Transmission Line Transfer Function from ABCD and S-parameters
Circuit designers and board designers like to use S-parameters to describe signal behavior as it passes through an interconnect. These important parameters tend to get overgeneralized (in my opinion), and there are other important quantities that may be easier to calculate if you use some different parameters. In particular, a transmission line transfer function is one important quantity used for signal integrity calculations and simulations, particularly when modeling an interconnect in lossy media.
A transmission line transfer function also allows you to simulate signal behavior for any input stimulus using an impulse response function, which is a critical aspect of high speed signal integrity simulations and modeling for modern signaling standards. The power of this method seems to be lost on many PCB engineers and has become the purview of IC engineers. PCB designers often default to simulation tools to examine this aspect of interconnect design, which inevitably produce the wrong results because they aren’t accounting for all high speed effects in a real channel.
Despite these drawbacks, there are some simple calculations you can perform to get an accurate view of how a signal will behave on real transmission lines with real load components and termination. Let’s look at a simple yet powerful way you can calculate the transfer function of your transmission lines and learn more about your system.
Transmission Line Transfer Function Equations
By far, the easiest way to calculate a transmission line transfer function is to use ABCD parameters or S-parameters. I prefer to use ABCD parameters because I work on modeling, and they are more easily generalized to any transmission line. After all, they are defined directly from the general solution for a transmission line. I personally think S-parameters get over-generalized and are misapplied to situations where they don’t fit so well conceptually. I think it’s also important to note that there are other equations for converting between different types of parameters (e.g., Z-parameters, Y-parameters, etc.), so you can always find a way to get to the transfer function.
If you’re unsure of why we need to get to a transfer function for a transmission line, I’ll go over it at the end of the article. For now, just know that regardless of which approach you want to take, ABCD parameters and S-parameters offer some specific advantages:
- Why use ABCD parameters: These parameters are defined directly from the general solution for any transmission line (as long as it’s an LTI system). They are far more generalizable than S-parameters, including to cases with copper roughness, causal dispersion, and fiber weave/skin effect losses. ABCD parameters are best for cascaded networks (e.g., line + stub + line + load branch type networks) as you can just multiple ABCD matrices together.
- Why use S-parameters: These are what you would normally measure in a standard setup for characterizing high-speed channels with multi-GHz bandwidths. Therefore, it’s natural to use these to calculate a transfer function as you don’t need to do some other complicated inversion from impedances. Both sets of parameters can be generalized to N-port networks, but ABCD parameters require building a transfer function matrix, while S-parameters are easy to extend to N ports.
If you can accept the above arguments for using ABCD parameters on the theoretical side and sticking with S-parameters on the experimental side, then we’re ready to jump into the important equations you’ll need.
From ABCD Parameters
The standard definition of the ABCD parameters is shown below. These equations apply to any transmission line as long as you know its impedance and propagation constant:
Note that the ABCD matrix, which is invertible, is defined “backwards” in that it relates the input voltage/current (i.e., looking towards the load) to the output voltage/current. This is okay; to create a relation that shows the output voltage/current as a function of input voltage/current, simply calculate the inverse matrix. You don’t need to do this to find the transfer function for a transmission line. You can use the ABCD parameters defined above with the following formula to get the transmission line transfer function:
Alternatively, you can approach this problem using S-parameters. As I mentioned above, this is great if you have some measurements of your channel’s S-parameters and you want to get the transfer function. In this case, your S-parameters are referenced to the load impedance, and you can just use a simple S-parameter to ABCD parameter conversion:
After converting, just plug these into the transfer function equation shown above, and you’re finished. Remember, Z in this equation is the reference impedance, which is usually taken as the load or characteristic impedance of the line.
Alternatively, you may want to calculate the S-parameters directly from ABCD parameters in the event you don’t have any S-parameter measurements. The formula below shows S-parameters defined from ABCD parameters assuming both ports have the same reference impedance. You can then use these to calculate a transmission line transfer function. Again, watch out for the reference impedance in the following formula:
If we have different port impedances, we have:
Finally, with either of the above equations, we can calculate the transfer function using the S-parameters and the reflection coefficients at the source and load:
Other Transfer Functions
Note that an S-parameter itself is a transfer function, but not in the sense that it provides a conceptually useful impulse response. The same can be said for Z-parameters and Y-parameters, which don’t have conceptually satisfying meanings. This is why a transfer function (in the sense of filters and amplifiers) is normally used for high speed channel characterization; its impulse response function does have a concrete meaning in a channel or circuit.
Once you’ve calculated the transfer function, remember that it is bandlimited, so you will need to apply a windowing function before you can calculate the channel’s response. Numerically, I think it’s easiest to just get the channel’s response using the inverse Fourier transform and the windowed transfer function H(f):
Alternatively, you can calculate the channel’s response using the convolution theorem, i.e., with the channel’s impulse response function. This now tells you exactly how the channel will respond when excited with an arbitrary stimulus. Once you’ve found the transmission line transfer function and you’re ready to layout your channel, use the design and layout tools in Altium Designer®. You’ll have the routing and layout features you need to route transmission lines and waveguide geometries with ease.
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