Computer modeling of the inverted L antenna shows a small directivity with the greatest signal pickup from the direction opposite that to which the open end points. I have given some thought as to why the inverted L might exhibit directional characteristics for the reception of ground (vertically polarized) waves, and present some ideas here.
First off, understand that I am not an antenna engineer and present these speculations to suggest a way to increase the directive gain of a small (compared to a 1/4 wavelength) antenna. For simplicity of discussion, wave propagation issues will not be delved into. We will consider that the operation of an inverted L antenna results from the sum (superposition) of two modes of operation.
The first mode is that of a capacitively loaded vertical antenna. See Fig. 1. Co represents the “top hat” loading capacity. Visualize the horizontal wire BCD shifted to the left so as to be symmetrical relative to the downlead AB (with point C of ABC directly above vertical downlead BA). The total capacitance, Co, of wire BD to ground acts as the top hat capacitance for the “vertical antenna” downlead BA (the additional capacitance between the upper and lower parts of down-lead BA can be ignored because the antenna is assumed to be short compared to 1/4 wavelength). The second mode is as single turn ‘virtual loop’ antenna.
Let X represent a crystal radio set with antenna and ground connections, a and g. Fig. 2 shows Co as a lumped capacitor C1, connected to ground at the center of the BCD horizontal element. The rectangular loop circuit ABCFG, consisting of the four sides AB, BC, CF and FG can be looked at as a single turn loop antenna of area A, oriented to pickup signals from the B<->D direction. Note that the path for the ‘displacement current’ through C1 makes up one side of the loop. The current flowing from the induced EMF in the loop will combine to current from the EMF picked up in the down lead BA. These two EMFs add when a signal comes from the direction opposite to that to which the open end points and tend to cancel when coming from the opposite direction. This gives a plausible explanation of the mild directivity of the inverted L antenna. This directivity has been theoretically demonstrated by using computer simulation. For clarity, Fig. 3 shows a top view of the antenna.
The following changes to the configuration may serve to increase the directional gain of the antenna (I haven’t tried them out.). The induced voltage in a loop antenna (small compared to a wavelength) is proportional to its area, among other things. If the effective location of capacitor C1 could be moved to point D, and its value kept the same, the area of the loop would be doubled to 2*A (ABDEG), thus doubling the current from its induced EMF and further increasing the directive antenna gain. Let us change the lumped representation of Co as one capacitor C1 into two capacitors, one from point B to ground and the other from point D to ground. They will each be equal to one-half of Co and both still will act as a top-tat capacitor for the vertical antenna BA, otherwise known as the lead-in. The one from D to ground will be called C2. See Fig. 4. One way to add more capacitance to ground at point D is to construct two lateral arms on the antenna. See Fig. 5. If each arm is the same length as one half the horizontal element BD (They can be of some other length.), the effective capacitance they add at point D is 2*Co/2=Cadded. The sum of the two capacitors C2 and Cadded is 1.5*Co = C4. The loop area is doubled and the current from the induced EMF is tripled (because C3 is one and one half times C1). The antenna directivity may thus be increased giving more gain in the direction in which the two effects add, and less in the opposite direction. If the directive effect can be made strong enough, a cardioid pattern should result, with a good null in the “opposite” direction. A more practical approach might be to have the two side arms droop down at an angle and be secured to ground with insulated guy wires. A single vertical wire, similar to the lead-in, might work as a substitute for the two arms if its bottom end does not get too near the ground (inverted U antenna).
Note there is no “free lunch” here. To the extent that the signal pickup is increased in the direction opposite to that to which the open end points, it is reduced in the other direction.