Hence, to simplify the analysis, we can bisect the circuit with grounds as shown (Fig 7.10b): V2oįor zino, notice that the load is a short circuit and the TL is  4 long (1/2 rotation around the Smith chart). Since the circuit is fed anti-symmetrically, V3o  V2o and the voltage = 0 at points A and B. Therefore, 2  z0,Q 2  2 (7)    2  z0,Q  3 2  2 Substituting V  from (6) into (5) and using (7) we find that V V1e   1       jV 2 j 1    7  Anti-symmetric excitation (odd mode): V2o  is the reflection coefficient at port 1 seen looking towards the normalized load of 2 /. Next, to find V1e we’ll use the TL equation V  x   V  e  j  x  e j  x Therefore, z0,Q  2 zine  1  Since zine  1, then by voltage division at the output port zine 1 e V2  e Vg 2  Vg 2  V zin  1 2 Consequently, z0,2 Q e (7.33),(1) zin  2 z0,Q  2 zine (2) or We want the output ports to be matched. Hence, V2e  V3e and we can bisect this circuit as shown to simplify the analysis (Fig. Notice that I  0 because we have symmetrical excitation. Hence, the Wilkinson power divider shown in the first figure above, and with matched terminations, can be drawn asĮven-Odd Mode Analysis of the Wilkinson Power Divider In the even-odd mode analysis for the S parameters, we will first excite this network symmetrically at the two output ports, followed by an anti-symmetrical excitation. For example, a TL with characteristic impedance 2Z 0 will be delineated as 2

To simplify matters, as in the text, we will: 1. We will now show that for a 1:1 Wilkinson power divider, Z 0,Q  2Z 0 and R  2 Z 0. This mathematical process is called an “even-odd mode analysis.” It is a technique used in many branches of science such as quantum mechanics, antenna analysis, etc. Specifically, we will excite this circuit in two very special configurations (symmetrically and anti-symmetrically), then add these two solutions for the total solution. There is much symmetry in this circuit that we can exploit to make the S parameter calculations easier. This is a popular power divider because it is easy to construct and has some extremely useful properties: 1.

The next three port network we will consider is the Wilkinson power divider, in particular the 1:1 power divider (Fig 7.8b): /4 Z0

Wilkinson divider smith chart