Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, Zi 1, seen looking into port 1 when port 2 is terminated with the image impedance, Zi 2, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical (or anti-symmetrical) with respect to the ports.
Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.
Derivation
As an example, the derivation of the image impedances of a simple 'L' network is given below. The 'L' network consists of a series impedance, Z, and a shunt admittance, Y. The difficulty here is that in order to find Zi 1 it is first necessary to terminate port 2 with Zi 2. However, Zi 2 is also an unknown at this stage. The problem is solved by terminating port 2 with an identical network: port 2 of the second network is connected to port 2 of the first network and port 1 of the second network is terminated with Zi 1. The second network is terminating the first network in Zi 2 as required. Mathematically, this is equivalent to eliminating one variable from a set of simultaneous equations. The network can now be solved for Zi 1. Writing out the expression for input impedance gives:
Z i 1 = Z + 1 2 Y + 1 Z + Z i 1 {\displaystyle Z_{i1}=Z+{\frac {1}{2Y+{\frac {1}{Z+Z_{i1}}}}}}
and solving for Z i 1 , {\displaystyle Z_{i1}\ ,}
Z i 1 2 = Z 2 + Z Y {\displaystyle Z_{i1}^{2}=Z^{2}+{\frac {Z}{Y}}}
Zi 2 is found by a similar process, but it is simpler to work in terms of the reciprocal, that is image admittance Yi 2,
Y i 2 2 = Y 2 + Y Z . {\displaystyle Y_{i2}^{2}=Y^{2}+{\frac {Y}{Z}}~.}
Also, it can be seen from these expressions that the two image impedances are related to each other by:
Z i 1 Y i 2 = Z Y . {\displaystyle {\frac {Z_{i1}}{Y_{i2}}}={\frac {Z}{Y}}~.}
Measurement Directly measuring image impedance by adjusting terminations is inconveniently iterative and requires precision adjustable components to effect the termination. An alternative technique to determine the image impedance of port 1 is to measure the short-circuit impedance ZSC (that is, the input impedance of port 1 when port 2 is short-circuited) and the open-circuit impedance ZOC (the input impedance of port 1 when port 2 is open-circuit). The image impedance is then given by,
Z i 1 = Z S C Z O C {\displaystyle Z_{i1}={\sqrt {Z_{\mathrm {SC} }Z_{\mathrm {OC} }}}}
This method requires no prior knowledge of the topology of the network being measured.
Usage in filter design When used in filter design, the 'L' network analysed above is usually referred to as a half section. Two half sections in cascade will make either a T section or a Π section depending on which port of the L section comes first. This leads to the terminology of Zi T to mean the Zi 1 in the above analysis and Zi Π to mean Zi 2.
Relation to characteristic impedance Image impedance is a similar concept to the characteristic impedance used in the analysis of transmission lines. In fact, in the limiting case of a chain of cascaded networks where the size of each single network is approaching an infinitesimally small element, the mathematical limit of the image impedance expression is the characteristic impedance of the chain. That is,
Z i 2 → Z Y {\displaystyle Z_{i}^{2}\rightarrow {\frac {Z}{Y}}}
… excerpt ends here. Continue reading the full article.
