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★ Heaviside condition - transmission lines ..



                                     

★ Heaviside condition

The Heaviside condition, named for Oliver Heaviside, is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the distortionless condition, it can be used to improve the performance of a transmission line by adding loading to the cable.

                                     

1. The condition. (Состояние)

The transmission line can be represented as a distributed-element model to its primary constants, as shown in the figure. The main constants are the electrical properties of the cable per unit length are: the capacity in farads per meter, inductance L in henries per meter, series R is the resistance in ohms per meter, and the shunt conductance G, in Siemens per meter. Series resistance and shunt conductance will cause losses in the line, for an ideal transmission line, R = g = 0 {\the style property display the value \scriptstyle R=g=0}.

The Heaviside condition is satisfied when

G C = R L. {\displaystyle {\frac {G}{C}}={\frac {R}{L}}. }

This condition without distortion, but not without losses.

                                     

2. Background. (Фон)

The signal on the transmission line can become distorted even if the constants and the resulting transmission function, are all quite linear. There are two mechanisms: first, the attenuation of the line can vary depending on the frequency, which leads to a change of the pulse shape transmitted by the line. Secondly, and usually more problematic, the distortion caused by the frequency dependence of the phase velocity of the transmitted frequency components of the signal. If different frequency components of the signal are transmitted with different velocities, the signal becomes "smeared" in space and time, a form of distortion called dispersion.

This was a major problem on the first transatlantic Telegraph cable led to theories about the causes of the scattering is studied first Lord Kelvin, then Heaviside who discovered how it can be reversed. Dispersion of Telegraph impulses, if severe enough, will cause them to overlap with adjacent pulses, the result is what is now called inter-symbol interference. To prevent intersymbol interference, it is necessary to decrease the transmission speed of the transatlantic Telegraph cable, equal to 1 ⁄ 15 BAUD. This is an extremely slow data transfer speed, even for a person who with great difficulty working the Morse key that slowly.

For voice telephone circuits in the frequency distortion, the answer is usually more important than the variance, whereas digital signals are very sensitive to the distortion of the dispersion. For any kind of analog image transmission such as video or facsimile of both types of distortions should be eliminated.

                                     

3. Derivation. (Деривация)

The transfer function of a transmission line is defined in terms of its input and output voltages when correctly, i.e. without reflection

V i n V o u t = e γ x {\displaystyle {\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}=e^{\gamma x}}

where X {\the style property display the value of x} represents the distance from the transmitter in meters and

γ = α + j β {\displaystyle \gamma =\alpha +j\beta \,}

the secondary line constants, and α is the attenuation in nepers per meter and β to be a constant change of phase in radians per meter. No distortion, α is required to be independent of the angular frequency ω, while β must be proportional to ω. This requirement of proportionality to the frequency because of the relationship between the speed V and the phase constant β is given,

v = ω β {\displaystyle v={\frac {\omega }{\beta }}}

and the requirement that phase velocity, V, be constant at all frequencies.

The relationship between primary and secondary line constants is given

γ 2 = α + j β 2 = R + j ω L G + j ω C {\displaystyle \gamma ^{2}=\alpha +j\beta^{2}=R+j\omega LG+j\omega C\,}

which should be in the form of A J in ω B 2 {\the style property display the value of \scriptstyle and in J\omega B^{2}} in order to satisfy the distortion condition. The only way it can be so, if R J ω l {\the style property display the value \scriptstyle R j\omega L} and G J in ω and C {\the style property display the value of \scriptstyle g j\omega s} differ by no more than a real constant factor. As there is the real and the imaginary, the real and imaginary parts must independently be related to the same factor, so that

R G = j ω L j ω C {\displaystyle {\frac {R}{G}}={\frac {j\omega L}{j\omega C}}}

and the Heaviside condition is proved.



                                     

3.1. Derivation. Line characteristics. (Характеристики линии)

The secondary constants of a line meeting the Heaviside condition are consequently, in terms of fundamental constants:

Attenuation

α = R G {\displaystyle \alpha ={\sqrt {RG}}} nepers / metre

The change in the phase constant,

β = ω L C {\displaystyle \beta =\omega {\sqrt {LC}}} radians / metre

The phase velocity,

v = 1 L C {\displaystyle v={\frac {1}{\sqrt {LC}}}} metres / second
                                     

3.2. Derivation. Characteristic impedance. (Характеристический импеданс)

The characteristic impedance of the transmission line with losses is given

Z 0 = R + j ω L G + j ω C {\displaystyle Z_{0}={\sqrt {\frac {R+j\omega L}{G+j\omega C}}}}

In General, it is not possible to match this resistance of the transmission line at all frequencies with any finite network of discrete elements, since such networks are rational functions in JW, but in General the expression for the characteristic impedance irrational because the square root concept. However, on the line which meets the Heaviside condition, there is a common factor in fraction, which extinguishes depending on the frequency conditions allow

Z 0 = L C, {\displaystyle Z_{0}={\sqrt {\frac {L}{C}}},}

which is a real number, and does not depend on frequency. Therefore, the line can be impedance matched with just a resistor at both ends. This expression for z 0 = L / C {\the style property display the value of \scriptstyle Z_{0}={\sqrt function {L / s}}} is the same as for the lossless line R = 0, g = 0 {\the style property display the value \scriptstyle R=0,\ g=0} with the same L and C, although the attenuation of R and G is of course also present.

                                     

4. Practical use. (Практическое использование)

A real line, especially using modern synthetic insulators, will G, which is very low and, as a rule, do not come close to achieving the condition of Heaviside. The normal situation is that

G C ≪ R L. {\displaystyle {\frac {G}{C}}\ll {\frac {R}{L}}. }

To make a line to execute the Heaviside condition one of the four primary constants needs to be adjusted and the question what. G can be increased, but this is highly undesirable, since increasing g will increase the loss. The decrease in R sends loss in the right direction, but this is not usually a satisfactory solution. R should be reduced by a large fraction for this wire size must be increased dramatically. This not only makes the cable much more bulky, but also significantly increases the amount of copper or other metal and therefore the cost. Reduction of capacity also makes the cable more bulky, since the insulation needs to be thicker, but not as expensive as the increase of copper content. This leaves the increase in L, which is usually the decision.

The required increase in L is achieved by loading the cable with a metal with high magnetic permeability. You can also download cable of conventional design by adding discrete loading coils at regular intervals. It is not identical to a distributed load, the difference is that with the coil load is transferred without distortion to a certain cut-off frequency beyond which the attenuation increases dramatically.

Load cables to fit the conditions of Heaviside already not a common practice. Instead, regularly spaced digital repeaters now housed in long queues to maintain a desired shape and duration of pulses for long-distance transmission.



                                     

5. Bibliography. (Библиография)

  • Nahin, Paul J, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age, JHU Press, 2002 ISBN 0801869099. See especially pp. 231-232.
  • Schroeder, Manfred Robert, Fractals, Chaos, Power Laws, Courier Corporation, 2012 ISBN 0486134784.
                                     
  • method Heaviside condition Heaviside layer or Kennelly Heaviside layer Heaviside lunar crater Heaviside Martian crater Heaviside s dolphin, named
  • Oliver Heaviside FRS ˈhɛvisaɪd 18 May 1850 3 February 1925 was an English self - taught electrical engineer, mathematician, and physicist who adapted
  • transmitted signal. The mathematical condition for distortion - free transmission is known as the Heaviside condition Previous telegraph lines were overland
  • circuit using loading coils. Campbell was aware of Heaviside s work in discovering the Heaviside condition in which the specification for distortionless
  • of loading add series inductance to the cable to try to meet the Heaviside condition for no signal distortion. Krarup cable consists of iron wires wound
  • Montana. Oliver Heaviside and Arthur E. Kennelly independently predict the existence of what will become known as the Kennelly - Heaviside Layer of the ionosphere
  • consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response
  • coefficient Heat transmission Heather Couper Heather Reid Heating pad Heaviside condition Heavy Rydberg system Heavy neutrino Heavy water Hedwig Kohn Heidi
  • positive numbers, and is the simplest non - constant step function. The Heaviside function H x which is 0 for negative numbers and 1 for positive numbers
  • off. Most common residential thermostats are bang bang controllers. The Heaviside step function in its discrete form is an example of a bang bang control
  • developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside and Thomas Bromwich. The current widespread use of the transform mainly
  • Maxwell s equations, Heaviside pronounced that longitudinal waves could not exist in a vacuum or a homogeneous medium. Heaviside did not note, however
                                     
  • between the angular brackets. In order to impose this condition Gribov proposed to introduce a Heaviside step function containing the above into the path
  • Equicontinuous Absolute continuity Holder condition condition for Holder continuity Dirac delta function Heaviside step function Hilbert transform Green s
  • the transmission line reached its fullest development with Oliver Heaviside Heaviside 1881 introduced series inductance and shunt conductance into the
  • fails to be a smooth function. For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to be 1 x a
  • often understated but still sharply realized songs and lyrics. Simon Heavisides stated: Isn t it great when your old favourites don t let you down? .
  • bounded support, then we can interpret integration as convolution with the Heaviside function and apply the convolution law. Computing the scalar products
  • Oliver Heaviside and Hertz further developed the theory and introduced modernized versions of Maxwell s equations. The Maxwell - Hertz or Heaviside - Hertz
  • atmosphere. This was the first direct indication of the reality of the Heaviside layer, proposed earlier but at this time largely dismissed by engineers
  • front had been theoretically predicted by the English polymath Oliver Heaviside in papers published between 1888 and 1889 and by Arnold Sommerfeld in
  • they make the Jellicle choice, deciding which cat will ascend to the Heaviside Layer and come back to a new life. The musical includes the well - known
  • Searle noted in 1896 that Heaviside s expression leads to a deformation of electric fields which he called Heaviside - Ellipsoid of axial ratio original
  • described until 1880 by English physicist, engineer, and mathematician Oliver Heaviside who patented the design in that year British patent No. 1, 407 Coaxial
                                     
  • equation in a vacuum using the modern method, we begin with the modern Heaviside form of Maxwell s equations. In a vacuum - and charge - free space, these
  • notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta
  • the logistic function with scaling is a smooth approximation of the Heaviside step function. The standard logistic function is the solution of the simple
  • delta prime function. It is analogous to the second derivative of the Heaviside step function in one dimension. It can be obtained by letting the Laplace
  • included no longer. The vector calculus formalism below, the work of Oliver Heaviside has become standard. It is manifestly rotation invariant, and therefore
  • Charles Proteus Steinmetz  AC mathematical theories for engineers Oliver Heaviside theoretical models for electric circuits During these decades use of

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