Note: skip to the "short version" for the payoff of my rambling!
Maybe this is an opportunity to help with the antenna, as I've been an amateur radio operator since I was 16, and I'm the son of an RF engineer. There is nothing special about car radio antennas, they are as they appear - a piece of wire hanging out there in the breeze. The wire does have two important properties - Its length will be a certain fraction of the free-space wavelength of the signal you intend to receive, and it is electrically insulated from the ground plane (which in this case is the body of the car). Its length is engineered so that its impedance will match the design impedance of the radio. In practice, you can construct circuitry to match any length you like to that impedance, but its cheapest and easiest for the manufacturers to simply make them the length that matches. Most older cars seem to have an antenna designed to be 1/4 wavelength on the FM band, which presents about a 75 ohm impedance (depending on the ground plane, the design could be anywhere from 34 - 90 ohms impedance). This matches to the 75 ohm coax and the input of the radio, making the energy transfer from the antenna to the radio optimal (mismatch causes loss). In case you are curious, the wavelength in feet is equal to the speed of light in feet per second over the frequency in hertz. That would make a full wave on the AM band (I used 1200) 820 feet, and a full wave on the FM band (I used 102) 9.6 feet. To be a 1/4 wave at the FM frequency, the antenna would be about 2.4 feet long. It suffices to say that cut for anywhere in the FM band, the antenna will be between 2.2 and 2.5 feet long as a car body isn't a perfect ground plane, so the dimensions could shift somewhat. That presents the problem that the antenna is very very short compared to the AM band wavelength, which means it will not present the same 75 ohm impedance on that band to the radio. There is usually some matching circuitry in the radio to match the very high impedance that the antenna would be on the AM band to the receiver. Some cars run preamplifiers and matching circuits outboard of the radio, and it seems to be very common with the short stubby antennas that are so common now.
I'm sure you all enthralled, but let me bring this back to hypermiling and your antenna, and the subject of using the defroster grid. All the radio really cares about is the impedance of your antenna. If the impedance right, at the size of our cars, the only other real variable is size. The bigger the better, up to a point (for reasons that would be equally as boring to you). Antennas can be thought of as having two halves.. I'm sure you've seen this on old TV rabbit ears. With the usual car antenna, the antenna makes up half of the system, and the body of the car makes up the other half. Clever eh? Radio waves don't care what it looks like. SO, when you see antennas in windows, you'll note that they have two halves.. they aren't using the body as another half.
A 1/2 wave dipole will have a feed impedance of about 50 ohms in free space. What's a halfwave dipole? It's like two antenna elements in a straight line facing away from each other and fed in the center. Each element is a 1/4 wave, so the total size of the antenna is 1/2 wavelength. Think TV rabbit ears spread as far apart as possible, so that they are flat. like this:
---------------------0 0-----------------------
This gives us two options for glass mounted antennas. You can make a single element, make it about 1/4 wavelength long, and connect that to the center of your radio antenna feedline (the coax going to the radio). Then connect the shield, which is the ground, to the car body.
The other option, is to make two elements, each 1/4 wavelength long, and connect to them in the middle. The center of the coax to one, the sheild to the other.
So, if you are using the defroster grid, attach to it and cut it to be an element of about the right length (2.2-2.5 feet should be about right). If you don't have a defroster grid, you can use cheap metal foil tape and stick it to the window. Size isn't really too important, a thin strip would work fine.
Since we are only talking about receiving, none of the dimensions are very critical. On the transmitter end, having a very small mismatch can cause a large amount of power to be used up as heat instead of transmitted, and since they are dealing with thousands of watts it is something to worry about. On our receiving end, the amount of power we are talking about is tiny.. a strong signal might impart a few microvolts to the input of your radio, which is why most of the radio is just high gain amplification.
One last note, Broadcast radio is almost always transmitted with vertical polarization. A vertical antenna will receive that signal with greater efficiency than a horizontal one, by about 6dB. Don't worry about the specifics, just know vertical is better for a car antenna.
So the short version? Stick some metal tape to one of your windows. If you attach the shield to the body, you can use 2.2-2.5 feet. If you don't, use two pieces, each 2.2 - 2.5 feet. Run the antenna vertically if possible. Connect to your new antenna using 75 ohm coax available just about anywhere (RG-59 is extremely cheap at home depot). I like copper tape, but I've used the metal tape sold for insulation purposes at home depot. The best way to connect to it is solder, but you could probably leave "tabs" to connect to with a twist-n-tape connection if you had to.
I've used the metal foil tape on my car windows to transmit 50 watts @ 144 Mhz before and had no issues, so it will work well for receiving. That same setup contacted the ISS
Should cost you less than 5 dollars.
Your chances of damaging the radio are just about nil. The worst thing that could happen is that you short the antenna center to the ground, which would only result in crappy reception, and would easily be fixed by getting rid of the short.
Hope I didn't bore you all to death!
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.....otherwise that sounds like it would work great.
