. The importance of impedance matching. 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. There are some advantages to using a microstrip transmission line over other alternatives. W. Impedance and line width calculator, DOS Transcalc (FREE) : TL calculator can calculate either electrical properties or physical parameters from the user inputsThe 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. The calculator utilizes the formula for calculating the impedance of a microstrip trace: Impedance (Z) = [87 / √(ε + 1. Figure [Math Processing Error] 7. This is read from the point where the circle you drew intersects the Re{ Z N} = 1 circle. We recommend the software from Polar Instruments if you want to get exact impedance calculation. The length of the stub line Lstub Any load impedance can be matched to the line by using single stub technique. The correct dimensions properties, therefore, ensure the characteristic. Single-stub matching is a very common method for impedance matching using microstrip lines at frequences in the UHF band (300-3000 MHz) and above. Fig. Impedance matching is, in simple terms, ensuring that a trace lines’ dimensions propagate the effective transmission of RF signals from load to source. Also, what is the ECE145A/ECE218A Impedance Matching Notes set #5 Page 13 Basis for distributed matching using transmission line segments: the equivalent circuit model of a short transmission line. If Z / Z 0 is inside the 1+jx circle on the Smith chart (i. This implies that the resulting transmission line resonators are one-quarter wavelength long at 2 f 0 (i. 1. Press Analyze to see the results. 5 - 0. The complex conjugate of z S is marked as point A on the Smith chart. Γ0 = Z01 − ZS Z01 + ZS Γn = Zn + 1 − Zn Zn + 1 + Zn ΓN = ZL − Z0N ZL + Z0N. L networks for narrowband matching a source or load impedance Z to a transmission line with characteristic impedance Z 0. Antenna Modelling with Numerical Electromagnetic Code. 3. If the line impedance is closer to the. Characteristic impedance is purely a function of the capacitance and. There are four different types of impedance used in characterizing differential trace impedances. Frequency span ±. In other words, if the load impedance is equal to the transmission line characteristic impedance, the input impedance will be likewise be equal to Z 0 regardless of the transmission line length A. Source voltage. In Smith Chart terms, we want to move the impedance ZL towards the center of the Smith Chart, where the reflection coefficient is zero. USB traces must be 90 Ohms differential. Critical length depends on the allowed impedance deviation between the line and its target impedance. 4. For example, should L2 return a value of -62 degrees, the correct result will be. Electronics, V12, N1. The input impedance of a short- or open-circuited lossless transmission line alternates between open- ( Zin → ∞ Z i n → ∞) and short-circuit ( Zin. 19. Assume microstrip line for which propagation occurs with wavelength 60% that. While commonly constructed using printed circuit boards, this structure can also be built using other materials as long as there is a conductor separated from. For arbitrary lengths of transmission line it is more complex, but if you. 5, determines the magnitude and phase of the reflected wave given the incident wave, the characteristic impedance of the transmission line, and the terminating impedance. , the resistive component is zero), the input impedance is: Z 0,β A ZL=Z0 in 0 ZZ=Example. 1- Assume the load is 100 + j50 connected to a 50 ohm line. The phase is mostly + 90 ∘ or − 90 ∘, indicating that Zin is mostly reactive. a. These quantities - source impedance, line impedance and load impedance, respectively - must equal one another (impedance matching]) to ensure the efficient transmission of power. Use the R, L, C, G parameters from the Telegrapher’s equations to calculate the impedance of the transmission line Build a model from experimental data of impedance vs. Reference: R. Lumped impedance matching design is of three basic types: series matching, parallel matching (or a combination of the two), and transmission line matching. L/2 L/2 C L C/ 2 C/ 2 Z0 , τ L = τ Z0 C = τ/ Z0 τ=A/vp Let’s approximate a shunt inductor with a transmission line section. Series Impedance –accounts for series voltage drops Resistive Inductive reactance c. 5 ȷ Ω to a load [Math Processing Error] Z L = 50 − 50 ȷ Ω, as shown in Figure [Math Processing Error] 6. 7 gives 49. 4. In Figure 3. e. 01x and 0. Recall that if an impedance is of the form z1 = 1 + iX, then we can exactly match it using a series inductor or capacitor. Frequency-dependent transmission line behavior can also be introduced by loss, as discussed in Section 8. This seems to be a point not fully resolved(inductors and capacitors), transmission line and impedance transformers. Shunt Capacitance –accounts for Line-Charging Currents d. 1. Enter values for W and L for a microstrip line to determine its Zo and Electrical Length. Impedance matching is important for the following reasons: maximizing the power transfer. The circuit got its name because the inductor and the capacitor form a Pi symbol (see schematic below). The input impedance is purely real since the line length is one quarter wavelength: This impedance is used to calculate the depth ( D) into the patch antenna. ; Z 0 is the characteristic impedance of free space (approximately 377 ohms). 1x the wavelength of the signal. There are four different types of impedance used in characterizing differential trace impedances. To determine the characteristic impedance of the parallel-plate waveguide first calculate the voltage of the top plate with respect to the bottom plate. 54 + 0. If the source impedance, load impedance and transmission line characteristic impedance are purely resistive, then reflection-less matching is the same as maximum power transfer matching. Note the stub is attached in parallel at the source end of the primary line. Every transmission line in your PCB has a particular characteristic impedance and it is the ratio of the transmission signal’s voltage amplitude to its current amplitude as it travels over a conductor. Figure 4-1: Impedance matching calculator from EEWeb. 4. 19. 6. With our handy calculator, you can choose from 50 Ohm, 75 Ohm, 93 Ohm and 95 Ohm coaxial cables. The result here reduces to any odd multiple of 1/8th the signal’s wavelength. Determination of the propagation constant from the input impedance. Frequency in Ghz. Most importantly, it computes the critical length of the trace. For a short-circuited stub, the. Calculate Monopole Impedance. matching and w will discuss some of them belo a QuarterW a v e T ransformer A quarter w a v. The impedance to be synthesized is reactive so the termination must also be lossless. Modeling approximation can be used to design the microstrip trace. That is, we want ZL=Z0 (or Zin=Z0). Maximum power is delivered to a load when the transmission line is matched at both. Calculates the resistor values, attenuation, minimum attenuation, 'impedance', reflection coefficient, VSWR and return loss of a matching Pi attenuator. The purpose of integrating output-damping resistors in line buffers and drivers is to suppress signal undershoots and overshoots on the transmission line through what is usually referred to as line-impedance matching (see Figure 1). 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. In Figure 3. By using the provided formula and examples, as well as addressing common questions, you can confidently work with transmission. On the wavelength scale, points z L and A correspond to 0. 45 for gold. Impedance (Z) at any single point on a lossless transmission line (for stub. I'm trying to understand how to calculate the total impedance of a 3-phase overhead transmission line. For example, Let’s assume an impedance of Z = 0. Assuming you chose a transmission line topology, then you would need to compute the electrical lengths and impedances to use. 39 for the effective dielectric constant. In practice, we must use matching networks at one or both ends of the transmission line to achieve the desired type of matching. 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. 331 1 2 5. Input impedance (Ohms) Desired output impedance (Ohms)Transmission-line impedance matching circuits are used at higher frequencies where the lumped elements become very small and impractical to use. Draw a circle through this point around the center. This allows us to use a single transmission line element as the impedance-matching network. Altium Designer’s differential line impedance calculator will set up your impedance-controlled differential pair routing widths for you. To arrive at the correct positive value, simply add 180 degrees to the calculated result. Design a match consisting of a transmission line in series with a single capacitor or inductor that matches a source impedance of 50Ω to a load impedance of 33. 98mil. 4 6. 338λ - 0. 6 Ω at 1. In this paper, the uncertainty and the impact of imperfect load calibration standard for on-wafer Through-Reflect-Match calibration method are analyzed with the help of 3D electromagnetic simulations. The input impedance is the ratio of input voltage to the input current and is given by equation 3. Single Stub Matching using parallel connection. They consist of shorted or opened segments of the line, connected in parallel or in series with the line at a appropriate distances from the load. antenna impedance, Za*SU, shunted by the inductance impedance, j X , of the short transmission line formed by the added gamma rod as seen at the input end of the gamma rod. 1: Cancellation of reflections on transmission lines. These include series, shorted shunt section, and open shunt section. Here, the antenna impedance matches the transmitter output impedance. If the impedance of the load matches that of the source, impedance matching is not required, and the balun has a 1 : 1 impedance ratio. To achieve perfect matching, we want the antenna or load impedance to match the transmission line. To locate this point the arcs corresponding to the real and imaginary parts of (z_{L}) are considered. 72mil. To verify the design, assemble a circuit using 50-Ohm microstrip transmission lines for the matching networks. The value of Z is determined by using the equation for the input impedance of a terminated transmission line. The characteristic impedance (Z o) of coplanar waveguide with ground or microstrip lines with signal side ground plane can be calculated using the active calculator or the equations at the bottom of the page. This online microstrip impedance calculator calculates the characteristic impedance of the microstrip (Ω), W/H ratio, and effective dielectric constant (εe) by entering the value of dielectric material relative permittivity (εr),. 1 Reflectionless and conjugate matching of a. If we add another infinitesimal section to this infinite ladder network, the input impedance should remain unchanged. The CST Cable Studio Impedance Calculator can be started from Cables > Impedance Calculator. This means that a transmission line can act as an impedance-matching. To find the length of the stub, we should consider the impedance of point A. Develop a two-element matching network to match a source with an impedance of [Math Processing Error] Z S = 12. 1. Insert an L or C to match impedance, Why no R? Avoid Ohmic Losses 2. *Optimal D/d ratio for power transfer is 1. 66, the trace height is 1. c is the speed of light in a vacuum (approximately 3 × 10^8 meters per second). It supports both balanced and unbalanced lines. Figure 2. The reflection at point z on the line for a taper segment of length Δz is (refer to Figure 7. An example follows. Value. 2000Ω (500 turns/1000 turns)2= 2000Ω (0. One of the simpler ap-plications is to determine the feed-point impedance of an antenna, based on an impedance measurement at the input of a random length of transmission line. 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. The Transmission Line Calculator is a powerful tool for understanding and analyzing transmission line properties, including characteristic impedance, resistance, inductance, conductance, and capacitance per unit length. ; W is the width of the conductor (in meters). The TEM mode exists down to DC. Use several of them and. Impedance matching is designing source and load impedances to minimize signal reflection or maximize power transfer. The functions reflcoef( ), lineleng( ), zin( ), yin( ), and vswr( ) are created. In this section the theory behind the synthesis of a taper is developed beginning with the theory of small reflections. (In fact, there are installations that use such transformers at both the shack and tower ends of long runs of 450-Ohm transmission line to reduce losses that would be incurred with an all-coax system. Even and Odd Mode Impedance. This causes reflections and results in a low return loss. A transmission line can be used instead (Fig. In this case, the line length will always be 1/8th of the signal wavelength: Complex-to-real impedance transformer line length when the matching line’s impedance equals the magnitude of the load impedance. Instructions. Single Stub Matching. When Z = Z2 is real, then the lengths of the matching sections. 4: Normalized even-mode and odd-mode effective. Here's a spreadsheet that performs via-hole inductance for microstrip circuits. At resonance the impedance takes on a minimal value. 16. Build a model from experimental data of impedance vs. F. transmission line impedance, ΓL= 0 and we have zero reflection. Example 3. This parameter is used for the loss calculations. 1: 300-to- 50 Ω match using an quarter-wave section of line. This free online interactive Smith chart tool is a calculator which can help you design matching networks and obtain maximum power transfer between your source and load. In this case, the line length will always be 1/8th of the signal wavelength: Complex-to-real impedance transformer line length when the matching line’s impedance equals the magnitude of the load impedance. With a two-section cascaded quarter-wave transformer an appropriate matching network is shown in Figure 7. Unless RZ L = 0, the resistor is mismatched to the line, and thus some of the incident power will be reflected. The term “impedance matching” is rather straightforward. zo depends upon the geometry and the material of. Once the dialog is open a segment to use for the impedance calculation can be defined by either: Selecting it in the dialog opened by "Segment List". The width of the conducting layer is important to calculate the radiation intensity since it directly affects. Commonly fabricated with printed circuit board (PCB) technology, a microstrip antenna calculator tool is an electrical transmission line that is. Feedline Impedance (Z (Feedline)) Ohms. Calculates the. 1. 19. Solved Example. 57 pF can be used. or any device with a web browser. If you look through search engine results, you’ll find plenty of links to calculators that can help you properly match the antenna tuner load impedance with your trace. 75. The Smith chart is commonly used to impedance match RF devices, such as amplifiers, antennas and transceiver devices. This transmission line calculator calculates the impedance at input and attenuation of a given transmission line once terminated with the given load. For better impedance control, use PCB design software you can trust. For example, if the source and load are both real impedances, then we can move from high/low impedance to low/high impedance by adding a quarter wave line. This calculator is designed to calculate the characteristic impedance of a broadside couple stripline (see diagram below). 1: A source with Thevenin equivalent impedance ZS Z S and load with impedance ZL Z L interfaced by a matching network presenting an impedance Zin Z in to the source. This calculator is designed to calculate the characteristic impedance of a microstrip - a flat conductor suspended over a ground plane with a dielectric between them (see diagram below). So for the special case of a quarter-wave section where you can pick the transmission line impedance, this is how you can match two. Coaxial Cable Impedance Calculator. This has been found practical for testing 50-ohm to 75-ohm transformers, for which matching pads are readily available. 1. The actual input impedance to the terminated line isSo, the dielectric at my 5-6GHz frequency range is 3. into the study transmission lines having voltage and current along the line in terms of 1D traveling waves. 6 Summary. Transmission line calculator. An open circuit, which is difficult to realize, can also be used in this. In Figure 6, we implicitly assumed that the impedance of the signal source (not shown) is matched to the line characteristic impedance. 25 Ω, for the purpose of providing broadband matching for an ultralow-noise SQUID amplifier. An example of an infinitely long transmission line. Quarter-wave impedance transformer placed between a transmission line with impedance Z0 and load with impedance ZL. Figure 5. ) S21 and S11 0. 5. At low frequencies near 0 GHz, the input impedance is inductive since. Transmission of electrical energy from the source to the load is done using a transmission line. Some systems use 75 Ω; this latter value is more appropriate for high-speed digital signals. 1. . 1 For Microstrip Fed Patch Antenna The Transmission Line model to represent the microstripCoax Impedance (Transmission Line) Calculator. Simons, "Coplanar Waveguide Circuits, Components, and Systems", Wiley. Figure 7 · Impedance at R S port for the two matching options of Fig. With a two-section cascaded quarter-wave transformer an appropriate matching network is shown in Figure 7. Topic 56: Characteristic Impedance One of the most basic parameters of a transmission line is zo, its characteristic impedance. Fig. They consist of shorted or opened segments of the. To verify the design, assemble a circuit using 50-Ohm microstrip transmission lines for the matching networks. Description. A great example is the surface microstrip which forms on the outer layers of the PCB. This example is to design a single section quarter-wave transformer to match the 100 Ω load to a 50 Ω transmission line at an operating frequency of 2 GHz. Relative Dielectric Constant ( εr ): Track Width: mm. Feel free to specify a transmission line with an impedance of 32 Ohms and a velocity factor of 27%. I used an online. This input impedance Zin can be altered by selection of the Z1, so that Zin=Z0 and the antenna is impedance matched. The input impedance, Zin, of the shorted microstrip line is shown in Figure 3. Figure 6. In transmitting systems with an antenna separated from the transmitter and connected to it by a long transmission line , there may be another matching network (ATU) at the antenna that matches the transmission line's impedance to the antenna. Single Stub Matching. Figure 25. When with an impedance matching calculator, you need to consider whether you will be working with a chip antenna, printed antenna, or an external. The first thing to know when calculating the impedance of a wire is whether it's a coaxial or twisted pair cable. ; H is the height of the dielectric layer (in meters). In this ideal scenario, no power is lost, and the VSWR value is 1 : 1, which signifies 100% transmission of the input power from the source. Step 2: Use a shunt (series) reactive element to resonate with (or cancel) the imaginary part of the impedance that results from Step 1. For example, if the load impedance is , the transmission-line impedance is , the magnitude of the reflection coefficient is 0. Transmission Line Calculator March 6, 2007. Stub Matching - Impedance (Z) over a 180 degree cycle along a lossless transmission line. Frequency. 2 (b). 41)] * ln. The line imped-ance is the characteristic impedance of the transmission line and is related to the physi-cal construction of the line. tline is intended to calculate various transmission line parameters. All the math involved is well explained and documented by Dan AC6LA on his site. 4 Comments. Example 3. Most RF systems are built around 50 Ω impedance. First, determine if the microstrip line is a suitable choice by analyzing the default microstrip transmission line at a design frequency of 1. Antenna Element Calculator. It’s simply defined as the process of making one impedance look like another. Impedance Matching Transmission Line. One of the features of this type of microstrip is the coupling between lines. It renders fairly accurate results suitable for use in circuit board manufacturing and engineering analysis. The coaxial cable, along with the balanced two-wire, is the most common type of transmission line used in RF communications. Some of its key features are; displays insertion. The coaxial cable, along with the balanced two-wire, is the most common type of transmission line used in RF communications. 23. com Antenna/filter, power combiner/divider, coupler. The matching network is ideally lossless, to avoid unnecessary loss of power, and is usually designed so that the impedance seen looking into the matching network is Z0. With the (antenna + impedance matching network) designed to match a target impedance of the feedline, the next step is to ensure the input impedance also matches 50 Ohms. Both cases will be considered. Calculate the nominal impedance of the loudspeaker required for maximum power transfer. Relative Dielectric Constant ( εr ): Track Width. This impedance taper was first described by R. Input Impedance of a Transmission Line with a Load Impedance The Smith Chart graphically maps S11=reflection coefficient= (ZL-Z0)/ (ZL+Z0) to Load Impedance (ZL), normalized to Z0 (source impedance = center of chart). This section discusses matching objectives and the types of matching networks. In Section 3. A simple equation relates line impedance (Z 0 ), load impedance (Z load ), and input impedance (Z input) for an unmatched transmission line operating at an odd harmonic. It’s particularly useful for determining the characteristic impedance. For question 2: your transmission line impedance should match both the load and the source impedances in order to deliver maximum power to the load. reducing reflected power toward generator. , one that does not use length), you only need to know impedance and velocity. Calculate the Transmission Line impedance Z0 for a large variety of mechanical structures. It consists of a quarter-wavelength transmission-line, whose input impedance can be easily calculated using the expression. Definition. 7. While transferring this energy, it is important to zero or minimize energy losses that occur. Can impedance matching? Impedance matching to minimize reflections is achieved by making the load impedance equal to the source impedance. The edge couple differential microstrip transmission line is a common technique for routing differential traces. Jan. The characteristic impedance and phase velocity of the transmission line are 50Ω and 0. 19. Pick a signal frequency for your taper. Impedance Analysis Calculator. 29-31. It’s not an arbitrary number; there are good technical reasons. 0. The following formula is used to calculate the characteristic impedance of a Quarter Wave Transformer. In later discussion there comes up the question as to whether the stepped up impedance should be Za*SU or Za*SU/2. 32GHz. 338λ - 0. Search for microstrip (or stripline) impedance calculator. L L ZjX= If the load is purely reactive (i. Example [Math Processing Error] 6. . Shunt Conductance –accounts for V2G losses due to leakage currents between conductors or between conductors and ground. 7). 4mil and the dielectric thickness to the first intermediate layer (this is a 4-layer board) is 6. . Single Stub Matching using parallel connection. The default units for all given values, except the. The wire pair is known as a transmission line. It presents at its input the dual of the impedance with. 4. The Excel based tool (see link below) calculates the capacitance and inductance per unit length as well as the. 33 mA; end: V=0, I=13. This will also provide the time delay the cable provides for a signal and also the capacitance and inductance per. 1 7. Then use an L-pad calculator like this: - You can double check the formula derivation on that site. e. . You can also specify if the circuit will pass direct current or block it. The impedance (50 Ohms) and velocity factor (66%) have been automatically filled in, but they can be edited if desired. trace geometry, and. 2. g. Specify the source (generator) impedance, the reference (transmission line) impedance and the load (antenna) impedance. 1 Reflectionless and conjugate matching of a. stubTL4 = rfckt. Access Rogers' calculators, literature, technical papers and request samples on your smart phone or tablet through the ROG mobile app. Using the characteristic impedance, we can define the current in terms of the voltage. Klopfenstein in a paper titled A Transmission Line Taper of Improved Design, published in the Proceedings of the IRE, page 31-35, January 1956. Figure 6. 86 L ≈ 9. , the resistive component is zero), the input impedance is: Z 0,β A ZL=Z0 in 0 ZZ=The impedance is to be measured at the end of a transmission line (with characteristic impedance Z0) and Length L. For complex impedance matching, the input impedance is important as this is the impedance seen by a signal as it enters the differential pair. Figure 1: An impedance matching network matching an arbitrary load impedance to a transmission line Impedance matching is needed according to [1, 5] for the following reasons: i. In coaxial cable or two-wire line applications. 1. This voltage is the integral of the electric field between the plates: [Math Processing Error] since [Math Processing Error] is a constant. Zin = ZS Z in = Z S. Good impedance matching decrease the. The chart is used during RF simulation by RF engineers and is displayed by vector network. on the line. The MWI-2017 Microwave Impedance Calculator software doesn’t. Why impedance matching is important here is if the load impedance and the cable's characteristic impedance are the same (Zload = Zo), theoretically, the transmitter sees the load as if the transmission line. end Resonances on shorted transmission line . This tool is javascript so it works on Windows, Mac, IOS, Android. Transmission line model: Calculating Impedance and S-parameters Differential impedance: 95Ωat 0. You can easily construct your own transmission lines with a. The radius of the circle is the reflection coefficient G , where the radius to the edge is 1. 2 Matching using Cascaded Transmission Lines and Constant Q Circles. 6. the transmission line characteristic impedance 0 Z. In addition to the characteristic impedance of a transmission line, the tool also calculates. Next, substitute the values into the formula: Z0 = sqrt ( (R + jwL) / (G + jwC)). Figure 7. 6c respectively. When the load impedance and source impedance are equal to the transmission line characteristic impedance, reflectionless matching functions as maximum power transfer matching as well. The radius of the circle is the reflection coefficient G , where the radius to the edge is 1. A quarter-wave impedance transformer is a useful and simple matching network that is used to match the impedance of a terminating load (Z L) to the characteristic impedance of a feeding transmission-line (Z 0) . Realization of Accurate Load Impedance Characterization for On-Wafer TRM Calibration. Transmission line v1 1 0 ac 1 sin rsource 1 2 75 t1 2 0 3 0 z0=75 td=1u rload 3 0 1u . The plots show the magnitude and phase of the input impedance. whatsapp no +923119882901If you want to design a project/need help/teach you email me [email protected]: Impedance-matching using a quarter-wavelength transmission line. Figure 25. Output-Damping Resistors. microstrip; analyze (stubTL4,freq); Z0 = stubTL4. One thing is for sure: When you are routing a high-speed PCB design you are going to be working with impedance-controlled routing and. L1 Z1, τ1 L1 = Z1τ1Z0/Za = Zb/Z0 Z 0 / Z a = Z b / Z 0. In effect, by choosing a source reactance that is equal but opposite to the load. 1. Transmission line equivalents—All designs using lumped elements may use transmission line elements, as. 1. Physical chemistry) Quantum mechanics. Figure 1-1: A load connected to a transmission line through a matching network. 16 a), filters, and other devices entirely from transmission lines, with fewer or. Most RF test and measurement equipment manufactured today has a characteristic impedance of 50 Ω. This can be set to zero, but the calculated loss will not include conductor losses. The series representation of the circuit in Figure 1. 16. If you try to find the admittance of the load with the series transmission line at this point, you get something like: YL = 1 50 + jK = 50 K2 + 2500 − jK K2 + 2500 Y L = 1 50 + j K = 50 K 2 + 2500 − j K K 2 + 2500. Whatever the value, it is known as the characteristic impedance of the wire pair. Regier [6] gave a generalization that permits matching a transmission line of (real) impedance Z1 to a complex load impedance Z = R+iX,whereR is the load resistance and X is the load reactance. The voltage reflection coefficient Γ, given by Equation 3. To calculate the quarter wave transformer impedance, multiply the load. The end of the stub is not connected to. e. e. At this point, it has been carried out a general analysis to determine the. 22. 1. b.