每日彙整: 2018-07-10

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰轉子【五】《電路學》三【電阻】III.上

何謂『線性系統』？ 假使從『系統論』的觀點來看，一個物理系統 S，如果它的『輸入輸出』或者講『刺激響應』滿足

設使 I_m(\cdots, t) \Rightarrow_{S} O_m(\cdots, t)，I_n(\cdots, t) \Rightarrow_{S} O_n(\cdots, t)，

那麼\alpha \cdot I_m(\cdots, t) + \beta \cdot I_n(\cdots, t) \Rightarrow_{S} \alpha \cdot O_m(\cdots, t) + \beta \cdot O_n(\cdots, t)

也就是說一個線線系統︰無因就無果、小因得小果，大因得大果 ，眾因所得果為各因之果之總計。

如果一個線性系統還滿足

\left[I_m(\cdots, t) \Rightarrow_{S} O_m(\cdots, t)\right] \Rightarrow_{S} \left[I_m(\cdots, t + \tau) \Rightarrow_{S} O_m(\cdots, t + \tau)\right]

，這個系統稱作『線性非時變系統』。系統中的『因果關係』是『恆常的』不隨著時間變化，因此『遲延之因』生『遲延之果』 。線性非時變 LTI Linear time-invariant theory 系統論之基本結論是

任何 LTI 系統都可以完全祇用一個單一方程式來表示，稱之為系統的『衝激響應』。系統的輸出可以簡單表示為輸入信號與系統的『衝激響應』的『卷積』Convolution 。

300px-Tangent-calculus.svg

220px-Anas_platyrhynchos_with_ducklings_reflecting_water

350px-PhaseConjugationPrinciple.en.svg

雖然很多的『基礎現象』之『物理模型』可以用  LTI 系統來描述。即使已經知道一個系統是『非線性』的，將它在尚未解出之『所稱解』── 比方說『熱力平衡』時 ── 附近作系統的『線性化』處理，以了解這個系統在『那時那裡』的行為 ，卻是常有之事。

科技理論上偏好『線性系統』 ，並非只是為了『數學求解』的容易性，尤其是在現今所謂的『雲端計算』時代，祇是一般『數值解答』通常不能提供『深入理解』那個『物理現象』背後的『因果機制』的原由，所以用著『線性化』來『解析』系統『局部行為』，大概也是『不得不』的吧！就像『混沌現象』與『巨變理論』述說著『自然之大，無奇不有』，要如何『詮釋現象』難道會是『不可說』的嗎？？

一般物理上所謂的『疊加原理』 Superposition Principle 就是說該系統是一個線性系統。物理上還有一個『局部原理』Principle of Locality 是講︰一個物體的『運動』與『變化』，只會受到它『所在位置』的『周遭影響』。所以此原理排斥『超距作用』，因此『萬有引力』為『廣義相對論』所取代；且電磁學的『馬克士威方程式』取消了『庫倫作用力』。這也就是許多物理學家很在意『量子糾纏』的原因！

俗語說『好事不出門， 壞事傳千里』是否是違背了『局部原理』的呢？？

蘇格蘭的哲學家大衛‧休謨 David Hume 經驗論大師，一位徹底的懷疑主義者，反對『因果原理』Causality，認為因果不過是一種『心理感覺』。好比奧地利‧捷克物理學家恩斯特‧馬赫 Ernst Mach  在《Die Mechanik in ihrer Entwicklung, Historisch-kritisch dargestellt》一書中講根本不需要『萬有引力』 之『名』與『因』 ，直接說任何具有質量的兩物間，會有滿足

m_1 \frac{d^2 {\mathbf r}_1 }{ dt^2} = -\frac{m_1 m_2 g ({\mathbf r}_1 - {\mathbf r}_2)}{ |{\mathbf r}_1 - {\mathbf r}_2|^3};\; m_2 \frac{d^2 {\mathbf r}_2 }{dt^2} = -\frac{m_1 m_2 g ({\mathbf r}_2 - {\mathbf r}_1) }{ |{\mathbf r}_2 - {\mathbf r}_1|^3}

方程組的就好了；他進一步講牛頓所說的『力』根本是『贅語』 ，那不過只是物質間的一種『交互作用』interaction 罷了！

當真是『緣起性空。萬法歸一，一歸於宗。』的嗎？？

─── 《【SONIC Π】聲波之傳播原理︰原理篇《四中》》

 

深化理解『線性系統』內函，容易掌握『網絡分析』竅門︰

Network analysis (electrical circuits)

A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are many different techniques for calculating these values. However, for the most part, the applied technique assumes that the components of the network are all linear. The methods described in this article are only applicable to linear network analysis, except where explicitly stated.

Definitions

Component A device with two or more terminals into which, or out of which, current may flow.
Node A point at which terminals of more than two components are joined. A conductor with a substantially zero resistance is considered to be a node for the purpose of analysis.
Branch The component(s) joining two nodes.
Mesh A group of branches within a network joined so as to form a complete loop such that there is no other loop inside it .
Port Two terminals where the current into one is identical to the current out of the other.
Circuit A current from one terminal of a generator, through load component(s) and back into the other terminal. A circuit is, in this sense, a one-port network and is a trivial case to analyse. If there is any connection to any other circuits then a non-trivial network has been formed and at least two ports must exist. Often, “circuit” and “network” are used interchangeably, but many analysts reserve “network” to mean an idealised model consisting of ideal components.[1]
Transfer function The relationship of the currents and/or voltages between two ports. Most often, an input port and an output port are discussed and the transfer function is described as gain or attenuation.
Component transfer function For a two-terminal component (i.e. one-port component), the current and voltage are taken as the input and output and the transfer function will have units of impedance or admittance (it is usually a matter of arbitrary convenience whether voltage or current is considered the input). A three (or more) terminal component effectively has two (or more) ports and the transfer function cannot be expressed as a single impedance. The usual approach is to express the transfer function as a matrix of parameters. These parameters can be impedances, but there is a large number of other approaches (see two-port network).

Equivalent circuits

Main article: Equivalent impedance transforms

A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing the actual components with other notional components that have the same effect. A particular technique might directly reduce the number of components, for instance by combining impedances in series. On the other hand, it might merely change the form into one in which the components can be reduced in a later operation. For instance, one might transform a voltage generator into a current generator using Norton’s theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load.

A resistive circuit is a circuit containing only resistors, ideal current sources, and ideal voltage sources. If the sources are constant (DC) sources, the result is a DC circuit. Analysis of a circuit consists of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to phasor analysis of AC circuits.

Two circuits are said to be equivalent with respect to a pair of terminals if the voltage across the terminals and current through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.

If \displaystyle V_{2}=V_{1} implies \displaystyle I_{2}=I_{1} for all (real) values of \displaystyle V_{1} , then with respect to terminals ab and xy, circuit 1 and circuit 2 are equivalent.

The above is a sufficient definition for a one-port network. For more than one port, then it must be defined that the currents and voltages between all pairs of corresponding ports must bear the same relationship. For instance, star and delta networks are effectively three port networks and hence require three simultaneous equations to fully specify their equivalence.

 

因為不打算離題太遠長篇連載，且以

Delta-wye transformation

Main article: Y-Δ transform

A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An n-terminal network can, at best, be reduced to n impedances (at worst nC2). For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or four node star (Y) network. These two networks are equivalent and the transformations between them are given below. A general network with an arbitrary number of nodes cannot be reduced to the minimum number of impedances using only series and parallel combinations. In general, Y-Δ and Δ-Y transformations must also be used. For some networks the extension of Y-Δ to star-polygon transformations may also be required.

For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations. The equations below are expressed as resistances but apply equally to the general case with impedances.

Delta-to-star transformation equations

\displaystyle R_{a}={\frac {R_{\mathrm {ac} }R_{\mathrm {ab} }}{R_{\mathrm {ac} }+R_{\mathrm {ab} }+R_{\mathrm {bc} }}}
\displaystyle R_{b}={\frac {R_{\mathrm {ab} }R_{\mathrm {bc} }}{R_{\mathrm {ac} }+R_{\mathrm {ab} }+R_{\mathrm {bc} }}}
\displaystyle R_{c}={\frac {R_{\mathrm {bc} }R_{\mathrm {ac} }}{R_{\mathrm {ac} }+R_{\mathrm {ab} }+R_{\mathrm {bc} }}}

Star-to-delta transformation equations

\displaystyle R_{\mathrm {ac} }={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{b}}}
\displaystyle R_{\mathrm {ab} }={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}}
\displaystyle R_{\mathrm {bc} }={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{a}}}

 

為範，說說不同方法旨趣︰

Choice of method

Choice of method[2] is to some extent a matter of taste. If the network is particularly simple or only a specific current or voltage is required then ad-hoc application of some simple equivalent circuits may yield the answer without recourse to the more systematic methods.

  • Nodal analysis: The number of voltage variables, and hence simultaneous equations to solve, equals the number of nodes minus one. Every voltage source connected to the reference node reduces the number of unknowns and equations by one.
  • Mesh analysis: The number of current variables, and hence simultaneous equations to solve, equals the number of meshes. Every current source in a mesh reduces the number of unknowns by one. Mesh analysis can only be used with networks which can be drawn as a planar network, that is, with no crossing components.[3]
  • Superposition is possibly the most conceptually simple method but rapidly leads to a large number of equations and messy impedance combinations as the network becomes larger.
  • Effective medium approximations: For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. Instead, the effective resistance and current distribution properties can be modelled in terms of graph measures and geometrical properties of networks.[4]

 

既然普遍認為『疊加原理』

無因就無果、小因得小果，大因得大果 ，眾因所得果為各因之果之總計。

Superposition

Main article: Superposition theorem

 

In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed and either short-circuited in the case of voltage generators or open-circuited in the case of current generators. The total current through or the total voltage across a particular branch is then calculated by summing all the individual currents or voltages.

There is an underlying assumption to this method that the total current or voltage is a linear superposition of its parts. Therefore, the method cannot be used if non-linear components are present. Note that mesh analysis and node analysis also implicitly use superposition so these too, are only applicable to linear circuits. Superposition cannot be used to find total power consumed by elements even in linear circuits. Power varies according to the square of total voltage or current and the square of the sum is not generally equal to the sum of the squares.

 

直覺簡單︰

Superposition theorem

The superposition theorem for electrical circuits states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

To ascertain the contribution of each individual source, all of the other sources first must be “turned off” (set to zero) by:

This procedure is followed for each source in turn, then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.

The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Norton equivalent or Thevenin equivalent.

The theorem is applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, linear passive elements (resistors, inductors, capacitors) and linear transformers.

Superposition works for voltage and current but not power. In other words, the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents.

 

就從此法開始。

但想早已過百年之理論，又有什麼人未曾說耶？

況自維基百科詞條

Y-Δ transform

The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.[1] It is widely used in analysis of three-phase electric power circuits.

The Y-Δ transform can be considered a special case of the star-mesh transform for three resistors. In mathematics, the Y-Δ transform plays an important role in theory of circular planar graphs.[2]

 

找到鏈結哩！

Resistors

Star and Triangle Conversion

The conversion between star and triangle configurations is a very useful solution to simplify complex resistor networks. In the figure below these configurations are shown.

With proper resistor values for each configuration, they behave exactly the same. Therefore one can replace one configuration with another, where it results in easier calculations. Below it is shown how to extract conversion formulas for start to triangle configuration and back. The final equations are enough for the conversion. But extracting these formulas is also a great practice to solve for resistor networks.

Like the solutions for series and parallel configurations, you can assume different voltages across each resistor and current through them. Then in order to get two similar circuits you can put the voltages and currents on terminals of two configurations equal and calculate for the resistor equations. Although this way works, it will take a tone of very complicated equations to solve. Always try to find the best way to solve a circuit as improper equations can make it so hard to solve a circuit.

In the case of any circuit, when you have the choice to assume voltages or currents for the circuit, it is better to make as many as you can constant, like zero value and solve for the remaining voltages and currents. This way you get fewer equations that are less complex. In the case of these configuration, we assume V2 and V3 to be equal to zero in both circuits, achieving the circuits below:

Now you can see how easier the circuits look having proper assumptions. Let’s solve the star circuit first to get its equivalent resistance between the remaining two nodes. Figure below shows the steps taken to get the equivalent resistance of the star configuration.

First, R2 and R3 are in parallel, therefore we replace them with their equivalent resistance shown as R2||R3. Then you see that two resistors in series remain. The equivalent resistance between the two node is equal to Req = R1 + R2||R3. Now for the triangle configuration, you can see that both sides of R’1 are connected to the same voltage, zero. Therefore the voltage across the resistor is zero and from the formula V=R.I, it means that there is no current going through this resistor. This leaves the resistor ineffective and we can eliminate it.

Note this: when two nodes are connected to the same voltage, it is exactly like shorting these two together. Because no matter what is between these two nodes, because of the lack of energy across them, no current goes true them and both nodes will have the same properties, similar as two shorted nodes.

This simply means that the ends of the resistor R’1 are shorted. Therefore as mentioned in the parallel equation above, we simply replace this with a short. Figure below shows the remaining circuit for the triangle configuration and its equivalent resistance, which is simply shown as R’2||R’3.

Now as mentioned before, both circuits must show the same properties. Therefore under the same conditions mentioned above (V2 = V3 = 0) the equivalent resistance of both circuits must be the same. This results in the first equation relating the resistances of both circuits together. Similarly under two more condition, V1 = V3 = 0 and V1 = V2 = 0, we get two more equations. They are all summarized below:

Now solving these equations gives us formulas to convert a star configuration to triangle and back. To convert from a start to triangle configuration, we use the formulas below:

To convert from triangle to star configuration, we use the formulas below:

Written by: Mehdi Sadaghdar

 

哈哈☺得來全不費工夫呦！

偏偏紙筆不好求解呀☻