# Why is the product of the $L$ and $C$ matrices for coupled transmission lines diagonal?

## Background - transmission line

$$\newcommand{\ket}{\left \lvert #1 \right \rangle}$$ A transmission line can be modeled as an infinite sequence of inductors and capacitors:

     z   z+dz
|   |
v   v
---L---L--- <-- signal
|   |   |
.. C   C   C ..
|   |   |
----------- <-- ground


Each $$L$$ and $$C$$ is the inductance and capacitance per length. The current and voltage in this circuit is represented by the so-called telegrapher's equations \begin{align} V(z + dz) = V(z) - L \, dz \, \dot{I}(z) \quad \text{and} \quad I(z + dz) = I(z) - C \, dz \, \dot{V}(z) \end{align} which can be rearranged, in the limit $$dz \rightarrow 0$$, to $$\frac{\partial V}{\partial z} = - L \frac{\partial I}{\partial t} \quad \text{and} \quad \frac{\partial I}{\partial z} = - C \frac{\partial{V}}{\partial t} \, .$$ If we take $$\partial/\partial z$$ on the left equation, and $$\partial/\partial_t$$ on the right equation, and substitude the right into the left, we get $$\left( \frac{\partial^2 V}{\partial z^2} \right) - LC \left(\frac{\partial^2 V}{\partial t^2} \right) = 0$$ which is the wave equation for waves moving with phase velocity $$v^2 = 1 / LC$$ (remember $$L$$ and $$C$$ are per-length quantities).

## Context - coupled transmission lines

Now suppose we have coupled transmission lines

    z      z+dz
|      |
v      v
------------------- <-- ground
|      |      |
..  Ca     Ca     Ca ..
|      |      |
-----La-----La----- <-- signal a
|      |      |
..  Cm Lm  Cm Lm  Cm ..
|      |      |
-----Lb-----Lb----- <-- signal b
|      |      |
..  Cb     Cb     Cb ..
|      |      |
------------------- <-- ground


We have two lines $$a$$ and $$b$$, each with inductance per length $$L_a$$ or $$L_b$$, and capacitance per length $$C_a$$ or $$C_b$$. The lines are coupled through $$C_m$$ and $$L_m$$; the mutual inductance is hard to draw in ASCII art, so think of $$L_m$$ as a mutual inductance between the inductors $$L_a$$ and $$L_b$$. It's pretty easy to show that the equations for these coupled lines are \begin{align} \frac{\partial V_a}{\partial z} &= - L_a \frac{\partial I_a}{\partial t} - L_m \frac{\partial I_b}{\partial t} \\ \frac{\partial I_a}{\partial z} &= -(C_a + C_m) \frac{\partial V_a}{\partial t} + C_m \frac{\partial V_b}{\partial t} \end{align} and similarly for line $$b$$.

The equations for the coupled lines can be expressed in a matrix form \begin{align} \frac{\partial}{\partial z} \ket{V} &= - [L] \frac{\partial}{\partial t} \ket{I} \\ \frac{\partial}{\partial z} \ket{I} &= - [C] \frac{\partial}{\partial t} \ket{V} \end{align} where $$\ket{V}$$ is the vector of voltages, $$\ket{I}$$ is the vector of currents, and $$[L]$$ and $$[C]$$ are matrices. The matrices are $$[L] = \begin{bmatrix} L_a & L_m \\ L_m & L_b \end{bmatrix} \quad \text{and} \quad [C] = \begin{bmatrix} (C_a + C_m) & -C_m \\ -C_m & (C_b + C_m) \end{bmatrix} \, .$$

## Question

In the book Fundamentals of Microwave Transmission Lines by Jon C. Freeman, it is written

It turns out that the velocity of propagation is $$[L][C] = \frac{1}{v^2} [I]$$ where $$v = \frac{1}{\sqrt{\mu \epsilon}}$$ (here $$[I]$$ is the identity matrix)

Why is the product of the $$L$$ and $$C$$ matrices diagonal, and why are the diagonal entries equal to each other ($$1/v^2$$)? It seems like the author must be making an assumption and I haven't figured out what that assumption is.

• If you just multiply together the general matrices, what are the componentwise expressions for the claimed matrix equation $LC = I/v^2$? Are they really messy? Jan 4 at 5:47
• It’s weird if the author suddenly jumps from only using abstract lumped-circuit elements to the microscopic physical material parameters $\epsilon$ and $\mu$, which is a pretty different “picture”. Jan 4 at 5:52

See Chapter 12 in Orfanidis: Electromagnetic Waves and Antennas, especially equation 12.1.35 where he shows that for a symmetrical pair of coupled lines in a homogeneous medium you $$(L_0+L_m)(C_0-C_m)= (L_0-L_m)(C_0+C_m)=\mu\epsilon$$and $$\frac{L_0}{C_0}=\frac{L_m}{C_m}=\frac{\mu\epsilon }{C_0^2-C_m^2}$$ from which the $$\mathbf{LC}=\frac{1}{\mu\epsilon}\mathbf{I}$$ follows. (In your notation this would be $$L_a=L_b=L_0$$ and $$C_a=C_b=C_0$$)