Impedance matching CAD analog circuit function?

[beta]

Is there a way to match impedance for a circuit given typical input and
output stages of transistors for square wave (and other) types of
inputs? Was wondering if there were such a function in spice type cad
simulators (e.g. orcad) that would compute the proper loads given
networks of R, C components, and later stages of transistors. Part of
issue is that I'm not certain how to do it without spice utils, so it
maybe the cart before the horse situation. Thanks.

 

[beta]

Why?

Three reasons:
1. I have a low power design that I'm trying to develop, and think
matched impedances will provide most efficient results.
2. The simulation model shows this odd transient (dampened ringing)
that I suspect is due to mis-matched impedances, which are not seen in
the design reference circuit.
3. Desire to understand basics of how to match them given source and
load situations/stages, how it translates into better design
performance, and what the manifestations/problems/characteristics are
of badly matched impedances.

For #3 I have read that when designing a communications bus, for
example, mismatched impedance results in standing waves, and
reflections. Although I am dealing with an analog design that has
nothing to do with a bus, clearly the importance of impedance matching
makes a big difference given these types of problems. I realize this is
only anecdotal, and therefore not necessarily meaning anything from a
technical point of view.

If an analog circuit is cut in half with the left side viewed as an
output impedance and the right with an input impedance, my
understanding is the max power possibly transferred is -3 db. I arrived
at this by looking at some texts and thinking in terms of voltage
dividers. The real design I have has a square wave input, and the
Fourier transform (I believe) is therefore infinite. So matching the
characteristics of the input transfer function to derive an output
transfer function such that -3 db is achieved is no trivial
accomplishment. I think that some simplification takes place when
trying to discover the optimal values of the R, L, C network to best
hit the -3 db target by neglecting the higher order terms. Again this
is all intuition, and I realize not a strong technical argument.
However I have several texts in front of me that I am reviewing
further.

For a low noise bipolar design, the optimum noise is given by around
about re=Rs/sqrt(hfe), and since ri=hfe.re, one can argue that there is
a matching being done.

What is re, Rs, if by hfe you mean DC Current Gain (or Small-Signal
Current Gain?). Also can you please elaborate on the meaning of
'ri=hfe.re'.
Thanks.

 

[beta]

The real design I have has a square wave input,
and the Fourier transform (I believe) is therefore infinite.

Real square waves are finite in their BW use.

I am trying to reconcile my understanding with your statement. How are
real sq waves finite in their BW use? This is important because I
thought the impedance for R, L, C networks changes for different input
signals?

Looking at sq wave Fourier derivations, it seems that the Fourier
Series of a square wave is appox sin(x) + (1/3)sin(3x) + (1/5)sin(5x) +
.... [ref google]. From Wolfram's math site I found the derivation as
follows. "The Fourier series for the square wave with period 2L, phase
offset 0, and half-amplitude 1 is therefore,

f(x) = (4/pi) * sum from n=1,3,5... to infinity of 1/n (sin (n * pi * x
/ L))"

If frequency is 100 kHz, then period is 10 u sec (T = 1/f). Solving
for L, to use in the above formula yields:

2L = 10us => L=5 us.

It would seem to imply that f(x) = (4/pi) * sin (pi * x / 5) + (4/pi) *
1/3 * sin ( 3 * pi * x / 5) + ...

As an approximation, I see that the series converges, thus at some
point the smaller signal contributions could be ignored. Looking at
different derivation of the sq wave, it says it is a square function
convolved with an infinite impulse train. The sinc function being the
Fourier transform of a single 0 centered square (or ideal low pass
filter) of length 2L. [Circuits, Signals and Systems, Siebert, pp 434],
resulting in the main lobe around zero containing the highest dB
contribution to the circuit response, and therefore the impedance value
(unless I've gone awry). In real life, I am using a sq wave that begins
at some time, then goes on for a while, then ends when the device is
turned off.

So
matching the characteristics of the input transfer function to derive
an output transfer function such that -3 db is achieved is no trivial
accomplishment. I think that some simplification takes place when
trying to discover the optimal values of the R, L, C network to best
hit the -3 db target by neglecting the higher order terms. Again this
is all intuition, and I realize not a strong technical argument.
However I have several texts in front of me that I am reviewing
further.

For a low noise bipolar design, the optimum noise is given by around
about re=Rs/sqrt(hfe), and since ri=hfe.re, one can argue that there
is a matching being done.

What is re, Rs, if by hfe you mean DC Current Gain (or Small-Signal
Current Gain?). Also can you please elaborate on the meaning of
'ri=hfe.re'.

re is the intrinsic emiiter impedance that sets the gm of the device.

ic = gm.vi

That is, the small signal output current is given by gm times the small
signal input voltage.

gm = 40IC, where IC is the DC bias current.

ri is input resistance. Rs is the source resistance. hfe is the small
signal current gain.

Its all here
http://www.anasoft.co.uk/EE/bipolardesign1/bipolardesign1.html

For impedance, it appears I should strive to make Rload >> Rinternal,
or Rinput >> Routput given a Thevenin equivalent circuit. [Art of
Electronics, pp 12]. This prevents input signal attenuation based on
the voltage divider relationship:

Vout = Vin * Rload / (Rload + Rinternal)

If Rload >> Rinternal, then Vout ~ Vin, meaning that the signal
strength is retained despite the additional load. But as you wrote,
ri=hfe.re (or as discussed in your paper approximately Ri=hfe.Re'),
means that the input impedance is high, about 200 * (25 + 100) = 25 k.
This would be OK if it is the Rload (and analog circuits were connected
to transistors in this fashion) which is what I think you were saying,
but how does it look if this is the source and I am connect an Rload? I
think you state it's 5 ohms, or just re the small signal effective
resistance, which is also good, but bad for the amp-speaker
relationship you described earlier.