Theoretical, Numerical Analysis, and Practical Guide to Dipole Antennas

They are used extensively. The rod-shaped antennas on Wi-Fi routers are exactly dipoles—well, more precisely monopoles using a ground plane, but the principle is the same. Half-wave dipoles are also standard for FM broadcasting and amateur radio antennas. Furthermore, they serve as the reference (0dBd = 2.15dBi) for measuring the gain of all other antennas. They are like the "meter prototype" of antenna engineering.

Two conductor rods (elements) are aligned in a straight line, and a high-frequency signal is fed from the center. One with a total length of half the wavelength ($\lambda/2$) is called a half-wave dipole. For example, for 2.4GHz Wi-Fi, the wavelength is 125mm, so you have two conductors each about 31mm long. It's a very simple structure, but whether you can properly explain why the input impedance becomes $73 + j42.5\,\Omega$ serves as a litmus test for your understanding of electromagnetism.

Good idea. Let's start with an infinitesimal dipole (Hertzian dipole). When a uniform current $I_0$ flows through a conductor that is sufficiently short compared to the wavelength ($l \ll \lambda$), the far-field electric field is:

Here, $\eta_0 = 120\pi\,\Omega$ is the impedance of free space, and $k = 2\pi/\lambda$ is the wave number. The important part is the $\sin\theta$ term, which is maximum at $\theta = 90°$ (direction perpendicular to the antenna) and zero at $\theta = 0°, 180°$ (along the antenna axis). In other words, it's a doughnut-shaped radiation pattern.

Integrate the Poynting vector from the far field over the entire solid angle to find the radiated power $P_{rad}$, and derive the radiation resistance from $P_{rad} = \frac{1}{2}R_{rad}|I_0|^2$:

For example, if $l = \lambda/10$, then $R_{rad} \approx 7.9\,\Omega$. The shorter the antenna relative to the wavelength, the smaller the radiation resistance, leading to poorer radiation efficiency. This is why "electrically short antennas have poor efficiency." This is also the challenge in designing antennas for mobile phones.

Good observation. In a half-wave dipole (total length $L = \lambda/2$), the current distribution is not uniform but sinusoidal:

Maximum $I_0$ at the center (feed point), zero at both ends. Calculating the far field from this current distribution yields:

Compared to the $\sin\theta$ of the infinitesimal dipole, $\cos(\frac{\pi}{2}\cos\theta)/\sin\theta$ results in a slightly "tighter" pattern. The direction of maximum radiation ($\theta = 90°$) is the same, but the nulls along the axis are deeper. This difference appears as a difference in directivity.

Exactly. It's the same boundary condition as the open end of a transmission line. The current is zero, but the voltage (electric field) is maximum. The fact that this voltage/current distribution forms a standing wave is the essence of the dipole, and the view that "an antenna is an opened transmission line" holds true.

Calculate the Poynting vector $\mathbf{S} = \frac{1}{2\eta_0}|E_\theta|^2 \hat{r}$ from the far field $E_\theta$ and integrate over the entire sphere to obtain the radiated power:

This integral cannot be solved analytically in closed form; numerical integration yields approximately 1.219. From $P_{rad} = \frac{1}{2}R_{rad}|I_0|^2$:

This calculation result is what's written as "approximately 73Ω" in electromagnetism textbooks. It's slightly mismatched with 50Ω coaxial cable but quite close to 75Ω cable. In fact, the 75Ω impedance for television has a history of being determined to match the radiation resistance of the half-wave dipole.

The antenna's near field has a reactive component that stores energy. Exactly at half-wavelength ($L = \lambda/2$), the stored electric field energy slightly exceeds the magnetic field energy, resulting in an inductive (positive reactance $j42.5\,\Omega$) component.

In practice, to cancel this reactance and make it purely resistive, the antenna length is shortened to about $0.47\lambda$～$0.48\lambda$. Then the reactance becomes zero and $Z_{in} \approx 73\,\Omega$ (pure resistance). This "resonant length" varies slightly depending on wire diameter and surrounding environment, so it's common to fine-tune it with simulation.

Directivity $D$ is the ratio of "radiation intensity in the maximum radiation direction" to "radiation intensity if radiated equally in all directions":

For a half-wave dipole, $D = 1.643$. Expressed in dB:

This means "approximately 64% stronger in the strongest direction compared to an isotropic antenna." Even with the exact same input power, because it's concentrated in the equatorial direction of the doughnut, it becomes stronger only in that direction. Yagi antennas (TV antennas) have gains like 10dBi or 12dBi because they further narrow the beam based on the dipole.

dBd is gain relative to a half-wave dipole as reference. That is, $0\,\mathrm{dBd} = 2.15\,\mathrm{dBi}$. dBd is often used in the amateur radio world, but dBi is standard in academic papers and CAE. Just remember the conversion: $G_{\mathrm{dBi}} = G_{\mathrm{dBd}} + 2.15$.

Approximate the unknown quantity using shape functions $N_i$:

Expressing this mathematically gives:

Expressing this mathematically gives:

Discretizing the governing equations of the continuum yields the following system of algebraic equations:

Here, $[K]$ is the global stiffness matrix (or equivalent system matrix), $\{u\}$ is the vector of unknown nodal variables, and $\{F\}$ is the external force vector.

• Full Integration: Integrates all terms accurately. Tends to overestimate stiffness (Locking).
• Reduced Integration: Reduces the number of integration points. Improves computational efficiency but risks hourglass mode generation.
• Selective Reduced Integration (B-bar Method): Separates and integrates volumetric and deviatoric terms. Avoids locking.

• h-refinement: Refine the mesh (reduce element size h) to improve accuracy.
• p-refinement: Increase the polynomial order of elements to improve accuracy.
• hp-refinement: Simultaneously optimize h and p.