NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 350

Next, equation (2-52) with µ r = 1 is used to calculate the relative dielectric

permittivity.

10 8
√ ε r

3 . 0 ×

10 8

ν p = = 1 . 75 ×

ε r = 2 . 9

Step 3: Calculate the characteristic impedance using the peak values of the
reflection coefficient. When the imaginary term is zero, the real term will peak
because the cosine term of equation (9-2) will equal 1 at frequencies predicted
by (9-3b). Therefore, the easiest way to calculate the characteristic impedance is
to use the value of the reflection coefficient measured at a real peak.
The first real peak at 0.775 GHz shows a maximum reflection coefficient
of 0.2. The characteristic impedance can be calculated by setting the reflection
coefficient equal to equation (9-2) at 0.775 GHz and solving for Z 0 .
cos 4 πf l √ LC =

10 9 )( 330 10 − 12 ) ] cos[4 π( 0 . 775 × × ≈− 1

sin 4 πf l √ LC =

10 − 12 ) ]

10 9 )( 330

sin[4 π( 0 . 775 × × ≈ 0

0 ( cos 4 πf l √ LC j sin 4 πf l √ LC)

R l − Z 0
R l + Z 0 [

0 . 2 = − = − 1]

− Z 0

50

=− → Z 0 = 75

50 + Z 0

Step 1 in Example 9-1 demonstrates a very useful relationship between the
periodicity of the input reflection coefficient looking into a network and the
propagation delay. If the distance between peaks ( f n = 3

− f n = 1 ) is represented as

and τ d = l √ LC , the time delay can be calculated using f

1
2 f

τ d = (9-4)

The utility of equation (9-4) will become apparent when analyzing S -parameters
in Section 9.2.2.
In summary, the reflection coefficient looking into a network is dependent on
(1) the impedance discontinuities, (2) the frequency of the stimulus, and (3) the
electrical length between discontinuities.

9.1.2 Input Impedance
Not surprisingly, if the reflection coefficient looking into a network is a function
of length, impedance discontinuities, and frequency, the input impedance looking
into the network must be a function of the same variables. Following a proce-
dure similar to that used to derive equation (9-1), the impedance looking into a