Yagi-Uda Antenna - Analysis

Analysis

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 V_j and a current I_j 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 Z_ij:

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:

and so

.

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:

where we have substituted Z₁₂ = Z₂₁. The ratio of voltage to current at this point is the driving point impedance Z_dp of the 2-element Yagi:

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 V_j=0 for all but the driven element, and solving for the currents in each element (and the voltage V₁ at the feedpoint).