# Re: [SI-LIST] : Differential Reflection Question

Alok Tripathi ([email protected])
Thu, 2 Apr 1998 13:36:30 -0800 (PST)

The formula given by: %Reflection=(Zo-Zs)/(Zo+Zs),
only describes reflection due to the input end
of a single transmission line. For a general transmission
line this formula will model reflection only when the output
end is terminated with the characteristic impedance, Zo,
of transmission line.

In general the S(w) in frequency domain
for a single transmission line is given by:

S(w)=(Z1(w)-Zs)/(Z1(W)+Zs) :

(NOTE: Modeling transmission line as one port device)

where Z1(w) is function of characteristic impedance, Zo,
electrical length and the termination
resistance at the receiving end of transmission line. Zs is
the impedance connected at the sending end of the transmission line.

In time-domain case: we can write reflection as:

----------------------------------------------------------------
reflection = rho(t)=(Z1(t)-Zs)/(Z1(t)-Zs)
^
|
The approximation made in this equation is: Multiple reflections
are neglected.
----------------------------------------------------------------

Case of n uniform coupled lines can be subdivided as:

a.) Coupled lines in homogeneous medium: The propagation velocity
of all the modes are equal.

b.) Coupled lines in homogeneous medium: The propagation velocity
of modes are unequal.

The [S(w)]_{nxn} for both the cases is given by:

-----------------------------------------------------------------
[S(w)]_{nxn}=([Zo]-[Zs])([Zo]+[Zs])^{-1} :
^
|
The assumption made in this equation is: The receding end
of n coupled transmission lines is terminated with a resistive
network corresponding to the characteristic impedance matrix, [Zo].

By definition: a resistive network corresponding to the
characteristic impedance matrix terminates all modes.

[Zs] is the impedance matrix associated with the resistive
network connected at the sending end of the transmission line.

------------------------------------------------------------------

For a general case:

[S(w)]_{nxn}=([Z1(w)]-[Zs])([Z1(w)]+[Zs])^{-1}

The expression for [Z1(w)] can be found in : Analysis of multiconductor
transmission lines, by: Clayton R. Paul
Wiley Series in microwave and optical engineering.

Regards,

Alok Tripathi,
Dept. of Elect. and Comp. Engr.,
Oregon State University,
Corvallis, OR 97331

On Wed, 1 Apr 1998, Lehew, John wrote:

> In the High Speed Digital Design book and in a few other places it
> states the percent reflection caused by a difference in impedance is:
>
> %Reflection=(Z1-Z2)/(Z1+Z2)
>
> This formula is typically used to calculate the reflection of a single
> line referenced to a ground plane. Does this formula also apply to
> differential/balanced lines where two lines carry one signal?
>
> Regards,
>
> John Lehew
> Design Engineer
> Compaq
>
>
>
>