Multi-dimensional Kirchoff Circuits and Wave Digital Networks

The 1D Transmission Line

• Described by the telegrapher's equations:
  

  $$l\frac{\partial i}{\partial t}+\frac{\partial u}{\partial x}+ri+e =$$ 0

  $$c\frac{\partial u}{\partial t}+\frac{\partial i}{\partial x}+gu+h =$$ 0

  
  
  
  
  
• Material parameters:
  • $l(x)=$ inductance/unit length
  • $c(x)=$ shunt capacitance/unit length
  • $r(x)=$ resistance/unit length
  • $g(x)=$ shunt conductance/unit length
• $e(x,t)$ , $h(x,t)$ are sources (possibly distributed, time-varying).

The 1D Transmission Line (cont'd)

• For $l$ , $c$ constant, $r=g=e=f=0$ , reduces to the wave equation

  $$\frac{\partial^{2} u}{\partial t^{2}}=\gamma^{2}\frac{\partial^{2} u}{\partial x^{2}}\hspace{0.5in}\gamma = \sqrt{\frac{1}{lc}}$$

  
  

  Otherwise, describes dispersive, lossy 1D wave propagation.

• A simple model problem.

First Attempt at a Kirchoff Circuit (cont'd)

We can write the telegrapher's equations down directly as a two-loop circuit, with currents $i$ and $\frac{u}{r_{0}}$ $r_{0}=$ scaling parameter, dimensions of resistance:

Problem: Not MD-passive.

Fix: Classical network theory manipulations.

Passive MD Circuit

To any T-junction corresponds a lattice or Jaumann equivalent:

Employing this equivalence gives a concretely MD-passive structure

Comments

• Energetically, the system of equations has been broken into several smaller interacting parts, each of which is passive on its own.
• The passive part of the circuit (RL) dissipates MD-power.

• Source-free case: with power-normalized waves, all operations (scattering, shifting) are norm-reducing (in finite arithmetic as well).

• With the judicious choice of $r_{0}= \sqrt{\frac{l_{min}}{c_{min}}}$ , the stability bound is

  $$v_{0}\geq \sqrt{\frac{1}{l_{min}c_{min}}}$$

  
  
  which is different from (worse than) the optimal bound

  $$v_{0}\geq \sqrt{\frac{1}{(lc)_{min}}}$$

  
  
  Fix: preconditioning of equations
• Memory requirements are double that of centered differences (a multistep method)

Simplifications

If $l$ and $c$ are constant, and we have no sources then we can derive a simplified form (a).

If in addition, we have $lg=cr$ , then the line is distorionless, and we have simple travelling waves, which are attenuated (b).

2D ``Parallel Plate''

• Equations are:
  

  $$l\frac{\partial i_{x}}{\partial t}+\frac{\partial u}{\partial x}+ri_{x}+e =$$ 0

  $$l\frac{\partial i_{y}}{\partial t}+\frac{\partial u}{\partial y}+ri_{y}+f =$$ 0

  $$c\frac{\partial u}{\partial t}+\frac{\partial i_{x}}{\partial x}+\frac{\partial i_{y}}{\partial y}+gu+h =$$ 0

  
  

• • $(i_{x},i_{y})$ = current density in plate
  • $u$ = voltage across plates
  • other quantities as before; in general, $l$ , $c$ , $r$ and $g$ all may be spatially varying.
• A useful, general model problem: identical to systems of
  • an ideal membrane (in forces and transverse velocity)
  • TE or TM mode
  • 2D linear acoustic medium (velocities and pressure)

A Passive Circuit and Wave Digital Network

Applying a coordinate transformation (for rectangular coordinates), we get the following circuit and network:

Other Systems

The circuit approach is applicable to a wide range of problems including:

• 3D Maxwell's Equations (inhomogeneous, lossy materials)
• 3D Linear Elasticity Equations (inhomogeneous)

  and with some tampering, to parabolic/borderline parabolic problems like

• Heat Equation (nD)
• Schrodinger's Equation (nD)
• Euler-Bernoulli Equations for a beam (1D) and plate (2D)

  Most interestingly, the method can also be applied to systems of nonlinear conservation laws, in particular

• Euler Equations (lossless nonlinear fluid dynamics), with an extension to the lossy case (Navier-Stokes).