Generalized Case of Impedance Transformation

Author: Edmund A. Laport

Any four-terminal network whose output power is equal to its input power must be composed of pure reactances. This was the basis of reasoning in all the preceding problem solutions.

It is of some academic value to look at the case where this power equality is not specified. Let us choose a scale of vectors for a given load impedance, and make a specified impedance transformation without equating input and output powers. This is possible since an impedance is merely a ratio of a potential to a current. In the preceding problems, we always solved for those particular values which not only gave the correct ratio but also represented the same power. We need discuss the present case only qualitatively to show that, instead of pure reactances, the network elements may be complex and that there are an infinite number of solutions possible.

Let us transform a low value Z₀ = R₀ - jX₀ to a large value

with random phase shift (Fig. 5.43). We start from the load as before and set up a vector statement of the problem at that point. We do the same for the input impedance, including phase angle between input and output potentials or currents, as shown in Fig. 5.44. The original problem is written vectorially in solid lines. The unknown current is I₁ (Fig. 5.43), which must connect I₀ with Iin. We draw this immediately. But when it comes to determining V₁ and V₃, we find an infinite number of choices.

Fig. 5.43:

Fig. 5.44:

Let us select a point P at random, anywhere in the entire plane of the diagram. Completing the diagram through P, we obtain

where V₁ leads I₀ less than 90 degrees,

where I₁ leads V₂ more than 90 degrees, and

where V₃ leads I_in less than 90 degrees.