Tag Archives: Antennas

Microstrip Antenna – Cavity Model

The following is an alternative modelling technique for the microstrip antenna, which is also somewhat similar to the analysis of acoustic cavities. Like all cavities, boundary conditions are important. For the microstrip antenna, this is used to calculated radiated fields of the antenna.

Two boundary conditions will be imposed: PEC and PMC. For the PEC the orthogonal component of the E field is zero and the transverse magnetic component is zero. For the PMC, the opposite is true.


This supports the TM (transverse magnetic) mode of propagation, which means the magnetic field is orthogonal to the propagation direction. In order to use this model, a time independent wave equation (Helmholtz equation) must be solved.


The solution to any wave equation will have wavelike properties, which means it will be sinusoidal. The solution looks like:


Integer multiples of π  solve the boundary conditions because the vector potential must be maximum at the boundaries of x, y and z. These cannot simultaneously be zero. The resonant frequency can be solved as shown:


The units work out, as the square root of the product of the permeability and permittivity in the denominator correspond to the velocity of propagation (m/s), the units of the 2π term are radians and the rest of the expression is the magnitude of the k vector or wave number (rad/m). This corresponds to units of inverse seconds or Hz. Different modes can be solved by plugging in various integers and solving for the frequency in Hz. The lowest resonant mode is found to be f_010 which is intuitively true because the longest dimension is L (which is in the denominator). The f_000 mode cannot exist because that would yield a trivial solution of 0 Hz frequency. The field components for the dominant (lowest frequency) mode are given.




Microstrip Patch Antennas Introduction – Transmission Line Model

Microstrip antennas (or patch antennas) are extremely important in modern electrical engineering for the simple fact that they can directly be printed to a circuit board. This makes them necessary for things like cellular antennas for GPS, communication with cell towers and bluetooth/WiFi. Patch antennas are notoriously narrowband, especially those with a rectangular shape (patch antennas can have a wide variety of shapes). Patch antennas can be configured as single antennas or in an array. The excitation is usually fed by a microstrip line which usually has a characteristic impedance of 50 ohms.

One of the most common analysis methods for analyzing microstrip antennas is the transmission line model. It is important to note that the microstrip transmission line does not support TEM mode, unlike the coaxial cable which has radial symmetry. For the microstrip line, quasi-TEM is supported. For this mode, there is a field component along the direction of propagation, although it is small. For the purposes of the model, this can be ignored and the TEM mode which has no field component in the direction of propagation can be used. This reduces the model to:


Where the effective dielectric constant can be approximated as:


The width of the strip must be greater than the height of the substrate. It is important to note that the dielectric constant is not constant for frequency. As a consequence, the above approximation is only valid for low frequencies of microwave.

Another note for the transmission line model is that the effective length differs from the physical length of the patch. The effective length is longer by 2ΔL due to fringing effects. ΔL can be expressed as a function of the effective dielectric constant.





The Helical Antenna

The helical antenna is a frequently overlooked antenna type commonly used for VHF and UHF applications and provides high directivity, wide bandwidth and interestingly, circular polarization. Circular polarization provides a huge advantage in that if two antennas are circularly polarized, the will not suffer polarization loss due to polarization mismatch. It is known that circular polarization is a special case of elliptical polarization. Circular polarization occurs when the Electric field vector (which defines the polarization of any antenna) has two components which are in quadrature with equal amplitudes. In this case, the electric field vector rotates in a circular pattern when observed at the target, whether it be RHP or LHP (right hand or left hand polarized).

Generally, the axial mode of the helix antenna is used but normal mode may also be used. Usually the helix is mounted on a ground plane which is connected to a coaxial cable using a N type or SMA connector.

The helix antenna can be broken down into triangles, shown below.


The circumference of each loop is given by πD. S represents the spacing between loops. When this is zero (and hence the angle of the triangle is zero), the helix antenna reduces to a flat loop. When the angle becomes a 90 degree angle, the helix reduces to a monopole linear wire antenna. L0 represents the length of one loop and L is the length of the entire antenna. The total height L is given as NS, where N is the number of loops. The actual length can be calculated by multiplying the number of loops with the length of one loop L0.

An important thing to note is that the helix antenna is elliptically polarized by default and must be manually designed to achieve circular polarization for a specific bandwidth. Another note is that the input impedance of the antenna depends greatly on the pitch angle (alpha).

The axial (endfire) mode, which is more common occurs when the circumference of the antenna is roughly the size of the wavelength. This mode is easier to achieve circular polarization. The normal mode features a much smaller circumference and is more omnidirectional in terms of radiation pattern.

The Axial ratio is the numerical quantity that governs the polarization. When AR = 1, the antenna is circularly polarized. When AR = ∞ or 0, the antenna is linearly polarized. Any other quantity means elliptical polarization.


The axial ratio can also be approximated by:


For axial mode, the radiation pattern is much more directional, as the axis of the antenna contains the bulk of the radiation. For this mode, the following conditions must be met to achieve circular polarization.


These are less stringent than the normal mode conditions.

It is also important to consider that the input impedance of these antennas tends to be higher than the standard impedance of a coaxial line (100-200 ohms compared to 50). Flattening the feed wire of the antenna and covering the ground plane with dielectric material helps achieve a better SWR.


This equation can be used to calculated the height of the dielectric used for the ground plane. It is dependent on the transmission line characteristic impedance, strip width and the dielectric constant of the material used.

Mathematical Formulation for Antennas: Radiation Integrals and Auxiliary Potentials

This short paper will attempt to clarify some useful mathematical tools for antenna analysis that seem overly “mathematical” but can aid in understanding antenna theory. A solid background in Maxwell’s equations and vector calculus would be helpful.

Two sources will be introduced: The Electric and Magnetic sources (E and M respectively). These will be integrated to obtain either an electric and magnetic field directly or integrated to obtain a Vector potential, which is then differentiated to obtain the E and H fields. We will use A for magnetic vector potential and F for electric vector potential.

Using Gauss’ laws (first two equations) for a source free region:


And also the identity:


It can be shown that:


In the case of the magnetic field in response to the magnetic vector potential (A). This is done by equating the divergence of B with the divergence of the curl of A, which both equal zero. The same can be done from Gauss Law of electricity (1st equation) and the divergence of the curl of F.

Using Maxwell’s equations (not necessary to know how) the following can be derived:


For total fields, the two auxiliary potentials can be summed. In the case of the Electric field this leads to:


The following integrals can be used to solve for the vector potentials, if the current densities are known:


For some cases, the volume integral is reduced to a surface or line integral.

An important note: most antenna calculations and also the above integrals are independent of distance, and therefore are done in the far field (region greater than 2D^2/λ, where D is the largest dimension of the antenna).

The familiar duality theorem from Fourier Transform properties can be applied in a similar way to Maxwell’s equations, as shown.


In the chart, Faraday’s Law, Ampere’s Law, Helmholtz equations and the above mentioned integrals are shown. To be perfectly honest, I think the top right equation is wrong. I believe is should have permittivity rather than permeability.

Another important antenna property is reciprocity… that is the receive and transmit radiation patterns are the same , given that the medium of propagation is linear and isotropic. This can be compared to the reciprocity theorem of circuits, meaning that a volt meter and source can be interchanged if a constant current or voltage source is used and the circuit components are linear, bilateral and discrete elements.


The Half Wave Dipole Antenna

The dipole is a type of linear antenna which commonly features two monopole antennas of a quarter wavelength in size bent at 90 degree angles to each other. Another common size for the dipole is 1.25λ. These sizes will be discussed later.

It is important for beginning the study of the dipole antenna to discuss the infinitesimal dipole. This is the dipole which is smaller than 1/50 of the wavelength and is also known as a Hertzian dipole. This is an idealized component which does not exist, although it can serve as an approximation to large antennas which can be broken into smaller segments. The mathematics behind this can be found in “Antenna theory:Analysis and Design” by Constantine Balanis.

More importantly, three regions of radiation can be defined: the far field (where the radiation pattern is constant – this is where the radiation pattern is calculated), the reactive near field and the radiative near field.


As shown in the image, the reactive near field is when the range is less than the wavelength divided by 2π or when the range is less than 1/6 of the wavelength. The electric and magnetic fields in this region are 90 degrees out of phase and do not radiate. It is known that the E and H fields must be in phase to propagate. The radiating near field is where the range is between 1/6 of the wavelength and the value 2D^2 divided by the wavelength. This is also known as the Fresnel zone. Although the radiation pattern is not fully formed, propagating waves exist in this region. For the far field, r must be much, much greater than λ/2π.

The radiating patterns of the dipole antenna is pictured below, with both the E and H planes. The E plane (elevation angle pattern) is pictured on the bottom right and the H plane (Azimuthal angle) beside it on the left. The plots are given in dB scale. The radiation patterns can be understood by considering a pen. While facing the pen you can see the full length of the pen, but if you look down on the pen you can only see the tip or end. This is analogous to the dipole antenna where maximum radiation is broadside to the antenna and minimum radiation on the ends, leading to the figure 8 radiation pattern. When this radiation pattern in extended to three dimensions, the top left image is derived.



Yagi-Uda Antenna/Parasitic Array

The Yagi-Uda antenna is a highly directional antenna which operates above 10 MHz and is commonly used in satellite communications, as well as with amateur radio operators and as rooftop television antennas. The radiation pattern for the Yagi-Uda antenna shows strong gain in one particular direction, along with undesirable side lobes and a back lobe. The Yagi is similar to the log periodic antenna with a major distinction between the two being that the Yagi is designed for only one frequency, whereas the log periodic is wideband. The Yagi is much more directional, so it provides a higher gain in that one particular direction that it is designed for.

The “Yagi” antenna has two types of elements: the driven element and the parasitic elements. The driven element is the antenna element that is directly connected to the AC source in the transmitter or receiver. A reflector element (parasitic) is placed behind the driven element in order to split the undesirable back lope into two smaller lobes. By adding directive parasitic elements in front of the driven element, the radiation pattern is stronger and more directional. All of these elements are parallel to each other and are usual half wave dipoles. These elements work by absorbing and reradiating the signal from the driven element. The reflector is slightly longer (inductive) than the driven element and the director elements are slightly shorter (capacitive).

It is well known in transmission line theory that a low impedance/short circuit load will reflect all power with an 180 degree phase shift (reflection coeffecient of -1). From this knowledge, the parasitic element can be considered a normal dipole with a short circuit at the feed point. Since the parasitic elements reradiate power 180 degrees out of phase, the superposition of this wave and the wave from the transmitter leads to a complete cancellation of voltage (a short circuit). Due to the inductive effects of the reflector element and the capacitive effects of the director antennas, different phase shifts are created due to lagging or leading current (ELI the ICE man). This cleverly causes the superposition of the waves in the forward direction to be constructive and destructive in the backwards direction, increasing directivity in the forward direction.

Advantages of the Yagi include high directivity, low cost and high front to back ratio. Disadvantages include increased sizing when attempting to increase gain as well as a gain limitation of 20dB.



Beamforming (spatial filtering) is a huge part of Fifth Generation wireless technology. Beamforming is basically using multiple antennas and varying the phase and amplitude of the inputs to these antennas. The result is a directed beam in a specific direction. This is a great method of preventing interference by focusing the energy of the antennas. Constructive and Destructive interference is used to channel the energy and increase the antennas’ directivity. The receiver receives the multitude of waves and depending on the receiver’s location will determine whether there is mostly constructive or destructive interference. Beamforming is not only used in RF wireless communication but also in Acoustics and Sonar.

An important concept to know is that placing multiple radiating elements (antennas) together increases the directivity of the radiation pattern. Putting two antennas side by side, creating a main lobe with a 3dB gain going forward. With four radiating elements, this becomes 6dB (quadruple gain). Feeding all of the elements with the same signal means that the elements are still one single antenna, but with more forward gain. The major issue here is that you only benefit from this in one single stationary direction unless the beam can be moved. This is where feeding the antennas with different phases and amplitudes comes in. The number of antennas becomes equal to the number of input signals. Having more separate antennas (and more input signals) creates a more directed antenna pattern. Spatial multiplexing can also be implemented to service multiple users wirelessly by utilizing space multiple times over.

Using electronic phase shifters at the input of the antennas can decrease cost of driving the elements quite a bit. This is known as a phased array and can steer the beam pattern as necessary but can only point in one direction at a time.

phased array


Fundamental Parameters of Antennas

To understand the details behind antennas, the vital interface between free space and a transmit/receive system, it is important to fully understand the basic properties of antennas in order to understand their performance.

One of the main properties of an antenna is its radiation or antenna pattern. This is defined as a mathematical function of the radiation properties of the antenna as a function of space coordinates. It is important to note that this pattern is determined in the far field region (there are three main regions when studying antenna radiation: reactive near field, radiating near field, and far field). This can be a trace of the Electric or magnetic field (field pattern) or the spatial variation of the power density (power pattern). These are generally normalized with respect to the maximum value and typically are plotted in decibel scale to accentuate minor lobes. Minor lobes are any lobes that are not the major lobe. In split beam antennas, there can be multiple major lobes. The following image shows a directive antenna’s radiation pattern. Side lobes are generally undesirable and should be minimized if possible.


The Half Power Beamwidth (HPBW or sometimes just beamwidth) can be determined by drawing two lines from the origin point to the -3dB (half power) point and seeing the resultant angle.

Antennas are generally compared to “isotropic” antennas. These are hypothetical antennas that radiate power equally in all directions. This is not to be confused with omnidirectional antennas, which radiate power equally in the azimuthal direction. The E and H planes are defined as the plane containing the electric field vector and direction of maximum radiation and the plane containing the H vector respectively.

The three main regions around an antenna are the reactive near field, radiating near field and far field. In the reactive near field, the radiation is reactive (eg. the E and H fields are out of phase by 90 degrees. Because the waves are not in phase and transverse, they do not propagate. In the radiating near field, the waves are not purely reactive and propagate, however the shape varies with distance. In the far field (where the radiation pattern originates from), the radiation pattern does not change with distance and the waves are transverse.

One of the major characterizing aspects of antennas is the directivity. This is equivalent to the ratio of the radiation intensity in a certain direction over the hypothetical isotropic radiator intensity.


The denominator represents the average power radiated in all directions. The function is the normalized radiation pattern as a function of both the elevation and azimuthal angles. It is also possible to calculate partial directivities in either the theta direction or the phi direction and total directivity is the sum of these two. For a highly directive antenna with a very narrow major lobe and negligible minor lobes, the solid angle can be approximated by the product of the half power beamwidths in two different planes.


Another important property is antenna efficiency, which is the product of reflection efficiency, conduction efficiency, and dielectric efficiency. This takes into account all possible loss: either from a VSWR greater than 1 due to an impedance mismatch between the feedline and the antenna and conductive losses due to Joule heating from both the dielectric and the conductive parts. The antenna gain can be defined as the product of the antenna efficiency and directivity.

Distributed Antenna Systems

Distributed antenna systems (DAS) provide a convenient, power efficient way to move RF signals within buildings (iDAS or indoor DAS) or in outdoor places such as stadiums or venues (oDAS or outdoor DAS). A DAS consists of two main parts: a signal source and a distribution system. The signal source can be an outdoor antenna or a local base station.


The principle idea is this: to replace a single, high power antenna with several power efficient antennas without losing any area coverage. There are many types of distribution systems. The main idea for each system is to propagate the signals in such as way that maximum power and signal coverage is utilized. In one architecture, a master unit is connected to the base station using RF coaxial cable. The frequency of this energy is then increased to the optical range and carried using fiber optic cables to each remote unit on each floor of the building. The energy is then reduced down to the RF frequency and passive splitters are used to distribute the signals to each antenna. Architectures can be purely passive, purely active or a hybrid combination of both. DAS is advantageous for increased coverage but can result in higher costs due to increased infrastructure required.