IEEE Solid-States Circuits Magazine - Spring 2021 - 19

phase is much easier to read off a
plot than slope is. A pure inductor
impedance has a phase angle of
90°. The closer your output impedance
gets to 90°, the more it will ring
when loaded with the appropriate
capacitor. At 90° exactly, it should
sustain continuous oscillation. Figure
3 is a plot of the output impedance
magnitude and phase for our
example block, and-sure enough-
the phase is maximum at 300 kHz,
where the loaded amplifier peaking
is worst.
You may be inclined to draw an
analogy between the impedance
phase of Figure 3 and the open-loop
phase of a Bode plot. That is, the phase
difference between the output impedance
and that of a pure inductor is a
kind of safety margin indicating how
far away you are from making an
oscillator. Is this the same as " phase
margin " in the familiar feedback
sense? (See " If It Quacks Like Phase
Margin.... " ) Surprisingly, the jury is
still out on that. Practically though, it
is close enough, and you can reliably
use your feedback intuition to gauge
what the numbers mean when looking
at this analysis. So go ahead, call
it " phase margin. " Or " load margin. " Or
" Chris margin. "
There is probably no need to
say this, but you should keep in mind
that everything here is small signal
analysis, which means that it is operating
point dependent. If your circuit
needs to function over a wide range
of operating conditions (as does a lowdropout
regulator, for example), make
sure you test the corners thoroughly.
Also, if you are operating with low
phase margin, the transients may
take you outside your intended signal
range, where conditions are different
than you chose for the ac sims. But
you knew that, right?
And, Now, the Showmanship!
Figure 3 is almost all you need. It tells
you the frequency range over which
the amplifier output will look inductive,
and it tells you what frequency
will have the worst ringing and how
bad it will be. The only thing left
is to associate each frequency with
the capacitor value that will cause
ringing at that frequency. That's
easy: use the impedance magnitude
curve at any frequency, and find the
capacitor that has that same impedance
magnitude at that frequency.
(Remember ZZ
CL is a condition
=
of resonance.) Done!
However, if you really
want to show off, you can
go a step further. Since the
impedance magnitude to
capacitance conversion is
a simple expression, why
not plot it for every point
on your ac sweep? Here's
a shortcut: plot the magnitude
and phase of the
capacitance-to-impedance equation
times the imaginary unit, j:
j/(2##r frequency × output_
impedance),
Once you know
what to look
for, you'll be
amazed at how
often you come
across inductive
nodes in practical
situations.
where output_impedance is the
complex output impedance of the
unloaded block as a function of
frequency. [Using the gray test circuit
in Figure 1, this would just be
v(vtest), but make sure your simulator
is using complex numbers for this
waveform.] The magnitude
of this expression
is the desired capacitor
value, and the j adds a
phase shift so that the
phase plots directly as
phase margin. This is displayed
in Figure 4(a).
At this point, we separate
the grandstanders, the real
showboats, from the pragmatists.
Using the XY plot features
of your simulator (MATLAB or Excel
can do this if your simulator can't),
plot the phase-margin versus capacitance,
as in Figure 4(b). Ta-da! You
If It Quacks Like Phase Margin...
Is the phase difference presented in Figure 3 really " phase margin " in the feedback sense? Mathematically,
it takes the same form. To illustrate this, consider an approximate model of a loaded
block, as shown on the left side of Figure S1.
Compared to the classical feedback equation, the term Zload
/Zout takes the same role as the
loop transmission. The closer Zload/Zout gets to −1, the greater the magnitude of the response, and
the more dramatic the phase, just as we expect with the complex loop transmission. In fact, if
you calculate the actual loop transmission of the circuit of Figure 1 when it is loaded by Zload
you will find that it is almost identical to Zload/Zout over the frequency range where amplifier gain
is declining... but not everywhere.
,
Output Impedance Model
Zout
+
-
Zload
+
A(s)
-
f
Feedback Loop
H(s) =
=
1 +
Zout
Zload
∴
FIGURE S1: A loaded block model.
Zout
Zload
≅ Af
Zload
H(s) =
Zload + Zout
1
=
A
1 + Af
11
f
.
1
1 +
Af
IEEE SOLID-STATE CIRCUITS MAGAZINE
SPRING 2021
19

IEEE Solid-States Circuits Magazine - Spring 2021

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Spring 2021