NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 398

R Port 1 Port 2 R

∼

R R

Port 3 Port 4

Figure 9-37 A pair of coupled transmission lines is a common example of a four-port
network that is conveniently described with a multimode S -matrix.

directly from the four-port S -parameters. For example, the S -parameters of the
four-port system pictured in Figure 9-34 are shown as follows:

b 1
b 2
b 3
b 4

S 11 S 12 S 13 S 14 a 1

a 2
a 3
a 4

=

S 21 S 22 S 23 S 24

(9-60)

S 31 S 32 S 33 S 34

S 41 S 42 S 43 S 44

An example of a four-port system is two coupled transmission lines, as shown
in Figure 9-37. To derive the multimode matrix, it is first necessary to calculate
the S -parameters for each mode.
The differential S -parameters, where the energy is being transmitted in the
odd mode, are calculated by driving the four-port network with

+ v on port 1

and − v on port 3 when ports 2 and 4 are not being driven. This allows (9-60)

to be simplified.

b 1
b 2
b 3
b 4

S 11 S 12 S 13 S 14 v

0
− v
0

=

S 21 S 22 S 23 S 24

(9-61)

S 31 S 32 S 33 S 34

S 41 S 42 S 43 S 44

The differential return loss at port 1 can then be calculated algebraically where
b d 1 = b 1 − b 3 and a d 1 = a 1 − a 3 = v − ( − v) = 2 v .

a 2 = a 4 = 0 =

b d 1
a d 1

1

S dd 11 =

2 (S 11

+ S 33 − S 13 − S 31 ) (9-62a)

S dd 11 is a measure of the differential energy reflected, or returned to the source.
Similarly, the differential insertion loss is calculated where b d 2 = b 2 − b 4 and

a d 1 = a 1 − a 3 = v − ( − v) = 2 v .

a 2 = a 4 = 0 =

b d 2
a d 1

1
2 (S 21

= + S 43 − S 23 − S 41 )

S dd 21 (9-62b)