Microstrip patch antenna calculator. An analysis problem refers to. Microstrip patch antenna calculator

 
 An analysis problem refers toMicrostrip patch antenna calculator Index Terms-Microstrip patch antenna, MATLAB calculator, CST, 2

Physical and effective lengths of Rectangular Microstrip patch. In fact, striplines often make components such as band-pass filter's more accurately, with more symmetric roll-off and no lower cutoff frequency. 45GHz. 2. Equation 11. As a result, the length and width of the Patch antenna can be calculated by either Online Microstrip Patch Antenna Calculator or experimental formulas [1]. The proposed design achieves polarization diversity by using two conventional truncated corner square patches. A circuit model is introduced to find the. When you want to determine the width and length of a microstrip line for any given characteristic impedance and electrical length, or vice versa, a microstrip calculator is the right tool to do so. The microstrip patch antenna is a low-profile and robust planar structure. You can start with simpler formula: L ≈ 0. In this work, the RMPA, CMPA and TMPA have been fabricated to operate at 28 GHz for the application in 5 G technology. php 5858 Bytes 31-05-2023 10:13:36. In 12, a dual-band antenna with an average gain of 2. This calculator is essential for engineers,. 1 depicts the microstrip antenna geometry. However, application specific patch shapes are often used to optimize certain aspects of MSA performance. The microstrip patch antenna can also be designed using an artificial neural network (ANN) modeling technique where size of the antenna is major limitation especially in mobile and wireless applications. 3 Answers. - MicroStrip Calculator - Cascaded Noise Figure Calculator - Resonant Frequency Calculator - Skin Depth Calculator. The Patch Antenna online calculator, which is based on simplified formulas for calculating microstrip transmission lines and is intended for preliminary calcula The microstrip patch antenna calculator determines the length (L) and width (W) of a rectangular microstrip patch antenna for a given resonant frequency or vice versa. **Note: Like our stripline impedance calculator, all of our RF calculators allow SI prefix input. This antenna patch calculator can simplify the calculation of antenna dimensions for rectangular, circular and triangular shapes. There are several methods to feed a microstrip patch antennas. microstrip patch antenna, also fulfils all requirements for antennas used in satellite communication system. Improving the gain performance of air substrate patch antenna array using the effect of conductive material thickness study for 5G applications. Feeding Methods 2. Figure 2. This online TEM wavelength calculator calculates the TEM wavelength (λ) by entering the value of frequency (GHz) and relative permittivity (εr) medium. 4 GHz with gain 11 d B for outdoor place. Calculation Dielectric Height: Operation Frequency: Result: Microstrip Patch Antennas (or simply patch antenna) are increasingly useful because the antenna is printed directly onto a circuit board. Pick out a standard part number from the data sheets listed below. 2 GHz, with a frequency range of 2. Design of a compact high gain microstrip patch antenna for tri-band 5G wireless communication. Additional benefits of. This calculator is essential for engineers, students, and researchers in the field of wireless communication. Cloverleaf antenna analysis and online calculator. Fringing of the fields, particularly the electric field, at the edges of the patch is an issue of concern because of the finite dimensions involved. They hadlarge in intensity and extravagant because of the province. d is proposed by Pozar in 1985, [2]. To determine the width (W), the microstrip patch antenna calculator was used to provide an initial starting point. TheA microstrip asymmetric folded dipole antenna on chip is proposed in this paper. 2 MHz (5. In the RF domain the antenna size also has reduced and thus the system has become more portable. 4 GHz, ε r = 4. In this paper, a novel microstrip patch antenna array based on a silicon-air photonic crystal substrate is designed and analyzed in the frequency range of 0. 1 – 5 Decreasing σ will decrease the boom length L. In its simplest form, it uses a conducting “patch” one-half wavelength long, so that the metal surface acts as a resonator similarly to the half-wave dipole antennas. 8 0. Note 688) Yagi Uda Antenna Designer (Rothammel/DL6WU) Filter Designer (Audio, AF) Lowpass Filters. It shows that the proposed antenna resonates at frequency equal to 5. Enter values for W and L for a microstrip line to determine its Zo and Electrical Length. Microstrip patch antennas is one of the most used antennas for wireless communication. 5e9,ForAI=true) pAI = AIAntenna with properties: Antenna Info AntennaType: 'patchMicrostripInsetfed' InitialDesignFrequency: 4. Microstrip patch antenna calculator; Antenna near field and far field distance calculator. 60mm. See full list on resources. The measured VSWR results are shown in Table 1. In this article, we will discuss the antenna design and performance. Ver licencia. 393 GHz with a bandwidth (BW) of 58 MHz and VSWR of 1. A stacked microstrip patch antenna with two parasitic patches, which is suitable for use in arrays and can be integrated with circuits, was proposed in [5]. For this purpose, in this. By default, the patch is centered at the origin with feed point along the. 2 permittivity and 1. A compact ultra-wideband patch antenna is presented in this paper. Whether you are a beginner or expert, this calculator can help you m achine your antenna design for optimal performance. The Octagon-Star geometry comprises three. The Square patch feed by 50 coaxial probe is positioned 1. Microstrip Patch Antenna Calculator . R w = Terminating resistance between two. 1 Transmission Line Method 3. Loss increases with distance, so understanding the FSPL is an essential parameter for engineers dealing with RF communications systems. 1, 14. The effect of the air gap distance on the gain of the dual layer microstrip patch antenna at the desired frequency is shown in Figure 4. Step 2: Calculation of the Effective Dielectric Constant. For example, if you wish to input "25000000", just type "25M" instead. The proposed four branches Yagi array of microstrip patch antenna can also achieve a gain above 10 dB and a high front-to-back (F/B) ratio as much as 13 dB. Calculates dimensions of circular microstrip patch antenna, provided frequency, relative permittivity and height of substrate. The default patch is centered at the origin. Helical or Helix Antennas transmit a circular polarized wave. In summary, the design process for the coaxial probe placement is simple: Select an operating frequency and stackup. A frequency range of 1. About Microstrip Patch Antenna Calculator The resource is currently listed in dxzone. Summary; Files; Reviews; Support;Microstrip patch antennas radiate primarily as wide open-open half-wave microstrip resonators. 2. The main category is Antenna design calculators that is about Antenna design calculators. 1. ” Journal of Electromagnetic Analysis and Applications. The dimensions of antenna can be found in the HFSS simulation file. 85mm×6. You can use this online calculator to find the approximate values. Find Microstrip Patch Antenna Calculator at CalcTown. You can perform full-wave EM solver based analysis on the regular patchMicrostripInsetfed antenna or you can create a patchMicrostripInsetfed type AIAntenna and explore the. xcy_antenna said: Hi, Kocha,Both a single patch antenna and patch antenna array were fabricated using the milling machine in Cal Poly’s chamber lab. 244 GHz which gives the measure of the wideband characteristic of the patch antenna. The radiation of the patch is. 1 GHz to 10. Matlab/Freemat file 1. The dimensions of antenna can be found in the HFSS simulation file. Fig. Figure 1. Figure 2. 3. 72 THz in order to improve its performance by suppressing undesirable surface waves that are found in the homogeneous substrate. Basic layout of rectangular patch antenna. 5: Online Microstrip Patch Antenna Calculator and Experimental Formulas . Data safety. A dual-band microstrip patch antenna (MPA) based on a polarization conversion metasurface structure was designed. Enter values for W and L for a microstrip line to determine its Zo and Electrical Length. 26 GHz to 2. Key Words: Microstrip Patch Antenna, Inset Feeding, Radiation Pattern, Bluetooth. T. . Hawkeye tag adopts inset-fed rectangular microstrip patch antenna with the patch, edge notch depth (i. The operation frequency of antenna is 15. They are suitable for vehicle based satellite link antennas, global positioning systems, radar for missiles and mobile handheld radios. Our RF calculators and converters will provide the figures you need for your radio frequency engineering needs. 186cm,substrate with height ℎ=0. 03 and 8. This paper proposes a design and fabrication of a floodlight-shaped microstrip patch antenna using flame-retardant (FR)-4 substrate within the frequency band of 1-6 GHz. The antenna is designed using 1. Output is radius and effective radius of the circular patch Microstrip Patch Antenna Calculator Formula. 235 mm × 0. In this chapter, analysis and synthesis problems for designing of microstrip patch antennas were discussed using the artificial neural network technique. 45 for gold. Advanced Circularly Polarised Microstrip Patch Antennas Adam Z. The patch cutout for the feed inset should be > 2 times the microstrip width. Microstrip Line Calculator Analyze or synthesize a. 04/02/2007: Ansoft Designer Planar EM Simulation of a microstrip transmission line added in the TutorialsTools section. The microstrip patch antenna spots much pertinence in wireless transmission. It also has a wide angle of beam in its radiation pattern. Coaxial Cable Impedance Calculator. 2 Cavity Method 3. Keywords: Circular Microstrip Patch Antenna Design. Developers can show information here about how their app collects. 0×15. The antenna is fed by a microstrip line designed for a frequency of 2. . 4 GHz, whichThe standard rectangular microstrip patch antenna has a narrow impedance bandwidth typically less than 5%. This paper provides details on how to investigate a new method of teaching microstrip patch antenna design made by Metamaterials using MATLAB. This paper proposes a method for designing a 77 GHz series-fed patch array antenna. The Transmission Line model method. [6] Putranto, Prasetyo, Mashury Wahab, Topik Teguh Estu, and A Flowchart. November 2023 ∴ Live And Learn. gold) [4]. There is also a dielectric material placed above the conductor. 15 to 5. The patch is very thin (t<<λ o where λ Superstrate Microstrip antenna with square patch (5. insetpatch = patchMicrostripInsetfed creates an inset-fed microstrip patch antenna centered at the origin. It was invented because of the undesirable effects on coaxial cable when signals in the microwave range propagate through it. Because of small size, light weight, less cost and good performance these types of antennas are also used in spacecraft, satellite and missile applications []. altium. This antenna does not take any much extra size and can be etched on the same mobile PCB itself. It is found that antenna resonates at two distinct. Figure 1a shows configuration of microstrip antenna with S coupled metamaterial superstrate. Z. The microstrip antenna, Part 2: Implementation. To be more specific, it is the ratio of the electric field intensity E to the voltage V induced across the antenna output terminals. Note that a quarter-wave length transformer was used to match the patch to a 50 Ohm feed line. 1. com 2Department of CSE, University Institute of. The patch is generally made of conducting material such as copper and gold (Wikipedia, 2010). An individual microstrip antenna consists of a patch of metal foil of various shapes (a patch antenna) on the surface of a PCB, with a metal foil ground plane on the other side. 6mm3and feed by aMicrostrip line feed of 50Ω. A 77 GHz series-fed. In this part of the workshop you will design a microstrip patch antenna. Alshamsi, Hadef Ahmed, "Design of a Multiband Stacked Microstrip Patch Antenna for Satellite Communications Application" (2019). This kind of antenna is easy to fabricate with involvement of very less capital. What’s New. MICROSTRIP PATCH ANTENNA OF RECTANGULAR DESIGN A Rectangular Microstrip Patch Antenna is designed to operate at 5. Fig. The gain of the antenna is above 7. Use the frequency and layer thickness/Dk value to determine the antenna size. The FR-4 dielectric board as well as the soldered SMA connectors were supplied by the chamber lab. 5 mm, and based on a ground plane as shown in Figure-1. The patch may be in a variety of shapes, but rectangular and circular are the most common. Analyze. A microstrip patch antenna can be designed, various methods are there: 1. The reactance reaches 0 Ohms at around 2. 56 mm and width is 17. The volume of the antenna is 75. 40, 5. The proposed antenna is obtained through optimizing the structural parameters of CSRR. Version 2. The aperture coupled technique exhibits reduced transmission line radiation and enhanced antenna. 微带贴片天线计算器. Here's a tip on an easy way to calculate the radius of a circular patch antenna from a rectangle patch antenna, from Dan: Lately I've been working on a dual band dual polarized AESA for a mobile weather radar for those guys who like to chase tornadoes. Updated on. 4. 2 and the loss tangent was 0. e. 264 0. 06X13. Without truncated corners, the array has a single resonant frequency at 2. 827db and bandwidth of 308 MHz at the operating frequency, The VSWR obtained is 1. 5 GHz and calculate its resonant frequency. 4. The proposed model also assesses the. 04/03/2007: Microstrip Patch Antenna: From Simulation to Realization added in the TutorialsTools section. Use our microstrip patch antenna calculator Patch antennas can also be circular or elliptical for a given operating frequency. 1,312. For example, if you wish to input "25000000", just type "25M" instead. They are mostly used at microwave frequencies. A novel periodic photonic crystal structure with through-holes drilled by a 4 × 7 array based on the line source analysis method was proposed in this paper to improve the gain and radiation of the microstrip antenna. At some point in the summer of 1984 we arrived at the idea of combining these two geometries, using a slot or aperture to couple a microstrip feed line to a resonant microstrip patch antenna. The antenna is fed by a 50 Ω microstrip line, and a quater wavelength transformer is used for the impedance matching. The patch cutout for the feed inset should be > 2 times the microstrip width. The resonant frequency (rf) of a circular patch antenna is approximately given without considering the effect of probe radius by [1]: e r nm r a K c f 2S H * (1) where a e = effective radius of the circular patch c = velocity of light in free. View full-text. Our RF calculators and converters will provide the figures you need for your radio frequency engineering needs. relation is used. Easy to form a large array, spaced at half-wavelength or less. GUI to calculate parameters and plot patterns of Microstrip Patch. This includes the remainder of the ground plane, the remainder of thePopular answers (1) Dear Medkour, The minimum size of the PCB is calculated with the following rule of thumb. (ii) Transmission Line equations: The proposed antenna is designed with the specifications as f o = 2. This is achieved by designing a friendly graphical user interface (GUI) for microstrip patch antennas through which antenna parameters and radiation pattern can be determined. Compared to waveguides, microstrip is generally has a lower power handling capacity, and higher losses due to the fact that it is not enclosed. 12BVO/0. The work is of practical importance for microwave researchers and engineers. microstrip antenna patch calc. As seen from figure 1. The patch has a compact structure of 6. Microstrip Patch Antenna Calculator. Microstrip Patch Antenna Calculator - This calculator calculates the length and width of a patch antenna based on the frequency, dielectric constant of the dielectric and the height of the thickness of the patch antenna substrate. When playing, some cookies are set and your privacy is futschikato. This is a tool I developed to assist us at the Cambridge University Spaceflight Club to design a patch antenna for telemetry. Tech Consultant Zach Peterson explores how to make patch antenna connections through the PCB substrate in order to avoid having traces on the same side of th. The geometry of the designed microstripline feed microstrip patch antenna is as shown in fig. The microstrip patch antenna model used for the numerical simulation in Ansoft HFSS is shown in Fig. We offer frequencies 1575. 4 shows the inset fed microstrip antenna. This parameter is used for the loss calculations. patch microstrip antennas," Antennas and Propagation, IEEE Transactions on, vol. 15:47:05. 5 GHz. It is simulated using by HFSS simulator. This effect can be done by changing the path of current. 1 mm. The Microstrip Patch Antenna Calculator is a powerful tool that enables engineers, students, and researchers to calculate the dimensions of a microstrip patch antenna quickly and accurately. 70 dB and −25 dB at frequency 300. Benefits of Microstrip Patch Antenna Transmission situated operations. A microstrip width calculator can calculate the width of a microstrip. 2. , the antenna feeding point [38]), and TL dimensions optimized to 50 Ω impedance matching. 9, pp. The input impedance of these antennas depends on their geometrical shape, dimensions, the physical properties of the materials involved, the feed type and location. In this case, each coordinate in the vector starts at the origin and lies along the specified points on the x-, y-, and z-axes. The antenna is designed withcircularly polarized antenna based on a circular microstrip patch. Index Terms-Microstrip patch antenna, MATLAB calculator, CST, 2. This link is listed in our web site directory since Thursday Jul 9 2015, and till today "Microstrip Patch Antenna Calculator" has been followed for a total of. 02. In telecommunication, a microstrip antenna (also known as a printed antenna) usually means an antenna fabricated using photolithographic techniques on a printed circuit board (PCB). Hence, for the proposed antenna's dimensions using ( 5 )–( 7 ) we obtain values of 13. The patch antenna is designed for 2. 86 mm and length x span) 9. 2 Design Specifications Before designing the antenna, the first step is to consider the specification of the antenna base on it application. An X-band, 12-element array of rectangular microstrip patches was demonstrated for the first time to reflect the main. This paper presents a bi-port, single-layered (planar), dual circularly polarized (CP) patch antenna with improved interport isolation for S-band satellite telemetry and telecommand applications. The general dimensions are /2, however due to the specific dynamics of the antenna and the fringing fields, the exact dimensions require more careful calculations. Metal-Only Antenna. For the best radiation characteristics, the proposed. 03/10/2007: Online microstrip patch antenna calculator added in the TutorialsTools section. Enter the resonant frequency or the substrate parameters and get the. Then you need to calculate patch sizes, you can do it by hand with formulas given in references below or. 3. The developed process started from designing the single element up to 1 × 8 arrays elements. 6 mm Table 1. The resonant frequency of an LC Circuit is the frequency at which the inductive reactance and capacitive reactance of the LC circuit are equal. The patched antennas with slots and slits is. The triangle patch microstrip antenna is design to operate for 2. The proposed antenna’s performance shows a good multiband operating at 2. There are a number of pcb transmission line typologies. The dimension of the feed line is 7. Microstrip Patch Antenna Calculator; PCB Loop Antenna Calculator; Wokatenna Design; Yagi Uda Antenna Designer (NBS Tech. Real Hams review new sites. By default, the patch is centered at the origin with feed point along the radius and the ground plane on the xy - plane at z = 0. E. 3 and 1. This design approach helps analyzer microstrip inset-fed patch antennas and helps to locate the exact inset length for 50-? input impedance. It consists of a rectangular radiator patch, which is fed by microstrip transmission line. 1. Introduction In recent years, the microstrip patch antenna has widely been used for various specific applications in satellite communications, aerospace, radars, biomedical applications, mobile communication for GSM, and remote sensing applications [1] because of its special features like ease of analysis and fabrication, low cost, light. This example shows how to create an AI model based inset-fed microstrip patch antenna at 4. Note that the "h", for dielectric thickness in this formula is omitted in some other Polar publications. If the feed is located closer to the edge of the patch the input impedance will be high, if the feed is located closer to the center of the patch the impedance will be low. While copper is overwhelmingly used as the radiated part in microstrip antenna design studies, the choice of dielectric material offers a wide range of possibilities. A microstrip is a transmission line used for signals in the microwave range. For more information visit my blogThis paper presents a high-band microstrip antenna of dimension (5 × 5 × 1) mm 3 for 5G wireless mobile communications ranging up to 40–45 GHz. Antenna design. RF. Use the above equation to calculate the x-coordinate for the probe location. The proposed antenna resonates at 27. The patch is generally made of conducting material such as copper and gold (Wikipedia, 2010). 1607 – 2. Product Categories; Custom Products. Basic Structure of Microstrip Patch Antenna Due to the fringing effects, the patch of the microstrip antenna looks greater than its physical dimensions as shown in Fig. 1 GHz is selected and 401The microstrip patch antenna is one of the most prominent antennas for THz band applications such future 5G communication systems and advanced wireless communication system. The substrate parameters (ε r and h) are required. The Microstrip Patch Antenna Calculation Process. 6K Downloads. Microstrip patch antenna calculator; Antenna near field and far field distance calculator; Free space path loss calculator . A new design procedure based on the odd-even mode method to calculate the input impedance of an asymmetric strip folded dipole antenna is presented. This paper review the different meth - ods used to calculate the input impedance of microstrip patch antenna along with different impedance match-ing techniques. Figure 1. The designed antenna is realized using microstrip line feed on a 1. 4GHz. By understanding the microstrip transmission line, designers can. 1 to 10. COAXIAL FEED TO MICROSTRIP ANTENNAS Patch size =8. Full ground plane microstrip antenna without slots and with vertical slots fabricated on FR-4 with a thickness of 1. antenna. In this design the substrate RT Duroid was used due to its advantages. An embedded microstrip is a microstrip transmission line or flat conductor separated from the ground place via a dielectric material. The ANSI-based circular microstrip patch antenna is designed to work at 7. 575mm is used in the simulation process. Using a 4 GHz reference antenna, the variations of input impedance and radiation patterns are compared, for (a) changing the feed positions to the non-radiating edge against (b) changing the. Matlab/Freemat file 1. The microstrip patch antenna calculator determines the length (L) and width (W) of a rectangular microstrip patch antenna for a given resonant frequency or vice versa. The microstrip antenna specifications include the following. This is a required field. Horn Antenna 2 Chapter Two: Microstrip Antenna 16 2. Regards. Microstrip patch antenna calculator; Antenna near field and far field distance calculator; Free space path loss calculator . With the help of this tool, you can estimate the length, width, frequency, and bandwidth of your patch antenna. where we apply the input to the microstrip patch antenna. The microstrip patch antenna is a low-profile and robust planar structure. And does any one have the idea. 46, no. These are considered to be standardized impedance values with cables readily available. The effectiveness of the proposed designs is confirmed through proper simulation results. The substrate parameters. In some cases, users require a custom impedance value. The design process started with the conventional rectangular microstrip patch antenna as shown in Figures 1a and 1b. 5 mm, respectively. Find: (a) The input impedance. 257, Evaluation of Zin has been explained in detail in a lucid way. 3 The. 225mm above the ground plane and the optimized distance of second layer is 11. 3, h = 1. Despite their limited bandwidth, the mm-wave microstrip patch antenna could be a promising choice to satisfy all the above needs in 5G [2,6]. Windows. Antenna “Flex” for 3G/4G & Wi-Fi. 57 mm thickness. The antenna patch is divided into four smaller self-similar segments as shown in Figure 3(a), with a scaling ratio of 1/3. It should be less than -10 dB for the acceptable operation. The microstrip antenna is a convenient, planar, easy-to-fabricate antenna with some attractive attributes and features, as well as. 4. Therefore, it has the advantage of low-cost manufacturing and a wide range of radiation patterns. Figure 2. 45 - empirical coefficient mainly for open end effect. This can be achieved by regulating the inner and outer diameter of the coaxial cable, along with the. 3. This online resonant frequency calculator is divided in to three calculators. The calculator enables users to determine the operating frequency, substrate thickness, dielectric. The default patch is centered at the origin. 0mm to 25. The design theory of antenna is studied by various researchers and different types of geometry models are invented with respect to their knowledge and applications [2] [3] [4]. The size of a microstrip antenna is inversely proportional to its frequency. Figure 2. This calculator uses the. Does anyone have an idea about the wave port or lumped port in hfss ,which should be used especially for microstrip patch antenna in uwb range i. A simple microstrip patch antenna with aperture coupler technique has been succesfully designed according to design specifications, simulated and analyzed. Therefore, in this paper, a low profile hexagonal microstrip patch antenna has been proposed for the technology to detect breast cancer. The substrate parameters (ε r and h) are required. For starter you can use standards like FR-4 substrate material with epsilon_r = 4. The main optimization goal is to reduce the maximum sidelobe level (SLL). Visión general. 04/03/2007: Microstrip Patch Antenna: From Simulation to Realization added in the TutorialsTools section. The reactance reaches 0 Ohms at around 2. A low return loss of -33 dB, a VSWR of 0. In the circular case, instead of the operating frequencies being related to rectangular resonances, they will be related to circular resonances defined by cylindrical Bessel functions. path. is dielectric constant of the substrate kept at 4. It is intended only as a very crude estimate, as I. 1. Low power handling. The aim of this work is to design an efficient and economical patch antenna for indoor and outdoor uses. 48. Based on the traditional genetic algorithm, the study explores different array topologies consisting of the same microstrip patches to optimize the design. As noted above, curved implementa-tions can be made to conform to aircraft hulls. We offer frequencies 1575. Electrical Length of Antenna. 23 GHz. The square microstrip antenna is designed on dielectric material FR-4 (lossy) and height from ground plane is sh = 0. A slot or gap with a conducting coating on a dielectric substrate makes up a slot line, a form of transmission line comparable to CPW. The proposed design is created with the substrate with FR4 epoxy under dielectric constant 𝜀𝑟=4. **Note: Like our microstrip line impedance. path. This design approach helps analyzer microstrip inset-fed patch antennas and helps to locate the exact inset length for 50-? input impedance. Four types of antennas were simulated using Computer Simulation Technology (CST) software. The radius of the circular patch is chosen as 7. 6 mm 3 which is highly compact sized antenna. Among them, widely used is theThe equation used will then be written in mat lab and then patch antenna parameter calculator will be developed. The patch antenna is constructed from a 2D rectangular sheet of width (y span) 11. 6 mm. The length of a half-wavelength dipole can be calculated by the given formula: L =. 57 GHz and 5. Williams, S. The rectangular microstrip antenna is represented as two slots or apertures (of width W and height h) separated by a low impedance transmission line of length L (see Figure 14.