## [SI-LIST] Re: Diode Termination

• From: "Howard Johnson" <howie03@xxxxxxxxxx>
• To: <si-list@xxxxxxxxxxxxx>
• Date: Thu, 13 Dec 2007 22:47:51 -0800

```Dear Ria,

I sense you are looking for a precise answer here, so I shall endeavor to
create one.

Prelude:
(1) When you talk about an amplitude level on a transmission line, you must
specify whether your amplitude is a "voltage" or a "current" amplitude. For
most digital PCB transmission structures we take the shortcut of assuming
that the characteristic impedance zc(f) is pretty flat over the frequency
band that interests us, so we just assign zc(f) a constant real value, like
50 ohms, which we give the name z0.  We further assume only two modes, one
going this-a-way and one going the opposite direction (that-a-way). In that
simplified case, we can talk about voltage waveforms v1(t) and i1(t)
representing a forward-moving mode of waveform propagation and make the
statement that v1(t,x)/i1(t,x)=z0 at any point x. The same is true for the
reverse-moving waveform number described by (v2,i2) but for one exception --
since it goes the opposite direction is has an opposite sense of current, so
at any particular point x we write v2(t,x)/i2(t,x)=-z0.

(2) I presume that much is familiar to you.

(3) If you wish to determine the total voltage at any given point x you must
sum the voltage amplitudes of the forward and reverse-travelling waves at
that point. For example, the total voltate (call it vt) at the endpoint of a
transmission line would be the sum of the forward wave v1(t,x) plus the
reflected wave, v2(t,x) at the point x=end-of-line.

(4) In the terminology of the article you cite, "Diode Terminations",
www.sigcon.com/Pubs/news/2_19.htm , the forward wave measured at the
endpoint I call "I" (for incident), the reverse-travelling wave measured at
the endpoint I call "R" (the reflection), and the sum of both voltages "T"
is the total voltage measured across the endpoint.

(5) It is tempting to think of the total voltage at the endpoint as somehow
"exiting" the line, because it is the voltage you measure at across the end
terminals.  Probably that's not very good language, though, as it seems to
have confused you.

(6) Now I want to throw you a curve.  What if I want to know the output
CURRENT (iout) flowing through the termination?  That would be the sum of
currents, i1(t,x)+i2(t,x), which you can derive from the voltages like this:
iout(t)=[v1(t,endpoint) - v2(t,endpoint)]/z0.

(7) If I give you an incident signal v1(t) (arriving at the endpoint), and
ask you to tell me the size of the reflected signal you'd probably whip out
the reflection formula G=(zL-z0)/(zL+z0), where G (gamma) represents the
reflection coefficient, and where zL is the terminating impedance at the
endpoint. Then you'd say that the reflected signal equals v1(t,endpoint)*G.
Pretty good.

(8) How about the current exiting the line? (Output current actually does
"exit" the structure, which is where I picked up the "exit" word). Well,
according to (4) you can calculate the exiting current from the voltages,
like this:

iout(t)=[v1(t,endpoint) - v2(t,endpoint)]/z0.

(9) But, we know a value for v2(t,endpoint), calculated using G, so use it:

iout(t)=[v1(t,endpoint) - v1(t,endpoint)*G]/z0.

(10) Simplify:

iout(t)=v1(t,endpoint)[1-G]/z0.

(11) Plugging in your value for G you get this:

iout(t)=v1(t,endpoint)*2/(zL+z0)

(12) Check: when zL equals infinity you get no current.  When zL=0 you get a
current equal to v1(t,endpoint)*2/z0 (that's correct).

(13) I brought up this whole discussion so you could see the relation of the
endpoint voltage to the incoming and reflected VOLTAGE amplitudes:

T = I + R

(14) If you re-arrange the terms, you get the relation in the article,  R =
T - I.  This is just a mathematical re-arrangement. If (14) is true, then so
is this.

(15) Sometimes I'm a little slippery with the details as to whether the
symbols R, T and I represent particular voltages, or they represent
"coefficients". I've never much cared to go into the difference.  I find
that as long as I stick with a convenient voltage amplitude, like 1V, it
hardly matters. If we are being complete, however, I should state that (13)
and (14) work if you consider each of R,T and I to be voltage waveforms. If
you want to work strictly with reflection coefficients, you can do this:

T/I = 1.00 + R/I

In this case we interpret "R/I" to be the voltage reflection coefficient at
the end of the line (I called it G earlier), and "T/I" to be another sort of
coefficient. It tells you the ratio of (A) what size VOLTAGE signal comes
out of the end of the line (there goes my bad terminology again -- I mean
what size VOLTAGE signal is measurable as the apparent voltage across the
load) to (B) the VOLTAGE of the incident wave.

THAT I find to be one of the more interesting points of the "Diode
Termination" article.

Sometimes I'm not sure how far to go with an answer. You asked a very good
question.

Did I make things better or worse for you?

Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
tel +1 509-997-0505,  howie03@xxxxxxxxxx
www.sigcon.com -- High-Speed Digital Design seminars, publications and films

-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of Ria R
Sent: Thursday, December 13, 2007 8:21 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Diode Termination

All,
Howard Johnson in one of his answers to a diode termination question says
the following:
One  equation  of interest regarding the  theory  of diode  terminations  is
the  relation  between  the incident   signal   amplitude   traveling   down
a transmission line (I), the amplitude of  the  signal reflected  at  the
end  of  a  line  (R),  and  the amplitude of the signal that exits the line
and  is apparent at the load (T).

I am clear about "I" and "R". But what does "T" mean? I thought there was
just signal incident and signal reflected. Is there a signal "Exiting"??

Also, if this is explained, the next question is, what does he mean by:

In  English,  the signal at the end of the  line  is the   superposition  of
whatever  came   in,   plus whatever  bounced back toward the driver. This
same equation  may  be  re-written to express  "R"  as  a function of "T"
and "I":

R = T - I

Can someone please clarify the above equation??

Regards,
Ria.

____________________________________________________________________________
________
http://www.yahoo.com/r/hs

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