Understanding Transmission Lines
Charles Eidsness
Back in the day (or so I'm told) they used to use a pencil and paper and maybe a slide-rule to do arithmetic. Now we have calculators and computers and can easily calculate anything we want, which I'm sure almost everyone will agree is superior. The one downside to our new-fangled calculating machines is that very few people can now perform arithmetic mentally. Why would you want to? If you're manipulating numbers yourself you can gage their relevance and correct errors in your initial assumptions. A calculator will tell you that 150% of $10 is $150, it won't tell you that a $150 tip is a little excessive and you probably only wanted to tip 15%. The human brain is exceptionally good at taking incomplete or incorrect input and still providing a reasonable response. The computer on the other hand is very bad at adapting but will provide a much more accurate response if the input is correct, i.e. the computer will almost always me more precise, but the brain can be more accurate. The best results can be obtained by combining the calculation ability of the computer with the data processing capabilities of the brain.
An alarming trend that I've noticed in Engineering in general and in High-Speed Digital Design in specific is that Designers are too quickly jumping to "application note design", and "cut-and-paste design". They are not fully developing an intuitive feel for the physical systems they are trying to manipulate. There can be no substitute to learning by playing around with stuff. I find the best tool for this is a low level simulator, like SPICE, or even better, models you build yourself using a tool like scipy and/or pencil and paper. These simulations and models may not model every aspect of a design but they can still be very useful, and by using them you can figure out how much accuracy is required, i.e. do I really need to model that extra via on that transmission line? Lab work is also a great way to develop a sense for the physical world. Once you have probed a couple thousand different transmission lines signal integrity should almost become second nature. Once you have developed a sense for how a system behaves you can use the great powers of the human mind to extrapolate that experience onto new systems that may be similar but not identical to ones you have seen in the past.
One method that I like to use for developing this intuition is to create simple models that are based on something that I'm familiar with, like Circuit Theory. Then I can expand them into a more complicated system. For example, a Transmission-Line can be viewed as several inductor/capacitor pairs arrayed in a circuit (the basis for the Loss-less Telegraphers Equation). This is a common approach, I didn't invent it, but other folks like to envision a T-Line in different ways. Like as a wave-guide using Electro-Magnetic Theory, or using a ladder diagram, they all do the job. Here's how I envision a Transmission Line operating:
When a voltage is applied to a Transmission Line the first capacitor will start to charge, i.e. dV/dt will be greater than zero so the current through the capacitor will be greater than zero. As the caps charge current begins to flow through the inductors, so dI/dt will be finite and there will be a finite voltage drop across the inductors. Given the direction of the current this voltage drop will work against the applied voltage. As the cap charges there is less current flowing through the inductor so the voltage drop is less, which means the cap charges more. This charging action is what creates the propagation delay of a signal down a T-Line. The capacitors demand current and the inductors limit the amount of current they can have. This balances out to a equilibrium which is defined by the characteristic impedance on the line, i.e. a 50Ohm line will require 50Ohms worth of current to change its voltage.
The reason I like envisioning a T-Line in this way is because it makes it easy to visualize what effect adding loads or stubs, etc.. For example, if I add a capacitive load to the end of the line the last capacitor of the line will be larger than all of the other capacitors so it will take more current to charge (effectively lowers the impedance). This extra current will create a larger dI/dt through the inductor which will create a larger voltage drop across the inductor. When the voltage drops across the last inductor in the chain, capacitor to its left; which is already charged, will see a voltage drop across it and will discharge some of its charge into the capacitor to its right, and so on back down the line until the current is zero again. This looks like a reflected voltage droop down the line.