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 that 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 $15, it won't tell you that a $15 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, 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 our own data processing capabilities.
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. One of the best tools for this is a low level simulator, like SPICE, or even better, models you build yourself using matlab (or scipy) and/or pencil and paper. These simulations 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, e.g. do I really need to model that extra via on a transmission line? Lab work is another great way to develop a sense for the physical world.
One method that can be used for developing this intuition is to start with simple models that are based on something that familiar, like Circuit Theory and expand them into a more complicated system model. For example, a Transmission-Line can be modeled as several inductor/capacitor pairs arrayed in a circuit (the basis for the Lossless Telegrapher's Equation ,this is a common approach, I didn't invent it). 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 can all do the job depending on what you're trying to figure out.
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, stubs, etc. For example, if I add a capacitive load to the end of the line the last capacitor on the line will be larger than the other capacitors so it will take more current to charge (effectively lowering 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.