Introduction to Transmission Line Circuits
Welcome to Radio Systems Design: Module 31 – Transmission Line Circuits.
In this module, we’re going to cover transmission lines as circuits and we’ll talk about some the typical constructs of transmission lines as circuit elements including impedance matching, RLC tanks, and transformers or couplers. We’ll then introduce the concept at Even and Odd Mode Analysis, which is particularly useful in analyzing transmission line circuits. We’ll talk about several of the applications of transmission line circuits, such as filters, dividers, combiners; in particular will go through the design of Wilkinson combiner. And then we’ll also talk about Couplers and we’ll talk in-depth about the Branch Line Coupler and then touch upon the Rat-race and the Lange Coupler.
Alright, so at the RF, transmission lines can be used in circuit elements instead of discrete lumped components; up to now, you probably thought about the coaxial cable in your lab just as a way to connect it, but it actually can be a circuit element itself. Some typical applications include transferring a signal from one location to another; so this is just the signal conductor that you’re used to, but it can also divide the power of signal; we’ll look at our come divider combiner about this. It can also generate a phase shift such as ninety degrees or 180 degrees and in earlier modules, we saw lots of uses for ninety degree and 180 degree phase shifts in quadrature modulators. Generally though, these types of applications can only be performed over limited frequency bandwidth and that has to do with the fact that we’re using the wave properties on the lines and oftentimes these are fractions of a wavelength.
And for a given the length of the line then if you want to be able to equate to quarter wavelength for instance, that’s only true at one specific frequency. But we usually can operate around that frequency and still have an approximate quarter wavelength, but you can see for instance if we change the frequency significantly, the length of the line is nowhere near the quarter wavelength that we needed.
So as I said, its performance depends upon lambda of the wavelength. And one last thing is that oftentimes, we impedance-match all of our input and output ports to 50 ohms. And so not only we can use a transmission line to do this impedance matching, but it’s also what we use to change that to route the signals around in our design.
Alright, so the next couple of slides are just an overview; this is a quarter wave transformers and if you remember, we can rate Zin that’s the impedance looking in terms of the characteristic impedance of the transmission line versus the load. And if we look at the quarter wavelength, can be reduced to and then goes to infinity. If we work to the math, we end up with a nice simple equation that Zin equals the characteristic impedance times RL and this is what a quarter wave transformer does as an appearance transformer. And I believe I mentioned before, we can also write Z1 equals the square root of Zin times RL. So the characteristic impedance is the geometric mean of the load and Zin impedance and that’s not something I think that’s easy to remember, but that’s true for the lambda over 4 transmission line.
So, as we mentioned that impedance transformer is a typical usage of a transmission line and we look at special cases in the past: what happens if you use a lambda over four with a short. And if you remember, it actually looks like a parallel LC tank because it takes a short and transforms it into an open at lambda over four just like a parallel LC looks like an open at and I should get rid of this and have this be an open and at resonance, this is going to be a short and that’s exactly the same thing series LC tank looks like a short. You can see kinda duality between a parallel and a series LC with a shorted transmission line and an opened transmission line. As I mentioned before, transmission line in its basic function has a phase shift; so we’re all familiar with ; so if we have a length of the transmission line, it’s gonna create a phase shift in the signals that are propagating. This can be useful if you want to generate a 90 degree or 180 degree phase shift.
Another typical usage of transmission lines which we haven’t covered because it gets a little bit into electromagnetic waves and modeling that is beyond this course, but we can actually use transmission lines as either a capacitive or inductive coupler. So here’s an example where we have a signal coming in and moving up to port two and part of that is coupled onto port 3 and we show this coupling are transformers effect with a capper RJ depending on whether it is voltage or current.
This is the slide we’ve seen before, but this is just reiterating that we can look at a transmission line version and that we have kind of a schematic equivalent that shows us what’s going on. And that here some circuit equivalents and so we’ve got our parallel LC that we saw before, with this being a short this is an open, and then we have a series LC. Here, we have a shorted line and then we have some coupling and we show the two-line links here with theta and theta because we’ve got theta on this one and theta on this one and the model for this is a transformer and then a parallel LC tank and this can also be done as a series LC model. We can then do it with two stubs if you’ve seen here and things just get more complicated as we move down. But, you can see basically, these are all LC tanks and transformers, where moving from here to here, we likely just change the values of the components to make these two equivalent. So, I really can’t go for transmission lines to sorted LC equivalent using this series and parallel tank as well as our coupler transformer.
And some applications of this is to build filters, where we can actually embody a different types the filter structures by doing different types of transformers. And this one’s kinda intuitive and I’m gonna have some capacitance between the two and then there’s some ground underneath this. I’m assuming this is a micro-strip line and I’d have some capacitors to that ground and I can intuitively see, “Okay, I’ve got my three capacitances here.” This inductor is just two transmission lines with a very thin wire; that thin wire looks a little bit inductive and you can go through and kinda envision with these others on how they work. This is a great one, where you’ve got a thin wire, which is an inductor and you’ve got a very thick wire which looks like a capacitor. So you’ve got inductor, capacitor inductor, and capacitor. In reality, this is a transmission line with very low characteristic impedance and this one is very high characteristic impedance. If you look at the math though, it looks like an inductor or looks like a capacity. And you see that you can build nice filters with these models.
Here’s a picture an actual low-pass filter that was built at 2 gigahertz and you could see that this is actually how you build it and you can see very nicely here the N launch connectors. This is likely a 50 ohm transmission line and then we have a little trace and another trace as a capacitor as we go on.
