A Yagi-Uda array, com­mon­ly known sim­ply as a Yagi anten­na, is a direc­tion­al anten­na con­sist­ing of a dri­ven ele­ment (typ­i­cal­ly a dipole or fold­ed dipole) and addi­tion­al par­a­sitic ele­ments (usu­al­ly a so-called reflec­tor and one or more direc­tors). The name stems from its inven­tors, as the Yagi-Uda array was invent­ed in 1926 by Shin­taro Uda of Tohoku Impe­r­i­al Uni­ver­si­ty, Japan, with a less­er role played by his col­league Hidet­sugu Yagi. How­ev­er the “Yagi” name has become more famil­iar with the name of Uda often omit­ted. The reflec­tor ele­ment is slight­ly longer (typ­i­cal­ly 5% longer) than the dri­ven dipole, where­as the so-called direc­tors are a lit­tle short­er. This design achieves a very sub­stan­tial increase in the anten­na’s direc­tion­al­i­ty and gain com­pared to a sim­ple dipole.

High­ly direc­tion­al anten­nas such as the Yagi-Uda are com­mon­ly referred to as “beam anten­nas” due to their high gain. How­ev­er the Yagi-Uda design only achieves this high gain over a rather nar­row band­width, mak­ing it more use­ful for var­i­ous com­mu­ni­ca­tions bands (includ­ing ama­teur radio) but less suit­able for tra­di­tion­al radio and tele­vi­sion broad­cast bands. Ama­teur radio oper­a­tors (“hams”) fre­quent­ly employ these for com­mu­ni­ca­tion on HF, VHF, and UHF bands, often con­struct­ing such anten­nas them­selves (“home­brew­ing”), lead­ing to a quan­ti­ty of tech­ni­cal papers and soft­ware. Wide­band anten­nas used for VHF/UHF broad­cast bands include the low­er-gain log-peri­od­ic dipole array, which is often con­fused with the Yagi-Uda array due to its super­fi­cial­ly sim­i­lar appear­ance. That design along with oth­er phased arrays have elec­tri­cal con­nec­tions on each ele­ment, where­as the Yagi-Uda design oper­ates on the basis of elec­tro­mag­net­ic inter­ac­tion between the “par­a­sitic” ele­ments and the one dri­ven (dipole) ele­ment.

Yagi-Uda anten­nas are direc­tion­al along the axis per­pen­dic­u­lar to the dipole in the plane of the ele­ments, from the reflec­tor toward the dri­ven ele­ment and the director(s). Typ­i­cal spac­ings between ele­ments vary from about 110 to 14 of a wave­length, depend­ing on the spe­cif­ic design. The lengths of the direc­tors are small­er than that of the dri­ven ele­ment, which is small­er than that of the reflector(s) accord­ing to an elab­o­rate design pro­ce­dure. These ele­ments are usu­al­ly par­al­lel in one plane, sup­port­ed on a sin­gle cross­bar known as a boom.

The band­width of a Yagi-Uda anten­na refers to the fre­quen­cy range over which its direc­tion­al gain and imped­ance match are pre­served to with­in a stat­ed cri­te­ri­on. The Yagi-Uda array in its basic form is very nar­row­band, with its per­for­mance already com­pro­mised at fre­quen­cies just a few per­cent above or below its design fre­quen­cy. How­ev­er using larg­er diam­e­ter con­duc­tors, among oth­er tech­niques, the band­width can be sub­stan­tial­ly extend­ed.

Yagi-Uda anten­nas used for ama­teur radio are some­times designed to oper­ate on mul­ti­ple bands. These elab­o­rate designs cre­ate elec­tri­cal breaks along each ele­ment (both sides) at which point a par­al­lel LC (induc­tor and capac­i­tor) cir­cuit is insert­ed. This so-called trap has the effect of trun­cat­ing the ele­ment at the high­er fre­quen­cy band, mak­ing it approx­i­mate­ly a half wave­length in length. At the low­er fre­quen­cy, the entire ele­ment (includ­ing the remain­ing induc­tance due to the trap) is close to half-wave res­o­nance, imple­ment­ing a dif­fer­ent Yagi-Uda anten­na. Using a sec­ond set of traps a “triband” anten­na can be res­o­nant at three dif­fer­ent bands. Giv­en the asso­ci­at­ed costs of erect­ing an anten­na and rotor sys­tem above a tow­er, the com­bi­na­tion of anten­nas for three ama­teur bands in one unit is a very prac­ti­cal solu­tion. The use of traps is not with­out dis­ad­van­tages, how­ev­er, as they reduce the band­width of the anten­na on the indi­vid­ual bands and reduce the anten­na’s elec­tri­cal effi­cien­cy.

Con­sid­er a Yagi-Uda con­sist­ing of a reflec­tor, dri­ven ele­ment and a sin­gle direc­tor as shown here. The dri­ven ele­ment is typ­i­cal­ly a λ/2 dipole or fold­ed dipole and is the only mem­ber of the struc­ture that is direct­ly excit­ed (elec­tri­cal­ly con­nect­ed to the feed­line). All the oth­er ele­ments are con­sid­ered par­a­sitic. That is, they rera­di­ate pow­er which they receive from the dri­ven ele­ment (they also inter­act with each oth­er).

One way of think­ing about the oper­a­tion of such an anten­na is to con­sid­er a par­a­sitic ele­ment to be a nor­mal dipole ele­ment with a gap at its cen­ter, the feed­point. Now instead of attach­ing the anten­na to a load (such as a receiv­er) we con­nect it to a short cir­cuit. As is well known in trans­mis­sion line the­o­ry, a short cir­cuit reflects all of the inci­dent pow­er 180 degrees out of phase. So one could as well mod­el the oper­a­tion of the par­a­sitic ele­ment as the super­po­si­tion of a dipole ele­ment receiv­ing pow­er and send­ing it down a trans­mis­sion line to a matched load, and a trans­mit­ter send­ing the same amount of pow­er down the trans­mis­sion line back toward the anten­na ele­ment. If the wave from the trans­mit­ter were 180 degrees out of phase with the received wave at that point, it would be equiv­a­lent to just short­ing out that dipole at the feed­point (mak­ing it a sol­id ele­ment, as it is).

The fact that the par­a­sitic ele­ment involved isn’t exact­ly res­o­nant but is some­what short­er (or longer) than λ/2 mod­i­fies the phase of the ele­men­t’s cur­rent with respect to its exci­ta­tion from the dri­ven ele­ment. The so-called reflec­tor ele­ment, being longer than λ/2, has an induc­tive reac­tance which means the phase of its cur­rent lags the phase of the open-cir­cuit volt­age that would be induced by the received field. The direc­tor ele­ment, on the oth­er hand, being short­er than λ/2 has a capac­i­tive reac­tance with the volt­age phase lag­ging that of the cur­rent. If the par­a­sitic ele­ments were bro­ken in the cen­ter and dri­ven with the same volt­age applied to the cen­ter ele­ment, then such a phase dif­fer­ence in the cur­rents would imple­ment an end-fire phased array, enhanc­ing the radi­a­tion in one direc­tion and decreas­ing it in the oppo­site direc­tion. Thus one can appre­ci­ate the mech­a­nism by which par­a­sitic ele­ments of unequal length can lead to a uni­di­rec­tion­al radi­a­tion pat­tern.

While the above qual­i­ta­tive expla­na­tion is use­ful for under­stand­ing how par­a­sitic ele­ments can enhance the dri­ven ele­ments radi­a­tion in one direc­tion at the expense of the oth­er, the assump­tions used are quite inac­cu­rate. Since the so-called reflec­tor, the longer par­a­sitic ele­ment, has a cur­rent whose phase lags that of the dri­ven ele­ment, one would expect the direc­tiv­i­ty to be in the direc­tion of the reflec­tor, oppo­site of the actu­al direc­tion­al pat­tern of the Yagi-Uda anten­na. In fact that would be the case were we to con­struct a phased array with rather close­ly spaced ele­ments all dri­ven by volt­ages in phase, as we posit­ed.

How­ev­er these ele­ments are not dri­ven as such but receive their ener­gy from the field cre­at­ed by the dri­ven ele­ment, so we will find almost the oppo­site to be true. For now, con­sid­er that the par­a­sitic ele­ment is also of length λ/2. Again look­ing at the par­a­sitic ele­ment as a dipole which has been short­ed at the feed­point, we can see that if the par­a­sitic ele­ment were to respond to the dri­ven ele­ment with an open-cir­cuit feed­point volt­age in phase with that applied to the dri­ven ele­ment (which we’ll assume for now) then the reflect­ed wave from the short cir­cuit would induce a cur­rent 180° out of phase with the cur­rent in the dri­ven ele­ment. This would tend to can­cel the radi­a­tion of the dri­ven ele­ment. How­ev­er due to the reac­tance caused by the length dif­fer­ence, the phase lag of the cur­rent in the reflec­tor, added to this 180° lag, results in a phase advance, and vice ver­sa for the direc­tor. Thus the direc­tiv­i­ty of the array indeed is in the direc­tion towards the direc­tor.

One must take into account an addi­tion­al phase delay due to the finite dis­tance between the ele­ments which fur­ther delays the phase of the cur­rents in both the direc­tors and reflector(s). The case of a Yagi-Uda array using just a dri­ven ele­ment and a direc­tor is illus­trat­ed in the accom­pa­ny­ing dia­gram tak­ing all of these effects into account. The wave gen­er­at­ed by the dri­ven ele­ment (green) prop­a­gates in both the for­ward and reverse direc­tions (as well as oth­er direc­tions, not shown). The direc­tor receives that wave slight­ly delayed in time (amount­ing to a phase delay of about 35°), and gen­er­at­ing a cur­rent that would be out of phase with the dri­ven ele­ment (thus an addi­tion­al 180° phase shift), but which is fur­ther advanced in phase (by about 70°) due to the direc­tor’s short­er length. In the for­ward direc­tion the net effect is a wave emit­ted by the direc­tor (black) which is about 110° retard­ed with respect to that from the dri­ven ele­ment (green), in this par­tic­u­lar design. These waves com­bine to pro­duce the net for­ward wave (bot­tom, right) with an ampli­tude slight­ly larg­er than the indi­vid­ual waves.

In the reverse direc­tion, on the oth­er hand, the addi­tion­al delay of the wave from the direc­tor (black) due to the spac­ing between the two ele­ments (about 35° of phase delay) caus­es it to be about 180° out of phase with the wave from the dri­ven ele­ment (green). The net effect of these two waves, when added (bot­tom, left), is almost com­plete can­cel­la­tion. The com­bi­na­tion of the direc­tor’s posi­tion and short­er length has thus obtained a uni­di­rec­tion­al rather than the bidi­rec­tion­al response of the dri­ven (half wave dipole) ele­ment alone.

A full analy­sis of such a sys­tem requires com­put­ing the mutu­al imped­ances between the dipole ele­ments  which implic­it­ly takes into account the prop­a­ga­tion delay due to the finite spac­ing between ele­ments. We mod­el ele­ment num­ber j as hav­ing a feed­point at the cen­ter with a volt­age Vj and a cur­rent Ij flow­ing into it. Just con­sid­er­ing two such ele­ments we can write the volt­age at each feed­point in terms of the cur­rents using the mutu­al imped­ances Zij:  V_1 = Z_{11} I_1 + Z_{12} I_2 V_2 = Z_{21} I_1 + Z_{22} I_2

Z11 and Z22 are sim­ply the ordi­nary dri­ving point imped­ances of a dipole, thus 73+j43 ohms for a half wave ele­ment (or pure­ly resis­tive for one slight­ly short­er, as is usu­al­ly desired for the dri­ven ele­ment). Due to the dif­fer­ences in the ele­ments’ lengths Z11 and Z22 have a sub­stan­tial­ly dif­fer­ent reac­tive com­po­nent. Due to reci­procity we know that Z21 = Z12. Now the dif­fi­cult com­pu­ta­tion is in deter­min­ing that mutu­al imped­ance Z21 which requires a numer­i­cal solu­tion. This has been com­put­ed for two exact half-wave dipole ele­ments at var­i­ous spac­ings in the accom­pa­ny­ing graph.

The solu­tion of the sys­tem then is as fol­lows. Let the dri­ven ele­ment be des­ig­nat­ed 1 so that V1 and I1 are the volt­age and cur­rent sup­plied by the trans­mit­ter. The par­a­sitic ele­ment is des­ig­nat­ed 2, and since it is short­ed at its “feed­point” we can write that V2 =0. Using the above rela­tion­ships, then, we can solve for I2 in terms of I1: 0 = V_2 = Z_{21} I_1 + Z_{22} I_2 and so  I_2 = — {Z_{21} over Z_{22}} , I_1 .

This is the cur­rent induced in the par­a­sitic ele­ment due to the cur­rent I1 in the dri­ven ele­ment. We can also solve for the volt­age V1 at the feed­point of the dri­ven ele­ment using the ear­li­er equa­tion:  V_1 = Z_{11} I_1 + Z_{12} I_2 = Z_{11} I_1 — Z_{12}{Z_{21} over Z_{22}} , I_1  qquadqquad = left( Z_{11} — {Z_{21}^2 over Z_{22}} right) , I_1

where we have sub­sti­tut­ed Z12 = Z21. The ratio of volt­age to cur­rent at this point is the dri­ving point imped­ance Zdp of the 2‑element Yagi: Z_{dp}= V_1 / I_1 = Z_{11} — {Z_{21}^2 over Z_{22}}

With only the dri­ven ele­ment present the dri­ving point imped­ance would have sim­ply been Z11, but has now been mod­i­fied by the pres­ence of the par­a­sitic ele­ment. And now know­ing the phase (and ampli­tude) of I2 in rela­tion to I1 as com­put­ed above allows us to deter­mine the radi­a­tion pat­tern (gain as a func­tion of direc­tion) due to the cur­rents flow­ing in these two ele­ments. Solu­tion of such an anten­na with more than two ele­ments pro­ceeds along the same lines, set­ting each Vj=0 for all but the dri­ven ele­ment, and solv­ing for the cur­rents in each ele­ment (and the volt­age V1 at the feed­point).

There are no sim­ple for­mu­las for design­ing Yagi-Uda anten­nas due to the com­plex rela­tion­ships between phys­i­cal para­me­ters such as ele­ment length, spac­ing, and diam­e­ter, and per­for­mance char­ac­ter­is­tics such as gain and input imped­ance. But using the above sort of analy­sis one can cal­cu­late the per­for­mance giv­en a set of para­me­ters and adjust them to opti­mize the gain (per­haps sub­ject to some con­straints). Since with an N ele­ment Yagi-Uda anten­na, there are 2N‑1 para­me­ters to adjust (the ele­ment lengths and rel­a­tive spac­ings), this is not a straight­for­ward prob­lem at all. The mutu­al imped­ances plot­ted above only apply to λ/2 length ele­ments, so these might need to be recom­put­ed to get good accu­ra­cy. What’s more, the cur­rent dis­tri­b­u­tion along a real anten­na ele­ment is only approx­i­mate­ly giv­en by the usu­al assump­tion of a clas­si­cal stand­ing wave, requir­ing a solu­tion of Hal­len’s inte­gral equa­tion tak­ing into account the oth­er con­duc­tors. Such a com­plete exact analy­sis con­sid­er­ing all of the inter­ac­tions men­tioned is rather over­whelm­ing, and approx­i­ma­tions are inevitably invoked, as we have done in the above exam­ple.

Con­se­quent­ly, these anten­nas are often empir­i­cal designs using an ele­ment of tri­al and error, often start­ing with an exist­ing design mod­i­fied accord­ing to one’s hunch. The result might be checked by direct mea­sure­ment or by com­put­er sim­u­la­tion. A well-known ref­er­ence employed in the lat­ter approach is a report pub­lished by the Nation­al Bureau of Stan­dards (NBS) (now the Nation­al Insti­tute of Stan­dards and Tech­nol­o­gy (NIST)) that pro­vides six basic designs derived from mea­sure­ments con­duct­ed at 400 MHz and pro­ce­dures for adapt­ing these designs to oth­er fre­quen­cies. These designs, and those derived from them, are some­times referred to as “NBS yagis.”

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