ECEN3613 Electromagnetic Fields

Important parameters of Patch Antenna:

Resonant Frequency (fr)

Input Resonant Impedance: (Rin)

Circuit Material

Width(w) of Patch Antenna

Effective Length (Leff)

Length(L) of Patch Antenna

Inset feed point (yo)

Directivity

Reference

Input parameters:

Resonant Frequency (fr): The resonant frequency parameter has to be entered by the user to calculate almost every parameter of microstrip antenna. The width and length calculation is directly related to this resonant frequency. The resonant frequency of microstrip antenna could be estimated by using this following formula:

Where μo = Permeability of free space

εo = Permittivity of free space

ΔL= Line extension

εeff= Effective permittivity

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Input Resonant Impedance: (Rin): This is the input impedance for the Patch Antenna which is supposed to be specified by the user. The resonant impedance takes in to consideration the mutual coupling between the radiating slots. So the equation to calculate input impedance of Antenna is:

Where G1 = self conductance

G12 = mutual conductance

+ = Odd resonant modes

-- = Even resonant modes

(+) Sign is used for modes with odd (antisymmetric) resonant voltage distribution beneath the patch and between the slots while the minus (-) sign is used for modes with even (symmetric) resonant voltage distribution [1]. G12 could be calculated using this following formula:

Where ko= (2π / λo)

Jo= First kind Bessel function of order 0

W= Width of Patch antenna

The Approximation of self conductance calculation is:

Where I1 is the integral defined by:

The Resonant input impedance can be changed by using inset feed, recessed a distance (yo). User can out any impedance they want. Normally the characteristic impedance for microstrip filter and the antenna should be same.

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Circuit Material: Users will have to choose one of the circuit materials listed in the dropdown menu as these are the only circuit materials available to print the circuit board once all the design parameters are calculated.

Relative permittivity will vary based on the dielectric substrate that the user chooses to manufacture the microstrip filter. The different boards used for the circuit layout all have a relative permittivity of 3. That is why the relative permittivity is fixed in the GUI. The user does not need to specify this field, but they will need to select which material they will use for the circuit layout. They will choose from the following: Rogers1 with a thickness of 0.50 mm, Rogers2 with a thickness of .75 mm, and Rogers3 with a thickness of 1.53 mm. The thickness of the material is imperative in calculating the width of the antenna elements.

Here is a list of the materials that are used in the GUI as circuit materials:

 Rogers1 RO3003 d=0.02 inch (0.50 mm) Er=3 Rogers2 RO3003 d=0.03 inch (0.75 mm) Er=3 Rogers3 RO3003 d=0.04 inch (1.52 mm) Er=3

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Output parameter:

Width(w): The Width of the patch Antenna totally depends on the dielectric constant and thickness of the substrate and on the resonant frequency which is specified by the users.

Here is the equation to calculate width of Microstrip Antenna:

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Effective Length (Leff): Because of the fringing effects, electrically the patch of microstrip antenna looks grater than in physical dimensions. For the principal E-plane (xy plane), this is demonstrated in the figure below where the dimensions of the patch along its length have been extended on each end by a distance L, which is a function of the effective dielectric constant and width to height ratio.

Figure 1: Physical and effective lengths of Rectangular microstrip patch

Effective permittivity could be calculated by this following formula:

For low frequencies the effective dielectric constant is essentially constant. At intermediate frequencies its values begin to monotonically increase and eventually approach the values of the dielectric constant of the substrate. The initial values (at low frequency) of the effective dielectric constant are referred to as the static values.

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Length(L): The length of microstrip filter depends on the effective permittivity of the substrate, the width and the desired resonant frequency.

Inset feed point (yo): Inset fed patch antenna is normally designed to have any desired input resonant impedance for better S11 response. The inset feed introduces a physical notch, which introduces junction capacitance. The resonance frequency gets slightly influenced by the physical notch and its corresponding junction capacitance. The resonant input impedance decreases monotonically and reaches zero at the center as the inset feed point moves from the edge towards the center of the patch antenna. To calculate yo the following equation could be followed:[1]

Where Zc = desired impedance

yo= inset feed point

Figure 2: Recessed microstrip-line feed

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Radiation Pattern: A microstrip antenna is basically broadside radiator that has a relatively large beamwidth and low gain characteristics.

An Antenna pattern describes the far field directional properties of an antenna when measured at a fixed distance from the antenna. In general, the antenna pattern is a three dimensional plot that displays the strength of the radiated field or power density as a function of direction, with being specified by the zenith angle q and the azimuth angle f. By the virtue of the reciprocity theorem, a receiving antenna has the same directional antenna pattern as the pattern that it exhibits when operated in the transmission mode. [4]

Each specific combination of the zenith angle q and the azimuth angle f denotes a specific direction in the spherical coordinate system. The normalized radiation intensity F(q, f) characterizes the directional pattern of the energy radiated by the antenna and a plot of  F(q, f), as a function of both q and  f constitutes a three dimensional pattern.

Often, it is of interest to characterize the variation of F(q, f) in the form of two dimensional plots in specific planes in the spherical coordinate system. The two planes most commonly specified for this purpose are the elevation and azimuth planes. The elevation plane (E-plane), also called the q-plane, is the plane corresponding to a constant value of f. For example f=90 defines the y-z plan, both of which are elevation planes. A plot of F(q, f) versus q in either of these planes constitutes a two-dimensional pattern in the elevation plane. The azimuth plane (H-plane), also called f-plane, is specified by q=90 and corresponds to the x-y plane.  The E and H planes are often called the two principal planes of the spherical coordinate system. [4]

Basically E-plane is defined as ‘The plane containing the electric field vector and the direction maximum radiation” and the H-plane is defined as “the plane containing the magnetic field vector and the direction of maximum radiation”. [1]

The equation for E-plane and H-plane are given below:[2]

where,

‘r’ is the distance between the far field and the origin for a single slot.

‘Vo is the voltage across the slot which is invariant with x over its width.

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Directivity: The directivity D of an antenna is defined as the ratio of its maximum normalized radiation intensity, Fmax(which is by definition is equal to 1), to the average value of F(q, f) over 4p space:[4]

According to the 1983 version of the IEEE standard definitions of Terms for Antennas, the directivity of antenna could be defined as “the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. The average radiation intensity is equal to the total power radiated by the antenna divided by 4p.  If the direction is not specified, the direction of maximum radiation intensity is implied.” [1]

Prad is the total power obtained by integrating the radiation intensity, as given by the following equation:

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Radiation intensity: Radiation intensity in a given direction is defined as the power radiated from an antenna per unit solid angle. The radiation intensity is a far field parameter, and it can be obtained by simply multiplying the radiation density by the square of the distance. The radiation intensity is related to the far zone electric field of an antenna by:

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Reference:

[1] Antenna Theory: Analysis and Design, Second Edition, Constantine Balanis, Chapter 14.

[2] Microstrip Antennas, I.J Bahl, P. Bhartia, Chapter 2.

[3] A theoretical investigation of the rectangular microstrip antenna element, Derneryd, A., Antennas and Propagation, IEEE Transactions on [legacy, pre - 1988] , Volume: 26 Issue: 4 , Jul 1978
Page(s): 532 -535

[4] Fundamentals of Applied Electromagnetics, Fawwaz T. Ulaby, Chapter 9.

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