Talk:Travelling Wave Line Model

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Footnotes

Derivation of Voltage Equation

The general solution to the transmission line wave equations are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(x, t) = i^{+}(x - vt) + i^{-}(x + vt) \, }
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V(x,t)=v^{+}(x-vt)+v^{-}(x+vt)\,}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{1}{\sqrt{LC}} \, } is the velocity of propagation (m/s).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{+}(t) \, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{-}(t) \, } are arbitrary current functions of time.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^{+}(t) \, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^{-}(t) \, } are arbitrary voltage functions of time.

The Laplace Transform of the above equations are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(x, s) = I^{+}(s)e^{-\frac{sx}{v}} + I^{-}(s)e^{\frac{sx}{v}} \, } ... Equ. (1)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x, s) = V^{+}(s)e^{-\frac{sx}{v}} + V^{-}(s)e^{\frac{sx}{v}} \, } ... Equ. (2)

Recall the Telegrapher's equation for current:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial I(x,t)}{\partial x} = - C \frac{\partial V(x,t)}{\partial t} \, }

Taking the Laplace Transform of this equation, we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial I(x,s)}{\partial x} = - sC V(x,s) \, } ... Equ. (3)

Substituting Equations (1) and (2) into Equ. (3):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial x} \left[ I^{+}(s)e^{-\frac{sx}{v}} + I^{-}(s)e^{\frac{sx}{v}} \right] = - sC \left[ V^{+}(s)e^{-\frac{sx}{v}} + V^{-}(s)e^{\frac{sx}{v}} \right] \, }

We can differentiate the left-hand side:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s}{v} \left[ - I^{+}(s)e^{-\frac{sx}{v}} + I^{-}(s)e^{\frac{sx}{v}} \right] = - sC \left[ V^{+}(s)e^{-\frac{sx}{v}} + V^{-}(s)e^{\frac{sx}{v}} \right] \, }

Equating the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{-\frac{sx}{v}} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\frac{sx}{v}} } terms on both sides, we get the following pair of equations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \frac{1}{v} I^{+}(s) = - C V^{+}(s) \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{v} I^{-}(s) = - C V^{-}(s) \, }

Solving for voltage yields the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{+}(s) = I^{+}(s) Z_{c} \, } ... Equ. (4)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{-}(s) = -I^{-}(s) Z_{c} \, } ... Equ. (5)

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{c} = \frac{1}{vC} = \frac{\sqrt{LC}}{C} = \sqrt{\frac{L}{C}} } is the characteristic impedance (Ohms)

We can use the above equations to re-write the voltage equation of Equ. (2) in terms of current as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x, s) = Z_{c} \left[ I^{+}(s)e^{-\frac{sx}{v}} - I^{-}(s)e^{\frac{sx}{v}} \right] \, }

Finally, taking the inverse Laplace Transform of the equation above, we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x, t) = Z_{c} \left[ i^{+}(x - vt) - i^{-}(x+vt) \right] \, }
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