Artificial Intelligence 呢？『人工』意味著人為的，它不是天生的，設想萬年前的人將如何區分現今的『人造物』或『自然物』呢？人能定義什麼是『人造物』嗎？反過來說人能定義何謂『自然物』的嗎？至於『智慧』一詞就更難講清的了，不如引用莊子在《莊子‧齊物論》──  文本摘自漢川草廬 ── 的一段議論吧︰

據聞今年六月八日，首度俄羅斯有一個命名為『尤金‧古斯特曼 』  Eugene Goostman 的『人工智慧聊天機器人』通過了圖靈測試。那麼是否一台機器通過了『圖靈測試』就能宣稱它真是具有『智慧』的呢？

二零零二年美國卡內基梅隆大學的 Luis von Ahn、Manuel Blum、Nicholas J.Hopper以及 IBM 的 John Langford 聯合提出了『CAPTCHA』── Completely Automated Public Turing test to tell Computers and Humans Apart ──。一般又叫做『驗證碼』，常常用於『□□下載』網頁，想要確認來下載的『真的是人』。它有著多種形式的設計，常見的是要使用者輸入『扭曲變形』的『文字』或是『數字』，這將使得大多數『OCR』這類圖像文字辨識軟體，無法自動判讀，所以很難寫個『軟體自動下載』。

雖說都是『圖靈測試』，一者想『不能分辨』，另者要『能夠區別』，不知最終的『AI』是『能』還是『不能』的呢？？

─── 《人工智慧！！》

 

『科學精神』意在求真，大概不需要『詭辯矛盾』之非吧！

『技術創新』追求圓滿，終究免不了『善惡美醜』之是哩！

故爾針對

Under-voltage detected! (0x00050005) … how to disable?

文本之『硬體』和『軟件』及其『偵測訊息』該與不該對誰顯示的問題，理當置於『是是非非』之外乎？

不巧人間『價值衝突』常有『是其所非，非其所是』之爭耶？？

所以借著

Under-voltage detected! (0x00050005) spams dmesg on new kernel 4.14.30-v7+ #2512

 

書己一二成見，非為議論呦！！

E3V3A commented Apr 16, 2018

After upgrade of kernel to 4.14. dmesg is now spammed by Under-voltage detected! (0x00050005)messages where no problem was shown previously. I’ve ran this device non-stop for months, without any problem until after update, so under-voltage level settings or other config must have changed.
Spamming dmesg or journlctl --system buffer certainly is not helping anyone.

kern  :crit  : [ 1701.464833 <    2.116656>] Under-voltage detected! (0x00050005)
kern  :info  : [ 1707.668180 <    6.203347>] Voltage normalised (0x00000000)

Also related to #2367

…

pelwell commented Apr 16, 2018

The kernel under voltage notification is new, but the threshold and detection mechanism is unchanged. You are now being made aware of the fact that your Pi is insufficiently powered for the load placed upon it. This is bad for performance and potentially harmful to system stability.

……

E3V3A commented Apr 16, 2018

I’m well aware of the working of the RPi power supply. But the fact of the matter is that:

• It didn’t happen with the earlier kernel, so what make you think this is an improvement?
  (The flash icon was good and annoying enough!)
• Spamming kernel messages (and all other related buffers) with repeated messages on a device where which has proven to be working just fine under those conditions before, is now risking excessive SD card wear and harder debugging because older and more relevant kernel messages get FIFO’d out eventually.
• The current crit level doesn’t even respect the printk settings and keep spamming even after setting dmesg -n 1 or using sysctl -w '1 1 1 1'. So AFAICT, this is neither critial, nor compliant to standard *nix behavior, and does not provide any improvement whatsoever.

………

ThomasKaiser commented Apr 18, 2018

In conclusion, the only serious solution for me (and you) seem to be to revert to kernel 4.9 and everyone will be happy again.

Simply create /etc/rsyslog.d/ignore-underpowering.conf with :msg, contains, "oltage" ~ and you can enjoy an instable system even with kernel 4.14 🙂

BTW: Just found it. There are SBC that allow for constant input voltage monitoring. What you can see here is a PSU that provided 5.25V in the beginning after approximately 1.5 years of constant operation:https://forum.armbian.com/topic/5699-how-to-provide-and-interpret-debug-output/?do=findComment&comment=44210 — DC-IN dropped as low as 4.2V with some light load (this board has also a good PMIC and a large battery and power circuitry uses boost converters to provide stable voltages to all subsystems, USB and SATA included)

………

E3V3A commented Apr 18, 2018

@ThomasKaiser
I edited the rsyslog.d config files as you mentioned in the default /etc/rsyslog.conf with and without tabs, like this:

:msg, contains, "oltage" ~

Indeed this removes the voltage related logs from the /var/log/*.log files. 👍
But apparently dmesg which is using /dev/kmsg and /proc/kmsg, seem independent of syslogd andrsyslogd settings, and thus still show all under-voltage entries as before with dmesg -e -x. But I guess I can live with that.

Regarding the input voltage, I am surprised that the detector is able to measure the voltage to the second decimal 4.63, but that there is no way to read it from /sys. What is that all about? How and what does the device actually measure when the voltage is lower than that threshold?

Either way I’ll report back, once I have the values. In the process of all this investigation I’ve unfortunately found a wide range of other unpleasant surprises coming from this update. All sorts of things, like overwriting ALSA configurations, starting services that was never ran before, automatically running apt upgrade, etc. 🙁

………

jacobq commented Apr 25, 2018

I don’t want to get mixed-up in this very long winded discussion, but FWIW I will say that I stumbled across it looking for a way to suppress kernel messages from the console (in my experience this has made bad problems worse as I’m trying to triage things and shutdown but get messages printed right over files in my editor, etc.) and there are some ways to do this, such as dmesg -n 1 see
https://superuser.com/questions/351387/how-to-stop-kernel-messages-from-flooding-my-console#answer-351402
A previous comment suggested that this does not work, but it seemed to work fine for my purposes (i.e. on RPi 3 B+ it stopped kernel messages from getting printed to my console though they still appear in the output of dmesg)

───

 

『知的權利』和『資訊管理』真是水火不相容嘛！？

『事件訊息』與『散布控制』果為對立的兩極嗎？！

『真』『假』時而並生，『善』『惡』歸結難料☆☆★★

宜乎尊重專業，信賴其人判斷耶？？！！

因此 Linux kernel 早有 sysctl 之設的焉！！？？

行或不行驗證容易也☆

pi@raspberrypi:~  more /etc/sysctl.conf 
#
# /etc/sysctl.conf - Configuration file for setting system variables
# See /etc/sysctl.d/ for additional system variables.
# See sysctl.conf (5) for information.
#

#kernel.domainname = example.com

# Uncomment the following to stop low-level messages on console
#kernel.printk = 3 4 1 3

##############################################################3
# Functions previously found in netbase
#

# Uncomment the next two lines to enable Spoof protection (reverse-path filter)
# Turn on Source Address Verification in all interfaces to
# prevent some spoofing attacks
#net.ipv4.conf.default.rp_filter=1
#net.ipv4.conf.all.rp_filter=1

# Uncomment the next line to enable TCP/IP SYN cookies
# See http://lwn.net/Articles/277146/
--More--(27%)

 

至於 的值，在選定

MF-MSMF-250/X ，

Resettable fuse

A resettable fuse is a polymeric positive temperature coefficient (PPTC) device that is a passive electronic component used to protect against overcurrent faults in electronic circuits. The device is also known as a polyfuse orpolyswitch. They are similar in function to PTC thermistors in certain situations but operate on mechanical changes instead of charge carrier effects in semiconductors. These devices were first discovered and described by Gerald Pearson at Bell Labs in 1939 and described in US patent #2,258,958.^[1]

Resettable fuses – PolySwitch devices

Operation

A polymeric PTC device is made up of a non-conductive crystalline organic polymer matrix that is loaded with carbon black particles^[2] to make it conductive. While cool, the polymer is in a crystalline state, with the carbon forced into the regions between crystals, forming many conductive chains. Since it is conductive (the “initial resistance”),^[3] it will pass a current. If too much current is passed through the device the device will begin to heat. As the device heats, the polymer will expand, changing from a crystalline into an amorphous state.^[4] The expansion separates the carbon particles and breaks the conductive pathways, causing the device to heat faster and expand more, further raising the resistance.^[5] This increase in resistance substantially reduces the current in the circuit. A small (leakage) current still flows through the device and is sufficient to maintain the temperature at a level which will keep it in the high resistance state. Leakage current can range from less than a hundred mA at rated voltage up to a few hundred mA at lower voltages. The device can be said to have latching functionality.^[6] The hold current is the maximum current at which the device is guaranteed not to trip. The trip current is the current at which the device is guaranteed to trip.^[7]

When power is removed, the heating due to the leakage current will stop and the PPTC device will cool. As the device cools, it regains its original crystalline structure and returns to a low resistance state where it can hold the current as specified for the device.^[6] This cooling usually takes a few seconds, though a tripped device will retain a slightly higher resistance for hours, slowly approaching the initial resistance value. The resetting will often not take place even if the fault alone has been removed with the power still flowing as the operating current may be above the holding current of the PPTC. The device may not return to its original resistance value; it will most likely stabilize at a significantly higher resistance (up to 4 times initial value). It could take hours, days, weeks or even years for the device to return to a resistance value similar to its original value, if at all.^[8]

A PPTC device has a current rating and a voltage rating.^[9]

 

之時，就已經受限，實在沒有多少餘裕度的呀◎

 

 

 

 

 

 

 

 

─── 《M♪O 之學習筆記本《丑》控制︰【白金西】時回流反》

 

閱讀『美國線規』詞條有何樂趣耶？

American wire gauge

American wire gauge (AWG), also known as the Brown & Sharpe wire gauge, is a logarithmic stepped standardized wire gauge system used since 1857 predominantly in North America for the diameters of round, solid, nonferrous, electrically conducting wire. Dimensions of the wires are given in ASTM standard B 258.^[1] The cross-sectional area of each gauge is an important factor for determining its current-carrying capacity.

Increasing gauge numbers denote decreasing wire diameters, which is similar to many other non-metric gauging systems such as British Standard Wire Gauge (SWG), but unlike IEC 60228, the metric wire-size standard used in most parts of the world. This gauge system originated in the number of drawing operations used to produce a given gauge of wire. Very fine wire (for example, 30 gauge) required more passes through the drawing dies than 0 gauge wire did. Manufacturers of wire formerly had proprietary wire gauge systems; the development of standardized wire gauges rationalized selection of wire for a particular purpose.

The AWG tables are for a single, solid, round conductor. The AWG of a stranded wire is determined by the cross-sectional area of the equivalent solid conductor. Because there are also small gaps between the strands, a stranded wire will always have a slightly larger overall diameter than a solid wire with the same AWG.

AWG is also commonly used to specify body piercing jewelry sizes (especially smaller sizes), even when the material is not metallic.^[2]

 

或可拿來尋幽探秘乎？？

比方藉著 Tables of AWG wire sizes 

Fusing current

索引，或能開拓視野，認識歷史裡人物事蹟也！

Preece

 

Onderdonk

 

爾後面對『紅色條規』 Code Red 時︰

勿將電壓源短路。

將會反思鍊結『高溫、熔斷、火災、爆炸 …』吧！！

故知，雖曾有此

並聯電路

電池

假設一個電池組是以幾個單電池以並聯方式連接成電源，則此電源兩端的電壓等於每一個單電池兩端的電壓。例如，假設一個電池組內部含有四個單電池並聯在一起，它們共同給出1安培電流，則每一個單電池給出0.25安培電流。很多年前，並聯在一起的電池組時常會被使用為無線電接收機內部真空管燈絲的電源，但這種用法現在已不常見。

 

之用法，需得借『開關』避免

 

無『負載』時的內耗哩☺

 

那麼仔細衡量 『範圍』及『安全顧慮』后，到底會不會定出

USB_(Physical)#Power

Where devices (for example, high-speed disk drives) require more power than a high-power device can draw,^[45] they function erratically, if at all, from bus power of a single port. USB provides for these devices as being self-powered. However, such devices may come with a Y-shaped cable that has two USB plugs (one for power and data, the other for only power), so as to draw power as two devices.^[46] Such a cable is non-standard, with the USB compliance specification stating that “use of a ‘Y’ cable (a cable with two A-plugs) is prohibited on any USB peripheral”, meaning that “if a USB peripheral requires more power than allowed by the USB specification to which it is designed, then it must be self-powered.”^[47]

 

Peripheral Power Consumption

Mandate: Required
Effective Date: Now

 

必須如是的呢☆★