The Telegrapher's Equations (or just Telegraph Equations) are a brace of beeline cogwheel equations which call the voltage and accepted on an electrical manual band with ambit and time. They were developed by Oliver Heaviside who created the manual band model, and are based on Maxwell's Equations.
Schematic representation of the elementary basic of a manual line.
The manual band archetypal represents the manual band as an absolute alternation of two-port elementary components, anniversary apery an infinitesimally abbreviate articulation of the manual line:
The broadcast attrition R of the conductors is represented by a alternation resistor (expressed in ohms per assemblage length).
The broadcast inductance L (due to the alluring acreage about the wires, self-inductance, etc.) is represented by a alternation inductor (henries per assemblage length).
The capacitance C amid the two conductors is represented by a blow capacitor C (farads per assemblage length).
The conductance G of the dielectric actual amid the two conductors is represented by a blow resistor amid the arresting wire and the acknowledgment wire (siemens per assemblage length).
The archetypal consists of an absolute alternation of the elements apparent in the figure, and that the ethics of the apparatus are defined per assemblage breadth so the account of the basic can be misleading. R, L, C, and G may aswell be functions of frequency. An another characters is to use R', L', C' and G' to accent that the ethics are derivatives with account to length. These quantities can aswell be accepted as the primary band constants to analyze from the accessory band constants acquired from them, these getting the advancement constant, abrasion connected and appearance constant.
The band voltage V(x) and the accepted I(x) can be bidding in the abundance area as
\frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)
\frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x).
When the elements R and G are negligibly baby the manual band is advised as a lossless structure. In this academic case, the archetypal depends alone on the L and C elements which abundantly simplifies the analysis. For a lossless manual line, the additional adjustment steady-state Telegrapher's equations are:
\frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0
\frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0.
These are beachcomber equations which accept even after-effects with according advancement acceleration in the advanced and about-face admonition as solutions. The concrete acceptation of this is that electromagnetic after-effects bear down manual curve and in general, there is a reflected basic that interferes with the aboriginal signal. These equations are axiological to manual band theory.
If R and G are not neglected, the Telegrapher's equations become:
\frac{\partial^2V(x)}{\partial x^2} = \gamma^2 V(x)
\frac{\partial^2I(x)}{\partial x^2} = \gamma^2 I(x)
where
\gamma = \sqrt{(R + j \omega L)(G + j \omega C)}
and the appropriate impedance is:
Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}.
The solutions for V(x) and I(x) are:
V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x} \,
I(x) = \frac{1}{Z_0}(V^+ e^{-\gamma x} - V^- e^{\gamma x}). \,
The constants V^\pm and I^\pm have to be bent from abuttals conditions. For a voltage beating V_{\mathrm{in}}(t) \,, starting at x = 0 and affective in the absolute x-direction, again the transmitted beating V_{\mathrm{out}}(x,t) \, at position x can be acquired by accretion the Fourier Transform, \tilde{V}(\omega), of V_{\mathrm{in}}(t) \,, attenuating anniversary abundance basic by e^{\mathrm{-Re}(\gamma) x} \,, advancing its appearance by \mathrm{-Im}(\gamma)x \,, and demography the changed Fourier Transform. The absolute and abstract locations of γ can be computed as
\mathrm{Re}(\gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,
\mathrm{Im}(\gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,
where atan2 is the two-parameter arctangent, and
a \equiv \omega^2 LC \left \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right
b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right).
For baby losses and top frequencies, to aboriginal adjustment in R / ωL and G / ωC one obtains
\mathrm{Re}(\gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,
\mathrm{Im}(\gamma) \approx \omega \sqrt{LC}. \,
Noting that an beforehand in appearance by − ωδ is agnate to a time adjournment by δ, Vout(t) can be artlessly computed as
V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x }. \,
Schematic representation of the elementary basic of a manual line.
The manual band archetypal represents the manual band as an absolute alternation of two-port elementary components, anniversary apery an infinitesimally abbreviate articulation of the manual line:
The broadcast attrition R of the conductors is represented by a alternation resistor (expressed in ohms per assemblage length).
The broadcast inductance L (due to the alluring acreage about the wires, self-inductance, etc.) is represented by a alternation inductor (henries per assemblage length).
The capacitance C amid the two conductors is represented by a blow capacitor C (farads per assemblage length).
The conductance G of the dielectric actual amid the two conductors is represented by a blow resistor amid the arresting wire and the acknowledgment wire (siemens per assemblage length).
The archetypal consists of an absolute alternation of the elements apparent in the figure, and that the ethics of the apparatus are defined per assemblage breadth so the account of the basic can be misleading. R, L, C, and G may aswell be functions of frequency. An another characters is to use R', L', C' and G' to accent that the ethics are derivatives with account to length. These quantities can aswell be accepted as the primary band constants to analyze from the accessory band constants acquired from them, these getting the advancement constant, abrasion connected and appearance constant.
The band voltage V(x) and the accepted I(x) can be bidding in the abundance area as
\frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)
\frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x).
When the elements R and G are negligibly baby the manual band is advised as a lossless structure. In this academic case, the archetypal depends alone on the L and C elements which abundantly simplifies the analysis. For a lossless manual line, the additional adjustment steady-state Telegrapher's equations are:
\frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0
\frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0.
These are beachcomber equations which accept even after-effects with according advancement acceleration in the advanced and about-face admonition as solutions. The concrete acceptation of this is that electromagnetic after-effects bear down manual curve and in general, there is a reflected basic that interferes with the aboriginal signal. These equations are axiological to manual band theory.
If R and G are not neglected, the Telegrapher's equations become:
\frac{\partial^2V(x)}{\partial x^2} = \gamma^2 V(x)
\frac{\partial^2I(x)}{\partial x^2} = \gamma^2 I(x)
where
\gamma = \sqrt{(R + j \omega L)(G + j \omega C)}
and the appropriate impedance is:
Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}.
The solutions for V(x) and I(x) are:
V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x} \,
I(x) = \frac{1}{Z_0}(V^+ e^{-\gamma x} - V^- e^{\gamma x}). \,
The constants V^\pm and I^\pm have to be bent from abuttals conditions. For a voltage beating V_{\mathrm{in}}(t) \,, starting at x = 0 and affective in the absolute x-direction, again the transmitted beating V_{\mathrm{out}}(x,t) \, at position x can be acquired by accretion the Fourier Transform, \tilde{V}(\omega), of V_{\mathrm{in}}(t) \,, attenuating anniversary abundance basic by e^{\mathrm{-Re}(\gamma) x} \,, advancing its appearance by \mathrm{-Im}(\gamma)x \,, and demography the changed Fourier Transform. The absolute and abstract locations of γ can be computed as
\mathrm{Re}(\gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,
\mathrm{Im}(\gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,
where atan2 is the two-parameter arctangent, and
a \equiv \omega^2 LC \left \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right
b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right).
For baby losses and top frequencies, to aboriginal adjustment in R / ωL and G / ωC one obtains
\mathrm{Re}(\gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,
\mathrm{Im}(\gamma) \approx \omega \sqrt{LC}. \,
Noting that an beforehand in appearance by − ωδ is agnate to a time adjournment by δ, Vout(t) can be artlessly computed as
V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x }. \,
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