Characteristic Impedance of a Cable

9 Key excerpts on "Characteristic Impedance of a Cable"

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  Electromagnetism for Engineers

  • Andrew J. Flewitt(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Normally, when we think of a simple elec- tric circuit component, like a resistor, the impedance is the ratio of the potential difference across the component to the current through the component. The current is moving across the potential difference and therefore the energy of the electrons is changing and energy is dissipated (as heat in the resistor). However, inspection of Figure 6.11 shows that for the characteristic impedance, the voltage is the potential difference between the two con- ductors, but the current is flowing along one conductor – it does not flow directly from one conductor to the other across the potential difference. Therefore, no energy is being dissipated. The characteristic impedance is purely a useful mathematical construction which relates the voltage to the current at a point in the transmission line at some moment in time. A good way of thinking about it is that if we were to connect an a.c. voltage source to one end of an infinitely long transmission line (so that there is only a unidirectional wave travelling in the forward direction) then the characteristic impedance would be the ratio of the voltage and current output by the voltage source. This is the situation shown in Figure 6.11, and so Z 0 = V (0, t)/I (0, t). It is also possible to derive a characteristic impedance for the lossy transmission line using Eq. (6.36) as the starting telegrapher’s equation. We need to substitute into this an expres- sion for generic forward and backward waves. This is simplest mathematically if the prop- agation constant is used (as in Eq. (6.45)). Therefore, the voltage wave can be expressed as V = V F e j𝜔t e −𝛾 x + V B e j𝜔t e 𝛾 x (6.57) Note that it is the sign of the 𝛾 x exponent that determines whether the wave is travelling in the positive x-direction (for e −𝛾 x ) or negative x-direction (for e 𝛾 x ). Similarly, the current expression is I = I F e j𝜔t e −𝛾 x + I B e j𝜔t e 𝛾 x (6.58) Substitution of Eqs.

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  The Global Cable Industry

  Materials, Markets, Products

  • Günter Beyer(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  Figure 12.3 ).

  Of the many various configurations of coaxial cables that are available, shown here are but a few (Figures 12.4 and 12.5 ).

  Figure 12.3

  Typical hardline coax cable used in CATV installation.

  Figure 12.4

  Hybrid coaxial cable.

  Figure 12.5

  Drop coaxial cable, QuadShield. Source: Courtesy of Amphenol Broadband Solutions.

  12.2.2 Principles

  Now that the components and a few of the more basic materials are understood, attention will be focused on the basic principles of the design and function of a coaxial cable. Basically, coaxial lines or cables are a transmission system wherein electromagnetic waves are transmitted through a dielectric insulator bound by two conductive coaxial cylinders. This is accomplished via transverse electromagnetic (TEM ) wave-field propagation within the coaxial cable. In this way, electrical energy, or signals, are transmitted from one location to another. The physics related to the function of coaxial cables is defined and explained in transmission line theory. For the purposes of this chapter, basic cable design formulary will be given and explained as related to general use and function.

  12.3 Characteristic Impedance (Zo)

  Characteristic, or surge, impedance is the combined opposition to the flow of electrical energy within the coaxial line or cable. Impedance is a complex value, derived from the combined equivalent of the resistance, capacitance, and inductive and conductive values of the cable. This parameter is how the coaxial cable is generally defined, partly because it is derived from all of the other electrical parameters of the cable design. Characteristic impedance (Z

  o

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  High-Speed Digital System Design

  • Justin Davis(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  The wire will draw a constant current from the source for an unlimited amount of time. In this way, the infinite wire will act like a simple 26 HIGH-SPEED DIGITAL SYSTEM DESIGN FIGURE 2.3: Distributed model of an infinite wire resistive load from the perspective of the voltage source. The resistance of this set of wires is called the characteristic impedance measured in ohms. Coaxial cable is often rated in 50 , or 75 , because of its characteristic impedance. This impedance is determined by the geometries and distance of the wires. The set of wires is collectively known as a transmission line. The characteristic impedance is set by the geometry of the two wires. If the separation of the conductors is increased, the capacitance is decreased and the inductance is increased. This will result in a reduced constant current being drawn from the voltage source, which acts like an increased resistance. If the separation is decreased, the opposite effect will occur and the characteristic impedance will decrease. While in this example the inductance and capacitance is a series of finite elements, the actual transmission line is measured in instantaneous capacitance and inductance. For a specific geometry, there will be a capacitance per unit length and inductance per unit length. The equation to determine the characteristic impedance is Z 0 = L l C l (2.3) where Z 0 is the characteristic impedance in ohms; C / l is the capacitance per unit length; L / l is the inductance per unit length. The unit length measurements will cancel out leaving the equation Z 0 = L C . (2.4) Note that for this equation the characteristic impedance is not dependent on the length of the transmission line. IDEAL TRANSMISSION LINES 27 Practical values for characteristic impedance on PCBs are usually either 50 or 75 . If the signals will be entering/leaving the PCB by means of cables or connectors, those connectors should have the same impedance.

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  Radio and Line Transmission

  The Commonwealth and International Library: Electrical Engineering Division, Volume 2

  • Dermot Roddy, N. Hiller(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The series resistance and inductance per unit length of line, r and L, and the shunt conductance and capacitance per unit length, G and C, are termed the primary constants, as these are determined by the dimensions of the lines, i.e. the radii of the conductors and their spacing, and by the conductivities, permittiv-ities, and permeabilities of the conductor material and the dielec-tric material. From these primary characteristics, two further characteristics, known as secondary characteristics, may be derived for any given frequency of operation. The secondary characteristics are the characteristic impedance Z 0 and the propagation coefficient γ (γ is the Greek letter gamma), and these are of more practical use in determining the value of load impedance TRANSMISSION LINES A N D CABLES 55 to be used with the line, and also the line losses. The propagation coefficient is dealt with in § 3.7. The characteristic impedance of a line is defined as the ratio of voltage to current at any section of an infinitely long line. Although such a line is not a practical possibility, it can be easily simulated in practice, as will be shown, and that doing so usually leads to the most efficient transmission of signals along the line. It is assumed throughout that the line is uniform, i.e. the primary characteristics do not change along the line. A voltage V applied to the input terminals of an infinitely long line will result in a current J flowing which is determined only by the primary constants of the line (i.e. the load terminating the line need not be considered). If now a section of the line is removed and voltage V applied to the remaining line, which is still infinitely long, the current must have the same value / as before. The ratio V/I is the characteristic impedance Z 0 of the line.

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  Advanced Signal Integrity for High-Speed Digital Designs

  • Stephen H. Hall, Howard L. Heck(Authors)
  • 2011(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  (3-25) :

  (3-32)

  Since the units of C are F/m = A · S/m and L is in units of H/m = V · S/A, the term has units of A/V. which is the same as siemens. Therefore, the quantity must be similar to the intrinsic impedance defined in Section 2.3.4. The characteristic impedance of a loss-free transmission line in ohms is given by

  (3-33)

  Just as the intrinsic impedance describes the relationship between the electric and magnetic fields, the characteristic impedance describes the relationship between the voltage and current on a transmission line. Thus, we can write the transmission-line equation for the current in terms of the voltage equation:

  (3-34)

  3.3.3 Effective Dielectric Permittivity

  As described in Section 3.3.1, electrical signals propagating along a transmission line will travel at a speed that is dependent on the dielectric permittivity of the surrounding medium and the geometry of the transmission-line cross section. Although this concept is relatively simple, it becomes complicated when a transmission line is built with a nonhomogeneous dielectric. The most common example of this is a microstrip transmission line. For example, consider Figure 3-11a , which shows the electric field of a signal on a microstrip transmission line built on an FR4 dielectric, which has a relative dielectric permittivity of approximately

  ε r =

  4.0. Note that a percentage the electric field is propagating simultaneously in the dielectric and in the air, where the latter has a relative dielectric permittivity of approximately

  ε r

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  Electromagnetic Theory for Electromagnetic Compatibility Engineers

  • Tze-Chuen Toh(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  □ Observe trivially from the above derivation that a transmission line can be represented by the distributed parameter model illustrated in Figure 4.3. Indeed, from the equivalence between ( v , i ) and ( E , B ), it follows that volt-age and current are subject to reflection at a boundary interface, just as electric and magnetic fields are. This topic is investigated in subsequent sections. R L R L G C FIGURE 4.3 A distributed lumped parameter model representing a pair of transmission lines. 108 Electromagnetic Theory for Electromagnetic Compatibility Engineers 4.3 Characteristic Impedance and the Smith Chart In this section, the concept of characteristic impedance is established. It has special significance in transmission line theory: to wit, the characteristic impedance is independent of the length of the conductors. This is established below. In Section 4.2, infinitely long conductors were studied. In contrast, finite conductors are investigated here. The motivation for this section runs as follows. • Under what circumstances do finite conductors appear to be infi-nitely long conductors to a propagating TEM wave? • What happens to an incident voltage and current at a boundary interface? • What is the impedance along a transmission line from the perspec-tive of an incident propagating wave? For simplicity, consider a propagating time-harmonic TEM wave. Then, phasor transforming ( v , i ) → ( V , I ), Equations (4.3) and (4.4) can be rewrit-ten as = --ω = -+ ω I z I z I z V z z R ( ) i L ( ) (R i L) ( ) d ( ) d (4.12) = --ω = -+ ω V z V z V z I z z G ( ) i C ( ) (G i C) ( ) d ( ) d (4.13) The equations are now a pair of coupled first-order ordinary differential equations.

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  Radio-Frequency Electronics

  Circuits and Applications

  • Jon B. Hagen(Author)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  C H A P T E R 10 Transmission lines We draw circuit diagrams with “lumped”components: ideal R’ s, C’ s, L’ s, tran- sistors, etc., connected by lines that represent zero-length wires. But all real wires, if not much shorter than the shortest relevant wavelength, are themselves com- plicated circuit elements; the current is not the same everywhere along such a wire, nor is voltage uniform, even if the wire has no resistance. On the other hand, when interconnections are made with transmission lines, which are well- understood circuit elements, we can accurately predict circuit behavior. In this section we will consider two-conductor lines such as coaxial cables and open parallel wire lines. “Microstrip lines” (conducting metal traces on an insulation layer over a metal ground plane) behave essentially in the same way, but they have some subtle complications, which are mentioned in Appendix 10.1. 10.1 Characteristic impedance The first thing one learns about transmission lines is that they have a parameter known as characteristic impedance, denoted Z 0 . How “real” is characteristic impedance? If we connect an ordinary dc ohmmeter to the end of a 50-ohm cable will it indicate 50 ohms? Yes, if the cable is very long, so that a reflection from the far end does not arrive back at the meter before we finish the measure- ment. Otherwise, the meter will simply measure whatever is connected to the far end, which could be short, an open circuit, or a resistance. However, using a pulse generator and an oscilloscope, you can easily make an ohmmeter set-up that is fast enough that, even for a short cable, you can determine V in and I in and then calculate V in /I in = Z 0 . To make a theoretical determination of Z 0 , we first model the transmission line as a ladder network made of shunt capacitors and series inductors, as shown in Figure 10.1. = Lδz Lδz Lδz Cδz Cδz Cδz Figure 10.1. Transmission line model – a ladder network of infinitesimal LC sections. 101

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  Principles of Communications Networks and Systems

  • Nevio Benvenuto, Michele Zorzi, Nevio Benvenuto, Michele Zorzi, Nevio Benvenuto, Michele Zorzi(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Figure 4.16b ). This configuration minimizes electrical and radio frequency interference. Coaxial cabling is primarily used by the cable television industry and is also widely used for computer networks. Although more expensive than standard telephone wire, it is much less susceptible to interference and can carry an higher data rate.

  A widely used model for transmission lines is reported in Figure 4.17 , illustrating a uniform line segment of length dx where x denotes the distance from the origin. The parameters that determine the line characteristics are the primary constants r , , c and g which define, respectively, resistance, inductance, capacitance and conductance per unit length of the line.

  Figure 4.17 Transmission lines: line segment of infinitesimal length d .

  The input and output line impedances coincide and are related to the characteristic impedance of the transmission line, given by

  (4.93)

  A commonly used expression for the two-port network frequency response is

  (4.94)

  where d (for distance) is the length of the line and where and are the real and imaginary parts of , with the attenuation constant , measured in neper per unit length, and the phase constant , measured in radians per unit length. So, if the simplified relation (4.43) holds between available power gain and system frequency response, from (4.94) the power gain becomes

  (4.95)

  All the above expressions are valid for both coaxial and twisted-pair cables insulated with plastic material, as long as we correctly choose the parameter values. Table 4.2 gives the characteristic quantities (experimentally measured 4 ) for some telephone transmission lines characterized by their diameter, which is usually indicated by a parameter called gauge .

  Table 4.2 Characteristic constants for some telephone lines.

  The most common model for takes into account the variations of r (resistance per unit length) with the frequency, due to the skin effect. With these assumptions, the attenuation and phase constants are of the form 3

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  Electronic Devices and Circuits

  In Three Volumes

  • G. J. Pridham, N. Hiller(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  If 2 V are applied to the sending end. calculate the voltage 10 miles away if the line is termi-nated in its characteristic impedance. Q.5. A length of loss-free coaxial cable of characteristic impedance 80 Ω is to be employed to provide an inductive reactance of 500 Ω at 1000/2π MHz. Determine the length of line required (a) if it is short-circuited, (b) if it is open-circuited. Q.6. A short-circuited transmission line of characteristic impedance 600 z_0° Ω has an attenuation coefficient of 01 nepers per mile and a phase change coefficient of 0Ό5 radians per mile. If the sending end current is 10 mA and ω = 5000 rads/sec, determine : (a) sending end voltage, (b) receiving end current, (c) resistance per unit length, (d) inductance per unit length, (e) capacity per unit length, (i) leakance per unit length. Q.7. A quarter wavelength line is short-circuited at the receiving end. If the resis-tance is 002O/cm the capacitance 0-2 x 10~ 12 F/cm the inductance 5 x 10 9 H/cm and the leakance is zero, determine the input impedance at a frequency of 100 MHz. Q.8. A line of characteristic impedance 100 Ω (resistive) is terminated in 100 — /Ί00 Ω. Determine the voltage standing wave ratio and the distance of the voltage node (or minimum) nearest the load. Q.9. A 300-Ω air-speed balanced line connects a 100 MHz transmitter to an aerial of impedance 75 L 0° Ω. If the power delivered is 300 W, determine the maxi-mum or minimum voltages and currents on the line and the voltage standing wave ratio. Q.10. The line described in Q.9 is to be matched by (a) a quarter-wave trans-former, (b) a short-circuited stub. Determine the characteristic impedance of the first and the position and length of the second.

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