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      <text>25.2.3.1 Transmission line equations</text>
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<h3> Transmission line equations</h3>
<p> In a long electrical cable or a telephone wire both the current and voltage depend upon position along the wire as well as the time (see figure below).</p>

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<p>It is possible to show, using basic laws of electrical circuit theory, that the electrical current \(i(x,t)\) satisfies the PDE </p>

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            <td>\(\qquad \dfrac{\partial^2i}{\partial x^2}=LC\dfrac{\partial^2i}{\partial t^2}+(RC+GL)\dfrac{\partial i}{\partial t}+RGi\)</td>
            <td>\[(5)\]</td>
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where the constants \(R,L, C\) and \(G\) are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage \(v(x,t)\) also satisfies \((5)\). Special cases of \((5)\) arise in particular situations.
For a submarine cable \(G\) is negligible and frequencies are low so inductive effects can also be neglected. In this case \((5)\) becomes

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            <td>\(\qquad \dfrac{\partial^2i}{\partial x^2}=RC\dfrac{\partial i}{\partial t}\)</td>
            <td>\((6)\)</td>
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which is called the <strong>submarine equation</strong> or <strong>telegraph equation</strong>. For high frequency alternating currents, again with negligible leakage, \((5)\) can be approximated by

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            <td>\(\qquad \dfrac{\partial^2 i}{\partial x^2}=LC\dfrac{\partial^2 i}{\partial t^2}\)</td>
            <td>\((7)\)</td>
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which is called the <strong>high frequency line equation</strong>.]]></text>
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