2-port network conversion from S-parameters to Voltages and Currents

For the general 2-P network in Fig. 1, considering the input/output lines (that model finite connection lengths) effect and assuming loss-less propagation with real Z_G, Z_L:

Fig.1

The incident/reflected waves and voltage/current at the ith (i=1,2) port are by definition and as a function of distance z from the source:

(1)

The average power associated with the incident wave on the input/output line (z=0, z=l₃) connected to the ith (i=1,2) port is:

(2)

Similarly, the reflected power at the same line is:

(3)

because the lines are loss-less, the incident and reflected power at port ith are the same as in (1) and (2).

If both input and output line are matched i.e. and Z_G= Z₀₁ and Z_L= Z₀₂ then:

(4)

and:

(5)

then P_AVS=P₁⁺(0) when Z_G= Z₀₁ and because the line are loss-less this is also the the power available from the source at port 1, which is independent of Z_IN. If Z_G≠ Z₀₁ then the expression for V₁(0) in (5) must be modified accordingly and the resulting incident power, which is provided by (2), will no longer be equal to the power available from the source. In the first case, i.e. Z_G= Z₀₁ and (5) is valid, because:

(6)

then subtracting the two gives the power delivered to port 1 (or to the line, because the line has no loss):

(7)

Then:

(8)

which shows that the generator sends to the input line or port (if the line is loss-free) the power P_AVS that is independent of the input matching condition (i.e. Z_IN). However if the input impedance is not matched to the input line ( Z_IN≠ Z₀₁) part of the incident power P_AVS=½|a₁(0)|²=½|a₁(l₁)|² is reflected and this reflected power is just ½|b₁(l₁)|² while the net power delivered to port 1 is P₁(0)=P₁(l₁)=P_AVS-½|b₁(l₁)|2 .

In the particular case with Z_G=Z₀₁=Z₀₂=Z_L= Z₀ the input/ouput lines have no-influence because the input impedance of a transmission line closed on a load impedance equal to its characteristic impedance is just equal to the load (i.e. the of characteristic impedance):

(9)

and in the input size Thevenin analysis shows that the generator through the input line is equivalent to a generator with internal resistance Z₀ and voltage V_G that differs from the original one only by a phase factor. Then the circuit in Fig. 1 simplifies to the one in Fig. 2:

Fig. 2

From the incident/reflected waves, and voltage/current definition:

(10)

and considering the matched termination at port 2:

(11)

therefore the input impedance and the current at each port as a function of the same port voltage are:

(12)

the port 2 parameters can be expressed as a function of port 1 voltage by noticing that:

(13)

hence all port voltages, currents and the total current flowing into the 2-port network I_tot=I₁+I₂ can be expressed as a function of port 1 voltage which is useful when one wants to extract I_tot and this is the current flowing in the shunt branch of a T-network such that the current might not be physically accessible while V₁ and the scattering parameters are generally measurable:

(14)

Notice that in this case (2-port network is simply a shunt Y) from (7) in the section on even/odd mode decomposition of symmetrical networks S₂₁-S₁₁=1 or S₂₁=1+S₁₁ thus (14) reduce to:

(15)

where particularly the first and the last one are intuitive from the shunt configuration forcing the same voltage on both ports. As pointed out in the section on Even/odd mode decomposition, if |Y| is small, which is the case when R is in the tens of kΩ range and C is in the pF range, then S₁₁≈ -YZ₀/2 and S₂₁≈1. Hence Y can be evaluated by reflection-type measurement if S₁₁ or the reflected power are available.