Antenna Theory Crash Course Part 4: Understanding How Antennas Radiate - Patterns and Directivity

Antennas serve as crucial transitional structures between guided electromagnetic energy (like in transmission lines) and the free space through which radio waves propagate [1, 2]. Understanding how they radiate energy is fundamental to antenna theory, and two key concepts in this understanding are **radiation patterns** and **directivity**.

Radiation Patterns

An **antenna radiation pattern**, also known as an **antenna pattern**, is defined as a **mathematical function or a graphical representation of the radiation properties of the antenna as a function of space coordinates** [3]. These properties are typically determined in the **far-field region** and are represented based on **directional coordinates** [3]. The radiation properties of concern include **power flux density, radiation intensity, field strength, directivity, phase, or polarisation** [3].

• The radiation pattern primarily describes the **two- or three-dimensional spatial distribution of radiated energy** as observed at a **constant radius** from the antenna [3]. A convenient coordinate system for antenna analysis is the spherical coordinate system [3, 4].
• A **trace of the received electric (magnetic) field at a constant radius** is termed the **amplitude field pattern** [3]. Conversely, a **graph of the spatial variation of the power density along a constant radius** is known as an **amplitude power pattern** [3]. Power patterns often represent a plot of the **square of the magnitude of the electric or magnetic field as a function of angular space** [4].
• Radiation patterns can be visualised as **two-dimensional cuts** or as **three-dimensional patterns** [5-8]. **Three-dimensional programs**, such as one designated as **Spherical** (a MATLAB-based program), can be used to produce these visualisations [7].
• A typical radiation pattern exhibits a **major lobe** (or main beam), which represents the direction of maximum radiation, and other lobes with lower radiation intensity, known as **minor lobes** and potentially a **back lobe** [9].
• **Beamwidth** is a significant parameter associated with the radiation pattern. It is defined as the **angular separation between two identical points on opposite sides of the pattern maximum** [6]. The **Half-Power Beamwidth (HPBW)**, a widely used measure, is the angle between the two directions where the radiation intensity is **one-half the maximum value** [6]. Another important beamwidth is the **First-Null Beamwidth (FNBW)**, which is the **angular separation between the first nulls of the pattern** [6].

Directivity

**Directivity** is a key **figure of merit** that describes **how well a radiator directs energy in a certain direction** [10]. The IEEE Standard Definitions of Terms for Antennas (IEEE Std 145–1983) define the **directivity of an antenna** as the **ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions** [11]. If the direction is not specified, the direction of **maximum radiation intensity is implied** [11].

• In simpler terms, the directivity of a nonisotropic source is the **ratio of its radiation intensity in a given direction over that of an isotropic source** [11]. Mathematically, directivity ($D$) can be expressed as $D = \frac{U}{U_0} = \frac{4\pi U}{P_{rad}}$, where $U$ is the radiation intensity in a given direction, $U_0$ is the radiation intensity of an isotropic source (equal to $\frac{P_{rad}}{4\pi}$), and $P_{rad}$ is the total power radiated [11]. The **maximum directivity** ($D_0$) occurs in the direction of maximum radiation intensity ($U_{max}$) and is given by $D_0 = \frac{4\pi U_{max}}{P_{rad}}$ [11].
• For an antenna where the radiation intensity $U$ is a function of the spherical angles $\theta$ and $\phi$, the directivity $D(\theta, \phi)$ as a function of these angles can be expressed [10, 12]. For example, if $U = A_0 \sin \theta$, the maximum directivity $D_0 = 1.27$, and the directivity as a function of direction is $D = 1.27 \sin \theta$ [12]. Similarly, if $U = A_0 \sin^2 \theta$, $D_0 = 1.5$, and $D = 1.5 \sin^2 \theta$ [10].
• For antennas with one narrow major lobe and negligible minor lobes, the **beam solid angle** ($\Omega_A$) is approximately equal to the product of the half-power beamwidths in two perpendicular planes: $\Omega_A \approx \Theta_{1r} \Theta_{2r}$, where the beamwidths are in radians [13]. In this case, the **approximate maximum directivity** can be calculated as $D_0 \approx \frac{4\pi}{\Omega_A}$ [14]. If the beamwidths are given in degrees ($\Theta_{1d}$ and $\Theta_{2d}$), an approximate directivity formula by Kraus is $D_0 \approx \frac{32400}{\Theta_{1d} \Theta_{2d}}$ [14, 15]. Other approximate formulas, such as those by Tai and Pereira, also exist [16-18].
• The directivity is a crucial parameter for various antenna types:
  • For a **small dipole**, the directivity $D_0 = 1.5$ or $1.761$ dB [19].
  • A **quarter-wavelength monopole** has a directivity of $3.286$ or $5.167$ dB [20].
  • The directivity of a dipole increases as its length increases, up to a point [21, 22].
  • Arrays of antennas are designed to achieve higher directivities than single elements [23, 24]. For example, the directivity of end-fire arrays can be improved using techniques like the Hansen-Woodyard condition [25, 26].
  • The directivity of aperture antennas depends on the field distribution over the aperture area [27, 28]. For a rectangular aperture with a constant field distribution, the directivity can be calculated using specific formulas [28, 29].
  • Horn antennas exhibit directivity that is related to their flare angle and length [30].
  • Microstrip patch antennas also have a defined directivity that can be estimated using approximate formulas or more complex models [31-33].
  • Reflector antennas, like paraboloidal reflectors, can achieve high directivities depending on their size and the feed antenna used [34, 35].
• **Computer programs** such as **Directivity**, **Arrays**, **Dipole**, **Yagi Uda**, **Log-Periodic Dipole Array**, and **Aperture**, available for Balanis' textbook, are valuable tools for computing and analysing the directivity of different antenna configurations [29, 36-38].

In summary, radiation patterns provide a spatial map of an antenna's radiated energy, while directivity quantifies its ability to focus that energy in a particular direction. These concepts are fundamental in characterising and designing antennas for various applications [39].