##### Design Guidelines for Skeleton Slot Antennas: A Simulation-Driven Approach

This article navigates the intricacies of **Skeleton Slot antennas**, exploring their sensitivity to geometric parameters and the transformative impact of a **simulation-driven methodology**. The Skeleton Slot is treated as an **array of two loop antennas** with a common feed point. We delve into the balance of loop perimeters, conductor radii, and aspect ratios, unraveling their influence on output parameters such as input impedance, VSWR, and gain. We present a **script-driven approach** to optimize designs, empowering engineers and enthusiasts to craft high-performance Skeleton Slot antennas. Bridging theory and application, the article showcases practical insights, making it an essential resource for anyone seeking to elevate their radio frequency design projects.

##### Table of Contents

#### Introduction

Bill Sykes (call sign G2HCG) is acknowledged as the innovator behind the **Skeleton Slot antenna**, having successfully deployed it in **VHF bands**. The inherently versatile Skeleton Slot principle extends its utility to **HF communication bands** by scaling dimensions based on wavelength, with the physical antenna dimensions remaining practical within the **14-28 MHz** bands. Noteworthy advantages of this design include its lightweight nature, ease of construction, low-angle radiation, bi-directional directivity, and the convenience of mounting it as a simple metal framework without the need for insulation.

The nomenclature “Skeleton Slot” is derived from the **slot antenna concept**. This **aperture** antenna is crafted by cutting a rectangular hole in a conducting sheet, essentially serving as a “photographic negative” of a dipole, where the slot functions as the radiating element. Reducing the metal sheet until it transforms into a rectangular wire frame results in the formation of the “skeleton slot.”

In our previous article, **“A Closer Look at the HF Skeleton Slot Antenna,”** we introduced a Skeleton model in **AN-SOF** and presented the results for the 15m (20 MHz) band. Expanding upon that analysis, this article delves into a comprehensive discussion of the skeleton slot from a general perspective, supported by the theory of **loop antennas**. This approach complements the insights provided by the inventor in the January 1955 issue of **The Short Wave Magazine** in the article titled **“The Skeleton Slot Aerial System”** (Vol. XII, No. 11, pp. 594-598). In that article, the author elucidates the antenna as an array of two closely positioned dipoles. Furthermore, we offer **dimensioning guidelines** for experimenters keen on venturing into antenna construction.

#### Geometry of Skeleton Slot and Loop Antennas

In Figure 1, a schematic representation of the Skeleton Slot antenna is presented, highlighting key dimensions:

**L:**Length of each loop.

**w:**Width of the antenna.

**a:**Wire radius.

**p = 2(L + w):**Loop perimeter.

**r = L/w:**Loop aspect ratio.

The skeleton slot, depicted in Figure 1, functions as a vertical antenna that can be conceptualized as an array comprising two identical, closely coupled loops—a **top loop** and a **bottom loop**. These loops share a **common feed point** located at the antenna’s center, where the feeding transmission line is connected.

In adherence to loop theory, when the loop contour spans approximately **half a wavelength**, it exhibits an input impedance transitioning **from inductive to capacitive**. This shift is characterized by high resistance and reactance values, indicative of a resonance akin to that observed in a **parallel RLC circuit**. Referencing the validation article **“Input Impedance and Directivity of Large Circular Loops”**, specifically Figure 2, illustrates the input impedance variation concerning the loop circumference measured in wavelengths, C/λ (see Figure 2 below). As the loop circumference approaches **one wavelength**, the capacitive (negative) reactance decreases in absolute value, reaching **resonance** similar to a **series RLC circuit** when the reactance approaches zero. Consequently, the resistance assumes manageable values in practice, approximately around 100 Ohms. The **“useful zone”** of the loop in practice is identified when **C/λ ≈ 1**, as shown in Figure 2.

While Figure 2 refers to circular loops, the analogous behavior is applicable to **rectangular loops** as well. Therefore, the practical utility of the loop is realized when its perimeter, denoted as ‘p,’ approaches **one wavelength** (p/λ ≈ 1).

A further observation drawn from loops with **circumferences comparable to the wavelength** is the pronounced **sensitivity of reactance to variations in the wire radius**. This sensitivity manifests in a logarithmic manner, specifically proportional to ln(C/a), where ‘a’ denotes the wire radius.

Given its configuration, as previously mentioned, the Skeleton Slot antenna can be viewed as comprising **two tightly coupled rectangular loops**. Consequently, we can anticipate a behavior analogous to that described for loops in general.

Maintaining **a constant loop aspect ratio** (r = L/w) and conducting numerous calculations while varying the **loop perimeter** (p = 2(L + w)), we observe that **the Skeleton Slot resonates when the loop perimeter is approximately one wavelength** (p ≈ λ), aligning with expectations for a single loop. However, it’s crucial to note that this perimeter isn’t precisely equal to one wavelength; its value fluctuates based on the aspect ratio (L/w) and the wire radius compared to the loop perimeter (a/p). While the specific results of these calculations fall beyond the scope of this article, we will concentrate on the behavior of the skeleton slot when the perimeter of each loop approximates one wavelength. In this condition, the antenna approaches **self-resonance**, obviating the necessity for an impedance matching network at the feed point.

In Sykes’ article, the author employs **the aspect ratio of the Skeleton Slot**, expressed as 2L/w = 2r, rather than that of each individual loop. Through multiple measurements, the conditions for achieving a self-resonant antenna are outlined as follows:

- An
**optimal aspect ratio**of 3:1, i.e., 2r = 3, leading to r = 3/2 = 1.5 based on our definition.

- The
**total length of the skeleton**must be 2L = 0.56λ, so the loop length is L = 0.28λ.

- The
**ratio of width to conductor diameter**must be 32:1, denoted as w/(2a) = 32.

Given L = 0.28λ and r = L/w = 1.5, the resulting perimeter is calculated as p = 2 (0.28λ + 0.28λ/1.5) = 0.93λ. This closely aligns with our simulation calculations, indicating **resonance when the loop perimeter approximates one wavelength**. However, it’s essential to note that this resonance condition varies with the ratio of perimeter to conductor radius, denoted as p/a, rather than the ratio w/(2a). Subsequent results, presented in the following sections, illustrate that **a thicker conductor** necessitates **an increased loop perimeter** for the antenna to be self-resonant with a given aspect ratio. Conversely, a **thinner conductor** requires a **decreased loop perimeter** for the same resonant condition.

#### Script for Varying the Loop Aspect Ratio

A pivotal inquiry in Skeleton Slot antenna design revolves around determining the **optimal aspect ratio**. Is there a specific aspect ratio that outperforms others? This section aims to delve into this question, with the pursuit of an “optimal” point focusing on achieving a **self-resonant** antenna, thereby obviating the need for a matching network. In Sykes’ investigation, a conductor with a radius of 4.76mm (rounded up to 5mm in our study) was employed, corresponding to a 3/16″ radius (3/8″ diameter).

For our exploration, we maintain **a fixed conductor radius** of 5mm, and we ensure that **the perimeter of each loop remains close to one wavelength**, as previously discussed. Simulations conducted using **AN-SOF** are set at a frequency of 20 MHz (15-meter band). Importantly, the conclusions drawn from these simulations hold true for any frequency band, contingent upon scaling the antenna dimensions proportionally with the wavelength. Naturally, the resulting physical dimensions at a given frequency must be practical for constructing the antenna in practice.

To perform calculations with **varying geometric parameters**, we can leverage the **“Run Bulk Simulation”** function in AN-SOF in conjunction with a script in **Scilab**. For those unfamiliar with script programming, a comprehensive tutorial on antenna-related scripts is available in the article **“Element Spacing Simulation Script for Yagi-Uda Antennas”**, specifically focusing on Yagis with variable element spacing.

##### Description of Script Elements

Below, the script is provided to generate multiple files in **.nec** format, where the loop aspect ratio, L/w, is systematically altered while maintaining a fixed perimeter, p. Through multiple simulations, we have determined that the “optimal” p value, rendering the antenna self-resonant for a broad range of L/w ratios, is p = 14.8m at 20 MHz, corresponding to **p = 0.99λ**.

When the perimeter p is held constant and the loop aspect ratio is varied (r = L/w), the antenna dimensions can be calculated using the following formulas:

**w = 0.5 p/(r+1)**

**L = 0.5 p r/(r+1)**

To expedite the task, consider creating a Skeleton Slot antenna model in AN-SOF or downloading the model provided in **this article**. Then, in AN-SOF, navigate to the File menu, select **“Export Wires,”** choose the file format “.sce,” and save the file. Subsequently, open the .sce file with Scilab and make the modifications as illustrated below:

*// Script for AN-SOF Professional*
*// Skeleton Slot Antenna with varying aspect ratio*
r_min = 1.0; *// Min loop aspect ratio*
r_max = 2.5; *// Max loop aspect ratio*
n = 20; *// Number of intervals between r_min and r_max*
f = 20.0; *// Frequency in MHz*
k = 0.987; *// Factor for loop perimeter*
p = k*299.8/f; *// Loop perimeter [m] (299.8/f = wavelength at f MHz*)
radius = 5; *// Wire radius in [mm]*
S = 11; *// Number of segments per wire (it must be odd)*
for i = 0:n,
r = r_min + i*(r_max-r_min)/n; *// Loop aspect ratio*
w = 0.5*p/(r+1); *// Loop width*
L = r*w; *// Loop length (total length of skeleton slot = 2L)*
antenna = [
CM('Skeleton Slot Antenna')
CM('Loop length-to-width ratio = ' + string(r))
GW(1, S, 0, -0.5*w, 0, 0, 0.5*w, 0, radius*1e-3)
GW(2, S, 0, 0.5*w, -L, 0, -0.5*w, -L, radius*1e-3)
GW(3, S, 0, -0.5*w, L, 0, -0.5*w, 0, radius*1e-3)
GW(4, S, 0, 0.5*w, 0, 0, 0.5*w, L, radius*1e-3)
GW(5, S, 0, -0.5*w, -L, 0, -0.5*w, 0, radius*1e-3)
GW(6, S, 0, 0.5*w, 0, 0, 0.5*w, -L, radius*1e-3)
GW(7, S, 0, 0.5*w, L, 0, -0.5*w, L, radius*1e-3)
GE(0)
FR(0, 1, f, 0.0)
EX(0, 1, (S+1)/2, 1.4142136, 0)
EK()
];
mputl(antenna,'C:/AN-SOF/Skeleton_Ratio' + string(i) + '.nec');
end

This simple script streamlines the process, allowing for efficient exploration of the Skeleton Slot antenna’s behavior under varying loop aspect ratios. This script comprises two main elements:

**1. Definition of Constants:**

– Fixed values for the extremes of the loop aspect ratio variation range.

– Number of intervals ‘n’ to be calculated (with ‘n+1’ discrete points).

– Loop perimeter ‘p’ and wire radius.

– Numerically adjusted perimeter ‘p’ within 3 significant digits at p = 0.987λ.

**2. ‘For’ Loop:**

– The script contains a “for” loop where the “antenna” matrix is defined. Each row contains commands (CM, GW, GE, FR, EX, EK) used to describe an antenna in **NEC format**.

– Each generated .nec file (n+1 files) is named “Skeleton_Ratio**i**.nec” with i = 0, 1, 2, …, n.

This script is complemented by a second script that reads the results from CSV files and represents them graphically in plots. Additionally, there is a third script that contains the functions associated with NEC commands. To download these three scripts, click on the button provided above.

##### Running the Scripts in Combination with AN-SOF

Here are the steps to run this script, along with the one displaying graphs with results, in combination with AN-SOF:

1. **Download the .zip file** containing the three necessary scripts: **NECcommands.sce**, **SkeletonSlot.sce**, and **SkeletonSlotResults.sce**.

2. **Unzip the file** and save the scripts in a folder to run them from **Scilab**.

3. Start Scilab and open the scripts.

4. Run **NECcommands.sce**, which contains functions that write **NEC commands**.

5. Create a folder C:\AN-SOF and run **SkeletonSlot.sce**. The n+1 “.nec” files will be saved in this folder.

6. In AN-SOF, go to the menu **Run > Run Bulk Simulation**, navigate to the C:\AN-SOF folder, and select all the generated .nec files (you can press Ctrl + A). AN-SOF will calculate them one by one, saving the corresponding results in **CSV** files.

7. Return to Scilab and run the **SkeletonSlotResults.sce** script. Three graphs will be displayed: the **gain**, the **input impedance**, and the **VSWR** as a function of the loop aspect ratio.

With these scripts, you can obtain results that will be analyzed in the subsequent sections for the input impedance, VSWR, and antenna gain as a function of the loop aspect ratio.

#### Input Impedance, VSWR, and Gain vs. Aspect Ratio

In Figure 3, the Skeleton Slot **input impedance** (R_{in} + jX_{in}) is depicted as a function of the loop aspect ratio, L/w. It’s crucial to note that **the loop perimeter remains constant at approximately one wavelength**, p ≈ λ, resulting in **variable antenna length and width** to uphold the constant perimeter. The relative sizes of the Skeleton Slot for three aspect ratios—**L/w = 1, 1.8, and 2.5**—are illustrated at the bottom of Figure 3. Note that, when L/w = 1, the loops form squares (L = w). These outcomes have been calculated for a conductor radius of 5mm.

The input impedance unveils an intriguing property: **commencing at an aspect ratio of 1.7, the antenna maintains self-resonance (X _{in} = 0) as the aspect ratio increases**. The reactance curve (X

_{in}) is notably flat with values that are practically manageable even when the loops form squares (L/w = 1). However, the

**input resistance**, R

_{in},

**exhibits a more pronounced variation**, initiating at 140 Ohms for L/w = 1 and steadily decreasing to approximately 30 Ohms for L/w = 2.5.

**The input impedance approaches 50 + j0 Ohms at L/w ≈ 1.8**. This suggests an

**optimal point**where the antenna achieves

**self-resonance**without requiring an impedance matching network.

Within the range of **L/w spanning from 1.6 to 2**, we observe a practical sweet spot with the input resistance falling between **40 and 60 Ohms** and **the reactance approaching zero**. Figure 4 (top) illustrates the **Voltage Standing Wave Ratio (VSWR)** as a function of the loop aspect ratio, considering a reference impedance of 50 Ohms. The “useful” range for VSWR falls within values of L/w between 1.6 and 2. Additionally, Figure 4 (bottom) showcases the gain of the skeleton slot, demonstrating a monotonic increase with the loop aspect ratio. Opting for L/w = 2 becomes advantageous if the design objective is to maximize gain.

*Fig. 4: (Top) VSWR of Skeleton Slot antenna as a function of the loop aspect ratio (reference impedance of 50 Ohms). (Bottom) Gain of Skeleton Slot antenna as a function of the loop aspect ratio.*

*Fig. 4: (Top) VSWR of Skeleton Slot antenna as a function of the loop aspect ratio (reference impedance of 50 Ohms). (Bottom) Gain of Skeleton Slot antenna as a function of the loop aspect ratio.*

*Fig. 4: (Top) VSWR of Skeleton Slot antenna as a function of the loop aspect ratio (reference impedance of 50 Ohms). (Bottom) Gain of Skeleton Slot antenna as a function of the loop aspect ratio.*

In the subsequent sections, we will delve into an analysis of the skeleton slot’s sensitivity to variations in loop perimeter and conductor radius. This exploration will contribute to the establishment of simulation-driven design guidelines, enabling a more informed and optimized design process.

#### Sensitivity to the Loop Perimeter Around One Wavelength

With the established **optimal loop perimeter** for achieving a self-resonant antenna at **p = 0.99λ**, we explore the impact of a **±1% change** in this perimeter. For a frequency of 20 MHz, corresponding to a wavelength of 15 meters, this adjustment would equate to a ±15 cm change in perimeter. Figure 5 (top) illustrates the input impedance as a function of the loop aspect ratio for three different perimeters: 0.99p, 1.00p, 1.01p, where p = 0.99λ.

*Fig. 5: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different loop perimeters: 0.99p, 1.00p, 1.01p, where p = 0.99λ is the self-resonance loop perimeter.*

*Fig. 5: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different loop perimeters: 0.99p, 1.00p, 1.01p, where p = 0.99λ is the self-resonance loop perimeter.*

*Fig. 5: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different loop perimeters: 0.99p, 1.00p, 1.01p, where p = 0.99λ is the self-resonance loop perimeter.*

Notably, the **resistive part** (R_{in}) demonstrates **minimal variation** with changes in perimeter, whereas the **reactive part** (X_{in}) undergoes a **significant alteration**. The sensitivity of the reactive part to changes in p is notably higher. An **increase in perimeter** results in an **augmented reactance** (X_{in}), while a **decrease in perimeter** leads to a **diminished reactance**. This observation suggests that **the perimeter of the loops can serve as a tuning parameter for the antenna**. If, for a given loop aspect ratio L/w, the antenna is not self-resonant (X_{in} ≠ 0), adjustments can be made by increasing the loop perimeter when X_{in} < 0 and decreasing it when X_{in} > 0. Consequently, **the antenna can always be tuned to a self-resonant state**, provided the ability to adjust its physical dimensions, manipulating both the loop perimeter and aspect ratio.

In the central part of Figure 5, the Voltage Standing Wave Ratio (VSWR) is presented as a function of the loop aspect ratio. The observed **variation in VSWR is predominantly attributed to changes in reactance** resulting from adjustments in the loop perimeter.

In our model, the antenna is considered in **free space** without the presence of a ground plane, and **no resistivity** has been added to the conductors, effectively eliminating power losses. This deliberate choice allows us to isolate the ideal behavior of the skeleton slot and analyze its parameters independently. In an **antenna devoid of ohmic losses**, the **resistive component** of its input impedance equals its **“radiation resistance.”** The gain of a lossless antenna is inversely proportional to the radiation resistance. If this resistance remains insensitive to changes in perimeter, **we can anticipate a corresponding insensitivity in gain**. This expectation is affirmed in Figure 5 (bottom), where the gain is depicted as a function of the loop aspect ratio for the three distinct values of loop perimeter used in the upper graphs.

#### Effect of Changing the Conductor Radius

The investigation explores the impact of changing the **conductor radius** on the input impedance, VSWR, and gain as a function of the loop aspect ratio. It is widely recognized that loops with a circumference close to one wavelength exhibit **higher reactance for thinner wire radii**. To quantify loop thickness, the loop perimeter to wire radius ratio, p/a, is commonly used. However, since the loop perimeter is held constant in our analysis, we solely vary the radius (a = 5mm) of the example model.

*Fig. 6: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different conductor radii: a = 2.5mm, 5mm, and 7.5mm.*

*Fig. 6: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different conductor radii: a = 2.5mm, 5mm, and 7.5mm.*

*Fig. 6: (Top) Input impedance, (middle) VSWR, and (bottom) gain of Skeleton Slot antenna as a function of the loop aspect ratio for three different conductor radii: a = 2.5mm, 5mm, and 7.5mm.*

At the top of Figure 6, the input impedance of the Skeleton Slot is depicted as a function of the loop aspect ratio for three different conductor radii: a = 2.5mm, a = 5mm, a = 7.5mm. As observed, **reactance increases with a thinner conductor and decreases with a thicker conductor**. This observation, combined with insights from Figure 5 in the previous section, leads to the conclusion that **the loop perimeter required for a self-resonant antenna is influenced by the wire radius**. Given a specific loop wire thickness, p/a, the exact value of p for self-resonance depends on p/a itself, so we can write **p(self-resonance) = k(p/a) λ**, where k(p/a) is near 1. While it is not within the scope of this study to generate curves illustrating the behavior of the factor k(p/a), simulation tools like AN-SOF allow for **a simulation-driven design**, a topic to be discussed in the next section.

In the middle of Figure 6, the VSWR behavior is presented, with variations predominantly attributed to changes in input reactance. The bottom graph in Figure 6 illustrates the antenna gain, demonstrating no sensitivity to the wire radius, as expected, since **the radiation resistance also remains insensitive to the wire radius**, as indicated in the top graph of Figure 6.

#### Simulation-Driven Design of a Skeleton Slot Antenna

Having established the **optimal relationships** among the **geometric parameters** of the **Skeleton Slot antenna**, conceptualized as two closely coupled loops sharing a feeding point, we can now outline a procedural approach for designing such antennas using simulation tools like **AN-SOF**.

In practical scenarios, it’s common to have a conductor or wire with a specific diameter. Therefore, our initial step will involve setting the **wire radius as a fixed parameter**, followed by running simulations with slight variations in the perimeter of each loop, starting with **p = λ** as a reference. The previously described script can be employed for this purpose, keeping the perimeter constant while adjusting the loop aspect ratio.

Following this, the subsequent step is to identify the **loop aspect ratio that maximizes gain within an acceptable VSWR range**. To illustrate this procedure, we will present example calculations for HF and VHF, namely for operating frequencies of 14 MHz and 145 MHz, respectively.

##### HF Skeleton Slot Antenna

We will take as a reference the same example presented in Sykes’ article in **“The Short Wave Magazine.”** For applications in the **HF band**, at an operating frequency of **14 MHz**, in accordance with the Sykes criterion (loop width to conductor diameter of 32:1), a 4¾-inch wire would be required—an impractical dimension. The author suggests using multiple wires (e.g., 6) to form a circular contour with the desired diameter. However, in our demonstration, we aim to show that achieving a self-resonant antenna with a 3/8″ diameter conductor is indeed feasible.

Figure 7 illustrates the results for input impedance and VSWR obtained from the script with the following input parameters:

`f = 14.0; `*// Frequency in MHz*
k = 0.987; *// Factor for loop perimeter*
p = k*299.8/f; *// Loop perimeter [m] (299.8/f = wavelength at f MHz)*
radius = (3/16)*25.4; *// Wire radius in [mm]*
S = 7; *// Number of segments per wire (it must be odd)*

*Fig. 7: (Left) Input impedance and (right) VSWR of HF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 14 MHz, for a wire radius of 3/16″.*

*Fig. 7: (Left) Input impedance and (right) VSWR of HF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 14 MHz, for a wire radius of 3/16″.*

*Fig. 7: (Left) Input impedance and (right) VSWR of HF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 14 MHz, for a wire radius of 3/16″.*

The gain is not displayed since it closely resembles the previously shown results. Figure 7 illustrates that the optimal loop perimeter is maintained with a factor **k = 0.987**, ensuring the antenna is **self-resonant at 14 MHz**. The chosen design point is **L/w = 1.825**, precisely where the **VSWR exhibits a dip**. Following this, we open the corresponding AN-SOF file (Skeleton_Ratio11.emm) and perform a frequency sweep around the central frequency of 14 MHz.

Figure 8 portrays the **VSWR as a function of frequency** for the Skeleton Slot with **L/w = 1.825**. The observation indicates an achieved **bandwidth** of almost **600 KHz** (for VSWR < 2), equivalent to **4.3%** in the 14 MHz band. The **gain** obtained is **5.5 dBi**.

If we were to choose L/w = 2.05 and conduct a frequency sweep, we would notice a reduced bandwidth of 500 KHz, signifying that making the skeleton slot slimmer is no longer advantageous, despite yielding slightly higher gain.

*Fig. 8: VSWR as a function of frequency around 14 MHz for the HF Skeleton Slot antenna with loop aspect ratio L/w = 1.825. The wire radius is 3/16″.*

*Fig. 8: VSWR as a function of frequency around 14 MHz for the HF Skeleton Slot antenna with loop aspect ratio L/w = 1.825. The wire radius is 3/16″.*

*Fig. 8: VSWR as a function of frequency around 14 MHz for the HF Skeleton Slot antenna with loop aspect ratio L/w = 1.825. The wire radius is 3/16″.*

##### VHF Skeleton Slot Antenna

For the operation of the Skeleton Slot at **145 MHz**, we maintain the same conductor diameter of 3/8″ (wire radius, a = 3/16″). In this case, by executing the script with the same perimeter factor as the one used before (k = 0.987, with the loop perimeter being p = kλ), we obtain a **negative input reactance**. As we learned in the previous sections, we will then need **to lengthen the loop perimeter** to increase the reactance and approach **resonance**.

Through several calculations (not shown here), we determined that the optimal value is **k = 1.03**, for which the results are shown in Figure 9. Therefore, a loop perimeter that is 3% longer than a wavelength is necessary in this case to obtain a self-resonant antenna in a wide range of loop aspect ratios. The chosen design point is **L/w = 1.9**.

`f = 145.0; `*// Frequency in MHz*
k = 1.03; *// Factor for loop perimeter*
p = k*299.8/f; *// Loop perimeter [m] (299.8/f = wavelength at f MHz)*
radius = (3/16)*25.4; *// Wire radius in [mm]*
S = 7; *// Number of segments per wire (it must be odd)*

*Fig. 9: (Left) Input impedance and (right) VSWR of VHF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 145 MHz, for a wire radius of 3/16″.*

*Fig. 9: (Left) Input impedance and (right) VSWR of VHF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 145 MHz, for a wire radius of 3/16″.*

*Fig. 9: (Left) Input impedance and (right) VSWR of VHF Skeleton Slot antenna as a function of the loop aspect ratio calculated at 145 MHz, for a wire radius of 3/16″.*

By opening the file corresponding to this aspect ratio (Skeleton_Ratio12.emm for L/w = 1.9) with AN-SOF and performing a frequency sweep around 145 MHz, we obtain the VSWR curve shown in Figure 10. In this case, the obtained **bandwidth** is **9.5 MHz** (for VSWR < 2), which represents **6.6%** with respect to the center frequency of 145 MHz. The **gain** obtained is **5.7 dBi**.

*Fig. 10: VSWR as a function of frequency around 145 MHz for the VHF Skeleton Slot antenna with loop aspect ratio L/w = 1.9. The wire radius is 3/16″.*

*Fig. 10: VSWR as a function of frequency around 145 MHz for the VHF Skeleton Slot antenna with loop aspect ratio L/w = 1.9. The wire radius is 3/16″.*

*Fig. 10: VSWR as a function of frequency around 145 MHz for the VHF Skeleton Slot antenna with loop aspect ratio L/w = 1.9. The wire radius is 3/16″.*

With this last example, we believe we have covered the design of Skeleton Slot antennas in a depth that perhaps has not been done before. We complete the information for the designer with a few words about the **number of segments**, set using the “S” variable in the script. Since each loop has a perimeter of **one wavelength**, the total number of segments used per wavelength is **4S**. Through comparisons with theoretical data for the loops, we have established that about 30 or 40 segments per wavelength are sufficient to reproduce theoretical data. Note that when S = 11 in the Skeleton Slot example for 20 MHz, we have 44 segments per λ, while with S = 7, we have 28 segments per λ. If you have measured data at hand, it is advisable that the number of segments be increased until the simulation model reproduces these experimental data.

#### Conclusions

In this comprehensive article, we conducted **an in-depth study of the Skeleton Slot antenna**, emphasizing its applicability across frequency bands by normalizing dimensions to the wavelength. While the theoretical analysis holds true for any frequency, practical construction considerations will be constrained by physical dimensions and installation space available in a specific frequency band.

The Skeleton Slot antenna, conceptualized as **an array of two loops with a common feed point**, was meticulously examined. We provided a **script** enabling the alteration of the antenna’s aspect ratio, generating multiple files for **bulk simulation** in AN-SOF. This facilitated the extraction of input impedance, VSWR, and gain as functions of the aspect ratio. The results highlighted the **antenna’s self-resonance** when the **perimeter of each loop is approximately one wavelength**.

We explored how the antenna’s behavior changes with variations in loop perimeter and conductor thickness. The **optimal design point** for the Skeleton Slot was identified as **the loop aspect ratio minimizing VSWR for a given conductor radius**. The **simulation-driven design methodology** can be summarized in the following steps:

- Choose the
**conductor diameter**for constructing the Skeleton Slot.

- Define the
**operating frequency**and determine the**optimal perimeter**using the provided scripts. The**self-resonance loop perimeter**is typically around**one wavelength**.

- Choose the
**design point**by selecting the**loop aspect ratio that minimizes VSWR**(or maximizes gain, depending on the objective). Conduct a**frequency sweep**to ascertain the obtained**bandwidth**.

This **simulation-driven design approach** is particularly valuable for amateur radio enthusiasts, antenna hobbyists, or RF professionals embarking on projects involving Skeleton Slot antennas. We trust that this article will serve as a valuable resource for those interested in exploring and implementing Skeleton Slot antenna designs.

About the Author

*PHYSICS Ph.D. & TECH INNOVATOR*. With over 20 years of experience in Computational Electromagnetics, I am a dedicated independent researcher specializing in radiating systems. Founder of a software company focused on innovative antenna modeling tools, I share insights on antenna simulation, theory, and numerical methods.

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Updated

##### AN-SOF 8.70: Enhancing Your Antenna Design Journey

We are thrilled to present **AN-SOF Version 8.70** 🚀. Packed with enhanced functionalities, this release is designed to empower you in the realm of antenna design. We have focused on both Input and Output Data Improvements to refine your experience and offer greater flexibility.

##### Input Data Improvements:

**✅ Solid Surfaces with Thickness:** We introduce the capability to model **solid surfaces**, not just wire grids. These solid surfaces are composed of **conductive strips** that can have a defined **thickness**. This enhancement opens the door for more realistic simulations, enabling you to model surfaces that are not infinitesimally thin. While, at present, these solid surfaces may appear as “skeletons” in the AN-SOF workspace, rest assured that we are diligently working on developing an interface to visualize them more realistically.

** ✅ Material Selection:** We’ve added a feature that allows you to select the material of wires, grids, and surfaces from a

**list of metals**. This selection automatically sets the resistivity. We’ve also included a ‘Custom’ option to set the resistivity manually.

** ✅ Substrate Dielectric Materials:** We have integrated a list of dielectric materials into the

**Substrate**ground plane option. Materials like

**FR4**,

**RT/Duroid**, and

**Rogers RO**slabs are now readily available. This means that the permittivity is automatically set for the chosen material, simplifying the design process.

##### Output Data Improvements:

** ✅ EIRP Compliance:** Compliance with electromagnetic field regulations is paramount. To assist you in this regard, we have added the

**Average and Peak Effective Isotropic Radiated Power (EIRP)**to the

**Power Budget**table. These values are presented in both Watts and dBW, and can be plotted against frequency. This feature enables you to evaluate whether an antenna model adheres to the maximum EIRP limits stipulated by regulations.

** ✅ AN-3D Pattern Enhancements:** Visualizing the 3D radiation pattern lobes just got easier. The

**AN-3D Pattern**app now includes up/down buttons, allowing you to effortlessly

**resize the antenna**relative to the radiation pattern. This feature enhances your ability to grasp the antenna’s directional properties.

** ✅ Axial Ratio:** Recognizing the importance of

**polarization analysis**, we’ve introduced the

**Axial Ratio**in both

**dB**and

**dimensionless**formats. This addition is available in

**2D rectangular and polar diagrams**,

**3D radiation patterns**, as well as in graphs depicting the

**far-field spectrum**. The Axial Ratio helps you determine the polarization of the field by providing the minor-to-major axis ratio of the polarization ellipse. Perfect circular polarization is indicated by an Axial Ratio of ±1 (with +1 representing right-handed and -1 representing left-handed polarization), while linear polarization is characterized by an Axial Ratio of 0. Additionally, AN-SOF offers both right and left circular components of the field.

** ✅ S_{11} in Decibels:** In response to the specific needs of users in microwave frequencies, we’ve added S

_{11}in decibels (representing

**return loss**) to

**plots**and

**tables**. This addition, along with the already available VSWR, offers valuable insights for

**RF analysis**of antenna bandwidth.

We are committed to continuously enhancing your experience and equipping you with the tools necessary for your success 🛠️. Please explore these features, and feel free to reach out with any questions or feedback.

##### Accurate Analysis of Solid Wheel Antennas at 2.4 GHz Using Cost-Effective Simulation

This article presents the calculated results for **return loss** and **gain** in the **2.4 GHz band** for **Wheel antennas**. By employing a simplified method to simulate planar antennas on ungrounded dielectric substrates with sufficiently high permittivity, we can accurately replicate the measured data published by the Wheel antenna manufacturer, making it a valuable resource for engineering applications. A summary of the obtained results is provided in Figure 1. Not only is this method straightforward, but it also facilitates the utilization and expansion of **cost-effective simulation software tools**.

##### Table of Contents

*Fig. 1: 2.4 GHz Wheel Antenna with return loss curves from the datasheet and AN-SOF simulation, alongside the AN-SOF model and gain pattern. Photo and measured data courtesy of Kent Electronics*

*(WA5VJB)*.#### Introduction

**Wheel antennas** derive their name from their circular configuration and the presence of spokes akin to a cartwheel. Among wheel antenna designs, **solid wheel antennas** feature flat metallic traces printed on a singular dielectric substrate. Typically, these substrates are circular and crafted from materials such as **FR4**. Solid wheel antennas are known for their compact dimensions and frequent application in **ISM (industrial, scientific, and medical) frequency bands** spanning from approximately 900 to 2400 MHz, primarily within **wireless network** contexts. In the plane of the wheel, these antennas offer **omnidirectional** coverage and exhibit **horizontal polarization**. Moreover, the radiation pattern in the plane perpendicular to the wheel closely resembles that of a magnetic dipole, taking on a distinctive donut-like shape.

This article centers on the examination of a 2.4 GHz wheel antenna manufactured by **Kent Electronics (WA5VJB)**. Our primary objective is to replicate the measured return loss data (S_{11}) for the **Big Wheel Rev B** antenna variant as provided by the manufacturer, available at **this link**.

#### Calculation Method

Given the **planar** nature of wheel antennas, fabricated on an **ungrounded dielectric substrate**, we can employ a straightforward method for simulating these microstrip antennas. This approach is outlined in the article **Simplified Modeling for Microstrip Antennas on Ungrounded Dielectric Substrates: Accuracy Meets Simplicity**, which offers a **cost-effective** means of modeling such antennas. This methodology capitalizes on the capabilities of wire antenna simulation software, such as **AN-SOF**.

The initial step involves defining the **frequency range** of interest, which, in this instance, spans from **2.2 to 2.6 GHz**. Subsequently, the antenna structure is created within AN-SOF. This process is relatively uncomplicated and entails the addition of **Line** objects to represent the straight metallic strips and **Arc** objects to replicate the curved sections of the antenna. This wheel antenna boasts a diameter of approximately **1.5 inches**. At the center of this circular structure, the feed point is positioned.

Radiating outward from the antenna’s center, there are **spokes** that connect to the arcs on the antenna’s periphery, situated above the dielectric substrate. Additionally, there are **spokes that return beneath the substrate** to close the electrical circuit of the antenna. For the sake of facilitating external connectivity, the manufacturer typically incorporates a coaxial connector at the antenna’s central point, enabling a straightforward connection to a coaxial cable.

AN-SOF, operating on the **Conformal Method of Moments**, mandates the division of wires into shorter **segments** relative to the wavelength. In the case of the antenna printed on FR4, which possesses a **dielectric constant of 4.6**, the applicable wavelength must be that of free space divided by **√(4.6) = 2.14**. Consequently, for the uppermost frequency within the specified range, 2.6 GHz, the wires have been partitioned into segments measuring 2% of the wavelength within the substrate. This aligns with the same criterion employed for planar dipoles, as detailed in **the previously referenced article**.

In accordance with the manufacturer’s datasheet, this antenna is tunable through the introduction of a **capacitor** with an approximate value of **1 pF**. This tuning capacitor has been incorporated into the model at the feed point, inserted in series with a **voltage source**. By configuring a **medium** characterized by a permittivity of 4.6, as depicted in Figure 2, and initiating the calculations with a **Ctrl + R** command, we have ascertained that **the antenna resonates at 2.43 GHz** when equipped with a **0.7 pF capacitor**, consistent with the datasheet’s specifications. Consequently, there is no necessity to determine the resonance frequency in free space, as elucidated in the simplified method. This instance exemplifies **an alignment between the effective permittivity and the substrate’s permittivity**.

*Fig. 2: Frequency and medium specification in the AN-SOF ‘Setup’ tab to calculate the antenna input impedance.*

Figure 3, included below, presents a photograph of the physical wheel antenna, sourced from the manufacturer’s website, alongside the corresponding AN-SOF model, illustrating the employed **segment density**.

*Fig. 3: Photo of the actual 2.4 GHz wheel antenna by Kent Electronics (WA5VJB) and model of the antenna in the AN-SOF’s workspace showing segmentation density. The wheel diameter is about 1.5″.*

#### Comparison with Measured Data

Initiating the **input impedance** calculation with a **Ctrl + R** command, within the scrutinized frequency span of 2.2 to 2.6 GHz and employing a medium permittivity of 4.6, we generate the S_{11} curve presented in Figure 4. This figure also overlays the measured S_{11} curve, extracted from the manufacturer’s **datasheet**. It is evident that the agreement between the simulated and measured outcomes is remarkably robust, particularly in the vicinity of the resonance frequency. It is noteworthy that this model’s **remarkable accuracy** persists despite the simplification that completely **disregards the dielectric substrate’s thickness and contour**.

*Fig. 4: Measured and Simulated Return Loss (S*

_{11}) for the ‘Big Wheel Rev B’ Antenna by Kent Electronics. Measured curve extracted from the antenna datasheet and simulated results obtained from AN-SOF.To calculate the radiation pattern, we must set the **permittivity of the medium (free space) to 1**, and then **rescale the antenna’s dimensions** by a factor of **√(4.6) = 2.14**. This can be achieved by first clicking on the “Selection Box” button within the AN-SOF toolbar. Subsequently, draw a box around the entire antenna using the mouse and then proceed to Edit > **Scale Wires** in the main menu. Here, input the scaling factor of 2.14, ensuring that you adjust the wire cross-section accordingly, as depicted in Figure 5. Furthermore, it is necessary to modify the value of the tuning capacitor, reducing it from 0.7 pF to **0.7/2.14 = 0.33 pF**.

*Fig. 5: Rescaling the antenna dimensions involves selecting the entire structure and navigating to Edit > Scale Wires in the AN-SOF main menu to enter the scaling factor.*

Once these adjustments are made, the calculation is tailored to the single frequency of **2.43 GHz**, since our primary interest lies in the radiation pattern at the resonance frequency. With this configuration, we proceed with the calculations by pressing **F11**. The outcome, illustrated in Figure 6 (left), portrays the **gain pattern** in decibels (dBi), which, as anticipated, exhibits **near-omnidirectional coverage** within the plane of the antenna. Furthermore, the polarization is predominantly horizontal, evident from the dominance of the **E _{φ}** (azimuthal) component over the

**E**(zenithal) component of the electric field, as shown in Figure 6 (right). The

_{Θ}**E**component of the electric field reveals that there are

_{φ}**three directions with radiation peaks**, corresponding to each wheel arch, indicating that the radiation pattern is not perfectly omnidirectional.

*Fig. 6: Gain pattern in dBi at 2.43 GHz (left) and electric field components on the horizontal plane (Θ = 90°, varying φ) for an input voltage of 1V at the antenna feed point.*

The calculated **peak gain** registers at **2.7 dBi**, slightly surpassing the manufacturer’s specification of 2 dBi. It is essential to acknowledge that this computed gain **does not account for all potential power losses** within the actual antenna, particularly within components such as the coaxial connector and the substrate (incorporating loss tangent). These factors have been omitted in our model, wherein solely a **resistivity** matching that of aluminum has been introduced for the metal traces.

#### Conclusions

In this article, we have introduced a simplified method for the modeling of **solid wheel antennas**, enabling the calculation of their **return loss** and **radiation pattern**. We have employed this method to simulate the performance of the Big Wheel Rev B antenna designed for the 2.4 GHz band, as provided by **Kent Electronics**. The results obtained through AN-SOF simulation have been compared with the measured data furnished by the manufacturer, and a high degree of agreement has been achieved.

This study serves as a clear demonstration of **AN-SOF’s efficacy in modeling planar antennas** that are printed on **ungrounded dielectric substrates**, specifically substrates like FR4. The method presented here not only offers **simplicity** but also demonstrates **remarkable accuracy**, highlighting its value for antenna engineers and researchers seeking **cost-effective**, reliable antenna design and analysis tools.

About the Author

*PHYSICS Ph.D. & TECH INNOVATOR*. With over 20 years of experience in Computational Electromagnetics, I am a dedicated independent researcher specializing in radiating systems. Founder of a software company focused on innovative antenna modeling tools, I share insights on antenna simulation, theory, and numerical methods.

Have a question?

Updated

##### Transmission Line Feeding for Antennas: The Four-Square Array

Since version 8.50, the **AN-SOF Antenna Simulator** has enabled us to implicitly model **transmission lines**. This capability allows us to define a transmission line by specifying its characteristic impedance, velocity factor, length, connection ports, and losses. One valuable application of this transmission line modeling technique is in modeling the feeding system of **phased arrays**. An illustrative example of a versatile phased array that utilizes transmission lines in its feeding system is the **four-square array**. This configuration consists of **four vertical elements**, each measuring **1/4-wavelength in height** and arranged in a **square pattern**. It serves as a powerful tool for both radio enthusiasts and professionals seeking a straightforward phased array for controlling the main lobe direction of the antenna radiation pattern.

The figure below illustrates the layout of the four-square array and its corresponding radiation pattern. When treating the four vertical elements as a 4-port network, calculations dictate the addition of an **18 Ohm resistor** at the base of each monopole to achieve the desired **directional radiation pattern**. Additionally, the feeding system of this array involves **six transmission lines**, meticulously configured for length and interconnections, all detailed in Chapter 8, Section “The Four-Square Array,” found in the 19th edition of the ARRL Antenna Book.

*Four-Square Array antenna configuration with a radiation pattern slice at θ = 65° (elevation 25°). The window in the bottom right corner displays settings for the transmission lines used in the antenna’s feeding system.*

Here are some of the properties that make the four-square array a compelling choice for antenna enthusiasts and professionals:

1) **Forward Gain:** 3.3 dBi, considering an average ground.

2) **Beamwidth:** The array offers a 3 dB beamwidth of 100°.

3) **Horizontal Front-to-Back Ratio:** 20 dB or better over a 130° angular range.

4) **Symmetry for Directional Switching:** Thanks to its symmetric design, the four-square array allows for directional switching in 90° increments.

By employing the feeding system outlined in this model, the four-square array showcases excellent performance characteristics, with any limitations primarily influenced by environmental factors. Furthermore, the array’s design lends itself to the implementation of a **remote switching mechanism**, enabling effortless adjustment of the array’s direction as needed.

Whether you’re a ham radio operator, a DXer, or a professional in the field, the four-square array is a fascinating option to consider for your next antenna project.

Updated

##### Simplified Modeling for Microstrip Antennas on Ungrounded Dielectric Substrates: Accuracy Meets Simplicity

In this article, we present a straightforward yet highly accurate model for calculating the input impedance and gain of **planar antennas** printed on **ungrounded dielectric substrates** with sufficiently high permittivity. We review existing models based on **effective medium theory** and introduce a novel approach that, instead of providing analytical formulas for effective permittivity, utilizes simulation software to determine antenna resonance frequencies. This approach leads to improved results compared to established effective medium formulas.

The model’s validity is confirmed through a comparison of simulation results with measured data for planar dipoles printed on **FR4 substrates**, which feature a standard dielectric constant of 4.5 and a thickness of approximately 1.6 mm—commonly used in practical applications.

##### Introduction

**“All models are wrong, but some are useful,”** a renowned saying attributed to the British statistician George E. P. Box. This statement closely aligns with a fundamental question: to what extent can one simplify a model without compromising its ability to capture what is truly essential? How minimal can a simplified model be while still retaining its “usefulness” despite its inherent inaccuracies?

We delve into this critical question within the context of accounting for the impact of dielectric substrates on planar antennas printed on **ungrounded FR4 slabs**. In this article, we present an **exceptionally simplified method** that yields results **closely aligned with measurements** for the input impedance and gain of planar antennas printed on FR4 dielectric substrates at microwave frequencies.

FR4 is a dielectric material with a standard relative permittivity of 4.5, readily available in the market as flat sheets of various thicknesses, with 1.6 mm being the most common. While these values may vary slightly in practice, the method we discuss is applicable to different variants of this material.

In the realm of microwave bands, antennas printed on dielectric substrates have gained immense popularity, primarily due to their practicality in supporting antenna structures measuring only a few centimeters. However, a pressing issue arises: **the dielectric constant’s influence on the effective length of traces** and, consequently, **the resonance frequency** of the antenna. To tackle this challenge, one could opt for full-wave methods, such as the Finite Element Method (FEM), which allow for the discretization of both the metallic antenna components and the dielectric substrate. Yet, this approach demands substantial computational resources, and the available software solutions often come with exorbitant costs.

*Fig. 1: Comparison of measured and calculated S*

_{11}for a planar Yagi-Uda antenna on ungrounded FR4 substrate. The graph displays*S*as a function of frequency, with calculations performed using HFSS software alongside results obtained through the presented simplified method, labeled as “Tony.” Measured and HFSS data sourced from Ref. [1]._{11}Figure 1 illustrates the precision attainable with our simplified method, which we will elucidate in the subsequent sections. It presents both simulation and measured results for a planar Yagi-Uda antenna designed for 2.4 GHz applications. The figure encompasses several components: the measured return loss^{1}, a photograph of the actual antenna fabricated on an FR4 epoxy dielectric, and graphs generated through HFSS software. These elements have been extracted from Ref. [1]. The red curve and radiation pattern labeled as “Tony” represent the results obtained after applying our simplified method. Here, it becomes evident that **our streamlined approach accurately predicts the resonance frequency and radiation pattern** of the planar Yagi-Uda antenna.

##### Effective Permittivity of Ungrounded Substrates

One of the most straightforward methods one might consider is replacing the dielectric substrate with an infinite medium characterized by an **effective permittivity**, denoted as ε_{eff}. In this approach, the entire antenna resides within this infinite medium. Notably, this method simplifies the model by disregarding the influence of substrate edges and corners, which typically give rise to **diffracted waves**. The effective permittivity relies on several factors, including the dielectric constant of the substrate, the medium surrounding the substrate (commonly air, treated as free space with a relative permittivity ε_{0} = 1), the substrate’s thickness, and the dimensions of the printed metal strips—essentially, the geometry of the printed antenna itself.

When a substrate incorporates a **ground plane**, typically composed of a metal plate, established theories on **effective mediums** offer closed analytical formulas for calculating the effective permittivity. In such cases, this is feasible because **the electric field predominantly concentrates just below the strips**. Utilizing techniques like **conformal mapping** or a **transmission line model**, researchers can derive closed-form expressions for the **capacitance** between the strips and the ground plane. Refer to Section “14.2.1 Transmission-Line Model” in Ref. [2] for further insights. However, when there is no ground plane, computing the effective permittivity becomes significantly more challenging.

In Ref. [3], researchers explored four analytical methods for estimating the **effective permittivity of ungrounded substrates**, ultimately concluding that two methods yield acceptable results:

**1) Insulated wire approach:** This method involves considering an infinitely long circular wire covered by a dielectric sheath with the same thickness as the planar substrate. The key is to calculate the propagation constant within the sheath, denoted as β_{s}, as illustrated in Fig. 2(a). Consequently, the effective permittivity depends solely on the dielectric constant of the sheath and its thickness. Importantly, it does not rely on the length of the strips.

**2) Coplanar strips technique:** In this method, a conformal mapping transformation is utilized to calculate the effective capacitance between two parallel strips situated within the same plane. From this capacitance, the effective permittivity can be derived. This approach considers the substrate’s permittivity, its thickness, as well as the dimensions (lengths and widths) of the strips, as depicted in Fig. 2(b). The expression for ε_{eff} incorporates a factor denoted as K_{2}/K_{1}, which relies on complete elliptic integrals of the first kind. Beyond the mathematical intricacies, it’s intuitive that as the substrate thickness, d, approaches infinity (d → ∞), K_{2}/K_{1} tends to 1, yielding an effective permittivity equal to the average of the substrate and free space permittivities, ε_{eff}(d → ∞) = (1 + ε_{r}) / 2. Conversely, when the thickness is zero (d = 0), K_{2}/K_{1} = 0, and the effective permittivity equals that of free space, ε_{eff}(d = 0) = 1. Therefore, the theory of the effective medium produces a permittivity value ranging between 1 and (1 + ε_{r}) / 2.

*Fig. 2: Two analytical methods to calculate the effective permittivity of ungrounded dielectric substrates. (a) Insulated Wire approach where the surface of the inner cylinder is analogous to the surface of a flat strip. (b) Effective Permittivity derived from Effective Medium Theory, where K*

_{2}/K_{1}is a function of the antenna dimensions (L, s, and w) and of the substrate thickness, d.Methods that rely on effective permittivity offer an enticing level of simplicity. However, when applied in practice, they often result in **a noticeable shift of the resonance frequency**, either toward higher or lower frequencies. This phenomenon was indeed observed and documented in Ref. [3], where a comparison between simulated results and measurements for a planar dipole printed on an ungrounded FR4 substrate slab demonstrated this effect.

In the realm of microstrip antennas operating at microwave frequencies, it’s quite common for the strip width to closely align with the thickness of the FR4 substrate. In such scenarios, both the “insulated wire” and “coplanar strips” methods typically yield an effective permittivity ranging from approximately 1.5 to 1.8. This specific scenario, where **strip width closely matches the substrate thickness**, forms the focal point of our discussion in this article. It’s worth noting that for many practical applications, the thickness of the substrate remains a fixed parameter and is not adjustable within our simplified model.

##### Modeling Method Based on Resonance Frequencies

Unlike the methods described in the previous section, our proposed approach does not rely on analytical formulas to determine the effective permittivity. Instead, it is rooted in **simulation techniques**, and its application comprises the following steps:

**1. Compute the Antenna’s Actual Resonance Frequency**

The method we propose is **heuristic in nature**, rooted in a fundamental observation: **in substrates with a permittivity high enough**, when the width of the printed metal strips is close to or lower than the substrate thickness (w ≈ d or w ≤ d), **the electric field in very close proximity to the strips is predominantly concentrated within the dielectric substrate**, rather than in free space (at microwave frequencies). This observation is crucial, as it implies that the propagation constant relevant for calculating the interaction between the strips is approximately β_{s} = 2π √(ε_{r}) / λ, where ε_{r} represents the relative permittivity of the substrate, and λ denotes the wavelength in free space.

This observation has paramount significance because it suggests that **the resonance frequency is primarily determined by ε _{r} itself**, rather than an effective permittivity (ε

_{eff}) as assumed in effective medium approaches. Therefore,

**our initial step involves calculating the resonance frequency of the antenna when it is immersed in an infinite medium with a permittivity equal to that of the substrate**. This frequency, denoted as

**f**, should closely align with the actual resonance frequency of the antenna on the original finite substrate, assuming a high substrate permittivity and either w ≈ d or w ≤ d. Fig. 3(a) illustrates the profile of the original antenna, while Fig. 3(b) depicts the antenna completely surrounded by a dielectric medium with permittivity ε

_{R}(ε_{r})_{r}, within which we must ascertain the actual resonance frequency.

*Fig. 3: Modeling method based on resonance frequencies to obtain the effective permittivity of a microstrip antenna on an ungrounded dielectric substrate. (a) Original microstrip antenna profile with finite dielectric substrate. (b) Antenna in an infinite medium matching substrate’s permittivity. (c) Antenna in free space. (d) Antenna in an effective medium.*

**2. Calculate the Effective Permittivity and Input Impedance**

The **radiation resistance** of an antenna relies on the **far-field radiation** it produces, where the propagation constant corresponds to that of free space (β = 2π/λ), rather than the dielectric substrate (β_{s} = 2π √(ε_{r}) / λ). In cases where power losses can be neglected, the **input resistance** of the antenna is, in fact, equal to the radiation resistance. A similar principle applies to the imaginary component of the input impedance, referred to as **input reactance**, which hinges on the balance between electric and magnetic energies in the vicinity of the antenna, where the dielectric substrate plays a role.

To compute the antenna’s input impedance, we must account for an **effective permittivity**, one that falls between the permittivity of free space and that of the substrate. Given that, from a radiation perspective, the propagation constant in free space takes precedence, our next step involves calculating **the resonance frequency when the antenna exists in free space**, devoid of the dielectric substrate, as depicted in Fig. 3(c). We denote this frequency as **f _{R}(1)**, with the “1” signifying the relative permittivity of free space.

It’s crucial to note that, since f_{R}(1) > f_{R}(ε_{r}), in order to lower the resonance frequency from f_{R}(1) to f_{R}(ε_{r}), we must divide it by the **square root of the effective permittivity**, i.e. f_{R}(1)/√(ε_{eff}) = f_{R}(ε_{r}). Consequently, the effective permittivity is calculated as **ε _{eff} = [ f_{R}(1)/f_{R}(ε_{r}) ]^{2}**. This value is the permittivity that we will employ to calculate the input impedance, treating the antenna as though it is immersed in an infinite medium with a permittivity of ε

_{eff}, as illustrated in Fig. 3(d).

**3. Rescale the Antenna to Obtain the Far Field**

Our final step involves calculating the radiated far field. In the far-field zone, only the free space propagation constant, β = 2π/λ, is relevant. When viewed from a considerable distance, the impact of the substrate on the radiation pattern is essentially **a dilation effect on the dimensions of the printed strips, scaled by a factor of √(ε _{eff})**. Consequently, to create an

**equivalent antenna in free space**without the dielectric substrate, we can increase the size of the antenna by multiplying all its dimensions—both strip lengths and widths—by the scaling factor √(ε

_{eff}). This results in an antenna of increased size

^{2}surrounded by free space, as illustrated in Fig. 4.

In this context, our primary focus commonly centers on the radiation pattern at the resonance frequency.

*Fig. 4: Method based on the effective permittivity to obtain the microstrip antenna Far-Field. (a) Original microstrip antenna on finite dielectric substrate. (b) Equivalent antenna from the Far-Field perspective, achieved by scaling antenna dimensions. Spherical coordinates, typically utilized for Far-Field calculations, are illustrated.*

##### Comparisons with Measured Data

In this section, we embark on a comparison between the data generated by the method we’ve elucidated and the measurements obtained for planar dipoles printed on FR4, sourced from Ref. [3]. From this reference, we’ve also extracted results calculated using an effective permittivity corresponding to the “insulated wire” method. This approach allows us to assess the outcomes of our proposed method in comparison to measurements and another method that relies on an effective medium.

For our calculations, we employ the **AN-SOF Antenna Simulator**, which offers the flexibility to manipulate the permittivity of the medium surrounding the antenna and define **flat strips**, not restricted to wires with circular cross-sections. The first dipole in our analysis possesses dimensions of L = 93.8 mm in length and w = 2 mm in width, with the dielectric substrate being a standard FR4 characterized by a permittivity of ε_{r} = 4.5 and a thickness of d = 1.6 mm. Our frequency range of interest spans from 1 GHz to 1.8 GHz.

To initiate the process, we set the medium’s permittivity to ε_{r} = 4.5 in the **Setup tab** > **Environment panel** of AN-SOF. We then execute a **frequency sweep**, stepping through frequencies with a granularity of 0.01 GHz to achieve three significant digits in the resonance frequency. AN-SOF operates on the **Conformal Method of Moments**, necessitating that the dipole be divided into **segments much smaller than the wavelength**. In this instance, we employ 25 segments. At the highest frequency of 1.8 GHz, each segment length constitutes approximately 5% of the wavelength within the dielectric medium. It’s important to note that while the wavelength at 1.8 GHz in free space is λ = 166.6 mm, the segments must be small relative to the wavelength within the dielectric medium, which is shorter than the free-space wavelength and can be calculated as λ/√(ε_{r}) = 78.5 mm.

After initiating the calculation by pressing **Ctrl + R** and identifying the point where **the imaginary part of the input impedance reaches zero**, we obtain a resonance frequency of **f _{R}(4.5) = 1.23 GHz**. Remarkably, with three significant figures, this result precisely aligns with the resonance frequency of the actual antenna reported by the authors of Ref. [3]. Thus,

**the error with this level of precision stands at 0%**.

The next step involves calculating **the resonance frequency of the antenna in free space**. To do this, we return to the Setup tab > Environment panel in AN-SOF and set ε_{r} = 1. Subsequently, we rerun the frequency sweep without needing to alter the number of segments for the dipole. As the wavelength nearly doubles when ε_{r} = 1, each segment now spans approximately 2% of λ. The computed resonance frequency in free space is determined to be **f _{R}(1) = 1.51 GHz**. Remarkably, this result closely aligns with the reported resonance frequency of the

**dipole in air**, which is

**1.5 GHz**, as documented in Ref. [3].

**The error with this precision is merely 0.67%**.

According to our model, the effective permittivity is now calculated as **ε _{eff} = (1.51/1.23)^{2} = 1.51**. Subsequently, we set ε

_{r}= 1.51 in the Environment panel of the Setup tab within AN-SOF and initiate the input impedance calculation either by pressing Ctrl + R or through the main menu > Run > Run Currents.

*Fig. 5: Comparison of measured and calculated S*

_{11}for a dipole on FR4 substrate (L = 93.8 mm, w = 2 mm). Results obtained using the Insulated Wire method and AN-SOF are contrasted with measured data. Additionally, the radiation pattern with a gain of 1.64 is presented. Photo and measurements sourced from Ref. [3].Figure 5 presents a photograph of the actual fabricated antenna, the measured return loss, and the results obtained using the insulated wire method, with all these elements extracted from Ref. [3]. Additionally, we’ve incorporated the curve obtained through AN-SOF^{3} into the graph of S_{11} (with a reference impedance of 50 Ohms). Notably, the predictions of the resonance frequency and the mismatch at resonance (the value of S_{11} at 1.23 GHz) generated by our proposed method **closely align with the measured data**, surpassing the results of the insulated wire method.

Furthermore, Figure 5 showcases the radiation pattern acquired **after rescaling the dipole** by multiplying its length L and width w by the factor √(ε_{eff}) = √(1.51) = 1.23, followed by conducting calculations in free space while setting ε_{r} = 1 in the Environment panel. This analysis yields a **donut-shaped radiation pattern** with a **maximum gain of 1.64**, which, to three significant figures, **coincides with the theoretical value** for a resonant dipole with a length approximately half a wavelength. We can confirm that the effective length of the dipole at 1.23 GHz is **0.47λ _{eff}**, where λ

_{eff}= λ/√(ε

_{eff}).

Now, let’s consider a second dipole, which possesses a length of 72.5 mm and the same width as the previous one, w = 2 mm. This second dipole has been obtained by rescaling the first one in order to achieve a resonance frequency of 1.5 GHz, although in the measurements, as reported by the authors of Ref. [3], a value of 1.49 GHz was obtained. By following the same procedure as the one outlined earlier, we determined **f _{R}(4.5) = 1.50 GHz** and

**f**for the scaled dipole. Consequently, the effective permittivity for this case is calculated as

_{R}(1) = 1.94 GHz**ε**. It’s worth noting that to obtain the resonance frequencies for this dipole, we had to employ 51 segments. In each instance, an analysis of result convergence based on the number of segments is imperative.

_{eff}= (1.94/1.50)^{2}= 1.67*Fig. 6: Comparison of measured and calculated S*

_{11}for a dipole on FR4 substrate (L = 72.5 mm, w = 2 mm). Results obtained using the Insulated Wire method and AN-SOF are contrasted with measured data. Additionally, the radiation pattern with a gain of 1.64 is presented. Photo and measurements sourced from Ref. [3].Figure 6 presents the results for the scaled dipole. We include a photograph of the dipole, the measured S_{11} curve, and the curve calculated using the insulated wire model, all sourced from Ref. [3]. Additionally, we’ve superimposed the curve obtained by AN-SOF onto the S_{11} graph. It’s evident that **the fit with the measured data is exceptionally satisfactory, surpassing the precision achieved through other methods**. The radiation pattern is also depicted, boasting **a gain of 1.64**, aligning with our expectations.

##### Conclusions

Recalling the introductory phrase of this article, “All models are wrong, but some are useful,” we have introduced a model here that indeed falls into the category of being “wrong.” It neglects the diffraction of waves at the edges and corners of the dielectric substrate and does not account for substrate thickness, necessitating sufficiently high permittivity and strip widths comparable to or lower than the substrate thickness.

However, it is an **exceedingly simple model**, arguably one of the simplest conceivable, and **it yields valuable results that closely align with measured data**. This model leverages the capabilities of simulation software tailored to wire antennas, such as AN-SOF, offering **a cost-effective solution for designing microstrip antennas on ungrounded dielectric substrates**.

In summary, the construction of the model involves the following steps:

- Determine the antenna’s
**actual resonance frequency**by assuming it is submerged in an infinite medium with the same permittivity as the dielectric substrate, denoted as**f**._{R}(ε_{r})

- Determine the antenna’s
**resonance frequency in free space**, in the absence of the substrate, denoted as**f**._{R}(1)

- Calculate the input impedance, from which the return loss (S
_{11}) can be derived, by simulating the antenna in an infinite medium with an effective permittivity of**ε**. The radiation pattern is obtained by rescaling the antenna in free space, multiplying its dimensions by the factor_{eff}= [ f_{R}(1)/f_{R}(ε_{r}) ]^{2}**√(ε**._{eff})

The model we’ve introduced considers **antenna dimensions**, including the lengths and widths of the printed strips, to determine effective permittivity through the resonance frequencies f_{R}(ε_{r}) and f_{R}(1). Therefore, **the influence of antenna geometry and its dimensions is inherently incorporated into our effective permittivity calculation**. In contrast, effective media methods frequently encounter challenges when dealing with complex antenna geometries, and analytical formulas are typically limited to simpler cases, such as the planar dipole. This distinction underscores the accuracy of our method, particularly in determining resonance frequencies for microstrip antennas.

##### References

[1] *“High Gain Improved Planar Yagi Uda Antenna for 2.4 GHz Applications and Its Influence on Human Tissues,”* by Claudia Constantinescu et al., Appl. Sci. 2023, 13, 6678.

[2] *“Antenna Theory Analysis and Design”* by Constantine A. Balanis, 4th Edition, Wiley 2016.

[3] *“A Robust Method of Calculating the Effective Length of a Conductive Strip on an Ungrounded Dielectric Substrate”* by M. Kanesan, D. V. Thiel, and S. O’Keefe, Progress In Electromagnetics Research M, Vol. 35, 57–66, 2014.

##### Footnotes

- In microstrip antennas, it is customary to present the reflection coefficient at the feed port, denoted as S
_{11}, rather than the input impedance, as S_{11}is the more frequently measured parameter. In technical terms, S_{11}expressed in decibels is often referred to as “return loss.” However, it’s important to clarify that the formal definition of return loss is the negative of S_{11}in decibels.

↩︎ - Instead of rescaling the antenna dimensions and reiterating the calculations for obtaining the radiation pattern, a simpler approach is to keep the dimensions constant and adjust the frequency by multiplying it by √(ε
_{eff}). This method allows us to leverage the calculations already performed in free space to determine the resonance frequency f_{R}(1).

↩︎ - AN-SOF provides us with the return loss based on its technical definition, which happens to be the negative of S
_{11}when expressed in decibels. To represent S_{11}as a function of frequency in the graphs, we reversed the sign of the return loss values in decibels obtained from AN-SOF.

↩︎

About the Author

*PHYSICS Ph.D. & TECH INNOVATOR*. With over 20 years of experience in Computational Electromagnetics, I am a dedicated independent researcher specializing in radiating systems. Founder of a software company focused on innovative antenna modeling tools, I share insights on antenna simulation, theory, and numerical methods.

Have a question?

Updated

##### Fast Modeling of a Monopole Supported by a Broadcast Tower

Are you fascinated by antenna design and eager to learn new techniques? Look no further! Our latest video tutorial, **Fast Modeling of a Monopole Supported by a Broadcast Tower**, is here to take your skills to the next level. 🚀

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Updated

##### Linking Log-Periodic Antenna Elements Using Transmission Lines

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Updated

##### Introducing AN-SOF 8.50: Enhanced Antenna Design & Simulation Software

We’re thrilled to introduce **AN-SOF version 8.50**, a comprehensive update that takes your antenna design and simulation experience to new heights. Let’s explore the remarkable new features:

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✅ **Expanded Compatibility:** AN-SOF now supports **current sources** defined via the **EX 6** command of NEC-4.2 when **importing NEC files**. Seamlessly import your preferred NEC files and take full advantage of AN-SOF’s capabilities.

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##### Get Ready for the Next Level of Antenna Design: AN-SOF 8.50 is Coming Soon!

##### Introducing AN-SOF 8.50: an exciting update that takes antenna design and simulation to new heights.

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✅ **Valuable Insights:** Gain valuable insights into your designs with integrated **Front-to-Rear** and **Front-To-Back ratios** as a function of frequency in the Power Budget table. This comprehensive analysis empowers you to optimize your antenna’s performance.

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✅ **Improved Far-Field Computation:** Enjoy precise results with AN-SOF’s advanced far-field computation, ensuring accurate **Directivity** and **Efficiency** calculations across various scenarios. Even with real ground planes or infinite substrate slabs, AN-SOF delivers reliable and accurate results.

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For experienced users, our comprehensive **Knowledge Base** provides in-depth descriptions of AN-SOF’s features and functions. Explore numerous **examples** and **articles** that validate AN-SOF’s results through comparisons with theoretical and experimental data. Additionally, our blog **Antennas and Beyond!** offers **advanced articles** for readers with a more in-depth background.

If you’re interested in antenna theory and numerical methods applied to **Computational Electromagnetics**, our blog categories **Antenna Theory** and **Numerical Methods** will provide valuable insights.

Furthermore, AN-SOF allows you to program **scripts** in the language of your choice to generate a sequence of files containing descriptions of an antenna model with variable geometric parameters. You can then run a massive calculation of all these files using the **Run Bulk Simulation** command. Explore examples of scripts in the language of **Scilab**, the open-source counterpart of the renowned Matlab, at **this link**.

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##### AN-SOF Implements James R. Wait Theory for Ground Losses of LF/MF Radio Masts

AN-SOF has incorporated an innovative method based on the **James R. Wait theory** to accurately compute the ground losses of LF/MF radio masts. The correct calculation of **power lost in the ground** is crucial for determining the **radiation efficiency** of monopole antennas. To minimize energy losses, radial wire ground screens are typically installed.

To illustrate the efficacy of this implementation, a simulation has been carried out on an antenna tower 1/4 wavelength in height, placed over an average soil with a conductivity of 0.005 S/m and a dielectric constant of 13. The simulation incorporated 60 buried radials and calculated the E-field at 3 MHz as a function of distance for an antenna input power of 1 kW.

To activate this calculation in AN-SOF for LF and MF band frequencies, users need to select the **Sommerfeld-Wait/Asymptotic** or **Radial wire ground screen** options as the real ground plane in the **Setup > Environment** panel. Here is an article detailing the validation of this model: **Enhanced Methodology for Monopoles Above Radial Wire Ground Screens**.

##### Who was James R. Wait?

James R. Wait was a Canadian engineer known for his academic qualifications and prolific contributions to electromagnetic propagation engineering. He was elected as a member of the National Academy of Engineering in 1977 and authored numerous papers and books. Born in Ottawa in 1924, Wait received his BS and MS in engineering physics and his PhD in electrical engineering from the University of Toronto. He held various research positions worldwide and became a professor at the University of Arizona. Those interested in learning more about Wait’s life may refer to the article *“James R. Wait—Remarkable Scientist”* by Ernest K. Smith, IEEE Transactions on Antennas and Propagation, Vol. 48, No. 9, pages 1278-1286, September 2000″.

Updated

##### Overcoming 7 Limitations in Antenna Design: Introducing AN-SOF’s Conformal Method of Moments

**AN-SOF Antenna Simulation Software** accelerates antenna design by addressing **seven limitations** encountered in the traditional **Method of Moments (MoM)**. The accompanying image provides a visual representation of the challenges overcome by AN-SOF.

Utilizing a **Conformal Method of Moments (CMoM)** with an **Exact Kernel**, AN-SOF overcomes these limitations, enabling accurate modeling and analysis of antennas with complex geometries. This includes high gain antennas with **grid reflectors**, broadcast towers with **elevated radial wires** close to the ground, **thick** dipoles, **stepped** cylindrical antennas, **curved antennas** such as helices, spirals, and loops, as well as any design involving curved, short, and closely positioned wires.

By overcoming these limitations, AN-SOF empowers engineers and researchers to design antennas with enhanced accuracy and reliability. The software not only improves the modeling and analysis of complex antenna geometries but also opens up new possibilities for antenna optimization and design across a wide range of applications.

##### Explore the Cutting-Edge World of AN-SOF Antenna Simulation Software!

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##### Linear Antenna Theory: Historical Approximations and Numerical Validation

This article provides a comprehensive review of the theoretical approximations to the current distribution on **linear antennas** that were analytically derived during the first half of the 20th century. While the advancements in computing and numerical calculation methods have enabled higher precision, the historical theoretical results continue to serve as vital references for **validating calculation methods** implemented in algorithms. By examining these approximations, researchers can ensure the accuracy and reliability of modern numerical techniques used in antenna analysis and design.

##### Historical Theoretical Results: Approximations to Current Distribution

In 1956, Ronold W. P. King published his monumental work titled *“The Theory of Linear Antennas,”* which laid the foundation for understanding linear antennas and became a benchmark for future research. This section explores the significant contributions of King’s work, specifically focusing on the first four approximations to the current distribution on cylindrical antennas derived through an iterative method.

**0th-Order Approximation > Perfect Sinusoid for Infinitely Thin Antennas**: The 0th-order approximation considers infinitely thin antennas (wire radius = 0) with a delta-gap source (zero gap width between the antenna terminals at the source position). This approximation involves a perfect sinusoidal current distribution.

**1st-Order Approximation > Accounting for Finite Radius Effect**: The 1st-order approximation incorporates the finite radius effect (wire radius > 0) into the current distribution calculation. In this case, it is necessary to incorporate an additional term to the perfect sinusoidal function in order to obtain the 1st-order current distribution.

**2nd-Order Approximation > Controversies Surrounding Finite Gap at the Source**: The 2nd-order approximation takes into account the finite gap at the source position, which has historically sparked controversies and debates.

**3rd-Order Approximation > Considering Feedline Effect**: The 3rd-order approximation considers the feedline effect, which accounts for the boundary conditions at the feeding point resulting from the connection of a transmission line to feed the antenna.

##### Current Distribution Along a Half-Wave Dipole

To illustrate the aforementioned approximations, this section focuses on the current distribution in amplitude along a center-fed half-wave dipole. Figure 1 provides a graphical representation of the normalized current, i(s), as a function of position in wavelengths, s/λ. It should be noted that the actual current distribution exhibits a sign change in its derivative, ∂i/∂s (the electric charge), at the dipole center due to the excitation source.

*Fig. 1: Normalized amplitude current distribution along a center-fed half-wave dipole. The figure illustrates the 0th, 1st, and 2nd order approximations, highlighting the discontinuity of the current derivative at the feed point.*

**0th-Order Approximation > Perfect Sine Function with Zero Derivative:**Figure 1 demonstrates that the 0th-order approximation, representing a wire radius of 0, yields a perfect sine function. Consequently, the derivative at the source position is zero, ∂i/∂s = 0. However, King’s analytical solution results in a finite input impedance of 73.1 + j42.5 Ω, which has been widely accepted and corroborated by other methods.

**1st-Order Approximation > Finite Wire Radius and Divergent Input Impedance:**The 1st-order approximation considers a finite wire radius and exhibits an infinite derivative, ∂i/∂s = ∞, at the source position. This singularity arises from the zero gap at the antenna terminals and has generated extensive debates throughout the history of linear antennas. As a consequence, the input impedance diverges.

**2nd-Order Approximation > Finite Source Gap and Converging Input Impedance:**Incorporating a finite source gap, the 2nd-order approximation yields a finite derivative at the source position. As a result, the input impedance converges to a finite value, dependent on the dipole wire thickness and the separation between its feeding terminals.

**3rd-Order Approximation > Consideration of Transmission Line Feed:**Although not visually distinguishable from the 2nd order on a graph, the 3rd-order approximation accounts for the characteristic impedance of the transmission line at the feed point. This effect, though small, can be accurately calculated using the**Method of Moments >**with an**exact Kernel >**.

##### Validating Numerical Methods: Impedance Convergence

Validating numerical methods is a critical step in ensuring their accuracy, achieved by examining the limiting cases predicted by theory. As demonstrated, the 0th-order input impedance (wire radius = 0) of a center-fed half-wave dipole is determined to be **73.1 + j42.5 Ω**. Consequently, this value should serve as a **horizontal asymptote** for the input impedance when the dipole length-to-radius ratio tends to infinity.

Figure 2 presents simulation results obtained using **AN-SOF**, which utilizes the **Method of Moments with an exact Kernel >**. The figures from King’s book illustrate the antenna terminals in detail, where a radial transmission line was considered to account for 3rd-order effects. **Notably, the calculated input impedance indeed converges to the theoretical value as predicted.**

*Fig. 2: Simulation results using the Method of Moments with an exact kernel, depicting the input impedance of a center-fed half-wave dipole as a function of its length-to-radius ratio. The figure also includes a comparison with the theoretical asymptotes. The cylindrical antenna illustrations are taken from King’s book.*

For a more comprehensive investigation into the impedance convergence of cylindrical antennas, a detailed study on the validation of AN-SOF can be accessed through **this link >**.

##### Conclusion

This article has reviewed the historical approximations of current distribution on linear antennas as presented in Ronold W. P. King’s book. The four approximations, namely the 0th, 1st, 2nd, and 3rd-order approximations, have been thoroughly examined. These approximations progressively refine the theoretical model of a cylindrical antenna by considering factors such as the finite wire radius, the finite gap at the feed point, and the incorporation of the connected transmission line.

Moreover, the article has highlighted the importance of **numerical validation** in establishing the reliability of modern methods. The validation process involved comparing the numerical results to the limiting cases predicted by theory. Through the **AN-SOF** simulation, which utilizes the **Method of Moments with an exact kernel >**, the calculated input impedance successfully demonstrated convergence to the theoretical values.

##### Further Reading

For further reading, we highly recommend the book *“The Theory of Linear Antennas”* by Ronold W. P. King, Harvard University Press, 1956. This seminal work provides a comprehensive understanding of linear antennas and serves as a benchmark for research in the field. In the paper *“Currents, Charges, and Near Fields of Cylindrical Antennas”* by R.W.P. King and Tai Tsun Wu, Radio Science Journal of Research NBS/USNC-URSI, Vol. 69D, No. 3, pp. 429-446, March 1965, the authors compare the sinusoidal current distribution with measured data and identify the need for an additional term in the model. To delve deeper into the source gap problem, we refer to *“The Influence of the Width of The Gap Upon The Theory of Antennas”* by L. Infeld, Quarterly of Applied Mathematics, Vol. V, No. 2, pp. 113-132, July 1947. This study provides valuable insights into the effects of gap width on antenna theory.

About the Author

*PHYSICS Ph.D. & TECH INNOVATOR*. With over 20 years of experience in Computational Electromagnetics, I am a dedicated independent researcher specializing in radiating systems. Founder of a software company focused on innovative antenna modeling tools, I share insights on antenna simulation, theory, and numerical methods.

Have a question?

Updated

##### Assessing Antenna Directionality: Front-to-Rear vs. Front-to-Back Ratio

⚠ Are you aware of the importance of calculating the front-to-rear ratio of your antenna? ⚠

✅ It is a crucial factor in determining the directional performance of your antenna system. AN-SOF can display this ratio in various forms, such as in **polar radiation patterns**, **tables**, and **graphs**, as a function of frequency.

✅ Discover the difference between the **front-to-rear** and **front-to-back** ratios in **this article >**.

*Front-to-Rear and Front-to-Back ratios can be displayed in polar plots and as a function of frequency in AN-SOF.*

Updated

##### Wave Matching Coefficient: Defining the Practical Near-Far Field Boundary

In this article, we explore the concepts of **absolute wave impedance** and **wave matching coefficient (WMC)** as practical alternatives to determine a useful boundary between the near and far field regions of an antenna. By utilizing these measures, we gain a better understanding of wave propagation as a function of distance from the source antenna, employing a decibel scale that enables clearer visualization of significant changes in wave impedance. As a general guideline, a **WMC value of 20 dB** proves to be an appropriate threshold for distinguishing between the **near and far field zones**. Through examples involving both elementary and large-scale antennas relative to the wavelength, we observe that the 20 dB boundary is consistently located at a distance of λ/3 for elementary antennas, while it takes on an irregular and non-spherical shape for antennas of comparable size or greater than the wavelength.

##### Introduction

The determination of the far-field region of an antenna has been a topic extensively discussed in books and texts on antennas for nearly a century. However, it continues to spark debates even today. Identifying the regions surrounding an antenna is crucial for various applications, such as near-field measurements in an anechoic chamber to predict the far-field radiation pattern or in electromagnetic compatibility (EMC) to optimize shielding in the near-field region, minimizing interference.

Based on the observation of **elementary electric or magnetic dipole fields**, three distinct regions can be identified in terms of the distance, r, from the dipole:

1) The **reactive near-field region**, where terms proportional to 1/r^{3} predominate.

2) A **transition region or Fresnel zone**, where terms proportional to 1/r^{2} predominate.

3) The **far-field region or Fraunhofer zone**, where terms proportional to 1/r predominate.

There is also a two-region model where the reactive near-field and the Fresnel zone are considered as part of the same near-field region. When antennas are more complex than elementary dipoles, it becomes nontrivial to identify the electromagnetic field zones. It is important to note that the definition of a boundary between the near-field and far-field regions is always **arbitrary** and depends on the acceptable **margin of error** in practice. There is no sharp edge or discontinuity between these regions; instead, the electromagnetic field initially behaves as a **quasi-static field** near the radiation source and gradually transforms into a **Transverse Electromagnetic (TEM) wave** with a spherical wavefront as the distance increases.

##### The Traditional Boundary Between Near Field and Far Field

In most textbooks, we can find that the far-field region begins at a distance from the antenna given by **2D ^{2}/λ**, where

**D**is the maximum dimension of the antenna and

**λ**is the wavelength. This boundary between regions works reasonably well for cases of electrically large antennas (D >> λ). However, there are many exceptions to this rule, such as in the case of parabolic antennas where this boundary must be extended twofold. For electrically small antennas (D << λ), the boundary between regions is located at

**λ/(2π)**regardless of the antenna size.

These calculations are based on placing the observation point of the field far enough away so that the antenna remains within a **sphere**, which, as it expands, approaches a **spherical wavefront** in the far-field zone. This allows us to develop the phase of the Green’s function of the problem in a Taylor series with respect to distance and retain the first terms. Depending on the number of terms retained, the different field zones will be delimited. For an antenna that is large compared to the wavelength, if we move away to enclose it within a sphere, we may have already moved too far and find ourselves in the far-field region, missing the details of what happens in the near field and where a boundary between both zones could be defined. Hence, these analytical formulas fail in many cases.

##### Definitions of Wave Impedance

Instead of using a single formula for all cases, which introduces a high level of uncertainty, a more convenient criterion for separating the near-field and far-field regions is to calculate the so-called **wave impedance**, which is calculated as the ratio of the electric and magnetic fields. Since fields are vectors, we can compute the ratio between their components. For example, when a wave is **vertically polarized**, at the wavefront, we consider the vertical component of the electric field, E_{v}, and the horizontal component of the magnetic field, H_{h}, omitting components in the direction of propagation (which rapidly diminish with distance from the emission source). We define the wave impedance as Z_{w} = E_{v}/H_{h}. This ratio involves two complex quantities with real and imaginary parts, so the wave impedance has both magnitude and phase. By decomposing the wave at the wavefront into its **right-handed** circular polarization components, E_{R} and H_{R}, and **left-handed** circular polarization components, E_{L} and H_{L}, we can define two complex wave impedances: a right-handed impedance, Z_{R} = E_{R}/H_{R}, and a left-handed impedance, Z_{L} = E_{L}/H_{L}.

Regardless of the chosen definition of wave impedance, it will have the following properties:

• Z_{w} is a function of the **distance** from the antenna measured in wavelengths, r/λ, and the **observation direction,** given by two angles (zenith and azimuth) when using spherical coordinates.

• In any chosen direction, as the distance increases (r >> D and r >> λ), Z_{w} tends to **377 Ω** in free space.

Therefore, in the far-field region, the wave impedance approaches the **intrinsic impedance of the medium,** which is 377 Ω for free space. For an ideal lossless and isotropic medium, the intrinsic impedance is given by Z_{i} = √(µ/ε), where µ is the magnetic permeability and ε is the dielectric constant. For vacuum, this value is approximately rounded to 377 Ω, often approximated as 120π Ω for convenience, with three significant digits.

##### Absolute Wave Impedance

The problem with defining wave impedance in terms of components of the **E** and **H** vector fields is that we have more than one definition, as we have just seen, and these definitions depend on the chosen coordinate system or frame of reference. A figure that allows us to identify the field regions should satisfy the following conditions:

• It should be calculated based on **observables**, i.e., quantities that can be **measured in practice**.

• It should be **independent of the frame of reference**, i.e., invariant under a coordinate transformation.

• It should be obtainable for **any polarization of the field**, even when it is unpolarized, as is the case when uncorrelated fields with random phases are summed.

A simple figure that meets these three requirements is what we will call the **absolute wave impedance**, which is given by the ratio of the root mean square (rms) values of the **E**(**r**) and **H**(**r**) vector fields,

Z_{w}(**r**) = E_{rms}(**r**)/H_{rms}(**r**), at each point **r** in space.

These are observables that are independent of the coordinate system. For example, when transforming from Cartesian to spherical coordinates, we have E_{rms} = |**E**| = √(|E_{x}|^{2} + |E_{y}|^{2} + |E_{z}|^{2}) = √(|E_{r}|^{2} + |E_{Θ}|^{2} + |E_{φ}|^{2}), where E_{x}, E_{y}, E_{z}, E_{r}, E_{Θ}, E_{φ} are complex components (if working with peak values, they should be multiplied by 1/√(2) to obtain rms values). The same applies to the rms value of the magnetic field, H_{rms} = |**H**|.

In general, E_{rms} and H_{rms} are invariant under any coordinate transformation. Therefore, Z_{w}(**r**) is defined at every point **r** in space outside the antenna surface because it is a quantity that can be calculated at any point **r** based on the measured fields, E_{rms}(**r**) and H_{rms}(**r**). Since this definition disregards the phase, it is also useful for unpolarized waves. Disregarding the phase of the wave impedance is not an issue since we will need to compare it with a real value, equal to 377 Ω (with zero phase), to determine if we are in the far-field zone.

##### Wave Matching Coefficient

Analogous to the definition of “return loss” used for transmission lines, if 377 Ω were the characteristic impedance of a line, we can define a coefficient in **decibels** that measures how well the wave impedance is “matched” to the intrinsic impedance of the medium. We will call this coefficient the **Wave Matching Coefficient (WMC)**, given by

where Z_{w} = E_{rms}/H_{rms} is the **absolute wave impedance**. We will not use the term “return loss” because in the propagation mechanism we are considering, there is no loss or wave returning by reflection to the source that originated it.

As Z_{w} approaches 377 Ω, the WMC always increases. In a transmission line, a return loss of 20 dB implies that 99% of the power is transmitted and 1% is reflected. Although we don’t have a reflection mechanism here, we could adopt the same tolerance and consider the limit of **WMC = 20 dB** as the **boundary between the near-field and far-field regions**. If this limit proves to be too strict or too lenient for a particular practical application, we are free to choose another boundary according to the acceptable tolerance. From an engineering standpoint, we would recommend placing the boundary between the near-field and far-field regions above WMC = 10 dB. In the examples we will consider next, we will use the 20 dB boundary.

##### Examples with a 20 dB Boundary

Figure 1(a) shows the absolute wave impedance as a function of distance for an **elementary electric dipole** and an **elementary magnetic dipole**, while Figure 1(b) shows the corresponding WMC. The direction along which the distance r/λ varies is perpendicular to the axis of the dipoles, where the far-field reaches its maximum value. In both cases, the 20 dB boundary is practically at r/λ = 0.33 ≈ 1/3. Additionally, we could divide the near-field region into two parts, one below the maximum at r/λ = 0.1 and one above.

*Fig. 1: (a) Absolute wave impedance for elementary electric and magnetic dipoles as a function of distance along the direction of maximum radiation. (b) The corresponding WMC and the separation between the near-field and far-field zones at a threshold of 20 dB.*

Figure 2(a) displays the absolute wave impedance for an **elementary electric dipole** along a direction at 45° from its axis, where the **power density drops 3 dB** compared to its maximum value, as well as the curve obtained in the previous Figure 1(a) along the direction at 90° from the dipole’s axis. Figure 2(b) shows the corresponding WMC. In this way, we observe the wave impedance and WMC for the directions that define the radiation maximum and the **beamwidth** of an elementary dipole.

We can observe that the boundary between the near-field and far-field regions moves away from the dipole when observed from a direction other than that of maximum radiation, and a **transition zone opens** around r/λ = 1/3.

*Fig. 2: (a) Absolute wave impedance for an elementary electric dipole as a function of distance along the direction of maximum radiation and the direction where the power density drops by 3 dB. (b) The corresponding WMC and the separation between the near-field and far-field zones at a threshold of 20 dB. A transition zone is opened.*

As an example of an antenna with a size of multiple wavelengths, Figure 3 presents the results for a **7-element Yagi-Uda antenna**, optimized to provide maximum front-to-back ratio. Figure 3(a) shows the absolute wave impedance in the direction of maximum radiation, perpendicular to the antenna elements, and in the direction where the maximum power density drops by half (-3 dB) and defines the edge of the beamwidth. Figure 3(b) shows the corresponding WMC curves.

In this case, the **20 dB boundary also shifts to a greater distance** when the observation direction is different from the direction of maximum radiation, similar to the elementary dipole. Additionally, the **transition zone** between the near-field and far-field zones is approximately half a wavelength.

We can see that the wave impedance can reach very high values, so representing **the WMC provides a more convenient decibel scale for comparing large and small values.**

*Fig. 3: (a) Absolute wave impedance for a 7-element Yagi-Uda antenna as a function of distance along the direction of maximum radiation and the direction where the power density drops by 3 dB. (b) The corresponding WMC and the separation between the near-field and far-field zones at a threshold of 20 dB. A transition zone of approximately half a wavelength is opened.*

Another interesting example of a long-wavelength antenna is the **axial mode helical antenna**. Figure 4 shows the results for a left-handed helical antenna with a diameter of 0.3λ, a pitch of 0.22λ, and 10 turns, resulting in a total length of 2.2λ from end to end. The helix reflector, which is necessary for it to operate in axial mode (with maximum radiation along the helix axis), has a diameter of 0.95λ.

Figure 4(a) shows the wave impedance along the axis of the helix, from the base, passing through the **interior of the helix** until it exits. It also shows the wave impedance along a direction corresponding to the -3 dB beamwidth edge. Figure 4(b) shows the corresponding WMC results. Here, too, a displacement of the boundary between the near-field and far-field regions can be observed. However, in this case, the displacement is opposite to the previous cases since the field inside the helix is always “close.” We can see that as we traverse the interior of the helix along its axis, **the boundary between the near and far-field zones begins at r/λ = 2.3**, which is just a distance of 0.1λ above the **top of the helix located at 2.2λ**. This is logical since the interior of the helix behaves like a **waveguide**.

*Fig. 4: (a) Absolute wave impedance for a helical antenna in axial mode as a function of distance along the helix axis, passing through its interior, and the direction where the power density drops by 3 dB. (b) The corresponding WMC and the separation between the near-field and far-field zones at a threshold of 20 dB. The near-field zones inside and outside the helix are indicated.*

From these examples, we can deduce that **the three-dimensional boundary between the near-field and far-field regions does not have a spherical or regular shape around antennas of a size comparable to or larger than the wavelength**. The absolute wave impedance, and especially the WMC, allows us to determine where the far-field region begins in each direction of space. We could then choose the farthest boundary resulting from this analysis as the radius of a limiting sphere from which the far-field region begins in all directions.

##### Conclusions

In this article, we have presented the concepts of **absolute wave impedance** and **Wave Matching Coefficient (WMC)** as alternatives for determining a practical boundary between the near and far field regions of an antenna. The WMC, in particular, enables a better visualization of the evolving wave propagation as a function of distance from the originating antenna, providing a decibel scale that enhances the observation of significant variations in wave impedance. As a general guideline, we have observed that a **WMC value of 20 dB defines an appropriate threshold for separating the near and far field zones**.

Through examples involving elementary antennas and antennas of significant size relative to the wavelength, we have observed that the 20 dB boundary remains at a distance of λ/3 for elementary antennas. However, for antennas of comparable or larger size than the wavelength, the boundary assumes an irregular and non-spherical shape. In this particular case, **the radius of a spherical boundary** separating the near-field and far-field regions around the antenna will be determined by **the maximum distance observed in all angular directions of space where the WMC reaches the 20 dB limit.**

##### Further Reading

The traditional separation between field regions is explained in detail in section “4.4 Region Separation” of the renowned book *“Antenna Theory, Analysis and Design”* by Constantine A. Balanis, 4th edition, 2016, John Wiley & Sons. For a compelling analysis utilizing wave impedance, refer to *“Near Field or Far Field?”* by Charles Capps, EDN, Design Feature, Aug. 16, 2001, pp. 95-102. A comprehensive examination can also be found in the paper *“Where Does the Far Field of an Antenna Start?”* by M. Abdallah, T. Sarkar, M. Salazar-Palma, and V. Monebhurrun, published in IEEE Antennas & Propagation, Vol. 58, Issue 5, Oct. 2016, pp. 115-124.

About the Author

*PHYSICS Ph.D. & TECH INNOVATOR*. With over 20 years of experience in Computational Electromagnetics, I am a dedicated independent researcher specializing in radiating systems. Founder of a software company focused on innovative antenna modeling tools, I share insights on antenna simulation, theory, and numerical methods.

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Updated

##### AN-SOF 8.20 Passes IEC Validation

The latest version **8.20** of the **AN-SOF calculation engine** has successfully completed validation for accuracy in accordance with the standard **IEC FDIS 62232**.

This standard outlines guidelines for calculating radio-frequency field strength and specific absorption rate (SAR) in the vicinity of radiocommunication base stations (RBS) to assess the potential human exposure.

Specifically, the validation of the AN-SOF calculation engine was conducted for the **“Antenna with Dipole Radiators”** in the aforementioned standard. This significant achievement enhances the reliability of the **AN-SOF Antenna Simulator**.

Updated

##### AN-SOF Mastery: Adding Elevated Radials Quickly

Are you seeking to expedite the process of incorporating **elevated radial wires** into your antenna model created using the AN-SOF Antenna Simulator? Elevate your proficiency in AN-SOF with this instructional video tutorial, spotlighting **two efficient methods** for swiftly integrating radial wires in a monopole antenna model.

The **first method** is exceptionally suitable when dealing with **a limited number of radial wires**. It entails the precise specification of each wire’s parameters using the **Start – Direction – Length** option. This technique facilitates the step-by-step drawing of radial wires, allowing for rapid adjustments by altering orientation angles in the horizontal plane.

For projects demanding the inclusion of a **substantial number of radial wires**, the **second method** shines as the most efficient choice. This approach entails the creation of a **disk** with a predetermined quantity of radial wires and subsequently eliminating its outline.

Uncover the intricacies of applying these techniques to rapidly incorporate elevated radial wires into your **monopole** or **broadcast tower** antenna models within the accompanying video tutorial:

Updated

##### Upgrade to AN-SOF 8.20 – Unleash Your Potential

**AN-SOF has released its latest version, 8.20**, which brings significant enhancements to improve the software’s accessibility and performance.

The new release builds on the improvements made in version 8, with a focus on two key areas. Firstly, it provides an intuitive way to access data, making it **easier than ever before to interact with graphs and results**. Secondly, the calculation engine has been improved, ensuring **greater accuracy and reliability**. In addition, math libraries are now embedded in the software, eliminating the need to install external packages.

With AN-SOF version 8.20, users can expect a faster, more powerful, and more accessible experience.

##### Modeling a J-Fed 5-Element Collinear Antenna for the 2 m Band

If you’re looking for a high-performing omnidirectional antenna, consider this design. It boasts **5 collinear radiant elements** that are connected by **phasing coils**. The feed point is conveniently located at the base of the antenna on a **J dipole**. By adjusting the position of the feed point along the J, you can minimize the SWR. At the top end of the antenna, you’ll find a **small loop** that serves as a tie point, measuring just 1/2″ in diameter. The total length of the antenna is 295″. Each coil features 32 turns and has a diameter of 5/8″.

To accurately model this design, we should use simulation software that is capable of modeling **curved wire segments and small, closely-spaced wires**. With AN-SOF, you can use the “Helix” object to perfectly model each coil. By using only one segment per turn, this model demonstrates the efficiency of AN-SOF in simulating complex designs.

*Modeling a J-Fed 5-Element Collinear Antenna for the 2 m Band.*

Updated

##### New Interactive User Guide

Explore **AN-SOF’s interactive User Guide >** for comprehensive guidance on using the Antenna Simulation Software.

##### AN-SOF 8: Elevating Antenna Simulation to the Next Level

✅ New **Plots tab** where we can quickly see the input impedance, VSWR, gain, Front-to-Rear, and Front-to-Back ratios as a function of frequency, with various visualization controls (grids, points, markers, etc.).

✅ The different parts of **tapered wires** are identified with alternating colors to better distinguish the wires of which they are composed.

✅ Click on a column header in the **Results tab** to display a plot immediately.

✅ New **3D Rotation button** on the toolbar to easily rotate the view by moving the mouse.

✅ **Combo-boxes now have “memory”.** Select the frequency or angles to display polar diagrams and the next time you do so the same values will be pre-loaded.

✅ Users who use the comma as the decimal symbol can now import NEC files that use the period as the decimal separator. Go to Tools > Preferences > Options and check the option **“The comma is set as the decimal symbol”.**

✅ **New Preferences window** in **AN-Smith** to change the display of graphics in the Smith chart (line width, fonts, background, points).

💡 **TIP 1:** Double-click on the AN-SOF or AN-3D Pattern workspace to center the view in the window.

**💡 TIP 2:** Check “Show Points” in the Preferences window of AN-Polar to see the points on the radiation pattern lobes.

💡 **TIP 3:** Check “Equal Scales in 2 Plots” in the Preferences window of AN-XY Chart to display the left and right axes with the same scale.

Enjoy it!

##### Simulating the Ingenious Multiband Omnidirectional Dipole Antenna Design

Presenting a formidable challenge, the creation of an omnidirectional antenna with the capability to function across diverse frequency bands from a singular feed point demands innovative engineering.

The image below illustrates a simulated **multiband dipole**, which is a composite of **five closely spaced parallel dipoles**. Converging at the antenna’s center, these dipole extremities meet at **a shared feed point**. Each dipole spans approximately half a wavelength, yielding resonance frequencies of **3.7, 7.05, 14.2, 18.1, 21.2, and 28.5 MHz.** The radiation pattern animation showcases the emergence of a donut-shaped omnidirectional pattern at lower frequencies, evolving into a more intricate lobed pattern as frequencies ascend.

This exemplifies a scenario where the indispensable utility of a simulation tool like **AN-SOF** comes to the forefront. AN-SOF exhibits an exceptional ability to simulate **closely spaced wires**, a feat that is often challenging. The remarkable capability of AN-SOF can be attributed to its implementation of a **calculation method** that triumphantly overcomes the **seven key limitations** prevalent in traditional methods. These intricacies are thoughtfully detailed in **this article**, highlighting the innovation driving AN-SOF’s prowess.

*Multiband dipole antenna showcasing shared feeding point and dynamic radiation pattern shift.*

Updated

##### The Loop on Ground (LoG): A Compact Receiving Antenna with Directional Capabilities

The **Loop on Ground (LoG)** is a small antenna with a **cardioid-shaped radiation pattern** in the horizontal plane. It is primarily used for **reception** purposes. The loop consists of a **110 Ω resistor** connected at its top, while the antenna terminals are positioned at the bottom. One of the terminals is connected to the ground through a vertical wire, forming a **monopole**. This clever arrangement enables the antenna to maintain directionality despite its small size.

The accompanying image illustrates the radiation patterns achieved with and without grounding.

Here are the specifications for the antenna configuration:

**Frequency:**4.5 MHz**Loop diameter:**1 m**Height above ground:**2 m (average real ground)

Thanks to the application of the **Conformal Method of Moments**, the LoG antenna model reveals a remarkable feature: it can be accurately represented with only **a minimal number of segments**. This efficiency is achieved by utilizing **curved segments** that precisely conform to the loop’s contour. The image provided clearly showcases these curved segments, which enable an effective simulation of the antenna’s behavior. This advantageous characteristic is a result of the antenna’s small size relative to the wavelength.

In conclusion, the Loop on Ground (LoG) antenna is a **compact** yet **directional** antenna with a cardioid-shaped radiation pattern. By incorporating a monopole and grounding, this antenna design achieves its distinctive characteristics.

*Comparison of radiation patterns achieved by the Loop on Ground (LoG) antenna with and without grounding.*