# Derivation of input admittance of a lossless T-Junction Power Divider that is matched at the input port

I'm trying to understand how the equation for the input admittance for a lossless T-Junction power divider was derived, which is only matched at the input.

Let $$\Z_0\$$, $$\Z_1\$$, $$\Z_2\$$ be the characteristics impedances of the 1st, 2nd and 3rd ports, respectively. 1st port is our input port. 2nd and 3rd ports are our output ports.(Ports 2 and 3 are not matched, only port 1 is).

The input admittance is said to be

$$Y_{in} = jB + \frac{1}{Z_1} + \frac{1}{Z_2}$$

Let us set $$\B = 0\$$ for simplicity.

$$Y_{in} = \frac{1}{Z_1} + \frac{1}{Z_2}$$

How was the above equation derived? Why is the input admittance equal to the sum of inverse characteristic impedances of the transmission lines at the 2nd and 3rd ports?

In my opinion, since the two transmission lines ($$\Z_1\$$ and $$\Z_2\$$) are open-circuited, the input impedances of the two lines are respectively: $$Z_{1, in} = -jZ_1\cot(\beta l_1)$$ and $$Z_{2, in} = -jZ_2\cot(\beta l_2)$$

Thus the input admittance should be: $$Y_{in} = \frac{1}{Z_{1, in}} + \frac{1}{Z_{2, in}}$$

What am I doing wrong?

Further, it makes perfect sense to me that $$\Y_{in} = \frac{1}{Z_0} \$$ as the input impedance for a matched transmission line with a characteristic impedance of $$\Z_0\$$ is: $$\Z_{in} = Z_0\$$

I'm following David M. Pozar's book titled "Microwave Engineering".

After a more careful examination of the book, I learned that the book says "In order for the divider to be matched to the input line of characteristic impedance $$\Z_0\$$, we must have...(above equation)"
Which really means: $$\S_{11} = 0\$$. $$\S_{11}\$$ is defined as: $$S_{11} = \left.\frac{b_1}{a_1}\right\vert_{a_2, a_3 = 0}$$ To achieve this the other two ports must be matched as well, to prevent any reflections from their ends. Thus we match the last two ports with their respective characteristic impedances $$\Z_1\$$ and $$\Z_2\$$ respectively.
The impedances thus seen from the junction towards the two ports are: $$\Z_1\$$ and $$\Z_2\$$ respectively (expected behaviour of load-matched transmission lines).
Since these two impedances are parallel to each other, we consider looking at the admittance $$\Y_{in}\$$:
$$Y_{in} = \frac{1}{Z_1} + \frac{1}{Z_2}$$
Which in turn must be equal to $$\\frac{1}{Z_0}\$$ to ensure that $$\S_{11}\$$ remains zero.