[SI-LIST] Re: capacitor impedance in time domain
- From: Dorin <dorin.oprea@xxxxxxxxxxx>
- To: weirsi@xxxxxxxxxx
- Date: Mon, 31 Jan 2005 08:55:02 -0500
Nice try! (dv/dt)/[d(dv/dt)] is not equal with dv/dt.
Thanks,
Dorin
Steve Weir wrote:
> 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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> > >>>>>>>>
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> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
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> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>>-list@xxxx>          
> > >>>>>>>>-LIST] Re: capacitor impedance in time
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>domain          
> > >>>>>>
> > >>;&#=
> > >>
> > >>>>>>160;
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>60I           
> > >>>>>>
> > >>=
> > >>
> > >>>>>>60;
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>          |
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>> >--------------------------------------------------------------
> > >>>>>>
> > >>---=
> > >>
> > >>>>>>----
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>--  -----------------------------------------| >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.  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:
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>//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:
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>           
> > >>>>>>
> > >>0;&=
> > >>
> > >>>>>>#160;
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>  http://si-list.org/wiki/wiki.pl?Si-List_FAQ List =
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>technical
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>documents are available at:
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>           
> > >>>>>>
> > >>=
> > >>
> > >>>>>>60;
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>> http://www.si-list.org List archives are viewable at:
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>           
> > >>>>>>
> > >>60;=
> > >>
> > >>>>>> 
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>           htt
> > >>>>>>
> > >>p:/=
> > >>
> > >>>>>>/www.
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>freelists.org/archives/si-list or at our remote archives:
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>           
> > >>>>>>
> > >>=
> > >>
> > >>>>>>60;
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>          http://gr
> > >>>>>>
> > >>oup=
> > >>
> > >>>>>>s.yah
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>oo.com/group/si-list/messages Old (prior to June 6, 2001) list
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>archives are
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>viewable at:
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>           
> > >>>>>>
> > >>60;=
> > >>
> > >>>>>> 
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>           htt
> > >>>>>>
> > >>p:/=
> > >>
> > >>>>>>/www.
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>>>qsl.net/wb6tpu
> > >>>>>>>>------------------------------------------------------------------
> > >>>>>>>>To unsubscribe from si-list:
> > >>>>>>>>si-list-request@xxxx with 'unsubscribe' in the Subject field
> > >>>>>>>>
> > >>>>>>>>or to administer your membership from a web page, go to:
> > >>>>>>>>//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:
> > >>>>>>>> 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:
> > >>>>>>>> //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
> > >>>>>>>>
> > >>>>>>>>
> > >>>>>>>
> > >>>>>>>=20
> > >>>>>>>=20
> > >>>>>>>------------------------------------------------------------------
> > >>>>>>>To unsubscribe from si-list:
> > >>>>>>>si-list-request@xxxx with 'unsubscribe' in the Subject field
> > >>>>>>>=20
> > >>>>>>>or to administer your membership from a web page, go to:
> > >>>>>>>//www.freelists.org/webpage/si-list
> > >>>>>>>=20
> > >>>>>>>For help:
> > >>>>>>>si-list-request@xxxx with 'help' in the Subject field
> > >>>>>>>=20
> > >>>>>>>List FAQ wiki page is located at:
> > >>>>>>> http://si-list.org/wiki/wiki.pl?Si-List_FAQ
> > >>>>>>>=20
> > >>>>>>>List technical documents are available at:
> > >>>>>>> http://www.si-list.org
> > >>>>>>>=20
> > >>>>>>>List archives are viewable at: =20
> > >>>>>>> //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
> > >>>>>>>
> > >>>>>>
> > >>>>>>------------------------------------------------------------------
> > >>>>>>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:
> > >>>>>>//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: =20
> > >>>>>> //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
> > >>>>>>=20
> > >>>>>>
> > >>>>>>------------------------------------------------------------------
> > >>>>>>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:
> > >>>>>>//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:
> > >>>>>> //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:
> > >>>>>//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:
> > >>>>> //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:
> > >>>//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:
> > >>> //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
> > >>>
> > >>>
> > >>>
> > >>>
> > >>
> > >>
> > >>
> > >>------------------------------------------------------------------
> > >>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:
> > >>//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:
> > >> //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:
> > > //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:
> > > //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
> > >
> > >
> > >
> > >
> >
> >
> >------------------------------------------------------------------
> >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:
> >//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:
> > //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:
> //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:
> //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
>
------------------------------------------------------------------
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:
//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:
//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
Other related posts:
