[SI-LIST] Re: capacitor impedance in time domain
- From: Peter Fekete <thefekete@xxxxxxxxx>
- To: si-list@xxxxxxxxxxxxx
- Date: Sun, 30 Jan 2005 00:59:49 -0800 (PST)
Steve,
You are right, the case of an unit step voltage (which is also the case
described in the two books mentioned in my previous message) helps
understanding what is happening, probably better then just some equations.
However when the pulse shape is more complicated and you want to know why the
pulse does not ?step instantaneously to zero? when there is a finite rise time
you need to look into details.
As group_delay wrote: ?The dip does go to zero for a zero rise time input, but
it does not go to zero for a finite rise time signal. i am trying to figure out
the best way to calculate the amount of this dip...?
Peter
Steve Corey <steven.corey@xxxxxxxxxxxxxx> 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 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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>>>>>>
>>>>>>
>>>>>>
>>>>>>-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
>>>>>>
>>>>>
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>>>>>>page is located at:
>>>>>>
>>>>>
>>>>> &=
>>>>>
>>>>
>>>>#160;
>>>>
>>>>
>>>>
>>>>>> http://si-list.org/wiki/wiki.pl?Si-List_FAQ List =
>>>>>>
>>>>>
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>>>>
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>>>>
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>>>>>
>>>>
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>>>>
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