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'''Iterative impedance''' is the input impedance of an infinite chain of identical networks. It is related to the [[image impedance]] used in [[filter design]], but has a simpler, more straightforward definition.

==Definition==

Iterative [[electrical impedance|impedance]] is the input impedance of one port of a [[two-port network]] when the other port is connected to an infinite chain of identical networks.<ref>Iyer, p. 340</ref> Equivalently, iterative impedance is that impedance that when connected to port 2 of a two-port network is equal to the impedance measured at port 1. This can be seen to be equivalent by considering the infinite chain of identical networks connected to port 2 in the first definition. If the original network is removed then port 1 of the second network will present the same iterative impedance as before since port 2 of the second network still has an infinite chain of networks connected to it. Thus the whole infinite chain can be replaced with a single [[Lumped element model|lumped impedance]] equal to the iterative impedance, which is the condition for the second definition.<ref>Bakshi & Bakshi, pp. 9.4-9.5</ref>

In general, the iterative impedance of port 1 is not equal to the iterative impedance of port 2. They will be equal if the network is symmetrical, however [[Antimetric electrical network#Physical and electrical antimetry|physically symmetry is not a necessary condition]] for the impedances to be equal.<ref>Bird, p. 594</ref>

==Examples==
[[File:L-section iterative impedance.svg|thumb|Iterative impedance of a simple generic L-circuit]]
A simple generic [[Topology (electrical circuits)#Simple filter topologies|L-circuit]] is shown in the diagram consisting of a series impedance ''Z'' and a shunt [[admittance]] ''Y''. The iterative impedance of this network, ''Z''IT, in terms of its output load (also ''Z''IT) is given by,<ref>Walton, p. 209</ref><ref name="lee2004"/><ref name="niknejad2007"/><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>

:<math> Z_ \mathrm {IT} = Z + Y \parallel Z_ \mathrm {IT} </math>

and solving for ''Z''IT,

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

Another example is an L-circuit with the components reversed, that is, with the shunt admittance coming first. The analysis of this circuit can be found immediately through [[Duality (electrical circuits)|duality]] considerations of the previous example. The iterative admittance, ''Y''IT, of this circuit is given by,

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

where,

:<math> Y_ \mathrm {IT} = {1 \over Z_ \mathrm {IT} } </math>

The square root term in these expressions cause them to have two solutions. However, only solutions with a positive real part are physically meaningful since passive circuits cannot exhibit [[negative resistance]]. This will normally be the positive root.<ref>Walton, pp. 209-210</ref>

==Relationship to image impedance==
[[File:Ladder iterative impedance.svg|thumb|upright=1.2|Iterative impedance of an infinite ladder of L-circuit sections]]
[[File:Ladder image impedance.svg|thumb|upright=1.2|Image impedance of an infinite ladder of L-circuit half-sections]]
Iterative impedance is a similar concept to [[image impedance]]. Whereas an iterative impedance is formed by connecting port 2 of the first two-port network to port 1 of the next, an image impedance is formed by connecting port 2 of the first network to port 2 of the next. Port 1 of the second network is connected to port 1 of the third and so on, each subsequent network being reversed so that like ports always face each other.

It is thus no surprise that there is a relationship between iterative impedances and image impedances. In the L-circuit example for iterative impedance, the square-rooted term is equal to the image impedance of a half section. That is, an L-circuit where the component values are halved. Designating this half-section image impedance as ''Z''IM we have for the L-circuit,<ref>Bakshi & Bakshi, pp. 9.55–9.56</ref>

:<math> Z_\mathrm {IT} = {Z \over 2} + Z_\mathrm {IM} </math>

The diagrams show this result: an infinite chain of L-sections is identical to an infinite chain of alternately reversed half-sections except for the value of the initial series impedance.

For a symmetrical network, the iterative impedance and image impedance are identical and are the same at both ports. This impedance is sometimes called the network's [[characteristic impedance]], a term usually reserved for [[transmission line]]s.<ref>{{multiref|Bird, pp. 594-595|Iyer, p. 345}}</ref> The model for a transmission line is an infinite chain of L-sections with infinitesimally small components. A transmission line characteristic impedance is thus the [[Limit (mathematics)|limiting case]] of a [[ladder network]] iterative impedance.<ref>Montgomery ''et al.'', pp. 112-113</ref><ref name="lee2004">{{cite book|title=Planar microwave engineering: a practical guide to theory, measurement, and circuits.|first=Thomas H.|last=Lee|author-link=Thomas_H._Lee_(electronic_engineer)|publisher=Cambridge University Press|year=2004|section=2.5. Driving-point impedance of Iterated Structure|page=44}}</ref><ref name="niknejad2007">{{cite book|title=Electromagnetics for high-speed analog and digital communication circuits|last1=Niknejad|first1=Ali M.|section=Section 9.2. An Infinite Ladder Network.|section-url=https://www.globalspec.com/reference/59926/203279/9-2-an-infinite-ladder-network|year=2007}}</ref><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 22-7. 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>

==References==
{{reflist}}

==Bibliography==

* Bakshi, U. A.; Bakshi, A. V., ''Electric Circuits'',
* Bird, John, ''Electrical Circuit Theory and Technology'', Routledge, 2013 {{ISBN|1134678398}}.
* Iyer, T. S. K. V, ''Circuit Theory'', Tata McGraw-Hill Education, 1985 {{ISBN|0074516817}}.
* Montgomery, Carol Gray; Dicke, Robert Henry; Purcell, Edward M., ''Principles of Microwave Circuits'', IEE, 1948 {{ISBN|0863411002}}.
* Walton, Alan Keith, ''Network Analysis and Practice'', Cambridge University Press, 1987 {{ISBN|052131903X}}.
* Feynman, Richard; Leighton, Robert; Sands, Matthew, ''The Feynman Lectures on Physics, Vol. II'', Chapter 22. AC Circuits, [https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S6 Section 6. A ladder network], California Institute of Technology – HTML edition.

[[Category:Analog circuits]]
[[Category:Filter theory]]
[[Category:Electronic design]]

Iterative impedance - Revision history

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