[SI-LIST] Re: Question about the Equation in Handbook of Black Magic
- From: "Dr. Howard Johnson" <howie03@xxxxxxxxxx>
- To: "Si-List@xxxxxxxxxxxxx" <si-list@xxxxxxxxxxxxx>
- Date: Fri, 22 Apr 2005 12:26:44 -0700
Dear Peter Zhu,
Regarding the reflections formula, suppose a wave of unit
amplitude propagates towards the right end of a long cable.
The cable impedance is Z0. The impedance loading the end of
the cable is ZL.
The incident waveform arrives with voltage vi(t) and
current ii(t) = vi(t)/Z0.
That ratio of voltage to current does not match what the
load requires, because in the load you must have
vload(t)/iload(t) = ZL.
What happens at the load is that a reflection entity is
created. The reflection has voltage vr(t) and current
ir(t)=vr(t)/Z0. Note carefully that current in the incident
wave moves to the right (assuming a positive voltage vr),
while current in the reflected wave moves to the left. The
current in the load must therefore be the superposition of
these two currents like this, iload(t) = ii(t) - ir(t).
This is the "current balance equation".
The voltage at the load equals the supoerposition of the
incident and reflected waveforms on the cable, like this:
vload(t) = vi(t) + vr(t). This is the "voltage balance
equation"
>From these basic equations you can derive both formulas 4.52
and 4.53.
To make the derivation, first substitute into the current
balance equation the expressions for current, as a function
of voltage. I have dropped the time-based arguments to
simplify the notation.
vload/ZL = vi/Z0 - vr/Z0
Now substitue on the left side what we know from the voltage
balance equation about vload, to get this:
(vi + vr)/ZL = vi/Z0 - vr/Z0
Multiply through both sides by ZL*Z0 and collect like terms:
vi(Z0 - ZL) = -vr(Z0 + ZL)
Express vr as a function of vi
vr = vi*(ZL - Z0)/(ZL + Z0) this is the reflection
formula.
now compute the voltage at the load
vload = vi + vr = vi*( 1 + (ZL - Z0)/(ZL + Z0)) = vi *
(2*ZL / (ZL + Z0))
Perhaps you are wondering why it is that a reflection entity
must be created? The cable supports only two modes of
propagation -- to the right, and to the left. With in each
mode, the equations of propagation are quite strict,
specifying the velocity of propagation, the attentuation,
and the rule that the ratio of voltage to current within
each mode is fixed and equal to Z0. The creation of a
reflected entity is the only possibility allowed by the
equations of propagation on a transmission media.
Lastly, I note that in some of your correspondence you have
referred to the equation T(w) = 2*ZL/(Z0 + ZL) as
representing the signal "absorbed by the load". This is not
quite correct, as loads absorb "power", not "signal
amplitude", therefore we must change your statment to say
that the voltage apparent across the equals vload(w) =
vi(w)xT(w), and the power absorbed by the load equals
PowerAbsorbedByLoad(w) = [vload(w)] x [vload(w)*] / zload
where the symbol vload(w)* denotes the complex conjugate
of vload(w)
I hope these brief comments are helpful to you.
Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
tel +1 509-997-0505, HYPERLINK
"mailto:howie03@xxxxxxxxxx"howie03@xxxxxxxxxx
http:\\sigcon.com -- High-Speed Digital Design seminars,
books, and articles
P.S., My business manager asks that I encourage you to sign
up for my all-new course based on my latest book, High-Speed
Signal Propagation. Check out the web site for times/dates
of public venues. The classes are a great way to interact
personally and get all your questions answered. If you've
already been, please tell a friend...
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx
[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Peter Zhu
Sent: Tuesday, April 19, 2005 6:57 PM
To: ray jiang; si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: Question about the Equation in
Handbook of Black
Magic
Ray:
The 4.53 is the reflection at the terminal, so it is
(Zl-Z0)/(Zl+Z0),
WHY?
Why do you think (Zl-Z0)/(Zl+Z0) has been reflect back to
source?
Thanks
Peter
-----Original Message-----
From: ray jiang [mailto:jlray2000@xxxxxxxxx]=20
Sent: Wednesday, April 20, 2005 12:26 AM
To: Peter Zhu; si-list@xxxxxxxxxxxxx
Subject: Re: [SI-LIST] Question about the Equation in
Handbook of Black
Magic
Peter,=20
I have some idea on this question. The 4.53 is the
reflection at
the terminal, so it is (Zl-Z0)/(Zl+Z0). The deduce can be
found in any
books about transmition line theroy. As 4.52, i agree with
you that it
is the fraction of propagating signal that is absorbed by
the load,
which should be 1-R2(w) =3D 2Z0/(Zl+Z0). If this deduce is
right, i
think the book has made a mistake.
Best Regards,
Ray Jiang
On 4/19/05, Peter Zhu <peter.zhu@xxxxxxxxxx> wrote:
> All:
>=20
> One question in Dr. Howard Johnson-High Speed Digital
Design:
>=20
> How to understand the Equation 4.52 and 4.53
>=20
> I believe the T(W) is the fraction of propagating signal
that is
> absorbed by the load, and the R2(W) is the fraction of
propagating
> signal that reflects back toward the source.
>=20
> But where do the two equations come from? How to deduce
it? I think
it's
> the base to fully understand the signal reflection.
>=20
> Thanks in advance.
>=20
> Peter
>=20
> UTStarcom
>=20
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