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

• From: Steve Weir <weirsi@xxxxxxxxxx>
• To: Alan Hilton-Nickel <alan@xxxxxxxxxxxxxxxxx>
• Date: Fri, 28 Jan 2005 15:35:35 -0800

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