NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 395

The input reflection in is the ratio of the reflected wave b 1 to the incident
wave a 1 :

b 1
a 1

S 12 S 21 L
1

in = = S 11 +

− S 22 L

If it is assumed that the error term is reciprocal, S 21 = S 12 , the input reflection

equation can be simplified:

S 12 L

b 1
a 1

in = = S 11 + (9-52)

1 − S 22 L

To calibrate the effect of the errors out of the measurement, the S -parameters
must be found. This is done by considering three known values for the load, Z L :
a short, an open, and a perfectly matched resistor. When the load is shorted, the
reflection at the load is L =−

1, which reduces (9-52) to

S 12

= S 11 −

in, short (9-53a)

1 + S 22

When the load is open and perfectly impedance matched, the reflection at the
load is L = 1 and L = 0 respectively, producing

S 12

= S 11 +

in, open (9-53b)

1 − S 22

in, matched = S 11 (9-53c)

By measuring the open, short and matched loads, the three equations above can
be solved simultaneously for S 11 , S 12 , and S 22 :

= ( in, matched S 12 = S 21 − in,short )( 1 + S 22 ) (9-54a)

2 in, matched − in,short − in, open

S 22 = (9-54b)

in,short − in,open

S 11 = in,matched (9-54c)

The equations above are then used to create a scattering matrix for the errors at
the ports:

S 11

S 22

AB
CD

S 12

short,open,load ⇒

(9-55)

S 21

error port

Finally, the measurements of the DUT are calculated by multiplying by the
inverse of the error terms:
AB
CD

AB
CD

− 1 AB

CD

AB
CD

DUT =

(9-56)

error port1 measured error port2