10.3 Parasitic Arrays, Yagi-Uda Antennas, and Quad Arrays

10.3 Parasitic Arrays, Yagi-Uda Antennas, and Quad Arrays

While omnidirectional antennas radiate radio frequency energy uniformly across the azimuth, establishing long-distance point-to-point contacts on congested High Frequency (HF) bands requires concentrating radiated energy into a narrow, intense beam. By focusing power in a single preferred direction, directional beam antennas provide forward gain on transmit and reject interference from competing directions on receive. Among directional systems, parasitic arrays—foremost the Yagi-Uda antenna and the cubical quad—represent the pinnacle of high-performance amateur station engineering.

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The Physics of Parasitic Arrays

Invented in 1926 by Shintaro Uda and Hidetsugu Yagi at Tohoku University, the Yagi-Uda antenna achieves high directivity without requiring complex feedline networks to drive every element. Instead, it consists of one driven element connected directly to the transmitter, flanked by one or more unpowered parasitic elements.

     Reflector             Driven element           Director
         |                                           
         |                       |                   
         |                       |                     |
         |                       o A                   |
         |                    Feed gap                 |
         |                       o B                   |
         |                       |                     |
         |                       |                   
         |                                           
        <------ d1 ------------><------ d2 ----------->

Elements are parallel; their supports lie along the boom.
A-B connects to the matching/feed system. Sketch is not to scale.
Reflector current lags its induced EMF; director current leads its
induced EMF. Rear radiation is suppressed; main beam points ---->

Mutual Electromagnetic Coupling and Phase Manipulation

The operation of a parasitic array relies upon mutual induction and vector wave interference:

1. The driven element radiates an expanding electromagnetic wave into space.
2. As this wavefront sweeps across adjacent parallel metallic elements, it cuts the conductors and induces an alternating radio frequency electromotive force (EMF).
3. The induced EMF drives an RF current along the parasitic element, causing it to re-radiate an electromagnetic wave.
4. The total far-field radiation in any spatial direction is the vector sum of the primary wave from the driven element and the re-radiated waves from all parasitic elements.

The elements' radiated fields combine constructively in the forward direction and reduce radiation to the rear. Two phase effects must be considered together:

• Spatial Phase Difference ($\beta d$): Element separation $d$ gives a direction-dependent propagation phase difference, with magnitude $2\pi d/\lambda$ along the boom.
• Induced-Current Phase: A parasitic element's complete impedance determines its current relative to the EMF induced by all the other elements. Inductive reactance gives lag and capacitive reactance gives lead relative to that EMF; neither implies a fixed phase relative to the driven-element current.

The Reflector: Inductive Reactance and Current Lag

• Physical Dimensions: Positioned behind the driven element at a spacing of $0.15\lambda$ to $0.25\lambda$, the reflector is cut approximately $5\%$ physically longer than the resonant half-wave driven element.
• Electrical Reactance: Because it is longer than half-wave resonance, the reflector operates above its natural fundamental frequency, exhibiting a net inductive reactance ($+jX_L$).
• Phase Mechanics: Inductive reactance makes current lag its induced EMF, by an angle set by the complete resistance and reactance. Mutual coupling and spatial phase then determine how the reflector field combines with the driven-element field. Suitable dimensions reinforce forward radiation and suppress rear radiation; a universal $90^\circ$ current shift or perfect rear cancellation is not implied.

The Director: Capacitive Reactance and Current Lead

• Physical Dimensions: Positioned in front of the driven element along the forward axis at a spacing of $0.10\lambda$ to $0.20\lambda$, the director is cut approximately $5\%$ physically shorter than the driven element.
• Electrical Reactance: Being shorter than half-wave resonance, the director operates below its natural fundamental frequency, exhibiting a net capacitive reactance ($-jX_C$).
• Phase Mechanics: Capacitive reactance makes current lead its induced EMF. The actual phase and amplitude, together with element spacing and coupling, help reinforce radiation along the forward axis.
• Multiple Directors: Additional directors extend the array along the boom. Their lengths and spacings are chosen together for gain, pattern, impedance and bandwidth. Successive directors need not become progressively shorter; practical designs can use equal or varied lengths.

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Performance Specifications: Gain, Front-to-Back, and Beamwidth

Forward Gain and the Primacy of Boom Length

Forward gain measures how effectively an antenna concentrates power in its main lobe compared to a reference standard:

• $\text{dBi}$: Decibels relative to a theoretical isotropic point source.
• $\text{dBd}$: Decibels relative to a resonant half-wave dipole in free space ($0\text{ dBd} = 2.15\text{ dBi}$).

                               Gain vs. Boom Length

         Boom Length (λ)        Typical Elements        Forward Gain (dBi)
         -----------------------------------------------------------------
             0.25 λ                  3 El                  7.0 -  8.5 dBi
             0.50 λ                  4 El                  8.5 - 10.0 dBi
             1.00 λ                  5 El                 10.5 - 11.5 dBi
             2.00 λ                  7 El                 13.0 - 14.0 dBi

Adding more elements to a boom of fixed length produces negligible additional gain; however, interleaving extra elements allows the designer to widen the operational SWR bandwidth and optimize the Front-to-Back ratio.

Front-to-Back ($F/B$) Ratio

The Front-to-Back ratio is the ratio of power radiated in the main forward lobe to the power radiated in the reciprocal $180^\circ$ rearward direction, expressed in decibels ($15\text{ to }30\text{ dB}$ in well-designed arrays).

• The Engineering Trade-Off: Dimensions chosen for maximum forward gain generally differ from those chosen for maximum $F/B$ ratio or impedance bandwidth. Optimizing one specification can reduce another. Practical Yagis balance gain, rear-lobe suppression and matching across the intended operating band; there is no universal $10\text{ dB}$ front-to-back limit for a maximum-gain design.

Azimuthal and Elevation Beamwidth

The half-power beamwidth ($\theta_{-3\text{dB}}$) is the angular span between the two directions where radiated power density is half ($-3\text{ dB}$) the main-lobe peak. Specify the plane of the measurement, such as azimuth or elevation:

• A standard 3-element HF Yagi has an azimuthal beamwidth of $60^\circ - 70^\circ$.
• A long-boom 5-element Yagi narrows the beamwidth to $45^\circ - 50^\circ$.
• As beamwidth narrows, off-axis noise and unwanted signals are rejected, substantially improving the signal-to-noise ratio (SNR) on receive.

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Driven Element Radiation Resistance and Matching Networks

When parasitic elements are brought into close proximity ($0.15\lambda$) with the driven element, the strong mutual impedance ($Z_{12}, Z_{13}$) reflects a large reactive load into the driven element feedpoint. This mutual coupling drastically depresses the driven element radiation resistance:

• While an isolated half-wave dipole exhibits $R_r \approx 73\ \Omega$, placing it in a 3-element Yagi drops its radiation resistance down to $15\ \Omega - 25\ \Omega$ (and under $12\ \Omega$ in ultra-high-gain arrays).
• Connecting standard $50\ \Omega$ coaxial cable directly to a $20\ \Omega$ element would produce an unacceptable SWR of $50 / 20 = 2.5:1$.

To transform this depressed impedance to $50\ \Omega$, specialized RF matching networks are utilized.

Gamma match connections:
  Coax center -- series capacitor -- gamma rod -- strap S
  Coax shield ----------------------------- center C
  Driven element: End A -------- C ---- S -------- End B
  The driven element is unbroken; the rod runs parallel to it.

Hairpin equivalent, viewed across the split feed terminals:
                  Ra                -jXa
  A o---+--------[ R ]--------------[ C ]---------+---o B
        |                                         |
        +-----------------[ +jXh ]----------------+
                    Shorted hairpin stub

A and B connect to opposite balanced terminals of a 1:1 current
balun. Its unbalanced port connects to the coax center and shield.

1. The Gamma Match

The Gamma Match is the most widely deployed matching system for amateur HF Yagis:

• Mechanical Elegance: The driven element consists of an unbroken, solid aluminum tube clamped directly to the metallic boom. This eliminates insulated element center plates, providing superior mechanical strength and allowing the entire antenna structure to be tied to DC ground for lightning protection.
• Electrical Topology: An unbalanced matching network. A parallel conductive rod (the gamma rod) connects to the center conductor of the coaxial feedline, running parallel to the driven element for a short distance before clamping to the element via a movable shorting strap. The coaxial shield connects to the center of the driven element at the boom.
• Impedance Transformation: The gamma rod and the driven element form a miniature parallel transmission line that steps up the low radiation resistance ($20\ \Omega$) to $50\ \Omega$. Because this transmission line loop is shorter than $\lambda/4$, it introduces an undesirable inductive reactance ($+jX_L$). A series variable capacitor ($C$) is inserted in line with the gamma rod to cancel this inductive reactance, presenting a purely resistive $50 + j0\ \Omega$ match.

2. The Hairpin Match (Beta Match)

The Hairpin Match (also termed a Beta Match) combines the shortened driven element’s series capacitive reactance with a shunt inductive transmission-line stub:

• Configuration: The driven element is severed at its center and cut physically shorter than half-wave resonance. Consequently, the driven element input impedance contains a low resistance and a capacitive reactance: $Z_{\text{in}} = R_a - jX_a$ (e.g., $20 - j24.5\ \Omega$).
• The Hairpin Inductor: A shorted parallel transmission line stub (resembling a large hairpin) is connected directly across the split feedpoint terminals. Being shorted and less than $\lambda/4$, the hairpin presents a parallel inductive reactance ($+jX_L$).
• L-Network Equivalence: Series capacitance and shunt inductance form a high-pass L-network. Treat the complete antenna impedance and hairpin as parallel branches. For $Z_a=20-j24.495\ \Omega$, $Y_a=0.020+j0.024495\text{ S}$. A $+j40.825\ \Omega$ hairpin adds $Y_h=-j0.024495\text{ S}$. Their total is $0.020+j0\text{ S}$, corresponding to $50+j0\ \Omega$ in the ideal model.

3. The T-Match

The T-Match is the balanced counterpart of the Gamma match, using symmetric matching conductors on the two halves of the driven element. Series capacitors may be used to tune out reactance. Conductor dimensions, spacing and connection points determine the transformed input impedance; it is not inherently $200\ \Omega$. A $200\ \Omega$ balanced input requires a 4:1 impedance transformation to $50\ \Omega$ coax. A 1:1 current balun is suitable when the balanced input is already about $50\ \Omega$.

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Quad and Loop Antennas

The Cubical Quad antenna, invented by Clarence Moore (W9LZX) in 1942 at HCJB in Ecuador, replaces linear dipole elements with full-wave closed perimeter wire loops.

            Driven loop                     Reflector loop
             Front view                       Front view
          +--------------+                 +--------------+
          |              |                 |              |
          |              |                 |              |
          +-----o  o-----+                 +--------------+
                A  B
         Feed gap: separate terminals

Each element is ONE planar loop. The two loop planes are parallel
and are separated along a boom perpendicular to those planes.
Driven perimeter: about 1.0 wavelength; reflector: about 1.05.

For the 112.5-ohm example:
A-B -> 75-ohm quarter-wave matching line -> 50-ohm feedline.
Use suitable current balancing/choking at the balanced loop feed.

Loop Geometry and Gain Advantages

• Element Configuration: Each element consists of a square loop with a total perimeter of approximately one wavelength ($1.0\lambda$ for the driven loop, $1.05\lambda$ for the reflector, and $0.95\lambda$ for directors). The loops are supported on non-conductive fiberglass or bamboo cross-arms (spreaders) mounted on a central horizontal boom.
• Loop Radiation and Array Gain: A full-wave square loop has two principal horizontal radiating sections about $\lambda/4$ apart. Their fields combine to give broadside directivity. Quad and Yagi arrays with comparable boom lengths can have similar performance; element dimensions, spacing, tuning and mounting height determine gain. A fixed $1.5$–$2.0\text{ dB}$ quad advantage does not follow from loop geometry.

Key Operational Characteristics

1. High Feedpoint Impedance: The natural feedpoint impedance of a 2-element quad on a typical boom ($0.15\lambda - 0.2\lambda$) is approximately $100\ \Omega - 120\ \Omega$. It is readily matched to $50\ \Omega$ coax using a quarter-wave matching transformer of $75\ \Omega$ coaxial cable ($Z_{\text{match}} = \sqrt{50 \times 112.5} = 75\ \Omega$) or a wideband matching transformer.
2. Precipitation Static: Corona discharges in strong atmospheric electric fields can produce broadband receiver noise. Height, conductor geometry and the feed system affect pickup. A closed loop alone does not establish a path to earth or guarantee immunity; grounded antennas can still suffer corona noise.
3. Elevation Pattern: Quad and Yagi elevation lobes depend on mounting height, ground and array geometry. Compare complete patterns at the same reference height; a quad does not inherently produce a lower takeoff angle.

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The Log-Periodic Dipole Array (LPDA)

The Log-Periodic Dipole Array (LPDA) provides continuous coverage over a broad designed frequency range. For example, an array covering $13$–$30\text{ MHz}$ spans a frequency ratio of $30/13\approx2.31:1$. Longer frequency spans require a design with enough appropriately scaled elements and boom length.

           Long-to-short feed order (not to scale)
                 Long                         Short
                 dipole                       dipole
                   |
                   |            |
                   |            |               |
                   A            B               A
                 feed          feed            feed
                   B            A               B
                   |            |               |

Balanced feeder conductors: A and B.
Successive split dipoles reverse which arm connects to A and B.
The feedline enters at the SHORT-element end.
                                  Maximum radiation ---->

Geometric Scaling and Frequency-Independent Operation

The LPDA uses geometric scaling. Number the elements from longest to shortest. Let $L_n$ be the full tip-to-tip dipole length, $s_n$ its distance from the virtual apex, and $d_n=s_n-s_{n+1}$ the spacing to the next shorter element. The design factor is

$$\tau=\frac{L_{n+1}}{L_n}=\frac{s_{n+1}}{s_n}=\frac{d_{n+1}}{d_n}<1 \qquad(\text{typically }0.82\text{–}0.95).$$

With this full-length convention, the relative spacing factor is

$$\sigma=\frac{d_n}{2L_n}.$$

If element half-length is used instead, the same definition has four times that half-length in the denominator. Keep the length convention consistent when using design tables.

The Transposed Central Feeder and Active Region

1. The Active Region: At a given frequency within the designed band, a small cluster of adjacent dipoles near half-wave resonance draws the strongest currents. This cluster, commonly illustrated by three or four elements, is the active region:
   • Elements behind the active region (longer than $\lambda/2$) act as reflectors.
   • Elements in front of the active region (shorter than $\lambda/2$) act as directors.
   • As the operating frequency is tuned upward, the active region smoothly slides along the boom toward the shorter elements!
2. Phase Transposition: The balanced feeder enters at the short-element end and reverses its connections to alternate dipoles, giving a $180^\circ$ polarity reversal between adjacent feeds. This transposition works with propagation phase along the feeder and mutual coupling to produce end-fire radiation toward the short elements. Polarity reversal alone does not establish the complete current phases or eliminate the rear lobe.

Performance Trade-Off

An LPDA trades peak gain for continuous broadband coverage. Typical forward gain is about $6$–$8\text{ dBi}$, comparable to a small monoband Yagi. A suitable design can maintain a useful input match across its operating range, but the exact SWR, gain and front-to-back ratio depend on its dimensions and feed system; the LPDA name alone does not guarantee SWR below $1.5:1$.

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Step-by-Step Worked Calculation: Hairpin (Beta) Match Design

An amateur antenna designer constructs a high-performance 3-element monoband Yagi for the 20-meter band at $f_0 = 14.200\text{ MHz}$. Due to close element spacing optimized for maximum front-to-back ratio, the driven element radiation resistance is depressed to $R_a = 20.0\ \Omega$. Design a Hairpin (Beta) matching network to step this impedance up to a perfect $50.0\ \Omega$ resistive match.

                  Equivalent antenna input
  A o---+--------[ 20 ohms ]----[ -j24.495 ohms ]---+---o B
        |                                           |
        +--------------[ +j40.825 ohms ]------------+
                         Hairpin stub

The two branches are in parallel across A-B.
A-B is also the balanced feed port; connect it through a 1:1
current balun to the 50-ohm coaxial feedline.

Step 1: Calculate the Matching Network Quality Factor ($Q$)

For this high-pass L match between real terminating resistances $R_{\text{high}}=50.0\ \Omega$ and $R_{\text{low}}=20.0\ \Omega$, the matching-network $Q$ is:

$$Q = \sqrt{\frac{R_{\text{high}}}{R_{\text{low}}} - 1} = \sqrt{\frac{50.0}{20.0} - 1} = \sqrt{2.5 - 1} = \sqrt{1.5} \approx 1.2247$$

Step 2: Determine Required Series Capacitive Reactance ($-jX_a$)

The driven element must be cut electrically shorter than resonance to supply the series capacitive reactance required by the L-network:

$$X_{\text{series}} = Q \times R_{\text{low}} = 1.2247 \times 20.0\ \Omega \approx 24.49\ \Omega$$

Thus, the driven element length is trimmed until an antenna impedance analyzer directly at the terminals reads:

$$Z_{\text{driven}} = 20.0 - j24.5\ \Omega$$

Step 3: Determine Required Parallel Inductive Reactance ($+jX_{\text{hairpin}}$)

The shunt inductive reactance bridged across the terminals must satisfy:

$$X_{\text{parallel}} = \frac{R_{\text{high}}}{Q} = \frac{50.0\ \Omega}{1.2247} \approx 40.82\ \Omega$$

Step 4: Calculate Inductance of the Hairpin Stub at $14.200\text{ MHz}$

$$L = \frac{X_{\text{parallel}}}{2\pi f_0} = \frac{40.82\ \Omega}{2\pi \times (14.200 \times 10^6\text{ Hz})} \approx 4.575 \times 10^{-7}\text{ H} \approx 0.458\ \mu\text{H}$$

Step 5: Physical Realization of the Hairpin Stub

A shorted parallel two-wire transmission line stub exhibits an inductive input reactance given by:

$$X_{\text{stub}} = Z_0 \tan(\beta l)$$

Using two $6\text{ mm}$ aluminum rods separated by $50\text{ mm}$ in air ($Z_0 = 120 \cosh^{-1}(50/6) \approx 336\ \Omega$):

$$\tan(\beta l) = \frac{X_{\text{stub}}}{Z_0} = \frac{40.82\ \Omega}{336\ \Omega} \approx 0.1215$$ $$\beta l = \arctan(0.1215) \approx 0.1209\text{ radians} = 6.927^\circ$$

At $14.200\text{ MHz}$, the wavelength is $\lambda = 299.792 / 14.200 \approx 21.112\text{ meters}$:

$$l_{\text{stub}} = \frac{6.927^\circ}{360^\circ} \times 21.112\text{ m} \approx 0.01924 \times 21.112\text{ m} \approx 0.406\text{ meters} = 40.6\text{ cm}$$

A shorted hairpin stub measuring $40.6\text{ cm}$ placed across the split feed terminals transforms the $20 - j24.5\ \Omega$ antenna impedance into an exact $50.0 + j0\ \Omega$ conjugate match!

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Comparison Table: Directional Array Architectures