Editing Characteristic impedance (section)

=== Using the telegrapher's equation ===
{{main|telegrapher's equation}}
[[File:Transmission line element.svg|thumb|Consider one section of the transmission line for the derivation of the characteristic impedance. The voltage on the left would be <math>\ V\ </math> and on the right side would be <math>\ V + \operatorname{d} V ~.</math> This figure is to be used for both the derivation methods.]]
The differential equations describing the dependence of the [[voltage]] and [[Electric current|current]] on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence <math>\ e^{j \omega t}</math>. Doing so allows to factor out the time dependence, leaving an ordinary differential equation for the coefficients, which will be [[phasor]]s, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent.<ref name="Miano">{{cite book | last=Miano | first=Giovanni | last2=Maffucci | first2=Antonio | title=Transmission Lines and Lumped Circuits | publisher=Academic Press | publication-place=San Diego | date=2001 | isbn=0-12-189710-9 | pages=130-135}}</ref><ref>{{cite book | last=Mooijweer | first=H. | title=Microwave Techniques | publisher=Macmillan Education UK | publication-place=London | date=1971 | isbn=978-1-349-01067-7 | doi=10.1007/978-1-349-01065-3 | pages=74-79}}</ref>

Consider a [[steady-state]] problem such that the voltage and current can be written as:
<math display="block">\begin{align}
v(x,t) &= V(x) e^{j \omega t}\\[.5ex]
i(x,t) &= I(x) e^{j \omega t}
\end{align}
</math>
Take the positive direction for <math> V </math> and <math> I </math> in the loop to be clockwise. Substitution in the telegraph equations and factoring out the time dependence <math>\ e^{j \omega t}</math> now gives:
<math display="block">\begin{align}
\frac{\mathrm{d}V}{\mathrm{d}x} &= -\left( R + j \omega L \right) I = -Z I,\\[.5ex]
\frac{\mathrm{d}I}{\mathrm{d}x} &= -\left( G + j\ \omega C \right) V = -Y V,
\end{align}</math>
with impedance <math>Z</math> and [[admittance]] <math>Y</math>. Derivation and substitution of these two [[first-order differential equation]]s results in two uncoupled second-order differential equations:
<math display="block">\begin{align} 
\frac{\mathrm{d}^2 V}{\mathrm{d}x^2} &= k^2 V,\\[.5ex] 
\frac{\mathrm{d}^2 I}{\mathrm{d}x^2} &= k^2 I,
\end{align}</math>
with <math> k^2 = Z Y = ( R + j \omega L)(G + j \omega C )= (\alpha + j \beta)^2 </math> and <math> k = \alpha + j \beta </math> called the [[propagation constant]]. 

The solution to these type of equations can be written as:
<math display="block">\begin{align} 
V(x) &= A e^{-k x} + B e^{k x}\\[.5ex] 
I(x) &= A_{1} e^{-k x} + B_{1} e^{k x}
\end{align}</math>
with <math>A</math>, <math>A_1</math>, <math>B</math> and <math>B_1</math> the [[constant of integration|constants of integration]]. Substituting these constants in the first-order system gives:
<math display="block">\begin{align} 
A_1 &= A \frac{k}{R+ j \omega L}\\[.5ex] 
B_1 &= -B \frac{k}{R + j \omega L}
\end{align}</math>
where
<math display="block">
\frac{A}{A_1} = -\frac{B}{B_1} = \frac{R + j \omega L}{k} = \sqrt{\frac{R+ j\omega L}{G + j \omega C}} = \sqrt{\frac{Z}{Y}} = Z_{0}.
</math>
It can be seen that the constant <math>Z_0,</math> defined in the above equations has the dimensions of impedance (ratio of voltage to current) and is a function of primary constants of the line and operating frequency. It is called the ''characteristic impedance'' of the transmission line.<ref name=":1"/> 

The general solution of the telegrapher's equations can now be written as:
<math display="block">\begin{align}
v(x,t) &= V(x) e^{j \omega t} = A e^{-\alpha x}e^{j(\omega t - \beta x)} +  B e^{\alpha x}e^{j(\omega t + \beta x)} \\[.5ex]
i(x,t) &= I(x) e^{j \omega t} = \frac{A}{Z_0} e^{-\alpha x}e^{j(\omega t - \beta x)} -  \frac{B}{Z_0} e^{\alpha x}e^{j(\omega t + \beta x)} 
\end{align}</math>
Both the solution for the voltage and the current can be regarded as a superposition of two travelling waves in the <math>x_{+}</math> and <math>x_{-}</math> directions.

For typical transmission lines, that are carefully built from wire with low loss resistance <math>\ R\ </math> and small insulation leakage conductance <math>\ G\ ;</math> further, used for high frequencies, the inductive reactance <math>\ \omega L\ </math> and the capacitive admittance <math>\ \omega C\ </math> will both be large. In those cases, the [[phase constant]] and characteristic impedance are typically very close to being real numbers:
<math display="block">\begin{align}
\beta & \approx \omega \sqrt{L C} \\[.5ex]
Z_{0} & \approx \sqrt{\frac{L}{C}} 
\end{align}</math>
Manufacturers make commercial cables to approximate this condition very closely over a wide range of frequencies.