Distributed constants Lumped constants are inadequate models of extended circuit elements at high frequency. Examples are telephone lines, guitar strings, and organ pipes. We shall develop the model for an electrical transmission line. Also a model for sound traveling in a pipe, and show how they are equivalent.

Ideal transmission line model A voltage v is applied between two wires comprising a transmission line. A current i + enters one wire. An equal current i - returns from the other. This voltage and current generate electric and magnetic fields around the wires, shown respectively as dashed and dot-dashed lines in a cross sectional view on the right. In turn, these fields manifest themselves as a series inductance L and a parallel capacitance C per length x of the transmission line, depicted by the circuit component overlay on the left.

An infinitesimal length of line As dx ® 0, the second term goes to zero, while the constant first term is simply L/C, the inductance and capacitance per unit length of the line. Thus, which is called the characteristic impedance of the transmission line. Consider a short segment dx of an infinite line. Its measurable input impedance Z 1 is identical to Z 2, that of the next segment. Thus we write, then let Z 2 = Z 1 = Z 0 and simplify:

Transmission lines with loss Wires (except superconductors) have some resistance R per unit length. Likewise, most insulators have some conductance G per unit length. Characteristic impedanceInfinitesimal line model

Signal propagation on a line (1) Take the first derivative of (1), then substitute (2) to get,

Signal propagation on a line (2) I A and I B are related to V A and V B as follows. From eq(1), The transmission line equations for sinusoidal signals are, where the explicit time dependence (usually ignored) is.

Traveling waves The propagation constant g = a + i b where a is the attenuation constant and b is the phase constant, provides a complete description of a wave on a transmission line. If V B = 0, we have a pure (forward) traveling wave. The amplitude of such a wave is plotted here over a 1 [m] length of transmission line for a = 2 [nepers/m] and b = 16 p [radians/m]. The neper is a dimensionless natural logarithmic unit of measure. Thus, a specifies the exponential decay rate of a wave, while b specifies its spatial angular frequency.

Boundary conditions V A is the amplitude of the forward traveling wave so if R s = Z 0, we can write. At the load, the voltage and current are related by the load impedance, Notice that the reverse traveling wave vanishes iff that is if the transmission line and the load impedances match.

Standing waves If Z l = Z 0 at the load, i.e., r v = 0, we have seen that no signal energy is reflected. Conversely, if Z l = ¥ (open circuit) or 0 (short circuit), r v = ±1, respectively; in either case, all the energy is reflected, resulting in a pure standing wave, which over time appears not to move, rather just to oscillate in place. For intermediate values of r v, the voltage standing wave ratio VSWR = (1 + | r v |)/(1 - | r v |) is the ratio between the max and min of the voltage envelope. A plot for r v = 0.5 is shown. Figure 2-3 from Matick, Transmission Lines for Digital and Communication Networks, Mcgraw Hill, 1969.

The wave equation

Acoustic wave in a pipe (1)

Acoustic wave in a pipe (2)

Acoustic wave in a pipe (3)

Acoustic plane wave propagation Same as wave in a pipe: c 2 = K/ r ; where c is the speed of sound. –c = 332 [m/s] at 0 [degC] at 50% RH. –dc/dT = [m/s/degC]. Solutions are of the form –where s = j w for temporal sine waves. – Factoring out time, just as for transmission lines.

Reflection of sound Wave impedance of a medium Reflection of a wave normal to an interface between media is, which is the same as r v for transmission lines

Homework problem A 10 [m] length of transmission line with 100 [pF/m] capacitance and 250 [nH/m] inductance in series with a 1 [ W /m] wire resistance is driven by a 10 [MHz] sine wave from a 1 [V] 100 [Ohm] Thevenin source, and loaded by a 318 [pF] capacitor. Calculate the complex input voltage V in of the transmission line. Extra credit: plot the magnitude and phase of the voltage as a function of position on the transmission line.