NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 357

From equation (9-12), the power going into a node can be calculated using the
power relation P = v 2 /R :

(v + ) 2
R

P input = (9-13a)

and the power coming out can be calculated as

(v − ) 2
R

P out = (9-13b)

meaning that the power delivered to the network is calculated as

P input − P out = P network (9-13c)

Equations (9-13) allow the definition of terms that describe the power wave
propagating into port j and the power wave propagating out of port j :

v(z) +
√ R = P input

a j = (9-14a)

P out

v(z) −
√ R b j = = (9-14b)

where a j is the square root of the power propagating into port j and b j is the
square root of the power propagating out of port j , as shown in Figure 9-7 for
a two-port network. Equations (9-14a) and (9-14b) are known as the scattering
coefficients . Since they are defined in terms of the square root of power, ratios of
the scattering coefficients simplify into ratios of voltage as long as the termination
impedance of each port ( R ) is the same.
S -parameters are derived from the ratios of scattering coefficients. For
example, referring to Figure 9-7, the term S 11 is calculated by the root of the
reflected and incident power ratio at port 1, which is written in terms of the
scattering coefficients:

b 1
a 1

S 11 = (9-15a)

Similarly, the term S 21 is calculated by injecting power into port 1 and measuring
at port 2.

b 2
a 1

S 21 = (9-15b)

Using the definition shown in equations (9-15a) and (9-15b), a set of linear
equations can be written to describe the network in terms of its scattering coef-
ficients:

b 1 = S 11 a 1 + S 12 a 2

= S 21 a 1 + S 22 a 2

b 2