Radio Waves (by )

I really love learning things, and recently I've finally been removing a long-standing thorn in my side - the fact that I don't really understand radio frequency electronics and the propagation of radio waves.

I've tried to fix this a few times in the past, but the resources I'd read never seemed to quite explain the whole picture - and I couldn't see how to piece the things they explained together into one coherent understanding of the electromagnetic world; they were clearly only shedding light on little corners of a totality that still remained mysterious to me.

Well, there are still gaps in my understanding... but I've made some progress, and in the hope that I can help others struggling with the same confusions as I was, I'd like to share my way of understanding it all.

One thing that bothered me was that explanations of transmission-line behaviour seemed to flip between talking about instantaneous voltages and currents at some point in the line, sampling the analogue signal travelling down the line - or talking about an RMS average voltage or current, and thereby causing me to struggle to make sense of what they were saying. But I think I now get transmission lines to some extent (although I'm still hazy on waveguides, because I've not gotten around to looking into them yet). And I was never quite sure what the impedance of a whole transmission line really meant, regardless of its length. If I had a transmission line and put a resistor over the end of it, and hooked it up to a battery and an ammeter, I knew that the current flowing would depend on the total resistance of the line and the resistor at the end - which would depend on the length of the line, as its resistance would be in ohms per meter. So what the heck was this impedance thing about? How did impedance mismatches cause reflections?

So, here's how I think about transmission lines now. The "DC model" of hooking a battery up to one end of a line and reading the current that flows into it is, of course, perfectly true - we can set the circuit up and test it; the reason it doesn't contradict with this weird parallel world of impedances is that the DC model is a steady state model of the system. When you first connect that battery to the line, current is going to start flowing into it, crawling along at a sizeable fraction of the speed of light; but until that current has reached the end, flowed through the terminating resistor, and flowed all the way back, it can't possibly have communicated any information about the total resistance of the line and its terminating resistor... So how much current initially flows from the battery, and why? Of course, the line can be thought of as two series of tiny inductors (with resistors in series, if we assume the inductors are perfect) with tiny capacitors connecting the two conductors, due to the inherent inductance of the wire and the inherent capacitance of the gap between them; you can imagine that the current from the battery has to charge the capacitors through the inductors for the voltage/current surge to propagate down the line. But what made "impedance" really click for me was going back to the basics of Ohm's Law and seeing it as a ratio of voltage to current. At any point along that transmission line, a certain instantaneous current will be flowing - and there will be a certain instantaneous voltage between the two conductors at that point; and the voltage divided by the current is the impedance.

So, if a 12 volt voltage source is connected suddenly to a 50 ohm impedance cable, an instantaneous current of 0.24 amps (12̣̣̣̣÷50) will flow. Now, as Kirchoff's current law tells us, the currents flowing into a node must sum to zero; so with the imaginary node at any point on the transmission line connecting two halves of it, the input current must equal the output current (although some current may be lost to resistive heating or leakage, that's not relevant in this case). So what happens when there's a change in impedance at some point in the transmission line? The current must remain the same, but the impedance changes - so the voltage must change, to make Ohm's Law still hold. If my 50 ohm cable is connected to a 75 ohm cable, that 0.24 amps flows into it and changes into a voltage of 18 volts (0.24×75). Which is why high impedance transmission lines are less lossy; a transmitter putting a watt of power down such a line (with proper impedance matching) will push out a higher voltage with less current that one putting a watt of power into a low impedance line - and resistive losses in the cable are worse for higher currents.

How about the reflections when impedance changes? I'm still a little hazy on this, but I think it's something along the lines of this: imagine a point just where the impedance changes in our example of moving from a 50 ohm cable to a 75 ohm one. A current is flowing into that point, but the voltage is higher after that point than before - which is going to create a current travelling back the other way. Where I'm hazy on is how this happens at junctions where the impedance falls (is it to do with the fact that the current flows alternately backwards and forwards, so the junction is traversed by current in both directions anyway, and the phases where the current travels from low to high impedance are what create the reflections? If so, isn't that a kind of rectifying action, that will create harmonics and intermodulation? But what is the "direction" of a signal travelling along a line, anyway? If we froze the signal in time, we'd just see a sine wave of voltage and a sine wave of current along the transmission line - if we restart time, how does it "know" what direction to propagate in? Something to do with the relative phase of the voltage and current waves?) So, yeah, I've a little more to learn there.

But this model of impedance does explain a lot. I wondered why the angles of the radials of a ground-plane monopole antenna affected impedance, but now it makes sense - the end of the transmission line basically spreads out to become a dipole, or a monopole and its ground plane; the electrical field of the travelling signal has to cross a larger region of space, so it makes sense that the voltage required to do so might vary depending on the amount of space crossed. All the mysterious constants, like the fact that a dipole trimmed to 0.48 times the wavelength has an impedance of 70 ohms are really down to the electromagnetic stretchiness of space: the impedance is the voltage required to push one amp along a transmission line (a dipole antenna just being an oddly-shaped transmission line, handing the signal over to the even weirder transmission line that is free space itself), and that is a function of the permittivity and permeability of that space.

This model also explains how impedance matching transformers work. A 1:2 transformer will transform X volts and Y amps on the "left" into 2X volts and Y/2 amps on the "right"; as the impedance is V/I, that means it converts R ohms on the left into R4 ohms on the right, simply through changing the voltages and currents. A 1:N transformer makes a 1:N^2 change in impedance.

Antennas with multiple elements are confusing, but I'm not sure anybody really understands them - as far as I can tell, the design process is almost always to mock it up in a finite-element computer simulation or build a prototype and tweak the design until the desired parameters are obtained experimentally; the mutual interactions between the elements (not to mention ground, support structures, and the transmission line feeding the antenna) are just too complicated to analyse.

I really don't get why there's a near field and a far field (or that funny one inbetween that, I think, is just a mixture of the two). Does the antenna both far and near fields at once, and the near field is stronger but doesn't spread out far, so the far field is negligible when close to the antenna? Or does the antenna create a near fields, which "decays into" the far field as it spreads out? Nothing I've found seems to explain.

I'm not very clear on why a balanced transmission line that's shorted at one end and open at the other end has varying impedance along its length, and can be used for impedance matching, but it doesn't create reflections from the ends.

But, I can understand how to run a cable to a dipole or monopole antenna, manage the impedance transitions, and make it radiate efficiently. That's progress!

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