Mechanics and Electrics Analogy

by allenlu2007

 

Why Mechanics and Electrics Analogy

1. Mechanics (F, v, x) 和 electrics (RLC) 都是 2nd order system.  互相類比可以 reinforce learning.  

2. Following 1, Mechanics and electronics Mechanics plant (mass, spring, damper) 在日常生活中可見,比較直覺。從 mechanics 所建立的直覺對於了解 electrics 行為有一定的幫助。就像 water dam and flow 可以用來了解 voltage and current.  但 mechanics 和 electrics 仍然是不同的系統,不要走火入魔。

3. 一些在 mechanics plant 常用的 controller (e.g. PID control, optimal control, robust control) 可以應用到 electrics plant 的 controller.

4. 最近快速發展的 MEMS (e.g. cantilever) 或是機電系統 (eletro-mechanics, electro-acoustic) 而言。需要把 mechanics 部份轉換為等效電路以 facilitate system design and integration.  事實上 crystal oscillator 的 crystal 等效 RLC 電路早已用在 oscillator 的設計中。

 

How Mechanics and Electrics Analogy

MIT 的 MEMS lecture note 是一個 good introduction.

基本上 MIT 的 approach 是把 mechanics element 轉為 lumped element, 再換成 electrical equivalents.

因此用 one representation for mechanics domain (mechanics, thermal, fluid) and electrics domain. 

Mechanics 基本上是 Newton’s law (似乎很少用 Lagrangian or Hamiltonian, 因為有 energy dissipation?)

Electrics 基本上是 KCL and KVL.  

兩者之間的連結 (mechanics 轉為 electrical equivalents) 是用 ODE 或 matrices 或是其他 representation.

 

每一個 lumped element (regardless mechanics or electrical element) 都包括一個或多個 ports.

每一個 port 都包含至少兩種 variables 見下圖:  (i) “through” variable (e.g. current I); (ii) “across” variable (e.g. voltage).   Power = “through” variable x “across” variable

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在 electrical circuits 非常直觀: voltage 是 “across”, and current 是 “through”

在 mechanics element 乍看也很直觀,從能量來看:

energy = 1/2 K x^2 + 1/2 M p^2   p = dx/dt   (mechanics)

energy = 1/2 q^2 / C + 1/2 L i^2   i = dq/dt   (electrics)

乍看 spring <=> capacitor (PE)  and    mass <=> inductor (KE) 

但 x/position <=> q/charge;  p/velocity <=> i/current   這是好的 analogy 嗎?  Yes but not exactly.

 

以下 direct analogy “force (position) <=> voltage”;  “velocity <=> current” 

因些 “velocity <=> current (inductor)” 部份一致。

那 “position/force <=> charge or voltage?”  因為 capacitor q = CV

因些 “position~force <=> charge~voltage” 也許更細是 “position<=>charge” and “force<=>voltage”

在 spring: position is proportional to force; 正如同 capacitor: charge is proportional to voltage

  

如何將 mechanics 轉為  electrical circuit? 兩種 approach

* 基於 ODE

* Element by element

 

Start with spring equivalent electrical model

Spring 的 mechanics equation 很簡單,儲存彈力位能。它有兩種型式如下。

可以是  force -> across (V);  velocity -> through (I)  :  capacitor

或是  force -> through (I);  velocity -> across (V) :  inductor

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那一種表式是正確呢? 兩者都正確。Beware!

* Force -> voltage (capacitor) 稱為 “direct” analogy

* Springs -> Capacitors (C ~ 1/K)

* PE in capacitor;  KE in inductor (比較接近一般的認知)

 

另一種表述稱為 “indirect” or “mobility” analogy

” Velocity -> voltage (capacitor)  and Force -> current (inductor)

* Springs -> Inductors (L ~ 1/K)  

* PE in inductor; KE in capacitor

* Based on reference, “indirect” representation is a cleaner match between mechanics system and circuit.  One reason is because velocity is naturally “across” (e.g. relative) variable.  why?

我們主要是用 “direct” representation.  

 

乍看之下會覺得很 confusing. 其實在電路中也有 duality representation. 是一樣的概念。flow (f) 和 effort (e) 可以互換。 

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Generalized Variables

我們把 mechanics 和 electrical 上的物理量抽象化如下:

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Displacement 對應 charge;  Velocity 對應 current; 這似乎還蠻直覺的。

Momentum 對應的量就比較 fuzzy (應該是 magnetic flux, Φ); 在 mechanics 時 momentum 正比於 velocity. 

 且  d(momentum)/dt = F (force).   在 electrical circuit 中 flux 正比於 current, 且 dΦ/dt = -v (多了負號).

Net power = (force) * (velocity) = e * f  in mechanics

Net power = (Voltage) * (Current) = v * i in electrics   因些 v 對應 force;  i 對應 velocity.  缺了 momentum 對應項。  p = integral(force)    =>   int(voltage) = ?

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如何建立 equivalent circuit

1. Need power sources

2. Add passive elements

3. Topology and connection rules

 

1. Source:  Effort source and flow source

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2. Add passive elements

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Spring 和 capacitor 之間的類比如前述。Mass 和 inductor 的類比仿照。

接下來是 resistor 和 damper 之間的類比如下。注意 in general resistor 可以是 nonlinear function of e and f (但不含 differentiation or integration).

 

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Energy and co-energy 是 Legendre transform.  如果 (V, q) 是 linear function (如一般的平板電容), both energy and co-energy function 都是拋物線。

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3. 再來是 connection or topology rule

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比較特別是 share charge or displacement 是串聯而非並聯。其實非常自然,因為 flow 是 charge or displacement 的微分。如果 charge or displacement 相同(或相反),微分後的 flow 自然也相同(或相反)。

Spring-mass-damper: F – kx – bv = m dv/dt   or  F = m dv/dt + bv + kx

RLC circuit:  F = m dI/dt + bI + kq

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如果用 through or across analogy  

 

force  -> across (V)

velocity -> through (I) 

OR

force -> through (I)

velocity -> across (V)

所以 V 永遠是 across;  I 永遠是 through.  這符合電路的直覺。

 

d(position)/dt = velocity   d(velocity)/dt = force

d(q)/dt = current   d(current) = voltage

Method 1 是 mechanics vs. electronics

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反過來由 electronics vs. mechanics

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model of RLC to damper, mass, and spring

damper <–>  R

mass  (kinetic energy) 0.5 m v^2 <–> inductor  0.5 L i^2    v = dx/dt;  i = dq/dt  m (mass) === L (inductance)

spring (potential)  0.5 k x^2 <–> capacitor  0.5 q^2/C         k (spring constant) === 1/C (1/capacitance)

 

以上 mechanics analogy 多半用於 AC power or signal.  

如果是 DC signal or power, 以上的 analog 就很奇怪。例如 switching regulator 持續 pump in inductor current 代表 DC offset.  比較好的 analogy 是 flywheel analogy.    

Inductor ~ flywheel

I (inductor current) ~ angular momentum

switch ~ on/off clutch

diode ~ anti-reverse rachet

capacitor ~ spring (angular deflection)

electrical energy (1/2Q^2/C)  ~ (angular deflection) potential energy (1/2kx^2)

Voltage ~ torque (force)  (V = Q/C,  F = kX  => k ~ 1/C)

 

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