[time-nuts] Collector current that minimizes BJT noise

[Matt Huszagh]

Hi,

In the the first edition of Low-Noise Electronic Design, Motchenbacher
states that minimal noise for a BJT is attained at low collector
currents (here we restrict ourselves to midband noise where 1/f noise
and noise terms that are appreciable near the transition frequency can
be neglected). To justify this, Motchenbacher gives the equation (eqn
4-23, for those with the text)

Fopt = 1 + sqrt(2rbb / (beta * re) + 1/beta)

However, this neglects the dependence of base-spreading resistance on
collector current. At high quiescent current, the emitter current crowds
out the base current such that the base current travels a shorter
distance through the base region, thus decreasing the effective base
resistance (for references see eg the 1963 paper by Hauser).

The gummel-poon model quantifies this effect as

rbb = rbm + 3(rb - rbm) ((tan(z) - z) / (z tan^2(z)))
z = (sqrt(1 + (12/pi)^2 * (ib / irb)) - 1) / (24/pi^2 * sqrt(ib/irb))

where rb is the zero-bias base resistance, rbm is the minimum base
resistance (at high current) and irb is the base current at which rbb is
halfway between rb and rbm.

I ran calculations on several (fairly random) parameter combinations and
the collector current that minimizes noise depends on the values
chosen. Does this rbb current dependence invalidate Motchenbacher's
simple prescription, or is the practical behavior of BJTs such that
minimizing collector current generally does minimize BJT noise
contribution?

This raises the more general question: how do people find the minimum
noise operating conditions for BJTs? Do you go straight to measurement,
or do you first attempt to estimate it from datasheet and SPICE values?
If so, can you shed some light on this process? I've attempted, for
instance, to estimate rbb from SPICE models (using the equations above)
and have had worse than bad success equating these to various measured
results (from Art of Electronics and various papers). It's worth
mentioning that measured results are also wildly different between
sources, so this whole thing may be an exercise in futility anyway.

As an aside, I feel this is on topic for this forum because BJT noise
and base-spreading resistance are very relevant for low residual phase
noise buffers, among other things. Additionally, I've seen some very
good discussions of BJT noise on here. But, let me know if you feel this
is off-topic and I can take this discussion elsewhere.

Matt