Part II 2
ITERATIVE IMPEDANCES
The concept of "iterative impedance" is intrinsically related to those of "load impedance" and "input impedance," which we now define.
Definition 15. A transmission network t' = xt -t-r will be said to be terminated in a load impedance ZL (load admittance YL) provided the output emf and current vectors satisfy e' = ZLi' for arbitrary i' (i' = YLe' for arbitrary e'), where ZL (Yr.) is an n X n matrix.
If termination of a given transmission network N in a load impedance Zt. (load admittance YL) implies that the input emf and current vectors satisfy an affine transformation of the form e = Z1i -b el for arbitrary i, then the n X n matrix Zt is called an input impedance of N corresponding to the load impedance ZL (load admittance YL). When no mention is made of the load impedance or admittance, it will be assumed that we are dealing with the case of a load admittance Yr. = 0. TM Uniqueness of the input impedance (when it exists) for a given load impedance (or admittance) is immediately evident.
Definition 16. If a passive transmission network N has input impedance h when terminated in a load impedance h, then h is called an iterative impedance of N.
In general, iterative impedances are not unique, since the unit element, for example, has any load impedance as an iterative impedance. Theorem 8. If a passive transmission network N in P has matrix then the iterative impedances of N are characterized as the solutions h of the n X n matrix equation h~x + h~ax -t~ = 0. (15)
Proof. It is easily seen that every iterative impedance X must satisfy (15), since e' = hi' and e = hi result in (hvh + h~ --~h --t~)i = 0 for arbitrary i.
On the other hand, suppose that h is a solution of (15). It then follows that a-hy is non-singular, since if it were singular we could find a non-zero row