Log-Periodic Tooth Antennas
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Log-Periodic antennas antennas are designed for the specific purpose of having a very wide bandwidth. The achievable bandwidth is theoretically infinite; the actual bandwidth achieved is dependent on how large the structure is (to determine the lower frequency limit) and how precise the finer (smaller) features are on the antenna (which determines the upper frequency limit). On this page, we will expand on the ideas related to wideband antennas presented on the Bow Tie Antennas) page. Then we will explain why Log-Periodicity in antennas will produce wide bandwidth.
What does Log-Periodic Mean?On the bow-tie antenna page, it was noted that wideband antennas are often defined by angles instead of lengths, so that they are more frequency independent (because angles are independent of wavelength for any frequency). As an alternative to this, what if the antennas had a self-similar structure, so that the properties at some frequency f2=k*f1 were the same as at the first frequency f1 (and k is some constant greater than 1). This isn't quite a fractal, just some sort of structure that has some repitition to it. I will explain the last paragraph in a bit more detail as it is the main idea behind log-periodic antennas. Suppose we design an antenna system that works at some frequency f_n (we are using a subscript n just so we can number different elements in the antenna). Suppose we setup an antenna system as in Figure 1, where there are wires of length L_(n-1), L_n, L_(n+1), ... each separated by a set of distances d_(n-1), d_n, d_(n+1):
 Figure 1. A Log Periodic Structure. We require that the ratio of the successive element lengths (L_(n+1)/L_n) be equal to some constant k, and that the distance between elements (d_(n+1)/d_n) also equal k. This is a log periodic structure (I'll explain the terminology later).
Now it is time to discuss the key property of log periodic antennas. Suppose that our antenna in Figure 1 radiates
well at frequency f_n (primarily due to the element L_n). Then the antenna must also radiate at f_(n+1) and f_(n-1),
because the antenna structure is electrically the same - it "looks" the same to wavelengths
 Does the last paragraph make sense? It is the reason we care about log-periodicity. We make a structure that repeats itself by a constantly increasing mulitplicative factor (k). Then if the structure radiates at some frequency, it will radiate at all the multiples of k. Reread the last two paragraphs if it doesn't make sense. Now, our radiation mode at frequency f_n will have some bandwidth (frequency range centered about f_n) where the antenna efficiency is good. Suppose we choose the expansion factor k such that the frequency band of the next radiation mode f_(n+1) overlaps the frequency band of the first radiation mode. Then we will essentially have an antenna where the radiation mode is good everywhere between frequencies f_n and f_(n+1). And since this is true for every n, we essentially have a very wideband antenna. It will only be limited in bandwidth by the number of elements in our antenna array. Is that cool? I think that is pretty awesome. Now, why do they call this "log-periodic"? I think that name is stupid. Really log-periodic is the same as what I just described - there are features that grow by a constant geometric multiple (k). Mathematically, due to the properties of logarithms, if all the elements grow by a constant multiple then the ratios of the logarithm will be constant:
Math junkies will try to make the reasoning more complicated than that, but it really isn't.
The Log Periodic Tooth AntennaNow we are ready to introduce a very beautiful antenna, the Log Periodic Tooth antenna. Let's first show a picture of the antenna so that you can see how cool it actually is:
 Figure 2. A Log Periodic Structure that Forms the "Tooth" Antenna. The above antenna is just awesome. We'll talk about why this is done and why it is log periodic, but take a moment and just check it out. The above antenna is actually an extension of the bow tie antenna. The bow tie antenna would be even wider if the currents that give rise to radiation died off faster along the structure. In order to do this, after a bunch of experimentation in the 1960s, it was proposed that the bandwidth of the bow tie antenna could be increased by adding the arms shown in Figure 2 in a log periodic manner. So let's define some of the parameters for our log-periodic tooth antenna and explain the geometry. We will define some an arm inner radius r_n and an outer radius R_n, along with two angles S_1 and S_2 as shown in Figure 3:
 Figure 3. Geometry of Log Periodic Tooth Antenna. As a note, check for yourself that if S1=180-S2 (degrees) then the geometry of Figure 3 reduces to the bow-tie antenna. For reasons we will discuss later, it is useful to set S1=S2. To keep the geometry similar to that used on the bow tie antenna page, we will use S1=S2=25.2 degrees. Now let's define some more design parameters (we've already specified S1 and S2). 
We are setting the log periodic expansion parameter k to 2, and setting the radius of our innermost arms to 4mm and 6mm (inner and outer edge). Now we have completely specified our log periodic tooth antenna. To break down the geometry of Figure 3, let's start with a series of concentric circles (with radii r1, r2,...,R1, R2, ...). Then we have:
 Figure 4. Simple Concentric Rings that will make up the LP Tooth Antenna. We will now mark in brown the areas where copper will go, and the non-copper areas will be green. First, let's fill in the center triangular regions of the bow-tie core of our LP antenna, and non-metallize the keep out zone that is on either side:
 Figure 5. Filling in the Center Section of the LP Tooth Antenna. Now, let's take the largest arm of the tooth antenna and fill that in (this is corresponds to R5=64mm or the outer circle). We also fill in the arm on the opposite edge one step down on the other side:
 Figure 6. Filling in the First Teeth of the LP Tooth Antenna. Now, we just fill in the teeth (or arms) of the antenna on opposite sides the whole way down the structure, and we have made our Log Periodic Tooth antenna (it's that simple):
 Figure 6. Filling in the First Teeth of the LP Tooth Antenna. Now we have our antenna, and we just add a feed (the radio or probe) across the center gap of the antenna, just as in the dipole or bowtie antenna case:
 Figure 7. The radio is connected across the center of the Log Periodic Tooth Antenna. We now have a complete Log Periodic Tooth antenna. From here, let's make a quick mocked up version and see how it performs:
 Figure 8. We Feed the LP antenna at the Center. Note that again we ground the feed coaxial cable along the exit path (the physical antenna structure). This minimizes the affect of the feed cable on the antenna (see baluns). We measure the VSWR of this antenna from 500 MHz to 8 GHz, and the result is shown in Figure 9:
 Figure 9. VSWR of LP tooth. The first thing to notice from Figure 9 is that there exists a strong resonance around 1 GHz, with a pretty wide bandwidth. The VSWR 3:1 bandwidth for this first mode from f_low=675 MHz to f_high = 1300 MHz, which is gives a Fractional Bandwidth of 63%, about twice as much as the bowtie antenna. However, the more interesting properties are at higher frequencies. For f>2GHz, we have the VSWR<4. The mismatch loss for VSWR<4 is less than 2 dB, so that the antenna efficiency will be greater than 50% (-3 dB) for this antenna frequencies between 2 and 8 GHz, which is a huge bandwidth. Further optimization on this antenna can smooth out the impedance variation and increase the overall bandwidth. All one has to do is optimize the angles (S1 and S2), expansion factor (k) and starting radius, along with the number of arms (teeth). However, for a first attempt, this is an excellent wideband antenna. Note that the lowest radiation frequency will be dictated by the longest (outermost) arm of the log-periodic tooth. The high frequency end will be dictated by how small the smallest (innermost) teeth are made.
Self-Complimentary Log Periodic Tooth AntennaThere is one more cool property of the Log Periodic Tooth Antenna, and this is the reason we set the angles S1 and S2 to be equal. Under the condition S1=S2, we will find that if we interchange the metallic and non-metallic areas of our antenna, we will end up with the exact same structure. This is known as the "self complimentary property". In Figure 10 we replace the metallic areas of our structure with air and vice versa, and we end up with the result as shown. If we then feed across the metallic gap as shown on the right, we have a log periodic tooth slot antenna. The properties of this slot antenna will be very similar to the standard dipole version of the antenna (same approximate radiation pattern, antenna gain, but the polarization will be changed).
 Figure 10. Self Complimentary Structure of LP Tooth Antenna. We see that the Log Periodic Tooth Antenna is an extension of the bowtie antenna. In the next section, we will expand on this tooth design and create the Log Periodic Dipole Antenna Array.
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as well as to 
and 
. Hence, if the antenna
radiates at frequency f_n it will radiate at all frequencies that are a constant multiple of f_n: