Editing Characteristic impedance (section)

==== Intuition ====
{{see also|Iterative impedance|Constant k filters}}

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Consider an infinite [[ladder network]] consisting of a series impedance <math>\ Z\ </math> and a shunt admittance <math>\ Y ~.</math> Let its input impedance be <math>\ Z_\mathrm{IT} ~.</math> If a new pair of impedance <math>\ Z\ </math> and admittance <math>\ Y\ </math> is added in front of the network, its input impedance <math>\ Z_\mathrm{IT}\ </math> remains unchanged since the network is infinite. Thus, it can be reduced to a finite network with one series impedance <math>\ Z\ </math> and two parallel impedances <math>\ 1 / Y\ </math> and <math>\ Z_\text{IT} ~.</math> Its input impedance is given by the expression<ref>{{cite book |title=The Feynman Lectures on Physics|title-link=The Feynman Lectures on Physics |volume=2 |first1=Richard |last1=Feynman |author1-link=Richard Feynman |first2=Robert B. |last2=Leighton |author2-link=Robert B. Leighton |first3=Matthew |last3=Sands |author3-link=Matthew Sands |section=Section 22-6. A ladder network |section-url=https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S6}} </ref><ref name=lee2004/>

:<math>\ Z_\mathrm{IT} = Z + \left( \frac{\ 1\ }{ Y } \parallel Z_\mathrm{IT} \right)\ </math>

which is also known as its [[iterative impedance]]. Its solution is:

:<math>\ Z_\mathrm{ IT } = {Z \over 2} \pm \sqrt { {Z^2 \over 4} + {Z \over Y} }\ </math>

For a transmission line, it can be seen as a [[Mathematical limit|limiting case]] of an infinite ladder network with [[infinitesimal]] impedance and admittance at a constant ratio.<ref name="feynman">{{cite book|title=The Feynman Lectures on Physics|title-link=The Feynman Lectures on Physics|volume=2|first1=Richard|last1=Feynman|author1-link=Richard Feynman|first2=Robert B.|last2=Leighton|author2-link=Robert B. Leighton|first3=Matthew|last3=Sands|author3-link=Matthew Sands|section=Section 22-7. Filter |section-url=https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S7 |quote=If we imagine the line as broken up into small lengths Δℓ, each length will look like one section of the L-C ladder with a series inductance ΔL and a shunt capacitance ΔC. We can then use our results for the ladder filter. If we take the limit as Δℓ goes to zero, we have a good description of the transmission line. Notice that as Δℓ is made smaller and smaller, both ΔL and ΔC decrease, but in the same proportion, so that the ratio ΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) as ΔL and ΔC go to zero, we find that the characteristic impedance z0 is a pure resistance whose magnitude is √(ΔL/ΔC). We can also write the ratio ΔL/ΔC as L0/C0, where L0 and C0 are the inductance and capacitance of a unit length of the line; then we have <math>\sqrt{\frac{L_0}{C_0}}</math>}}.</ref><ref name=lee2004>{{cite book |first=Thomas H. |last=Lee |author-link=Thomas H. Lee (electronic engineer) |year=2004 |title=Planar Microwave Engineering: A practical guide to theory, measurement, and circuits |publisher=Cambridge University Press |section=2.5 Driving-point impedance of iterated structure |page=44 }}</ref> Taking the positive root, this equation simplifies to:

:<math>\ Z_\mathrm{IT} = \sqrt{ \frac{\ Z\ }{ Y }\ }\ </math>