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

• From: Alan Hilton-Nickel <alan@xxxxxxxxxxxxxxxxx>
• To: weirsi@xxxxxxxxxx
• Date: Fri, 28 Jan 2005 15:04:22 -0800

```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,
>>
>>=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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>>>>
>>>>-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
>>>>
>>>>
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