一個諧振子 harmonic oscillator 是一個物理系統，當它從平衡位置發生位移時，會受到一個正比於位移量 $x$ 的恢復力 $R$ ── 虎克定律──︰ $R = -k x$ ，其中 $k$ 是一個正值常數。假使這個系統不受其它的外力影響，通常稱作『簡諧振子』Simple harmonic oscillator；如果此系統同時遭受到與速度成正比的『摩擦力』 $F_f = -c \frac {dx}{dt}$ ，一般叫做『阻尼振子』Damped harmonic oscillator；要是這個系統還有著跟時間相關的外力 $F(t)$ 的作用，那麼就稱之為『受驅振子』Driven harmonic oscillators。

依據牛頓第二運動定律，一個簡諧振子的方程式為

$F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x$ ，它的解是

$x(t) = A\cos\left( \omega t+\phi\right)$ ，此處 $\phi$ 是『相位角』，

$\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}$ ，式中 $\omega$ 是『角頻率』， $T$ 是『周期』。

也就是說簡諧振子是一種『頻率』為 $f = \frac {1}{T}$ ，『振幅』為 $A$ 的週期運動。假設 $t = 0$ 的初始時， $x_0 = A, \ v_0 = 0$ ，得到

$x(t) = A\cos\left( \omega t\right)$

$v(t) = -A\omega\sin\left( \omega t\right)$

動能 = $\frac{1}{2} m v^2$ ，位能＝ $\frac{1}{2} k x^2$ ，系統總能量

$E = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 = \frac{1}{2} k A^2$

由此可以知道簡諧振子的系統總能量是一個常數，這稱之為『能量守恆量』定律，它和『振幅』的平方成正比。它的『頻率』 $f = \frac {\omega}{2 \pi}$ 只依賴於系統『固有』的 $k$ 與 $m$ ，也是一個不變的常量。

依據牛頓第二運動定律，一個受驅振子的方程式為

$F(t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}$ ，一般將之改寫為

$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{F(t)}{m}$ ，此處 $\omega_0 = \sqrt{\frac{k}{m}}$ 稱為『無阻尼』角頻率，而 $\zeta = \frac{c}{2 \sqrt{mk}}$ 叫做『阻尼比率』。如果『外力』 $F(t) = 0$ ，那個方程式就成了『阻尼振子』的方程式

$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = 0$ ，當 $\zeta \leq 1$ 它的解是

$x(t) = A \mathrm{e}^{-\zeta \omega_0 t} \ \sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \phi \right)$ ，此處 $\phi$ 是相位角。

假使 $\zeta > 1$ 它的解是

$x(t) = C_1 \ \mathrm{e}^{\left(- \zeta + \sqrt{\zeta^2 -1}\right) \omega_0 t} + C_2 \ \mathrm{e}^{\left(- \zeta - \sqrt{\zeta^2 -1}\right) \omega_0 t}$ 。

如果我們從 $\zeta$ 的值來看『阻尼振子』的系統行為，當 $\zeta > 1$ 時，這一個系統已經『振動』不起來了，通常叫做『過阻尼』，負數的『指數項』使得系統的能量隨時間逐漸減少， $\zeta$ 的數值愈大能量減少將慢愈遲回到平衡。當 $\zeta = 1$ 時，這一個系統也『振動』不起來了，通常稱之為『臨界阻尼』，此時系統會用最快的方式設法回到平衡，這個可是『關門』系統的『最佳解』！！。當 $\zeta < 1$ 時，這樣的諧振子系統稱作『低阻尼』，這時系統用著『低於無阻尼』的『頻率』振動，系統的『振幅』隨著時間以 $\mathrm{e}^{-\zeta \omega_0 t}$ 為比率逐漸減小以至於『不振動』為止。事實上從自然界中來的一般現象都會比『理論值』更快的到達『停止點』，就像說不只有『動摩擦力』與『靜摩擦力』之區分，摩擦力的『速度相關性』也不是這麼『簡單的正比』之假設，更別說理論上還有著『摩擦生熱』的問題必須要考慮。我們也許可以說為著追求『基本現象的理解』，通常都會『假設』了一些數學上『解答問題』的『理想條件』。

現在談談受外力影響下的受驅振子︰

階躍 Step 外力

假設此系統的 $\zeta < 1$ ，初始位置 $x_0 = 0$ ，在 $t = 0^+$ 時受到如下的階躍外力︰

${F(t) \over m} = \begin{cases} \omega _0^2 & t \geq 0 \\ 0 & t < 0 \end{cases}$

它的解是

$x(t) = 1 - \mathrm{e}^{-\zeta \omega_0 t} \frac{\sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \varphi \right)}{\sin(\varphi)}$ ，此處相位角 $\varphi$ 由 $\cos \varphi = \zeta$ 所決定。

這個系統因為零點時刻突然受到固定大小的外力 $m \ {\omega_0}^2$ 所驅動，震盪以 $\mathrm{e}^{-\zeta \omega_0 t}$ 為比率逐漸增大，一般用 $\tau = \frac{1}{\zeta \omega_0}$ 為時間尺度來衡量這個變化，每一 $\tau$ 單位時間，系統將以 $\mathrm{e}^{-1}$ 為比率改變振幅，在物理上稱之為『弛豫時間』Relaxation Time，工程上常用多的 $\tau$ 單位時間，來談震盪達到預期大小的『安定時間』settling time。果真是『風吹枝擺』，待其風歇『搖曳而止』！！

頻率為 $\omega$ 的正弦驅動力

此時系統的方程式為

$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t)$

$F_0$ 是驅動力的振幅大小。在線性微分方程式如 $\hat{L} x(t) = F(t)$ 的『求解』裡，如過『 $\Box$ 』是 $\hat{L} x(t) = 0$ 的一個解，『 $\bigcirc$ 』是 $\hat{L} x(t) = F(t)$ 一個『特解』，那麼『 $c \ \Box + \ \bigcirc$ 』就是該方程是的『通解』。我們已經知道 $F(t) = 0$ 的『低阻尼振子』之解在若干個弛豫時間後數值將變得太小了，所以它對於系統長時間之後的『行為』沒有太多的貢獻。因此我們說這個系統的『穩態解』steady-state solution 是

$x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \phi)$ ，此處

$Z_m = \sqrt{\left(2\omega_0\zeta\right)^2 + \frac{1}{\omega^2}\left(\omega_0^2 - \omega^2\right)^2}$

是『響應阻抗』函數。而 $\phi$ 是驅動力引發的相位角，可由

$\phi = \arctan\left(\frac{\omega_0^2-\omega^2}{2\omega \omega_0\zeta}\right)$

所決定，一般它表達著相位『遲滯』 lag 現象。

如果使用不同頻率 $\omega$ 的驅動力，當 $\omega = \omega_r = \omega_0\sqrt{1-2\zeta^2}$ 時，系統的響應振幅最大，這稱之為『共振』resonant，這一個頻率就叫做『共振頻率』。

請參考左圖。

物理上所說的『慣性』是指一個系統遭受外力時，它會發生『抵抗變化』的作為。這就是『響應阻抗』和『相位遲滯』的物理原由與命名由來。假使考察穩態解，我們是否可以講︰『原因』── $F_0 \sin(\omega t)$ ── 產生成正比之『結果』── $x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \phi)$ ── 的呢？？

─── 《【SONIC Π】聲波之傳播原理︰振動篇》

僅會『求解』方程式，知道『解答』，恐不足乎？

就像『自然現象』通常需要『物理解釋』勒！

舉例而言，借著 lcapy ，『求解』下面這個 $RLC$ 電路接上

【衝激】

【階躍】

【正弦】

電壓源之『電流表現』，那個『解答』最好還是『改寫』耶？！

RLC circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

The circuit forms a harmonic oscillator for current, and resonates in a similar way as an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping. The resistor also reduces the peak resonant frequency. Some resistance is unavoidable in real circuits even if a resistor is not specifically included as a component. An ideal, pure LC circuit exists only in the domain of superconductivity.

RLC circuits have many applications as oscillator circuits. Radio receivers and television sets use them for tuning to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a band-pass filter, band-stop filter, low-pass filter or high-pass filter. The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis.

The three circuit elements, R, L and C, can be combined in a number of different topologies. All three elements in series or all three elements in parallel are the simplest in concept and the most straightforward to analyse. There are, however, other arrangements, some with practical importance in real circuits. One issue often encountered is the need to take into account inductor resistance. Inductors are typically constructed from coils of wire, the resistance of which is not usually desirable, but it often has a significant effect on the circuit.

A series RLC network: a resistor, an inductor, and a capacitor

Animation illustrating the operation of an LC circuit, an RLC circuit with no resistance. Charge flows back and forth between the capacitor plates through the inductance. The energy oscillates back and forth between the capacitor’selectric field ( $E$ ) and the inductor’s magnetic field ( $B$ ). RLC circuits operate similarly, except that the oscillating currents decay with time to zero due to the resistance in the circuit.

Series RLC circuit

Figure 1: RLC series circuit

$V$ , the voltage source powering the circuit
$I$ , the current admitted through the circuit
$R$ , the effective resistance of the combined load, source, and components
$L$ , the inductance of the inductor component
$C$ , the capacitance of the capacitorcomponent

In this circuit, the three components are all in series with the voltage source. The governing differential equation can be found by substituting into Kirchhoff’s voltage law (KVL) the constitutive equation for each of the three elements. From the KVL,

\displaystyle V_{R}+V_{L}+V_{C}=V(t)\,,

where $V R$ , $V L$ and $V C$ are the voltages across R, L and C respectively and $V (t)$ is the time-varying voltage from the source.

Substituting $\displaystyle V_{R}=RI(t)$ , $\displaystyle V_{L}=L{dI(t) \over dt}$ and $\displaystyle V_{C}=V(0)+{\frac {1}{C}}\int _{0}^{t}I(\tau )\,d\tau$ into the equation above yields:

\displaystyle RI(t)+L{\frac {dI(t)}{dt}}+V(0)+{\frac {1}{C}}\int _{0}^{t}I(\tau )\,d\tau =V(t)\,.

For the case where the source is an unchanging voltage, taking the time derivative and dividing by $L$ leads to the following second order differential equation:

\displaystyle {\frac {d^{2}}{dt^{2}}}I(t)+{\frac {R}{L}}{\frac {d}{dt}}I(t)+{\frac {1}{LC}}I(t)=0\,.

This can usefully be expressed in a more generally applicable form:

\displaystyle {\frac {d^{2}}{dt^{2}}}I(t)+2\alpha {\frac {d}{dt}}I(t)+\omega _{0}^{2}I(t)=0\,.

$α$ and $ω 0$ are both in units of angular frequency. $α$ is called the neper frequency, or attenuation, and is a measure of how fast the transient response of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be nepers per second, neper being a unit of attenuation. $ω 0$ is the angular resonance frequency.^[3]

For the case of the series RLC circuit these two parameters are given by:^[4]

\displaystyle {\begin{aligned}\alpha &={\frac {R}{2L}}\\\omega _{0}&={\frac {1}{\sqrt {LC}}}\,.\end{aligned}}

A useful parameter is the damping factor, $ζ$ , which is defined as the ratio of these two; although, sometimes $α$ is referred to as the damping factor and $ζ$ is not used.^[5]

\displaystyle \zeta ={\frac {\alpha }{\omega _{0}}}\,.

In the case of the series RLC circuit, the damping factor is given by

\frac {R}{2}}{\sqrt {\frac {C}{L}}}\,.

The value of the damping factor determines the type of transient that the circuit will exhibit.^[6]

Transient response

The differential equation for the circuit solves in three different ways depending on the value of $ζ$ . These are underdamped ( $ζ < 1$ ), overdamped ( $ζ > 1$ ) and critically damped ( $ζ = 1$ ). The differential equation has the characteristic equation,^[7]

\displaystyle s^{2}+2\alpha s+\omega _{0}^{2}=0\,.

The roots of the equation in $s$ are,^[7]

\displaystyle {\begin{aligned}s_{1}&=-\alpha +{\sqrt {\alpha ^{2}-\omega _{0}^{2}}}\\s_{2}&=-\alpha -{\sqrt {\alpha ^{2}-\omega _{0}^{2}}}\,.\end{aligned}}

The general solution of the differential equation is an exponential in either root or a linear superposition of both,

\displaystyle I(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}\,.

The coefficients $A 1$ and $A 2$ are determined by the boundary conditions of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time.^[8]

Overdamped response

The overdamped response ( $ζ > 1$ ) is^[9]

\displaystyle I(t)=A_{1}e^{-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)t}+A_{2}e^{-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)t}\,.

The overdamped response is a decay of the transient current without oscillation.^[10]

Underdamped response

The underdamped response ( $ζ < 1$ ) is^[11]

\displaystyle I(t)=B_{1}e^{-\alpha t}\cos(\omega _{\mathrm {d} }t)+B_{2}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t)\,.

By applying standard trigonometric identities the two trigonometric functions may be expressed as a single sinusoid with phase shift,^[12]

\displaystyle I(t)=B_{3}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t+\varphi )\,.

The underdamped response is a decaying oscillation at frequency $ω d$ . The oscillation decays at a rate determined by the attenuation $α$ . The exponential in $α$ describes the envelope of the oscillation. $B 1$ and $B 2$ (or $B 3$ and the phase shift $φ$ in the second form) are arbitrary constants determined by boundary conditions. The frequency $ω d$ is given by^[11]

\displaystyle \omega _{\mathrm {d} }={\sqrt {\omega _{0}^{2}-\alpha ^{2}}}=\omega _{0}{\sqrt {1-\zeta ^{2}}}\,.

This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, $ω 0$ , which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it.^[13]

Critically damped response

The critically damped response ( $ζ = 1$ ) is^[14]

\displaystyle I(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\,.

The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. $D 1$ and $D 2$ are arbitrary constants determined by boundary conditions.^[15]

Laplace domain

The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform.^[16] If the voltage source above produces a waveform with Laplace-transformed $V (s)$ (where $s$ is the complex frequency $s = σ + jω$ ), the KVL can be applied in the Laplace domain:

\displaystyle V(s)=I(s)\left(R+Ls+{\frac {1}{Cs}}\right)\,,

where $I (s)$ is the Laplace-transformed current through all components. Solving for $I (s)$ :

\displaystyle I(s)={\frac {1}{R+Ls+{\frac {1}{Cs}}}}V(s)\,.

And rearranging, we have

\displaystyle I(s)={\frac {s}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V(s)\,.

Laplace admittance

Solving for the Laplace admittance $Y (s)$ :

\displaystyle Y(s)={\frac {I(s)}{V(s)}}={\frac {s}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}\,.

Simplifying using parameters $α$ and $ω 0$ defined in the previous section, we have

\displaystyle Y(s)={\frac {I(s)}{V(s)}}={\frac {s}{L\left(s^{2}+2\alpha s+\omega _{0}^{2}\right)}}\,.

Poles and zeros

The zeros of $Y (s)$ are those values of $s$ such that $Y (s) = 0$ :

\displaystyle s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \,.

The poles of $Y (s)$ are those values of $s$ such that $Y (s) \to \infty$ . By the quadratic formula, we find

\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}}}\,.

The poles of $Y (s)$ are identical to the roots $s 1$ and $s 2$ of the characteristic polynomial of the differential equation in the section above.

General solution

For an arbitrary $V (t)$ , the solution obtained by inverse transform of $I (s)$ is:

In the underdamped case, $ω 0 > α$ :

$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left(\cos \omega _{\mathrm {d} }\tau -{\frac {\alpha }{\omega _{\mathrm {d} }}}\sin \omega _{\mathrm {d} }\tau \right)\,d\tau \,,$
$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }(1-\alpha \tau )\,d\tau \,,$
In the overdamped case, $ω 0 < α$ :

$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left(\cosh \omega _{\mathrm {r} }\tau -{\alpha \over \omega _{\mathrm {r} }}\sinh \omega _{\mathrm {r} }\tau \right)\,d\tau \,,$

where $ω r = \sqrt α 2 - ω 02$ , and $cosh$ and $sinh$ are the usual hyperbolic functions.

Sinusoidal steady state

Sinusoidal steady state is represented by letting $s = jω$ , where $j$ is the imaginary unit. Taking the magnitude of the above equation with this substitution:

\displaystyle {\big |}Y(j\omega ){\big |}={\frac {1}{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\omega C}}\right)^{2}}}}\,.

and the current as a function of $ω$ can be found from

\displaystyle {\big |}I(j\omega ){\big |}={\big |}Y(j\omega ){\big |}\cdot {\big |}V(j\omega ){\big |}\,.

There is a peak value of $| I (jω) |$ . The value of $ω$ at this peak is, in this particular case, equal to the undamped natural resonance frequency:^[17]

\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}\,.

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: 2018-08-15

STEM 隨筆︰古典力學︰轉子【五】《電路學》五【電感】 V‧上