page_title: Image impedance
section_title: Derivation
hierarchical_section_title: Image impedance / Derivation
context_page_description: 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 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.
context_section_description: 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;
  and solving for ,
  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,
  Also, it can be seen from these expressions that the two image impedances are related to each other by;
caption: Showing how a T section is made from two cascaded L half sections. Zi 2 is facing Zi 2 to provide matching impedances