Impossibility of a completely matched three-port network. When a wave is sent into any one of the ports of the four-port beam splitter shown in this section, there is no reflected wave directly back out the same port, so that ry = 0 for all i. In transmission line jargon this system is said to be matched looking into each of the four ports. Show that it is impossible in principle to devise any kind of lossless three-port network in which all three ports are similarly matched.

A matched network is one in which all ports are matched ports are matched to the respective port.

So considering the above definitions lets us consider a network a 3 port device with a following scattering matrix.

S11 S12 S13              

S= S21 S22 S23

S31 S32 S33

Assume the device is passive and made of isotropic materials so the device will be reciprocal

S21=S12, S31=S13 ,S23=S32

Likewise if it matched means S11=S22=S33=0

Thus a looseless reciprocal device has a scattering matrix will be in the form of

0   S12 S13              

S= S21 0   S23

  S31 S32 0

Thus as can be seen above in the scattering matrix it is defined by 3 non Zero

If we wish the matrix to be looseless the scattering matrix must be unitary and therefore

S21^2+S31^2=1 and S31*S32=0

S21^2+S32^2=1 and S21*S32=0

S31^2+S32^2=1 and S21*S31=0

As each complex value of s is represented by two real numbers equation gives 9 real equations , thus 3 complex values of S21,S31 and S32 are given by 6 real unkowns , thus we can conclude there is no solution for it thus it is impossible to design any systems