[SI-LIST] Re: capacitor impedance in time domain
- From: steve weir <weirsp@xxxxxxxxxx>
- To: matthias-bergmann@xxxxxx, si-list@xxxxxxxxxxxxx
- Date: Wed, 26 Jan 2005 17:20:25 -0800
matthias, in the time domain we would solve the differential equations for
the network, or more likely using a computer program we would solve the
difference equations over a series of discrete time steps. Now in either
case we could express impedance as dv/dt / di/dt. But I don't know how
useful it would be towards either visualizing behavior, or solving the
equations. Let's take the trapezoidal wave for instance. An effective
impedance is pretty easy to come by on each: the rising, and falling
portions of the waveform from the capacitance expression C = i/dv/dt, Z =
dv/dt / di/dt = 1/(dv/dt * C ). The flat portions are troublesome as are
the vertices, since dv/dt theoretically goes to zero and the impedance from
the formula jumps to an infinite value. Intuition should tell us that this
is wrong, as coupling capacitors routinely pass high frequency pulses.
In the frequency domain, we have this nailed. We don't have
discontinuities at the vertices. The vertices and flat portions follow
curves formed by the frequency components, and rather than a flat section
containing DC and no HF, quite the opposite is true: the flatter we want
the pulse tops to be, the higher the frequency content required. This
aligns with our intuition. But when we transform the representation back
to the time domain, those piecewise linear segments are now curved solving
the discontinuities at the vertices and eliminating the flat slopes with
theoretically infinite Z between the edges.
So if someone wanted to look only at the rising and falling edges, an
impedance in the time domain is reasonable, and possibly even useful. But
it really gets awkward when dealing with the whole waveform unless we first
perform frequency limiting operations, most easily performed in the
frequency domain.
I am not an expert on algorithms, so I really can't say from an error
analysis and computational efficiency standpoint what is really the best
way to perform a transient analysis. But in my naivete, I would be
inclined to transform everything into the frequency domain, compute the
solution and transform back. In my feeble mind, this would avoid some of
the discontinuity and convergence problems in SPICE and more closely
follows nature. But since people a whole lot better at math than I have
worked long and hard on those algorithms, I suspect either the
computational overhead, or error build-up of my naive approach would be
unacceptably high. Maybe what this world needs is a five cent, 256 bit
floating point, matrix solver!
Steve.
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@xxxxxxxxxxxxx>          
>-LIST] Re: capacitor impedance in time
>domain           
>60I            
>          |
> >---------------------------------------------------------------------
>--  -----------------------------------------| >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 = v(s)/i(s) or
>v(f)/i(f), which makes it a strictly frequency domain parameter. Regards,
>Andy ------------------------------------------------------------------ To
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