I got a response from TI about my question. A senior application engineer said that it is not a well written application note and it is not possible to find impedance matching by just using line calculator and smith chart.
The advice was to simulate the circuit on ADS, Microwave or similar software, as the behaviors of components changes by the brand, part model, size, frequency, etc.. The manufacturers provide simulation libraries for their components for the aforementioned SWs.
I used ADS and found two types of simulation to find the correct impedance matching.
One is to simulate the circuit on the schematic view, then you need to define exact component you use, the trace length, trace angle, property of the PCB, etc. According to another application note of TI, it says you can get the correct result about 10% accuracy.
One can get more precise result on the layout simulation due to interaction with the signals from the other components or traces on the board. ADS is a complicated software for a beginner, I could not find a good tutorial how to do it, therefore I simulated on the schematic part and my problem is solved. I don't have network analyzer, therefore I don't know how correct match I got.
In this situation, I'd build an approximately -20dB probe, and measure the voltage at the input directly.
The wavelength of 2.1GHz is 142mm (air), or \$\lambda/10\$ (plastic) is around 10mm. So it's straightforward to make 'short' connections with care, and small components.
simulate this circuit – Schematic created using CircuitLab
The 470 ohm resistor should be a good RF type. Its value is a compromise between giving an adequate signal, not loading the circuit too much, and having a good frequency response. The loading is reasonable compared to the 2x34ohms, and you can compute a small loading correction.
Although the coax appears to be mismatched, this is a flat probe. The 50 ohm input impedance of your power meter is transferred to the end of the coax, where it still looks like 50 ohm resistive. Then you get a (roughly) 10:1 pot-down of the probe resistor into that load.
You can calibrate the probe easily by dabbing the sharp end on the connector of calibrated source, or across a well-driven load resistor.
The small size and the low cost of 470 ohm resistors means you can solder one to the measurement node, leave it there, and forget about it. When you come to attach and detach the probe coax (usually use RG178 or smaller), it's soldered to the resistor, protecting the terminals from wear.
Best Answer
The 25 Ω resistor doesn't do anything when the hybrid is acting as a power splitter, in this case, the input is Port A and the outputs are Port B and Port C. The center-tap can be terminated with any impedance, shorted to ground, or kept as an open-circuit. Similarly, when the circuit is being driven as a power combiner with a pair of ideal 0-degree and 180-degree differential signals at Port B and Port C, 100% of the power is delivered to Port A with matched impedance, again, the resistor is not doing anything.
The resistor provides termination when (1) the hybrid is used as a power combiner, and (2) Port B and Port C are being driven by an in-phase common mode signal. When the power combiner is driven by two in-phase inputs, no power is delivered to Port A because common-mode signal is suppressed and goes to Port D, the center-tap. Because the two 50 Ω signal sources are in-phase, they are effectively identical voltage sources in parallel, two 50 Ω parallel resistors give the output impedance of 25 Ω - here, the 25 Ω resistor at the center-tap acts as a dummy load and provides impedance termination for any common-mode signal/noise presented in the differential input, ensuring power is never reflected.
Source of the circuit is Practical Radio-frequency Handbook by Ian Hickman.
I was looking for a suitable implementation for power splitter and first saw this picture in the Power dividers and directional couplers from Wikipedia with no explanation given, and I kept thinking the circuit as a power splitter and missed the explanation in the book.