Reflection coefficient and a transmission line taper

Hello all.

I am working on designing a transmission line which involves the use of a taper. One of the issues I am struggling to understand conceptually is this reflection coefficient often shown for your standard exponential, Klopfenstein, hyperbolic tapers like the one attached to this post.

So what does not make sense to me here is the low frequency behavior shown in that image. I thought you only had to worry about impedance mismatching/control for higher frequencies. At DC-Low frequency the only thing of concern should be resistivity no? What I mean to say is if I attached one end of the taper to a dc/low freq power supply and the other end to a 50 ohms terminator and measured the voltage across the terminator, I should just get a voltage divider no?

So I guess my question is how am I supposed to interpret that graph? At high frequencies it makes sense to me, because losses and reflections are governed by the characteristic impedance where ωL>>R. But I'm not sure what it means at low frequencies, is it really a high pass filter? If not at what point does impedance begin to dominate? I've read some places that impedance control only really becomes important when the trace approaches λ/8.

What you are forgotten is that the system impedance stuff works down to DC. So you feed your line DC via a 50 ohm resistor and you get Vo, this equates to 1 on your graph. Yes there will be a series resistance to reduce your DC output, likewise there will be insulation losses to reduce it further, but in the scheme of things they can be ignored UNLESS you are a telephone engineer where your lines are miles long, then very extreme measures are required to keep the lines to their nominal impedance.
Frank

So I guess the real issue is how long the taper is compared to the wavelength you are interested in? So for low frequency we don't really think about characteristic impedance (lab setting) because the wavelength is so much longer than anything we work with. This might be a difficult question to answer is a general sense, but at what point do you have to begin considering impedance control? I've read when the trace becomes shorter than lambda/8 other places lambda/16.

I can not comment other then to say it all depends on the precision of you measurement/application.
Frank