R S T U V W Y Z Microstrip Click here to go to our page on microstrip loss Click here to go to our page on bends in transmission lines Click here to go to our main. Quarter Wave Transformer Impedance Calculator. 1. 8. The designed balun exhibits good performance. The phase is mostly + 90 ∘ or − 90 ∘, indicating that Zin is mostly reactive. by . Coplanar Waveguide With Ground Characteristic Impedance Calculator. The microstrip coupling lines provide the input/output coupling and its strength are determined by the gap width and length (S 2, L 2 and S 3, L 8). TransLine. For an embedded (internal) microstrip, the formulas provided in paragraph 4. on1. The edge couple differential microstrip transmission line is a common technique for routing differential traces. Introducing the newest version of the Microwave Impedance Calculator from Rogers’ Advanced Connectivity Solutions. SAVE & ACCEPT. Relative Dielectric Constant ( εr ): Track Width: mm. Added calculation for Electrical Length as Air Line equivalent; Added calculation for Delay time ; Known Issues: Norton (Symantec) false positive. TransLine. A microstrip transmission line consists of a narrow metallic trace separated from a metallic ground plane by a slab of dielectric material, as shown in Figure 7. This tool calculates the matching network necessary to terminate a line of the specified characteristic impedence (Z o) in a specific complex load impedence (R L + jX L) at a specified frequency. AppCAD Free Design Assistant Tool for Microsoft Windows. It was found that the microstrip showed more benefit than the CBCPW when comparing a smooth copper to a rough copper surface for insertion loss. With hundreds of coaxial RF cables in-stock and ready to ship worldwide immediately, our coax capacitance calculator lets you calculate the impedance of a coax by allowing you to enter the outer diameter dielectric width and inner conductor diameter. 8 Updates & Additions: Go to our microwave calculator page, our microstrip calculator does this calculation for you! It's time for a Microwaves101 Rule of Thumb! For pure alumina (ε R =9. This is a required field. Reset. **Note: Like our microstrip line impedance calculator, all of our RF calculators allow SI prefix input. Versión: 1. Color-Code. Microstrip Impedance Calculator. Overview. Wr is a correction factor defined as Wr = W + 1. VSWR Calculator. Advertisement. Select the box with the geometry that you would like to calculate. Assuming I am thinking about this correctly, this is looking really unlikely: Set embedded microstrip to match the embedded stripline using 1. The right-hand side of Equation (5. A. The differential impedance will be a bit less than 2x the single-ended impedance. A study was done comparing microstrip and CBCPW transmission lines with different copper types, which had significantly different surface roughness. If the dielectric constant of the material is not known, you can choose to put in the velocity of. Input. ( 0. Microstrip Line Calculator Analyze or synthesize a microstrip line based on substrate parameters and frequency. You need to enter the substrate parameters, the frequency, and the physical. Conductor is assumed to have zero thickness and to have uniform height above ground. Dimensions are in meters. The quality factor indicates how selective or non-selective the band pass filter is. This is the basic principle of microstrip and stripline directional couplers. Impedance in Ohms. Tech. BLACK FRIDAY: 🎉 El 24 de Noviembre habrá 50% de DESCUENTO en todos los Productos de mi TIENDA 🎉 Descartar. 2 of the Microstrip Calculator, h=H1 and b=H2-H1. Transmission-Line Calculator. For an embedded (internal) microstrip, the formulas provided in paragraph 4. Microstrip Transmission Line Models The impedance Z 0 of a transmission line consisting of a circular signal conductor having a diameter d and centered between parallel ground planes separated by a distance h is given by: This web page allows you to calculate the width and length of a microstrip line for a given impedance, electrical length, or vice versa. 0 (modified)) A microstrip line is single-ended 1 in the sense that the conductor geometry is asymmetric and the one conductor – namely, the ground plane – also normally serves as ground for the source and load. Microstrip Filter Design Tool. Microstrip Filter Design PathWave Advanced Design System (ADS) Background Microwave filters play an important role in any RF front end for the suppression of out of band signals. Access technical papers, white papers, calculators, tools and more for circuit laminates, prepreg and bonding materials. Compute the height, width ratio, impedance and dielectric constant of a microstrip transmission line using Pasternack's online tool. The primary difference between microstrip and stripline configurations is the placement of the signal conductor. 72mil. h = rfckt. But the choice of transmission-line technology, such as microstrip or grounded coplanar waveguide. One of the features of this type of microstrip is the coupling between lines. You can build an RC low-pass filter with a cutoff frequency of 1 kHz using a 3. J. In radio technology, crosstalk is a phenomenon wherein a signal in a conductor jumps to an adjacent conductor. Part 2 looked at two types of RF/microwave filters, but there are many more, including ceramic coaxial filters and helical filters. Version 2. i am talking about the small gap between the patch and microstrip. This picture is true. 1 Simple Transmission Line Model This transmission line model [8–10] was the first technique employed to analyze a rectangular microstrip antenna by Munson in 1974 [8]. Enter the target impedance, trace thickness, dielectric thickness and relative dielectric constant to get the width of the microstrip and its radiation intensity. w h = 2 π[(dεr − 1) − ln(2dεr − 1)] + (εr − 1) πεr [ln(dεr − 1) + 0. tr = 0. λ = Signal wavelength in meter (m) = c/f ( where f = signal frequency, C = speed of light = 2. The Differential microstrip impedance calculator is used to compute the differential (Zd) and single-ended impedance (Zo) of an edge-coupled microstrip line just by entering the width, separation and thickness of the trace along with the dielectric thickness and constant of the coupled line. This calculator calculates the reactive near field distance, radiating near field distance (Fresnel region) and the far field distance. Width (w) Trace Thickness (t) Dielectric Thickness (h) Relative dielectric constant (εr) Calculate Reset Result Single Ended Impedance Ω Propagation Dealy ps/inch This calculator helps you compute the characteristic impedance of a microstrip, a flat conductor suspended over a ground plane with a dielectric between them. Metal loss is reduced by thicker substrates, and is almost inversely proportional. Going Forward with a Trace Impedance Calculator. The MWI-2017 Microwave Impedance Calculator software is able to calculate key parameters for most common microwave transmission lines, including microstrip, stripline, and coplanar-waveguide transmission lines. These include dielectric thickness (h), target impedance (Zo), and relative dielectric. 60. Edge Coupled Stripline Differential Impedance Calculator. A cross-section of a microstrip transmission line is shown below. Rectangular Waveguide Sizes. In the lumped and distributed form, they are extensively used for both commercial and military applications. 3%, are. Go to the. Go to the. This includes design capture, simulation, and displaying simulation results. 2. Users need only specify the material properties for. Download a free microstrip taper calculator worksheet. Figure 3. Passband insertion loss measured well within the target specification. 3 Easy to analyze using transmission line or cavity models Most common type of patch or microstrip antenna Transmission line model The rectangular microstrip antenna is represented as. The capacitance calculated using this tool is the capacitance between a flat conductor and the ground plane over the conductor length. 3 kΩ resistor and a 47 nF capacitor (which are standard resistor and capacitor values). Microstrip Patch Antenna Calculator Calculate dimensions and edge impedance for desired resonant frequency. 1) describes the series connection of short- and open-circuited stubs having characteristic impedances of Z 0 / 2 and half the original electrical length. Access technical papers, white papers, calculators, tools and more for circuit laminates, prepreg and bonding materials. “468” is the constant used to convert frequency in MHz to feet. Stripline Calculator. USB traces must be 90 Ohms differential. This makes Wheeler’s equations discontinuous and brings their accuracy into question when the microstrip width is similar to the height of the microstrip above the conducting plane. Here's microstrip on 10 mil quartz. W (Line width) : unit. If the ratio (L/W) is close to unity, the radiation pattern will be symmetric but may not provide a resonable input impedance. Microstrip Line Calculator Analyze or synthesize a microstrip line based on substrate parameters and frequency. Basic structure of microstrip junction circulator consists ground plane, dielectric slab and ferrite material. Metal loss is reduced by thicker substrates, and is almost inversely proportional. 0046 meters. The advantages of coplanar waveguide are that active devices can be mounted on top of the circuit, like on. 2. That still may not mean the formulae are valid for that range, just that the calculators have not implemented catches for this unexpected range of use. Basic Online Calculator. 5 oz copper (18 um), the resultant copper thickness is 45 um (-3 um. used to determine the radius of circular microstrip antenna. with electrodes very close to the microstrip. The "optimum" mitered bend equations for microstrip were found empirically way back in the 1970s. C is the capacitance of the microstrip. *Solder resist PCB coating will have the effect of lowering the impedance (due to the 30um or so of Er 3. This technique doesn’t work for wide band requirements, but is a simple way to achieve this at a specific. In addition to the characteristic impedance of a transmission line, the tool also calculates. To use it, first select the tolerance of the resistor: 10%, 5% or equal or smaller than 2%. Calculate the impedance of a microstrip transmission line with a PCB using this online tool. Since according to a popular saying a picture reportedly paints a thousand words , here are a few thousand words for you - in picture format. If the dielectric constant of the material is not known, you can choose to put in the velocity of. The length of a half-wavelength dipole can be calculated by the given formula: L =. Calculadora de Líneas Microstrip S. The input impedance, Zin, of the shorted microstrip line is shown in Figure 3. - Pi Attenuator Calculator. Transcalc. R w = Terminating resistance between two. Users just have to provide the required filter specifications as well as the material description. f if the frequency of the RF signals. The dielectric material and the air above the microstrip makes it a transmission line with the inhomogenous dielectric. 2. Figure 1. For example, if u = 1 and ε r = 10, then from the figure, Z 0 ε e = 126 Ω and ε e / ε r = 0. Modeling approximation can be used to understand the impedance of. T-Match Topology. Enter the values for trace thickness, substrate heights, trace width and substrate dielectric and get the output impedance in ohms. II. Microstrip routing is a transmission line trace routed on an external layer of the board. Calculators. So two estimates have been calculated for the frequency at which the first higher-order. Use our microstrip patch antenna calculator; Patch antennas can also be circular or elliptical for a given operating frequency. Enter the relative permitivity, width of the trace, the ground plane spacing, and the substrate thickness to calculate the characteristic impedance. RADAR tutorial Radar Measurements Doppler Radar FMCW RADAR RADAR BANDs RADAR SCOPE Weather Radar Ground Penetrating Radar System. 71. Coplanar wave guide. Low power handling. 6 GHz f Higher - Microstrip = c / ( 4 h ε r − 1) = 10. Both formulas of the dielectric, as well as the total losses, agree. microstrip = microstripLine (Name=Value) sets properties using one or more name value pair arguments. 35/fmax. Generate Kicad footprintGenerate SVG file. PI Network Impedance Matching. example. A microstrip line is a microwave transmission line composed of a single conductor strip supported on a dielectric substrate. 1 of the Microstrip Calculator using H2 in place of h, εr,eff can be computed as specified in Eq. High-quality circuit boards as prototype and series at reasonable prices. A single microstrip was analyzed results obtained for a parallel coupled microstrip band pass and then used for coupled line sections by making some filter for WiMAX applications at. Ω/mΩ/in. 66, the trace height is 1. For example, a maximum frequency of 100 MHz corresponds to a. There are a lot of online trace impedance calculators specifically designed for differential traces. λ is the wavelength of the signal in free space. In both cases, you’ll need to enter your stackup information into the calculator to get accurate results. Equation 11. The propagation delay of the microstrip line can also be calculated, as per Eq. Sorted by: 1. 1kOhm, 1% tolerance: Brown Black Black Brown Brown. Overview. Changed the microstrip impedance calculator formula to a complex version for accuracy. Phone: +44 (0)1425 489 111 / +49 (0) 8104 628 0. 93 ohms. Dielectric Thickness: mm. Microstrip antennas fluctuate in impedance hanging on the plangency frequency, polarity, prototype, and explicit patch composition. Use this calculator to analyze or synthesize microstrip transmission lines with different substrate parameters, such as relative permittivity, loss tangent, conductor resistivity,. If the reflection coefficient (Γ) is known, then the VSWR. It consists of rectangular substrate and the metal trace layer as shown in Fig. Typical realisation technologies are printed circuit board (PCB), alumina coated with a. One option for impedance matching is to use an inset, as shown in the image below. The coax is a good way to create a transmission line. Grieg and H. Each of the transmission lines consists of a signal trace and reference planes. Microstrip Impedance Calculator Enter the Height, Width and Permittivity to get the Impedance of the Microstrip Note : The unit of the height and width entered must be the same. In some cases, users require a custom impedance value. 1. Each Port is matched to the characteristic impedance Z 0 of the single, isolated microstrip transmission line. The transmission line types and the reference of their mathematical models are listed below: M. The. 39 for the effective dielectric constant. Then, multiply the values of L and l by 0. Amongst them are a Smith chart s-parameter plotter, a lumped element balun designer, a microstrip calculator, a mixer spurious product calculator, and thermal dissipation calculator. beammy. Resonant Frequency Calculator. Microstrip filters are built as a carefully defined pattern on a PC board. This calculator is designed to compute the maximum allowable current that can flow through a stripline. 11) the required electrical length of the taper at 8 GHz is. 0. Fig 1: Equilateral triangular patch antenna with inset feeding Abstract: Triangular microstrip antenna is commonly used in wireless communication. This spreadsheet is used to calculate inductor and capacitor values for Chebyshev filters. This hairpin must be printed on one side of a standard double sided 1. , vol. If the parameters exceed the allowable ranges, the curve-fit models blow up. 5 GHz. Rogers Advanced Connectivity Solutions (ACS) has introduced an updated design program that is free to download called the MWI-2017 Microwave Impedance Calculator, a transmission line modeling tool for electronics engineers (setting up an account is required). Such a stripline is constructed with two traces referenced to the same reference planes above and below the traces with a dielectric material between them. The substrate parameters (ε r and h) are required. Some are based on look-up tables, some completely hide their algorithms. Dielectric Constant. Height: Height of the substrate. Characteristic Impedance of the System (Z0) Ohms. Coax Impedance (Transmission Line) Calculator. Differential microstrip impedance calculator is an online electronic tool to measure the opposition to alternating current. Assumptions and Limitations. The maximum PCB track length is then calculated by multiplying tr by 2 inch/nanosecond. To use this tool, enter the values for. Calculate the width of a microstrip transmission line with this online tool. A microstrip lowpass filter based on transmission line elements for UWB medical applications is proposed in this paper. Design, development and small scale manufacture of simple RF and Audio items. L-Match Topology. Cross-hatched copper plane in a flex stack-up for controlled impedance. This is the 7th chapter in a series of tutorials that introduce the basic structure of ADS workspaces, libraries and cells. 81 Updates & Additions: Corrected S/H and W/H formula for the Edge Coupled Embedded differential pair calculator. Free Space Path Loss Calculator. Some research groups have studied about the equations for the formulation of microstrip radial stubs [27, [29] [30] [31]. 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. 4. 720. where Z 0,surf can be computed as specified in Eq. Calculate the width of a microstrip transmission line with this online tool. MTT-25, pp. To use it, first select the tolerance of the resistor: 10%, 5% or equal or smaller than 2%. They can be used with SDR to improve reception. Note : It is quite possible, that other calculators deliver slightly different results. 4. The characteristic impedance (Z o) of a microstrip transmission line can be calculated using the active calculator or the equations at the bottom of the page. It covers low pass and bandpass rf filter calculators used in microstrip designs. Choose from different models of microstrip, edge coupled, wire, embedded, or stripline,. 0 (modified)) A microstrip line is single-ended 1 in the sense that the conductor geometry is asymmetric and the one conductor – namely, the ground plane – also normally serves as ground for the source and load. Here is the link to the Microstrip line simulation in HFSS (Please watch the ADS design and simulation before!) fabricated microstrip UWB bandpass filter is then measured using a vector network analyzer and results are presented. The two side edges are translated from the edge a distance of 1/7 the microstrip. The lateral dimensions, however, are well controlled being photolithographically defined and then etched. Loss increases with distance, so understanding the FSPL is an essential parameter for engineers dealing with RF communications systems. Ground plane is assumed to be perfectly conducting and of infinite extent. The filter is designed to exhibit an elliptic function response with equal ripple in the passband and the rejection band. 1. The required inputs are the Dk value for the dielectric constant of the PCB substrate, and the. Their flexible layout is excellent, giving microstrip lines optimal versatility. Such a microstrip is constructed with two traces referenced to the same reference plane with a dielectric material between them. This calculator computes the trace width of a stripline if the maximum current, trace thickness, temperature rise, ambient temperature and length are given. i hope you understand. Description. The electrical parameters may be determined from specified physical parameters, or the physical parameters required to meet a given set of electrical parameters may be found. Latest News 04/20/2011: Noise Figure calculator added. ANSOFT CORPORATION HFSS FIELD CALCULATOR COOKBOOK INTRODUCTION feature of Ansoft’s HFSS Version 7. ice. All that is needed is to input the desired cutoff frequency, the passband, the impedance, and the ripple. However, these calculators are using curve-fitting models tracing back to the Wheeler and Getsinger era (60's 70's etc). This calculator helps determine the correct values of the inductance (L) and capacitance (C) of an inductor and capacitor to be used in a Butterworth LC bandpass filter. A Clemson ECE635 Project by Adam Schreiber and Jason Byers. You need to enter the substrate parameters, the frequency, and the physical parameters of the microstrip line and get the results in the form of a table. PCB Trace Max Current Calculator (External) This tool, based on the formulas and graphs contained in the standard document [1], calculates the maximum allowable current that can flow through a copper printed circuit board external trace (also called as microstrip), keeping the temperature increase of the trace itself below the specified input. The trace resistance of a microstrip is important in determining how much power it dissipates. 0300 meters. So, the dielectric at my 5-6GHz frequency range is 3. The length of a half-wavelength dipole can be calculated by the given formula: L =. 39 for the effective dielectric constant. When. The microstrip line is a commonly used. There is also a dielectric material placed above the conductor. Go to. 1. The shape factor is defined as the ratio of two separate attenuation levels used to calculate the cutoff frequency. 7 characteristic impedance 50 Ω . It include aggregate noise figure calculator, circular waveguide lowest cutoff frequency frequency calculator, dBm to dBW converter, dBm to Watt converter, Watt to dBm converter, microstrip line impedance calculator, PI attenuator resistance. Microstrip Crosstalk Calculator. If the ratio (L/W) is close to unity, the radiation pattern will be symmetric but may not provide a resonable input impedance. 3678) of its value near the surface. Coplanar Waveguide With Ground Characteristic Impedance Calculator. f c = 1 / (2π × 3. Here's a simple spreadsheet that will help you understand the concept of frequency translators Saturn PCB - Toolkit for Impedance Calculation. This works by taking advantage of coplanarity between the antenna and the feedline, which produces. This parameter is used for the loss calculations. Bookmark. You may enter custom values or choose from a list of commonly. Calculates the impedance of a microstrip. 1. Ground plane is assumed to be perfectly conducting and of infinite extent. Using this formula, you can calculate the characteristic impedance of a. Width (w) Trace Thickness (t) Dielectric Thickness (h) Relative. The waveguide width determines the lower cutoff frequency and is equal (ideally) to ½ wavelength of the lower cutoff frequency. The twisted pair helps create a more uniform inductance and capacitance per unit length of wire to ensure a constant impedance, by keeping the return path as close to the signal as possible. Discrete and Microstrip Coupler Design - Chapter 7. Dimensions are in meters. 2 of the Microstrip Calculator, h=H1 and b=H2-H1. Modeling approximation can be used to understand the impedance of. Ignoring (for the moment) the E r component, loss at 110 GHz would be 0. 0075e-6) creates a microstrip transmission line with thickness of 0. Description. Simons, "Coplanar Waveguide Circuits, Components, and Systems", Wiley. The electrical parameters may be determined from specified physical parameters, or the physical parameters required to meet a given set of electrical parameters may be found. In the lumped and distributed form, they are extensively used for both commercial and military applications. The magnitude of the coupled voltage between adjacent traces is also provided. So two estimates have been calculated for the frequency at which the first higher-order. If there is something in the microwave universe that you need a different calculator for, drop us a note and we'll see what we can. Crosstalk is unwanted coupled energy between traces. A usable directional. Microstrip directional couplers are easy to integrate into RF circuits constructed on a single substrate. This article features a JS generator for hairpin microstrip band-pass filters in the GHz band. Many conductor properties are frequency dependent and dynamically affect electromagnetic characteristics of circuitry and systems. To use the calculator, place values in either the velocity of propagation field or the dielectric constant field. The synthesis portion calculates the required physical parameters to meet desired electrical. dBm to Watt converter Stripline Impedance calculator Microstrip line impedance Antenna G/T Noise temp. 3 shall be used. Using the Microwaves101 microstrip calculator, we see that 50 ohms on 10 mil alumina corresponds to 9. 37mm for a 50 ohm. 7. Therefore doubling the thickness cuts the metal loss in half, but does nothing to loss tangent loss. Output. Transmission Lines – Microstrip, Coplanar, and Coax. 2. ±1% in h, ±2% in ±1% in C. to NF. structural defects in the etched microstrip lines. $egingroup$ So basically your answer shows that the JLCPCB impedance calculator results are generally in the same ballpark as the proven field simulators. The transmission line geometries are cross sections and the values calculated are the inductance per unit length. 6 mm thick PCB. Results indicate that perhaps the basic parameter is the normalized thickness of the antenna. 1. The IPC-2141 trace Impedance calculator will help make initial design easier by allowing the user to input basic parameters and get a calculated impedance according to the IPC-2141 standard. 4 GHz, microstrip line feed. Following is the list of useful converters and calculators. Visit to learn more about our other electrical engineering tools and resources. - Reflection Attenuator Calculator. Ground plane is assumed to be perfectly conducting and of infinite extent. Embedded Microstrip. In most practical cases, the dielectric substrate is electrically thin, that is, h ≪ λ. Description. These radio frequency calculators help with unit conversion, attenuator design, antenna design, radars, and various other basic calculators. 3048 to convert them to meters. The microstrip width calculator uses the following parameters to calculate: Dielectric thickness (h) Target impedance (Zo) Relative dielectric constant. hi balanis antena theory book thay have give example problem. High and low pass filters are simply connected in series. Se requieren los parámetros del sustrato: constante dieléctrica (εr), altura dieléctrica (h) y la frecuencia de interés. TEM wavelength is the distance between two adjacent peaks or troughs of adjacent cycles of the TEM wave. AsPasternack's RF calculators and conversions section provides engineers valuable and easy-to-use tools ranging from complex mathematical formulas to simple conversions. Impedance in your traces becomes a critical parameter to consider during stackup. Where L is the length of the dipoles. 3: Normalized characteristic impedance and normalized effective permittivity of a microstrip line as a function of u = w / h. This calculator provides accurate microstrip transmission line analysis and synthesis. Coplanar wave guide with ground plane. More complex vari-ations on the basic shapes are frequently used to meet particular design demands and in terms of polarization,a microstrip section. The filter is constructed of two symmetric shunt open-circuited stubs and three series unit elements. A rectangular microstrip patch antenna is designed and simulated with CST software using 0. The calculator below uses Wadell’s. 06/11/2010: Apple Claims Iphone 4 Has Novel Antenna Design added. The spacer material is. 998 × 10 8) G = Antenna gain (Linear value) The effective antenna aperture can be calculated by the following formula when the gain of the antenna is specified in dB (decibels). Spread the love Microstrip Propagation Delay Calculator Microstrip Propagation Delay Calculator Trace Length (in meters): Calculate Propagation Delay: nanoseconds FAQs How do you calculate propagation delay? Propagation delay is calculated by dividing the distance traveled by a signal by its propagation velocity. HFSS Waveport Size Calculator Figure out the correct waveport size to use in HFSS for a microstrip line feed. Reference: R. 1. Fairview Microwave's Microstrip Calculator computes a microstrip's height/width ratio, impedance and relative dielectric constant for a microstrip transmission line. The stripline impedance calculator provided below is useful for gaining an initial estimate of trace impedance for striplines. Joined Feb 22, 2008 Messages 34 Helped 4 Reputation 8 Reaction score 3 Trophy points 1,288Asymmetric Stripline Impedance Calculator online. For example: Yellow Violet Red Gold: 4 7 x100 ±5% = 4700 Ohm, 5% tolerance. High-frequency circuit designers must often consider the performance limits, physical dimensions, and even the power levels of a particular design when deciding upon an optimum printed-circuit-board (PCB) material for that design. 3 shall be used. This calculator finds both odd and even transmission line impedance. By determining its trace resistance, the contribution of a microstrip to the overall circuit resistance can be evaluated. W is the width of the microstrip trace. Stripline is a transmission line trace surrounded by dielectric material suspended between two ground planes on internal layers of a PCB. For example, rfckt. This tool is designed to find the pcb impedance. It is similar in function to capacitors and inductors but only for very high frequencies. This is a natural way to implement a transmission line on a printed circuit board, and so accounts for an important and expansive range of applications.