Clear definition of Maximum Available Gain of 3-port Balun?

In RF transformer characterization, maximum available gain (Ga) is widely used to indicate the power transfer performance.

I find in book as "Microwave and Rf Design of Wireless Systems" by Pozar, the Ga is obtained by conjugately matching a two port network.

However, Balun is usually used as a three port device in real circuits. If follow the 2 port network analysis, should Ga of either two ports defined by all three ports conjucated matched? I guess the derivation of Ga in terms of S parameters will be startingly complex.
Or just leave the third port open/short/terminated for example, by50 Ohm? (maybe the termination of the third port has marginal influence on the Ga)

I find the following paper does not give clear definition of this Balun Ga.

maximum available gain definition

Reply my topic to receive more attention here.

Another interesting problem is that, I think what impedance we use to conjugately match one port is depentdent on the load of other ports for passive networks.

In this respect, the derivation of maxium available gain of two port network in the book I mentioned is very interesting, the method that envolving wise mathematical treatment.

balun differential common mixed-mode

"However, Balun is usually used as a three port device in real circuits. If follow the 2 port network analysis, should Ga of either two ports defined by all three ports conjucated matched? "

The correct way to look at balun device is as a 3 port device in which the ports are : 1 differential, 3 common mode, 2 single end. these are Mixed mode S-parameters.
If you use a balun is because you have a differential source or circuit and you want to connect to a single end load (or viceversa). So the quantity you are interested in are the differential-to-single-end Gain (S21), the Differential input impedance (s11) and so on. Of course you would like to know the common mode rejection (S31) or Output match (S33).

I hope this will help you.
Mazz

p.s.: if you are an agilent ADS user, I remember that there is an available example that explain well the matter.

available gain

Thank you, Mazz for your attention.

I would like to discuss further with you on this.
I agree that balun are used with one single port and two differential port in real circuits. However, in device characterization or S parameter measurements, we treat balun as a three port device. The port transmission S21,S31 are properties of three port devices(perhaps only the 180 degree phase will indicate the real purpose of port 2 and port3).

So, in my view the maximum available gain (Ga) should be the Ga of a three port device, i.e. with all the three ports conjugated matched and then giving Ga21, Ga31.

maximum available gain s-parameters

Yeah, OK, if you prefer to see it like this.
Let me point out only that there is no missing informations if you see it as a 3 port device, whose ports are differential (1), common mode (2) and single end (3).
So max available gain, in a classical 3 port device, is not only one, but there are at least 2 Ga, Ga21 and Ga31, agree with you.

I haven't thought at the problem of max avail gain of an N-port device, so I can't tell you if the unused port have to be terminated by 50 Ohm or conjugated matched. By intuition I would say that the second hypotesis is the right one.

Mazz