LPDA
Design Curves for a Log-periodic Dipole Antenna (LPDA)

Description of Antenna:

The geometry of an LPDA is shown in the diagram below, with relevant parameters.

Geometry of an LPDA

The relationship between element spacing and element length are given by the scale factor tau, as shown in equation 1.

tau = Ln+1/ Ln = dn+1/ dn ..................................................1

Truncating coefficents are used to find the minimum value for L1 (longest dipole element) and the maximum value for LN (shortest dipole element) as shown in equations 2 and 3.

L1 = 0.5 lambda max .........................................................................2

LN = 0.25 lambda min ......................................................................3

Tau can be calculated using the following equation, if the length and spacing of individual elements is required.

tau = (LN / L1)1/(N-1)............................................................4

The Form Factor (FF) is defined as:

Form Factor = Boom Length / lambda max ................................5

The geometry of the antenna is fully defined by the number of dipole elements (N), the lengths of the shortest and longest dipole elements (LN and L1), and the form factor (FF).

Other parameters that affect the performance of the LPDA:

* Bandwidth Ratio (BWR): ratio between maximum and minimum operating frequencies.
* Boom Impedance (BoomImp): characteristic impedance of transmission line feeding dipole elements.
* Thickness Factor (TF): ratio between length and radius of an element.

Design Curves:

The following three design curves relate number of dipole elements, input impedance and gain to the form factor of the antenna.

Figure 1 shows the minimum number of elements (for different thickness factors and bandwidth ratios) needed to produce a stable design for an antenna of particular form factor. Note: The antenna will behave with a high degree of predictability and stability at optimum element number unless the following conditions are present simultaneously. * Form Factor < 1 * Thickness Factor > 500 * Bandwidth Ratio > 3 * Boom Impedance < 200
	Figure 1: Number of Elements vs Form Factor
This curve may be used to find the number of dipole elements required for a certain form factor (boomlength / ?max). In order to increase the BWR and/or the TF the number of elements will increase for any particular form factor. The curve may also be used to find the form factor of the antenna if the design is limited to the number of dipole elements it may have. If a BWR or TF other than those shown is required, a rough linear interpolation between two curves can be done. Figure 2 shows the ratio of the geometric mean of input resistance to the boom impedance as a function of form factor.
	Figure 2: Geometric Mean of Input Resistance ч Boom Impedance vs Form Factor

Using this figure the geometric mean of the input impedance can be found for a particular form factor (for different thickness factors and boom impedances) and vice versa.

Figure 3 shows how the gain of the LPDA is affected with changing form factor, bandwidth ratio and boom impedance.
	Figure 3: Gain vs Form Factor for varying BWR and Boom Imp

Gain increases with increased form factor and increased boom impedance. The bandwidth ratio decreases as the gain increases.

Example:

An LPDA of the following specifications is required:

* operating frequency: 200 ~ 2000Mhz
* boom impedance of 75 ohm
* input impedance of 50 ohm
* VSWR of 1:1.5 typical, 2:1 maximum.

From the required operating frequency it is clear that a bandwidth ratio (BWR) of 10 is needed.

The ratio of the input impedance to the boom impedance is 0.67. Using figure 2 the form factor required is approximately 1.2 (assuming a thickness factor of 20).

Using figure 1 the minimum number of dipole elements required for the antenna to be stable is approximately 31 (FF=1.2, TF=20, BWR=10). The average VSWR when the antenna is in this stable region is less than 1.5:1, with the maximum VSWR less than 2:1.

From figure 3 the gain of this antenna is approximately 8dBi.

The length of the longest element can be calculated using equation 2, and is 750mm. The length of the shortest element can be calculated using equation 3, and is 37.5mm. If the length and spacing of individual elements is required, the scale factor can be calculated as 0.9050 using equation 4.

The LPDA therefore has the following characteristics:

* operating frequency of 200 ~ 2000Mhz
* a bandwidth ratio (BWR) of 10
* a thickness factor (TF) of 20
* a form factor of 1.2
* a boom length of 1.8m
* 31 dipole elements, the longest being 750mm and the shortest being 37.5mm
* a scale factor (tau) of 0.9050

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Note: these design curves provide only a rough estimate of a particular antenna's dimensions and performance. In order to fully specify the antenna it should be simulated, using the values obtained from the curves if so desired.

Reference:
Fourth Year Report, "Design Curves for a Log-periodic Dipole Antenna" prepared by Dale Wright for the School of Electrical and Information Engineering, University of the Witwatersrand, August 2002.