[SI-LIST] Re: capacitor impedance in time domain

Steve, I am not sure why you would feel uncomfortable with Z = dv/dt / 
di/dt.  The miracle of linear superposition makes all the math that I am 
familiar with work.  One of the eye openers for me in much younger clothes 
was looking at passive filter networks as feedback networks.  This was 
described in a feedback and control systems text by MacDonald.  I'll dig it 
out if you are interested.  But basically we just end up with a Vo = Vin * 
1/( 1+G/H ) equation where Vin is the original pulse and Vo is the voltage 
waveform at the tx-line capacitor junction.  If you think about it, this is 
pretty much what Peter's math is a simplified representation of.

As to the derivation here we go:

1. i = C * dv/dt
2. Z = dv/dt / di/dt

Z = dv/dt * 1/( d/dt ( i ) )
Z = dv/dt * 1/ ( d/dt( C * dv/dt ))
Z = dv/dt * 1/(d(dv/dt)) * 1/C
Z = 1/( dv/dt * C )

Now if we compare this to the frequency domain definition:

Z = 1/jwC, recognizing that jw is just an expression of dv/dt, this all 
works.  It is just a matter of what representation is the most convenient 
to work with.

Regards,


Steve.

At 07:17 PM 1/30/2005 -0800, Steve Corey wrote:
>Steve -- perhaps I'm missing the point of your analysis, or not
>understanding how you're using Z.  If I combine your equation below [Z =
>1/( dv/dt * C )] with the defining equation for the capacitor
>i=C*(dv/dt), I get Z=1/i.  Perhaps you have a typo in there.
>
>Typos aside, I'm uncomfortable calling f = (dv/dt)/(di/dt) an impedance.
>     It's obviously time-dependent, but I would only feel comfortable if
>the time-dependent impedance transforms to the frequency-domain
>impedance.  Furthermore, the above funtion f is input-dependent, so it's
>not an intrinsic function of the device, which a system function such as
>impedance should be.  For example, consider a 1 farad capacitor.  If we
>excite it with v=sin(t), we get i=cos(t), and f=-cos(t)/sin(t).  If we
>excite it with v=cos(t), get i=-sin(t), and f=sin(t)/cos(t).
>
>The preceding arguments are really just nomenclature, about whether f
>should be called impedance, which obviously isn't important to solving
>the problem.  I may have missed the point of your prior analysis when I
>read it the first time by getting tripped up on the impedance part of
>it, so what I'm really interested in knowing is the following:
>
>Given a device's description (for example, i=C*(dv/dt)) how would you
>use f=(dv/dt)/(di/dt) to solve the problem of that device within a
>circuit such as that postulated in the initial post of this thread?  Or
>maybe more along the lines of your earlier analysis, how would f be used
>to lend rapid insight short of solving the problem rigorously (as was
>done by another list member)?
>
>I'm not trying to be contrary here -- just trying to understand if
>you're pointing out a circuit analysis technique to which I haven't been
>exposed.
>
>    -- Steve
>
>-------------------------------------------
>Steven D. Corey, Ph.D.
>Time Domain Analysis Systems, Inc.
>"The Interconnect Analysis Company."
>http://www.tdasystems.com
>
>email: steven.corey@xxxxxxxxxxxxxx
>phone: (503) 246-2272
>fax:   (503) 246-2282
>-------------------------------------------
>
>
>Steve Weir wrote:
> > Steve, actually the equation Z = 1/( dv/dt * C ) holds just fine in the
> > time domain.  You just have to remember that V and therefore dv/dt at the
> > node depends on Z.  The feedback equation using Z for the capacitor yields
> > the correct result whether we ground the capacitor, or simply use it to
> > couple two transmission line segments as is commonly done in a SERDES
> > backplane application
> >
> > Regards,
> >
> >
> > Steve.
> > At 11:54 PM 1/29/2005 -0800, Steve Corey wrote:
> >
> >>In my opinion, the easiest way to solve the problem is to think of what
> >>you would expect when TDR'ing an ideal 50-ohm T-Line of delay Td
> >>terminated by an ideal capacitance C, using an ideal 50-ohm TDR with a
> >>total voltage step of vstep.  The capacitor initially appears as a short
> >>circuit (voltage goes to zero, gamma=-1) and finally appears as an open
> >>circuit (voltage goes to vstep, gamma=+1).  Now think of the view from
> >>the perspective of the capacitor.  When it looks up the line, all it
> >>sees is 50 ohms to ground, tline energy storage notwithstanding -- the
> >>capacitor doesn't care if it's a terminated tline or a resistor.  As a
> >>result, the controlling time constant to charge the capacitor is
> >>(C*50ohms).  When we mix all this together, we come up with a TDR
> >>waveform that starts at zero, steps up to vstep/2, is flat for 2*Td,
> >>steps instantaneously to zero, then "decays" up to vstep with a time
> >>constant given by C*50ohms.
> >>
> >>We review this case, and a number of related cases, in our TDR quick
> >>guide, although the pictures do include parasitic effects such as finite
> >>risetime and a nonideal capacitor:
> >>http://www.tdasystems.com/library/appnotes/tdr_iconnect_quick_guide.PDF
> >>
> >>For a actual measurements on a power/ground plane pair, you can take a
> >>look at figure 4 in the following app note:
> >>http://www.tdasystems.com/library/appnotes/PDNA-0703.pdf
> >>
> >>Regarding Steve's earlier post -- replacing jw with dv/dt to map Z =
> >>1/jwC to Z = 1/(dv/dt * C) is really no different from a frequency
> >>domain analysis since the latter expression is only valid for a
> >>single-frequency sinusoidal v.  Looking for flat and steep places in a
> >>non-sinusoidal v and applying that relationship will not give valid 
> results.
> >>
> >>   -- Steve
> >>
> >>-------------------------------------------
> >>Steven D. Corey, Ph.D.
> >>Time Domain Analysis Systems, Inc.
> >>"The Interconnect Analysis Company."
> >>http://www.tdasystems.com
> >>
> >>email: steven.corey@xxxxxxxxxxxxxx
> >>phone: (503) 246-2272
> >>fax:   (503) 246-2282
> >>-------------------------------------------
> >>
> >>
> >>Steve Weir wrote:
> >>
> >>>Alan, well if we short a transmission line at the far end with a
> >>
> >>capacitor,
> >>
> >>>then Z still equals 1/( dv/dt * C ).  Hopefully this is obvious as a
> >>>reexpression the sinewave response Z = 1/jwC.  With a big "perfect"
> >>>capacitor, this will just look like a shorted line with a -1 reflection
> >>>coefficient. As the capacitor gets smaller, the pulse top reflection will
> >>>decay, and with further reductions in C the edge reflections will decay.
> >>>
> >>>The problem with viewing this as an RC is the stored energy in the line.
> >>>
> >>>Regards,
> >>>
> >>>
> >>>Steve
> >>>At 03:04 PM 1/28/2005 -0800, Alan Hilton-Nickel wrote:
> >>>
> >>>
> >>>>Steve, I don't think you and Chris are talking about the same 
> topology. My
> >>>>understanding is that Chris is looking at
> >>>>"a lossless transmission line terminated with a lossless capacitor", 
> which
> >>>>to me looks like:
> >>>>
> >>>>driver => txline => capacitor => GND
> >>>>
> >>>>I believe Arpad is correct in looking at this topology as an RC
> >>>>connection, especially in the lossless case.
> >>>>
> >>>>Alan
> >>>>
> >>>>Steve Weir wrote:
> >>>>
> >>>>
> >>>>
> >>>>>Arpad, this would be a good time to use Jon Powell's si-draw tool.  But
> >>>>>let me try this without:
> >>>>>
> >>>>>driver => txline => coupling cap => txline => terminator
> >>>>>
> >>>>>Now, let's suppose the driver has a Tr of 100ps, a Tf of 100ps, and a
> >>>>>pulse width of 2.4ns, an output impedance of 0.0001 ohms, and a drive
> >>>>>level of 2.5V.  Let's try three different values of coupling capacitors,
> >>>>>all in the same 0402 body and 50 ohm txlines.
> >>>>>
> >>>>>10pF
> >>>>>1nF
> >>>>>100nF
> >>>>>
> >>>>>Now, what sort of wave form do you expect to see at each point?
> >>>>>
> >>>>>I know what my answers are:  10pF lots of reflection towards the source
> >>>>>and droop towards the terminator.  1nF and 100nF very little reflection
> >>>>>or droop.  Plug it into your favorite simulator and see what you get.
> >>>>>
> >>>>>Regards,
> >>>>>
> >>>>>
> >>>>>Steve
> >>>>>At 02:03 PM 1/28/2005 -0800, Muranyi, Arpad wrote:
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>>You will get full reflection, because the impedance
> >>>>>>you need to use for the equation you quoted is the
> >>>>>>small signal (AC) impedance.  Referring to my previous
> >>>>>>posting a short time ago, the constant current source
> >>>>>>equivalent during the ramping portion of your trapezoid
> >>>>>>waveform has a (finite dV) / (dI =3D 0) =3D> infinite impedance.
> >>>>>>
> >>>>>>The question I have for you is this:  where do you mean
> >>>>>>that your waveform is trapezoid?  At the beginning of
> >>>>>>the T-line, or at the end, where the capacitor is?
> >>>>>>If the first, be prepared for a non trapezoid waveform
> >>>>>>at the capacitor, because the T-line and the cap forms
> >>>>>>an RC circuit, who's response is an exponential waveform.
> >>>>>>If the ramp is faster the RC constant you will see
> >>>>>>an exponential waveform, if it is slower, you will see
> >>>>>>a more or less trapezoid waveform.
> >>>>>>
> >>>>>>I hope this helps,
> >>>>>>
> >>>>>>Arpad
> >>>>>>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
> >>>>>
> >>3D=3D=
> >>
> >>>>>>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
> >>>>>
> >>3D=3D=
> >>
> >>>>>>=3D=3D=3D=3D=3D
> >>>>>>
> >>>>>>
> >>>>>>-----Original Message-----
> >>>>>>From: si-list-bounce@xxxxxxxxxxxxx
> >>>>>
> >>[mailto:si-list-bounce@xxxxxxxxxxxxx] =
> >>
> >>>>>>On Behalf Of group_delay
> >>>>>>Sent: Friday, January 28, 2005 1:48 PM
> >>>>>>To: si-list@xxxxxxxxxxxxx
> >>>>>>Subject: [SI-LIST] Re: capacitor impedance in time domain
> >>>>>>
> >>>>>>
> >>>>>>hi all,
> >>>>>>what i really want to do is find out how much waveform gets reflected
> >>>>>>
> >>>>>>
> >>>>>
> >>>>>>from the end of a lossless transmission line terminated with a
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>>lossless capacitor, assuming the input waveform is a trapezoidal
> >>>>>>signal. I know this can be computed using: gamma =3D (Zl-Zo)/(Zl+Zo),
> >>>>>>but this requires you to calculate Zl for the time domain signal. If I
> >>>>>>wanted to avoid it and use time domain analysis, how would I setup the
> >>>>>>equation?
> >>>>>>
> >>>>>>thanks,
> >>>>>>chris
> >>>>>>
> >>>>>>
> >>>>>>--- In si-list@xxxxxxxxxxxxxxx, steve weir <weirsp@xxxx> wrote:
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>matthias, in the time domain we would solve the differential
> >>>>>>>
> >>>>>>
> >>>>>>equations for=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the network, or more likely using a computer program we would solve =
> >>>>>>>
> >>>>>>
> >>>>>>the=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>difference equations over a series of discrete time steps.  Now in
> >>>>>>>
> >>>>>>
> >>>>>>either=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>case we could express impedance as dv/dt / di/dt.  But I don't know =
> >>>>>>>
> >>>>>>
> >>>>>>how=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>useful it would be towards either visualizing behavior, or solving 
> the =
> >>>>>>>
> >>>>>>>
> >>>>>>>equations.  Let's take the trapezoidal wave for instance.  An =
> >>>>>>>
> >>>>>>
> >>>>>>effective=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>impedance is pretty easy to come by on each:  the rising, and 
> falling=20
> >>>>>>>portions of the waveform from the capacitance expression C =3D
> >>>>>>>
> >>>>>>
> >>>>>>i/dv/dt, Z =3D=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>dv/dt / di/dt =3D 1/(dv/dt * C ).  The flat portions are troublesome
> >>>>>>>
> >>>>>>
> >>>>>>as are=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the vertices, since dv/dt theoretically goes to zero and the
> >>>>>>>
> >>>>>>
> >>>>>>impedance from=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the formula jumps to an infinite value.  Intuition should tell us
> >>>>>>>
> >>>>>>
> >>>>>>that this=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>is wrong, as
> >>>>>>>
> >>>>>>
> >>>>>>coupling capacitors routinely pass high frequency pulses.
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>=20
> >>>>>>>In the frequency domain, we have this nailed.  We don't have=20
> >>>>>>>discontinuities at the vertices.  The vertices and flat portions =
> >>>>>>>
> >>>>>>
> >>>>>>follow=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>curves formed by the frequency components, and rather than a flat
> >>>>>>>
> >>>>>>
> >>>>>>section=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>containing DC and no HF, quite the opposite is true:  the flatter we
> >>>>>>>
> >>>>>>
> >>>>>>want=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the pulse tops to be, the higher the frequency content 
> required.  This =
> >>>>>>>
> >>>>>>>
> >>>>>>>aligns with our intuition.  But when we transform the representation
> >>>>>>>
> >>>>>>
> >>>>>>back=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>to the time domain, those piecewise linear segments are now curved
> >>>>>>>
> >>>>>>
> >>>>>>solving=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the discontinuities at the vertices and eliminating the flat slopes
> >>>>>>>
> >>>>>>
> >>>>>>with=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>theoretically infinite Z between the edges.
> >>>>>>>=20
> >>>>>>>So if someone wanted to look only at the rising and falling edges, 
> an=20
> >>>>>>>impedance in the time domain is reasonable, and possibly even
> >>>>>>>
> >>>>>>
> >>>>>>useful.  But=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>it really gets awkward when dealing with the whole waveform unless
> >>>>>>>
> >>>>>>
> >>>>>>we first=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>perform frequency limiting operations, most easily performed in the=20
> >>>>>>>frequency domain.
> >>>>>>>=20
> >>>>>>>I am not an expert on algorithms, so I really can't say from an 
> error=20
> >>>>>>>analysis and computational efficiency standpoint what is really the
> >>>>>>>
> >>>>>>
> >>>>>>best=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>way to perform a transient analysis.  But in my naivete, I would be=20
> >>>>>>>inclined to transform everything into the frequency domain, compute =
> >>>>>>>
> >>>>>>
> >>>>>>the=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>solution and transform back.  In my feeble mind, this would avoid
> >>>>>>>
> >>>>>>
> >>>>>>some of=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>the discontinuity and convergence problems in SPICE and more 
> closely=20
> >>>>>>>follows nature.  But since people a whole lot better at math than I
> >>>>>>>
> >>>>>>
> >>>>>>have=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>worked long and hard on those algorithms, I suspect either the=20
> >>>>>>>computational overhead, or error build-up of my naive approach would =
> >>>>>>>
> >>>>>>
> >>>>>>be=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>unacceptably high.  Maybe what this world needs is a five cent, 256 =
> >>>>>>>
> >>>>>>
> >>>>>>bit=20
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>floating point, matrix solver!
> >>>>>>>=20
> >>>>>>>Steve.
> >>>>>>>=20
> >>>>>>>At 10:13 PM 1/26/2005 +0100, Matthias Bergmann wrote:
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>>Hello, I don`t understand why impedance should be limited to =
> >>>>>>>>
> >>>>>>>
> >>>>>>Frequency
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>domain. What impedance are we speaking about ? For example the
> >>>>>>>>characteristicimpedance Z of a transmission line also exists in
> >>>>>>>>
> >>>>>>>
> >>>>>>time domain.
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>If you look along a transmission line, v(t) / i(t) have got
> >>>>>>>>
> >>>>>>>
> >>>>>>singularities
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>(undefined, infinite), these are called short and open ?!?!?
> >>>>>>>>
> >>>>>>>
> >>>>>>Furthermore
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>mostof the simulation programs use the time domain because it permits
> >>>>>>>>non-linearities. I don`t know how what happens when your impulse is
> >>>>>>>>trapezoidal, but if it was a rectangular and your load is a
> >>>>>>>>
> >>>>>>>
> >>>>>>capacitance, you
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>are answer would look like an exponential function, with your
> >>>>>>>>
> >>>>>>>
> >>>>>>reflection
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>co-efficient as initial value. Regards, Matthias Bergmann P.S.:
> >>>>>>>>
> >>>>>>>
> >>>>>>Yes, use
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>SPICE or ADS ! _m |---------+---------------------------------->
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
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> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
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> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
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> >>>>>>>>
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> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>-list@xxxx>&#160;&#160;&#160;&#160;&#160;&#160;&#160&#160;&#160;&#160
> >>>>>>>>-LIST] Re: capacitor impedance in time
> >>>>>>>>
> >>>>>>>
> >>>>>>>domain&#160;&#160;&#160;&#160;&#160;&#16&#160;&#160;&#160;&#160;&#160
> >>>>>>
> >>;&#=
> >>
> >>>>>>160;&#1
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>60I&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;
> >>>>>>
> >>&#1=
> >>
> >>>>>>60;&#16
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;|
> >>>>>>>>
> >>>>>>>
> >>>>>>>&#160;>--------------------------------------------------------------
> >>>>>>
> >>---=
> >>
> >>>>>>----
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>-- &#160;-----------------------------------------| >I could be
> >>>>>>>>
> >>>>>>>
> >>>>>>wrong >but
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>tome >impedance is a concept strongly related to Frequency domain.
> >>>>>>>>It is
> >>>>>>>>meaningful just in that domain. Absolutely. If you define impedance =
> >>>>>>>>
> >>>>>>>
> >>>>>>as
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>voltage/current, then you run into great difficulties if you try to
> >>>>>>>>
> >>>>>>>
> >>>>>>do it in
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>the time domain.&#160; In general, with any complex impedance,
> >>>>>>>>
> >>>>>>>
> >>>>>>v(t)/i(t) has
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>singularities (undefined, infinite). I consider impedance =3D
> >>>>>>>>
> >>>>>>>
> >>>>>>v(s)/i(s) or
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>v(f)/i(f), which makes it a strictly frequency domain parameter.
> >>>>>>>>
> >>>>>>>
> >>>>>>Regards,
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>Andy
> >>>>>>>>
> >>>>>>>
> >>>>>>------------------------------------------------------------------ To
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>unsubscribe from si-list: si-list-request@xxxx with 'unsubscribe'
> >>>>>>>>inthe Subject field or to administer your membership from a web
> >>>>>>>>
> >>>>>>>
> >>>>>>page, go to:
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>http://www.freelists.org/webpage/si-list For help:
> >>>>>>>>si-list-request@xxxx with 'help' in the Subject field List FAQ wiki
> >>>>>>>>page is located at:
> >>>>>>>>
> >>>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160&#160;&#16
> >>>>>>
> >>0;&=
> >>
> >>>>>>#160;
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>&#160;&#160;http://si-list.org/wiki/wiki.pl?Si-List_FAQ List =
> >>>>>>>>
> >>>>>>>
> >>>>>>technical
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>documents are available at:
> >>>>>>>>
> >>>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#1&#160;&#160;&#160;
> >>>>>>
> >>&#1=
> >>
> >>>>>>60;&#16
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>&#160;http://www.si-list.org List archives are viewable at:
> >>>>>>>>
> >>>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#1
> >>>>>>
> >>60;=
> >>
> >>>>>>&#160;
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;htt
> >>>>>>
> >>p:/=
> >>
> >>>>>>/www.
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>freelists.org/archives/si-list or at our remote archives:
> >>>>>>>>
> >>>>>>>
> >>>>>>>&#160;&#160;&#160;&#1&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;
> >>>>>>
> >>&#1=
> >>
> >>>>>>60;&#16
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;http://gr
> >>>>>>
> >>oup=
> >>
> >>>>>>s.yah
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>oo.com/group/si-list/messages Old (prior to June 6, 2001) list
> >>>>>>>>
> >>>>>>>
> >>>>>>archives are
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>viewable at:
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> >>>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#1
> >>>>>>
> >>60;=
> >>
> >>>>>>&#160;
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;htt
> >>>>>>
> >>p:/=
> >>
> >>>>>>/www.
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>>>qsl.net/wb6tpu
> >>>>>>>>------------------------------------------------------------------
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> >>>>>>>
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> >>>>>>Old (prior to June 6, 2001) list archives are viewable at:
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> >>>>>>=20
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> >>>>>>
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> >>>>>The weirsp@xxxxxxxxxx e-mail address will terminate March 31, 2005.
> >>>>>Please update your address book with weirsi@xxxxxxxxxx
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> >>>
> >>>
> >>>
> >>
> >>
> >>
> >>------------------------------------------------------------------
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> >>si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
> >>
> >>or to administer your membership from a web page, go to:
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> >>
> >
> >
> > The weirsp@xxxxxxxxxx e-mail address will terminate March 31, 2005.
> > Please update your address book with weirsi@xxxxxxxxxx
> >
> >
> > ------------------------------------------------------------------
> > To unsubscribe from si-list:
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> >
> > or to administer your membership from a web page, go to:
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> >
> > For help:
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> > List FAQ wiki page is located at:
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> > or at our remote archives:
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> > Old (prior to June 6, 2001) list archives are viewable at:
> >               http://www.qsl.net/wb6tpu
> >
> >
> >
> >
>
>
>------------------------------------------------------------------
>To unsubscribe from si-list:
>si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
>
>or to administer your membership from a web page, go to:
>http://www.freelists.org/webpage/si-list
>
>For help:
>si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
>
>List FAQ wiki page is located at:
>                 http://si-list.org/wiki/wiki.pl?Si-List_FAQ
>
>List technical documents are available at:
>                 http://www.si-list.org
>
>List archives are viewable at:
>                 http://www.freelists.org/archives/si-list
>or at our remote archives:
>                 http://groups.yahoo.com/group/si-list/messages
>Old (prior to June 6, 2001) list archives are viewable at:
>                 http://www.qsl.net/wb6tpu
>

The weirsp@xxxxxxxxxx e-mail address will terminate March 31, 2005.
Please update your address book with weirsi@xxxxxxxxxx


------------------------------------------------------------------
To unsubscribe from si-list:
si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field

or to administer your membership from a web page, go to:
http://www.freelists.org/webpage/si-list

For help:
si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field

List FAQ wiki page is located at:
                http://si-list.org/wiki/wiki.pl?Si-List_FAQ

List technical documents are available at:
                http://www.si-list.org

List archives are viewable at:     
                http://www.freelists.org/archives/si-list
or at our remote archives:
                http://groups.yahoo.com/group/si-list/messages
Old (prior to June 6, 2001) list archives are viewable at:
                http://www.qsl.net/wb6tpu

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