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Transmission Line Theory
The electrical characteristics of the media used to send network datagrams partly define the physical layer: The determine the maximum transmission rate, the longest straight run of cable, and other constrains of the network. This are all products of the transmission line theory, a study how signals behave when they are transmitted over long distances. While the extremely lower-level theory doesn’t have any direct implications for higher-level protocols, violating the constraints imposed by transmission line theory can lead to intermittent and puzzling network failures that appear to the higher-level protocol breakdowns. A transmission line is any signal path that is long compared to the wavelength of the signal travelling the path. Signals of higher frequencies have shorter wavelengths, so higher frequencies signals require transmission line analysis over much shorter path lengths. For example, low-speed AC line voltage going from a power company generator to a substation or transformer is affected by transmission line problems over a distance of several miles. On the other end of the spectrum, high-speed integrated circuits that produce pulses in the nanosecond range require transmission line treatment for signals that are a few centimetres long. Signals on the Ethernet have wavelengths of about one meter, so transmission line theory applies to every network with at least two stations on it, assuming the machines aren’t located on top of each other. Every signal conductor has some inherent capacitance and inductance. The inductance comes from the fact that any conductor must have a real non-zero thickness, the capacitance is due to coupling with the ground plane and other nearby wires. Ethernet backbones are limited in length partly because of these capacitive loading effects: The longer the cable, the greater its capacitance. As the capacitance increases, each signal must charge up the line for a longer time, and after some critical value, the time required to charge the line’s capacitance is significant compared to the time required to send the packet’s preamble. At low frequencies, the non-ideal characteristics of the wire may be ignored, but at the Ethernet data transmission frequency of 10 MHz, the become important.
Figure 133 shows a drawing how a real-world Ethernet cable looks. In figure 133, the series of inductors/capacitor pairs define an AC impedance for the cable. Impedance is usually a function of the frequency of the signal encountering the L/C pairs. Ethernet packets are sent with a constant frequency (not the frequency of the packets themselves, but the frequency of the modulated signal representing the packet), fixing the AC impedance of the cable. The fixed impedance is why you can put a fixed-value resistor on the Ethernet as a terminator, the rest of this discussion explores the transmission line theory underpinnings that determine the value of that terminator. On a non-ideal wire, the voltage at an endpoint can’t change instantaneously, due to the capacitive and inductive effects described earlier. When a signal is impressed on a line (when a host sends a packet on the Ethernet), the voltage at the end of the wire must go from 0 to -2.5 volts. A packet rolling down the Ethernet cable is represented as a series of voltage changes, each with a corresponding change in current as defined by Ohm’s law. The endpoint of the wire appears to be a signal load, for this discussion assume that the load has an arbitrary value.
Figure 134 shows the signal on an Ethernet. The endpoint of the wire, represented as the load above, is initially at 0 volts. In order to satisfy Ohm’s and Kirkhoff’s laws, a reflected signal must be created. • Kirkhoff’s law dictates that the current flowing into a node must equal the current leaving it. The incident, load, and reflected currents obey the following equation:
• Kirkhoff’s law states that the loop voltage around a circuit must add up to zero. We can use this form of Kirkhoff’s law to express the relationship of the voltages in the circuit:
• Ohm’s law is used to describe the relationship of the line impedance, Z, and the current:
Substituting for VL and IL, we get:
Apply Ohm’s law again, with VR = IR . ZO, since the reflected signal sees the same impedance as the incident signal:
Rearranging terms, we can express the amplitude of the reflected signal as a function of the original signal:
Now let’s revisit our assumption that the load impedance, ZL, is some arbitrary value. An unterminated cable endpoint has an infinite load impedance, so with ZL infinite, the fraction’s value is approximately unity and VO = VR. The reflected current becomes a signal that looks electrically similar to the incident packet, travelling in the opposite direction. Again, the non-ideal physical characteristics of the wire prevent the reflected signal from being a mirror image of the incident signal. At the same time, the end point of the line starts to charge to -2.5 volts, so the voltage V at the endpoint of the wire isn’t precisely 0 volts. The combination of these two effects makes the reflected signal a slightly attenuated version of the original. After several trips down the length of the cable, the reflected signal is damped out completely. During the voltage rise time, however, reflected signals are making the line ring. The fairly obvious solution is to make the reflection coefficient (the numerator in the fraction above) equal to zero, so that there is no signal reflection. By placing a terminating resistor between the cable and ground, the incident signal is caught and any reflection is suppressed. Ethernet cabling has a characteristic impedance of 50 ohms, which is precisely the value used for termination. Note that the line impedance is seen by AC signals only, and that DC testing of the line itself, without the terminators, should show a DC resistance of a fraction of an ohm. However, this fact can be exploited to perform a simple cable test: With a multimeter set on ohms, measure the DC resistance between the centre conductor of the Ethernet and the ground shield on a network with no traffic. Do not measure resistance on a live network. The network activity will cause the ohmmeter to give an inexact reading. You may inadvertently create a short on the network, possibly damaging some transceivers equipment. The multimeter should read 25 ohms, half of the terminating resistor value, for a properly terminated Ethernet. The resistance of the entire cable is 25 ohms because it is the effective resistance of the two 50 ohm terminators wired in parallel, joined by two conductors of the Ethernet cable:
Figure 135 shows the terminators on an Ethernet cable. Sometimes the most perplexing network problems stem from a failure in the physical layer. This theoretical discussion may not help you debug open circuits or locate bad transceivers by watching waveforms, but it should help you build a mental checklist of potential problems to be used when examining network cabling.
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