User:Alfred Centauri/sandbox

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1 TL
2 Capacitor Weirdness
3 Theoretical basis for impedance
4 Hydraulic Capacitor Analogy

[edit] TL

(Note: In the following, subscript notation is used for 1st and 2nd partial derivatives.)

For an ideal lossless two conductor transmission line (TL), the voltage across the conductors and the current through the conductors must satisfy the following equations:

$V_x + L I_t = 0 \,$

$I_x + C V_t = 0 \,$

where L and C are the inductance and capacitance per unit length of the TL.

These equations lead to the following wave equations for the voltage and current:

$V_{xx} - LC V_{tt} = 0 \,$

$I_{xx} - LC I_{tt} = 0 \,$

Consider such a TL of unit length that is unterminated at both ends. The current at the ends of the TL is constrained to be zero. With these boundary conditions, the solution to the current wave equation is:

$I(x,t) = \sum_{n=1}^\infty a_n(t) \sin(n \pi x) \,$

where

$a_n(t) = 2 \left [ \left ( \cos(\frac{n \pi t}{\sqrt{LC}}) \int_{0}^{1} I(x,0) \sin(n \pi \tau)\, d\tau \right ) + \left ( \sin(\frac{n \pi t}{\sqrt{LC}}) \int_{0}^{1} I_t(x,0) \sin(n \pi \tau)\, d\tau \right ) \right ]$

For

$I(x,t) = 0 \,$

the only solution is the trivial solution:

$a_n(t) = 0 \,$

which can only be true when the initial conditions are:

$I(x,0) = I_t(x,0) = 0 \,$

Thus:

$V_x = 0 \Rightarrow \ V(x,t) = constant \,$

That is, for a finite length ideal lossless TL that is unterminated at both ends, there is no possible superposition of waves that give the solution V(x,t) = constant, I(x,t) = 0.

works for me. Pfalstad 02:23, 15 October 2005 (UTC)

[edit] Capacitor Weirdness

Over the years, many scientists and engineers have questioned whether or not displacement current "causes" magnetic fields. The following simple derivation shows how displacement current and charges interact to form magnetic fields.

The capacitance of a parallel plate capacitor with plates of area A and plate separation of d may be expressed as:

$C=\varepsilon_{R}\epsilon_{O}\frac{A}{d}$ (1)

Where $ε R$ and $ε 0$ are dielectric and free space permittivity, respectively.

Another basic equation relates the uniform charge, Q stored in a capacitor, to the capacitance C, and the Voltage across the plates, V.

$Q=CV \,$ (2)

Divide both sides of equation (2) by the distance between the plates, d:

$\frac{Q}{d}=C\frac{V}{d}$ (3)

Assuming a uniform electric field directed in the +z direction, $E_z = \frac{V}{d}$ , we may rewrite equation (3) as follows:

$\frac{Q}{d}=CE_z$ (4)

Substitute the value for C from equation 1 into equation (4):

$\frac{Q}{d}=\varepsilon_{R}\epsilon_{O}\frac{A}{d}E_z$ (5)

Multiply both sides of equation 5 by d, and take the partial derivative of equation 5 with respect to time to yield:

$\frac{\partial Q}{\partial t} = \varepsilon_{R}\epsilon_{O}A \frac{\partial E_z}{\partial t}$ (6)

Using the constituitive relation $\varepsilon_{R}\epsilon_{O} \vec E = \vec D$ , equation (6) may be written as:

$\frac{\partial Q}{\partial t}=A\frac{\partial D_Z}{\partial t}$ (7)

The point form of Ampere’s law, as modified by Maxwell, is often written as:

$\nabla \times \vec H =J +\frac{\partial \vec D}{\partial t}$ (8)

Applying Ampere's law to our capacitor, we have:

$\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y} = J_z + A^{-1}\frac{\partial Q}{\partial t}$ (9)

{What follows is nonsense}

But $A^{-1}\frac{\partial Q}{\partial t}$ is another way of expressing J. In this case, since the electrical field is Transverse to the plane of the conductor, we define a new term,

$J_{T}=A^{-1}\frac{\partial Q}{\partial t}$ (10)

The term “J” has traditionally been taken to mean “longitudinal conduction current.” To differentiate it from $J T$ , let us rename J as $J L$ standing for longitudinal current flow. With these re-definitions, we can now rewrite equation (9) in a form that is consistent with the spirit of Ampere:

$\nabla \times H=J_{L}+J_{T}$ (11)

Conclusion Equations 8, 9 and 11 may be used interchangeably, but only if one understands that the source of the magnetic fields is the motion of charges, $\frac{\partial Q}{\partial t}$ .

[edit] Theoretical basis for impedance

For a system such as an electric circuit that is described mathematically by a linear system of non-homogeneous differential equations, the solutions to these equations are in general of a different form than the driving functions. For example, consider a simple circuit consisting of a voltage source and a capacitor. The current through the capacitor proportional to the derivative of the the voltage across the capacitor.

$i_C(t) = C\frac{dv_C(t)}{dt}\,$