My book covers complex impedance of inductors and capacitors and looks a series and parallel circuits involving them—it even considers complicated non-linear models that have components for modeling loudspeakers (with ) and electrodes (with ). But I don’t consider models of distributed inductance and capacitance, commonly called *transmission lines.* For the low-frequency work we do in bioelectronics, these more sophisticated models are not needed.

Model of a transmission line using 3 LC elements. The transmission-line model takes the limit of N elements as N goes to infinity, with each L_i= L/N and each C_i = C/N.

But I’ve been thinking about them sometimes at night, doing the algebra in my head as a way to get to sleep, so I thought I would make a record of my thinking. It is fairly easy to determine the impedance of an infinite transmission line, by looking at what happens if you add a single inductor and capacitor at the front:

Treating the infinite transmission line as a single unknown impedance Z, we can solve an equation with Z as both the input impedance and the load impedance.

We can write the input impedance as . If is an infinite transmission line, then and we can simplify to the quadratic equation , which has solutions . But we are taking the limit as with and , so this simplifies to just , which is the standard result for the impedance of transmission lines.

If we have a finite-length transmission line, but connect a load resistor to one end with resistance , matching the impedance of the transmission line, it looks at other end just like a resistor . But what happens if we attach a different load (say a short-circuit, an open-circuit, or an arbitrary impedance )?

I found it too difficult to do the algebra in my head for concatenating multiple LC units, until I remembered a trick I used in my very first paper (A proof of the isomorphism of wxyz-transformals and 2×2 integer matrices under multiplication. *Computers and Mathematics with Applications*, 7(5):425–430, 1981. scanned copy). We can represent a fraction as a vector of the numerator and denominator, and multiply by 2×2 matrices to do simple transformations. For example, if an impedance is represented as , we can represent series connection as and parallel connection as . If we look at putting one of our LC units in front of Z, we get , and N of them is .

That doesn’t look simpler, until you rearrange to get and then take limits as to get .

We can make the mental math easier if we make the matrix a root of unity:

. We can then use the well-known limit , which works with matrices as well as it does with numbers. We now have .

Because the square of our matrix is the identity matrix, the even powers accumulate on the diagonal and the odd powers accumulate in other two corners, so we can do exponentiation to get .

We can do a simple sanity check by plugging in , which yields , which yields the input impedance , as desired for a transmission line with a matched load.

We can now plug in for a short-circuit load or for an open-circuit load to get input impedances of and , respectively. At very low frequency (), where we would not normally use transmission-line models, we can use the approximation , which gives the low-frequency approximations for the short-circuit load, and for the open-circuit load, which are just the approximations we would use if we were doing lumped models of a loop of wire or a pair of unconnected wires.

It is even more interesting to see frequencies where the input impedance goes to 0 or to infinity, which correspond to the periods for reflections off the far end of the transmission line. We should be able to use the formulas for capacitance between two long parallel wires and inductance of a loop of two long parallel wires to get the L and C values, and then use the periods to determine how fast waves travel in the transmission line. I’ve not done those calculations yet, as I have not memorized the formula for inductance or capacitance of parallel wires, nor the values of the permittivity of free space and the magnetic constant. Ignoring the values of the constants, I think that dimensions all work, with both inductance and capacitance proportional to the wire length and with everything cancelling except the relative dielectric constant $\latex{\kappa}$, the permittivity of free space, and the magnetic constant. I should work it out more carefully, though, to make sure I get the standard result that the velocity in a transmission line is .

Writing up this description of transmission lines probably took me about as long as working out the matrix equations in my head did (a few hours).

### Like this:

Like Loading...