A half-wave dipole illuminating a parasitic array will often require a "step-up" in feedpoint impedance to properly match the surge impedance of popular transmission lines. Extending the physical length of a monopole or dipole causes a slow rise in radiation resistance. Contracting the length has the opposite effect. This phenomenon is illustrated in the Brown-Woodward curves published in Chapter 4 of the Antenna Engineering Handbook:

This often ignored property of monopole and dipole antennas makes it possible to achieve a wide spectrum of feedpoint impedances suitable for virtually any application.

The only consequence of modifying the radiator's physical length beyond the point where natural resonance occurs is that the feedpoint impedance becomes highly reactive. In fact, the reactance varies more rapidly in magnitude than does radiation resistance as the length of the radiator is moved away from resonance.

The feedpoint impedance's reactive component is easily cancelled by placing conjugate reactances in series with the feedpoint of the radiator. When the added reactance is made equal in magnitude and opposite in phase of that presented by the radiator, a pure resistive impedance equal to that predicted by the Brown-Woodward curves is realized.

A shunt reactance can be placed in parallel with the feedpoint to cancel the reactive component of the driven element's impedance as well. However, what happens in the process is rather complex, and is not as trivial as simply making the shunt reactance equal and opposite to that of the driven element's reactance. Instead, a parallel resonant RLC circuit is formed,

Provided a step-up in feedpoint impedance is desired, a wide range of impedances can be obtained simply by selecting the radiator's physical length and shunt reactance appropriately. In fact, it almost doesn't matter whether the driven element is adjusted to a physical length above

The shunt capacitive reactance, X

After applying the shunt capacitive reactance, X

Note that all resistances and reactances are expressed in ohms, with
X_{C} expressed as negative in magnitude.

The shunt inductive reactance, X_{L}, required to resonate a
complex load exhibiting a pure resistance, R, in series with capacitive
reactance, X_{C}, can be computed as follows:

After applying the shunt inductance, X_{L}, across the driven
element to bring the system into resonance, the resulting feedpoint
resistance, R_{feedpoint}, can be computed as follows:

*In other words, much like the shunt capacitance impedance matching
technique, a step-up in feedpoint impedance is realized by applying
a shunt inductance across a capacitive load, despite the reduction
in radiation resistance caused by decreasing the length of the radiating
element to produce the needed capacitive reactance.*

Shunt inductance impedance matching is often employed in HF mobile antenna
installations when the radiating element is less than a quarter-wavelength
in height and an impedance step-up to match a 50 ohm transmission line is
desired. A "loading coil" of suitable inductance is connected
in series between the bottom of the radiating element and ground. If the
element length and the inductance of the coil are properly chosen, not
only is resonance established, but a 50 ohm feedpoint impedance across
the coil is also achieved.

Another example of where shunt inductance impedance matching is employed is in "Hairpin" Impedance Matching Networks.

For many years, the ARRL Handbook published a design for a 5-element 50 MHz yagi that employed a "hairpin" across the antenna's dipole driven element to raise its impedance to 200 ohms. Connection to a 50-ohm unbalanced feed-line was made through a 4:1 half-wave coaxial balun. The description of the antenna cautions the reader that the driven element is the shortest element of the entire array. The article states, "While this may seem a bit unusual, it is necessary with the hairpin matching system".

The reason these impedance matching techniques produce an impedance
step-up in either configuration is because they are really L-networks with
their high-impedance, shunt reactance side positioned at the feedpoint,
and their low-impedance, series reactance side connected in series with
the driven element's radiation resistance:

Another interesting relationship occurs with these networks. When the network is properly adjusted to the point of resonance, the ratio of radiation resistance to series reactance is always equal to the ratio of shunt reactance to feedpoint resistance. Knowing this relationship makes the calculation of feedpoint resistance significantly easier than previously described once the shunt reactance is first determined.

If the driven element is lengthed to 34.450 cm (below resonance), the feedpoint impedance rises to 42.43+j20.06 ohms. Using the equations illustrated above, when a capactive reactance of 109.8 ohms (3.35 pF) is placed across the feedpoint of the driven element, a 51.9 ohm purely resistive feedpoint impedance is obtained.

Going in the opposite direction, if the driven element is shortened to 31.6 cm (above resonance), a feedpoint impedance of 26.07-j25.95 ohms is obtained. Using the equations illustrated above, when an inductive reactance of 52.14 ohms (19.2 nH) is placed across the feedpoint of the driven element, a 51.9 ohm purely resistive feedpoint impedance is again obtained.

A balun is required to symmetrically feed a balanced driven element with an unbalanced coaxial transmission line. 4:1 half-wave coaxial baluns are often used, and require that the feedpoint of the Yagi be 200 ohms in order to yield a 50 ohm output impedance.

If the driven element is lengthed to 39.57 cm, a feedpoint impedance of 102.03+j100.11 ohms is obtained. A shunt capactive reactance of 204.1 ohms (1.8 pF) across the feedpoint yields a purely resistive impedance of 200.2 ohms. Note that under this tuning condition, the driven element is physically 13%

Going in the opposite direction, if the driven element is shortened to 29.665 cm, the feedpoint impedance becomes 18.71-j58.30 ohms. A shunt inductive reactance of 64.3 ohms (23.67 nH) yields a resistive feedpoint impedance of 200.3 ohms. Under this tuning condition, the driven element is

The shunt inductance method has the advantage of providing a DC (static discharge) path across the radiator, while the shunt capacitance method utilizes a radiator having a higher radiation resistance, which results in a matching network having a lower 'Q' and a wider operating bandwidth.

John MagliacaneAmateur Radio Operator: KD2BD |