SCHELKUNOFF POLYNOMIAL METHODSwapnil Bangera(ME-EXTC)

OVERVIEW• What is antenna synthesis?• Steps involved in antenna synthesis• Various methods of antenna synthesis• Schelkunoff Polynomial Method• Design Technique• Design Steps• Design example

ANTENNA SYNTHESIS• In general, analysis of antenna is done by selecting a particular antenna

model and its various radiation characteristics such as pattern, directivity, impedance, beamwidth, efficiency, polarization, and bandwidth are analysed using standard procedures usually by specifying its current distribution.

• In practice, it is often necessary to design an antenna system that will yield desired radiation characteristics. Common requests to design an antenna are for pattern to exhibit a desired distribution, beamwidth, size of side lobes, minor lobes, etc.

• The task, in general, is to find not only the antenna configuration but also its geometrical dimensions and excitation distribution.

• The designed system should yield, either exactly or approximately, an acceptable radiation pattern.

• This method of design is known as antenna synthesis.

ANTENNA SYNTHESIS STEPS• Antenna synthesis usually requires that first an approximate analytical

model is chosen to represent, either exactly or approximately, the desired pattern.

• The second step is to match the analytical model to a physical antenna model.

ANTENNA SYNTHESIS METHODSAntenna synthesis can be classified into three categories:One group requires that the antenna patterns possess nulls in desired directions.

Schelkunoff Polynomial MethodAnother category requires that the patterns exhibit a desired distribution in the

entire visible region. This is referred to as beam shaping. Fourier Transform Method Woodward Lawson Method

Third group includes techniques that produce patterns with narrow beams and low side lobes.

Binomial Method Dolph-Tschebscheff Method Taylor line-source

SCHELKUNOFF POLYNOMIAL METHOD

• Schelkunoff polynomial method is conductive to the synthesis of arrays whose patterns possess nulls in desired directions.

• To complete the design, this method requires information on the number of nulls and their locations.

• The number of elements and their excitation coefficients are then derived.

DESIGN TECHNIQUE• The array factor for an N-element, equally spaced, non-uniform

amplitude, and progressive-phase excitation is given by:AF = .........(1)

where accounts for the non-uniform amplitude excitation of each element. The spacing between the elements is and is the

progressive phase shift• Letting

= .........(2)• Rewriting (1) as

AF = = …......(3)

which is a polynomial of degree . From the mathematics of complex variables and algebra, any polynomial of degree has roots and can be expressed as a product of linear terms. Thus we can write (3) as

AF = .........(4)where are the roots, which may be complex, of the polynomial

• The magnitude of (4) can ne expresses as………(5)

• The complex variable of (2) can be written in another form as.........(6)………(7)

It is clear that for any value of , and the magnitude of lies always on a unit circle; however its phase depends upon , and For lets plot the value of z, magnitude and phase, as takes values of 0 to π rad.

• For all the physically observable angles of θ, only exists over a part of the circle.

• Any values of z outside that arc are not realizable by any physical observation angle θ for the spacing.

• The realizable part of the circle is referred to as the visible region and the remaining as invisible region.

• It is obvious that the visible region can be extended by increasing the spacing between the elements.

• It requires a spacing of at least to encompass, at least once, the entire circle. Any spacing greater than leads to multiple values of z.

• To demonstrate the versatility of the arrays, let us plot the values of z for the same spacings but with a .

• A comparison between the corresponding figures indicates that the overall visible region for each spacing has not changed but its relative position on the unit circle has rotated counter clockwise by an amount equal to β.

• We can conclude that the overall extent of the visible region can be controlled by the spacing between the elements and its relative position on the unit circle by the progressive phase excitation of the elements.

• These two can be used effectively in the design of the array factors.

• To demonstrate all the principles, lets consider an example along with some computations.

Roots of array factor Roots of array factor on unit circleand within visible region

• If all the roots are located in the visible region of the unit circle, then each one corresponds to a null in the pattern of because as θ changes z changes and eventually passes through each of the ’s.

• When all the zeros (roots) are not in the visible region of the unit circle, then only those zeroes on the visible region will contribute to the nulls of the pattern.

• If no zeros exist in the visible region of the unit circle, then that particular array factor has no nulls for any value of θ.

• However, if a given zero lies on the unit circle but not in its visible region, that zero can be included in the pattern by changing the phase excitation so that the visible region is rotated until it encompasses that root.

DESIGN STEPS1. For given spacing, phase shift and null locations plot the visible region.2. Find the roots of AF corresponding to the desired null locations.3. Check whether the roots lie on the unit circle within the visible region.4. Find the array factor with respect to the roots present on the unit circle

within the visible region.5. Find array coefficients

EXAMPLE• Design a linear array with a spacing between the elements of such that it

has zeros at . Determine the number of elements, their excitation, and plot the derived pattern.

Array Coefficients

Radiation Power Pattern

Amplitude Radiation Pattern

Thank You!