生於公元十年卒於公元七十年之古希臘數學家亞歷山卓的希羅 Ἥρων ὁ Ἀλεξανδρεύς 居住在埃及托勒密時期的羅馬省。希羅是一名活躍於其家鄉的工程師，他被認為是古代最偉大的實驗家。在亞歷山大大帝征服波斯帝國後之希臘化時代的文明裡，他的著作於科學傳統方面享負盛名。由於希羅大部份的作品 ── 包含了數學、力學、物理和氣體力學 ── 都以講稿的形式出現，因此人們認為他曾經在繆斯之家教學，可能也在亞歷山大圖書館授課。

希羅的發明林林總總，有人說其中最著名的是『風琴』，這或許是最早利用『風能』的裝置。另一是稱作『汽轉球』的蒸汽機，這個『蒸汽機』可比『工業革命』早了二千年。在其著作《機械學與光學》中，描述了世界上第一部『自動販賣機』︰使用者將硬幣投入機器頂上的槽，槽接受了硬幣後，這台機器就會分配一定份量的『聖水』給投幣者。一般認為希羅是一位『原子論』者，他的一些思想源自於克特西比烏斯 Ctesibius 的著作，從他的各種發明來看，他的創造具有時代之『超越性』！

據聞希羅也是第一個體認到『虛數』 imaginary number 的人！！

$e^{i \pi} + 1 = 0$

一五七二年義大利數學家拉斐爾‧邦貝利 Rafael Bombelli 是文藝復興時期歐洲著名的工程師，也是一個卓越的數學家，出版了《代數學》 L’Algebra 一書，他在書中討論了『負數的平方根』 $\sqrt{- a}, \ a>0$ ，這在歐洲產生了廣泛影響力。

一六三七年笛卡爾在他的著作《幾何學》 La Géométrie 書中創造了『虛數』imaginary numbers 一詞，說明這種『真實上並不存在的數字』。

瑞士大數學家和物理學家李昂哈德‧尤拉 Leonhard Euler 傳說年輕時曾研讀神學，一生虔誠篤信上帝，並不能容忍任何詆毀上帝的言論在他面前發表。一回，德尼‧狄德羅 Denis Diderot ── 法國啟蒙思想家、唯物主義哲學家、無神論者和作家，百科全書派的代表 ── 造訪葉卡捷琳娜二世的宮廷，尤拉挑戰狄德羅說︰『先生， $e^{i \pi} + 1 = 0$ ，所以上帝存在，請回答！』。作者以為這或許只是個『杜撰』。然而尤拉是位多產的作家，一生著作有六十到八十巨冊。一七八三年九月十八日，晚餐後，尤拉邊喝著茶邊和小孫女玩耍，突然間，煙斗從他手中掉了下來。他說了聲：『我的煙斗』，將彎腰去撿，就再也沒有站起來了，他祇是抱著頭說了一句：『我死了』。法國哲學家馬奎斯‧孔多塞 marquis de Condorcet 講︰..il cessa de calculer et de vivre，『尤拉停止了計算和生命』！！

一七九七年挪威‧丹麥數學家卡斯帕爾‧韋塞爾 Caspar Wessel 在『Royal Danish Academy of Sciences and Letters』上發表了『Om directionens analytiske betegning』，提出了『複數平面』，研究了複數的幾何意義，由於是用『丹麥文』寫成的，幾乎沒有引起任何重視。一八零六年法國業餘數學家讓-羅貝爾‧阿爾岡 Jean-Robert Argand 與一八三一年德國著名大數學家约翰‧卡爾‧弗里德里希‧高斯 Johann Karl Friedrich Gauß 都再次『重新發現』同一結果！！

虛數軸和實數軸構成的平面稱作複數平面，複平面上每一點對應著一個複數。

─── 《【SONIC Π】電聲學補充《二》》

如果人已知廣義『複變正弦波』 Complex Sinusoids

$\displaystyle y(t) = \mathcal A e^{st}, \ where$

$\displaystyle \mathcal A = A \cdot e^{j \phi}$

$\displaystyle s = \sigma + j \omega$

可以統攝多種常用『積分變換』，故可將線性非時變 LTI 系統之『微分方程式』轉化成『代數方程組』來求解也。

而且能夠舉一反三，明白『慣性』 $m$ 之於『牛頓力學』 $\vec{F} = m \cdot \vec{a}$ ，度量『抵抗運動變化』的能力，或可比擬為『電路電阻』 $R$ 之於『歐姆定律』 $V = R \cdot I$ ！

那麼將怎麼看待這個『動量』

$\frac{d}{dt} \vec{p} = \vec{F}$

表達形式呢？難到『速度快』非運動改變之『阻抗』耶！？

炎日讀文，心靜自然涼乎？☆

Electrical impedance

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of a sinusoidal voltage between its terminals to the complex representation of the current flowing through it.^[1] In general, it depends upon the frequency of the sinusoidal voltage.

Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.

The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law. In multiple port networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix.^[2]

Impedance is a complex number, with the same units as resistance, for which the SI unit is the ohm ( $Ω$ ). Its symbol is usually $Z$ , and it may be represented by writing its magnitude and phase in the form $| Z | ∠θ$ . However, cartesian complex number representation is often more powerful for circuit analysis purposes.

The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.

Introduction

The term impedance was coined by Oliver Heaviside in July 1886.^[3]^[4] Arthur Kennelly was the first to represent impedance with complex numbers in 1893.^[5]

In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by the magnetic fields (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.

Impedance is defined as the frequency domain ratio of the voltage to the current.^[6] In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency $ω$ .

For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular:

The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude;
the phase of the complex impedance is the phase shift by which the current lags the voltage.

Quantitatively, the impedance of a two-terminal network is represented as a complex quantity $\displaystyle \scriptstyle Z$ $\scriptstyle Z$ , defined in Cartesian form as

\displaystyle \ Z=R+jX.

Here the real part of impedance is the resistance $\displaystyle \scriptstyle R$ $\scriptstyle R$ , and the imaginary part is the reactance $\displaystyle \scriptstyle X$ $\scriptstyle X$ .

The polar form conveniently captures both magnitude and phase characteristics as

\displaystyle \ Z=|Z|e^{j\arg(Z)}

where the magnitude $\displaystyle \scriptstyle |Z|$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument $\displaystyle \scriptstyle \arg(Z)$ (commonly given the symbol $\displaystyle \scriptstyle \theta$ gives the phase difference between voltage and current. $\displaystyle \scriptstyle j$ is the imaginary unit, and is used instead of $\displaystyle \scriptstyle i$ in this context to avoid confusion with the symbol for electric current.

Where it is needed to add or subtract impedances, the cartesian form is more convenient; but when quantities are multiplied or divided, the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers.

Complex impedance

A graphical representation of the complex impedance plane

The impedance of a two-terminal circuit element is represented as a complex quantity $\displaystyle \scriptstyle Z$ and the term complex impedance may also be used.

The polar form conveniently captures both magnitude and phase characteristics as

\displaystyle \ Z=|Z|e^{j\arg(Z)}

where the magnitude $\displaystyle \scriptstyle |Z|$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument $\displaystyle \scriptstyle \arg(Z)$ (commonly given the symbol $\displaystyle \scriptstyle \theta }$ gives the phase difference between voltage and current. $\displaystyle \scriptstyle j$ is theimaginary unit, and is used instead of $\displaystyle \scriptstyle i$ in this context to avoid confusion with the symbol for electric current.

In Cartesian form, impedance is defined as

\displaystyle \ Z=R+jX

where the real part of impedance is the resistance $\displaystyle \scriptstyle R$ and the imaginary part is the reactance $\displaystyle \scriptstyle X$ .

Complex voltage and current

Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.

To simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as $\displaystyle \scriptstyle V$ and $\displaystyle \scriptstyle I$ $\scriptstyle I$ .^[7]^[8]

\displaystyle {\begin{aligned}V&=|V|e^{j(\omega t+\phi _{V})},\\I&=|I|e^{j(\omega t+\phi _{I})}.\end{aligned}}

The impedance of a bipolar circuit is defined as the ratio of these quantities:

\displaystyle Z={\frac {V}{I}}={\frac {|V|}{|I|}}e^{j(\phi _{V}-\phi _{I})}.

Hence, denoting $\displaystyle \theta =\phi _{V}-\phi _{I}$ , we have

\displaystyle {\begin{aligned}|V|&=|I||Z|,\\\phi _{V}&=\phi _{I}+\theta .\end{aligned}}

The magnitude equation is the familiar Ohm’s law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation

This representation using complex exponentials may be justified by noting that (by Euler’s formula):

\displaystyle \ \cos(\omega t+\phi )={\frac {1}{2}}{\Big [}e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}{\Big ]}

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

\displaystyle \ \cos(\omega t+\phi )=\Re {\Big \{}e^{j(\omega t+\phi )}{\Big \}}

Ohm’s law

The meaning of electrical impedance can be understood by substituting it into Ohm’s law.^[9]^[10] Assuming a two-terminal circuit element with impedance $\displaystyle \scriptstyle Z$ is driven by a sinusoidal voltage or current as above, there holds

\displaystyle \ V=IZ=I|Z|e^{j\arg(Z)}

The magnitude of the impedance $\displaystyle \scriptstyle |Z|$ acts just like resistance, giving the drop in voltage amplitude across an impedance $\displaystyle \scriptstyle Z$ for a given current $\displaystyle \scriptstyle I$ $\scriptstyle I$ . The phase factor tells us that the current lags the voltage by a phase of $\displaystyle \scriptstyle \theta \;=\;\arg(Z)$ (i.e., in the time domain, the current signal is shifted $\displaystyle \scriptstyle {\frac {\theta }{2\pi }}T$ later with respect to the voltage signal).

Just as impedance extends Ohm’s law to cover AC circuits, other results from DC circuit analysis, such as voltage division, current division, Thévenin’s theorem and Norton’s theorem, can also be extended to AC circuits by replacing resistance with impedance.

Phasors

A phasor is represented by a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from Ohm’s law given above, recognising that the factors of $\displaystyle \scriptstyle e^{j\omega t}$ cancel.

日	一	二	三	四	五	六
« 7 月				9 月 »
			1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30	31

FreeSandal

每日彙整: 2018-08-02

STEM 隨筆︰古典力學︰轉子【五】《電路學》五【電感】 III‧阻抗‧A‧上