As an example, assume that during
development the mean and sigma of the
open-loop bandwidth are µ = 55.0 kHz
and s = 0.750 kHz. Then, during normal
operation, we periodically identify the
0-dB bandwidth by exciting the system
at frequencies around the last bandwidth
estimate and adjust the measurement frequency
until a 0-dB gain is found.
This process of detecting the 0-dB crossover
is repeated four times, resulting in the
values [56, 58, 53, 55] kHz. The average
value is 55.5 kHz. To determine with 95%
confidence whether the mean has changed,
assign za/2 to be 1.96. Then the interval
k is 1.96 × 0.750/v(4) = 0.735 kHz and
the confidence interval is [54.2650 kHz,
55.7350 kHz]. Since 55.5 kHz is within
this interval, we can say with 95% confidence
that the mean has not changed.
Health Metrics
Using the aforementioned system ID
techniques and applying the confidence
intervals borrowed from statistical process
control, we can define a set of metrics
upon which to make decisions about the
power supply’s health.
Phase margin: This is one of the most
important parameters relating to the
closed-loop system’s behavior. If the
voltage-regulation circuits don’t have sufficient phase margin, the response to
changes in commanded voltage or load
current will be a large ringing disturbance
in the regulated output voltage. If severe
enough, the result could damage the
circuits powered by the regulator. This
makes phase margin a strong candidate for
a health metric.
To calculate phase margin, the loop
gain is measured and the magnitude of the
measured values is inspected to find the
frequency at which the magnitude of the
gain is equal to 1.0. The distance of the
measured loop phase response at this frequency
from 180º is the phase margin.
Power stage ?0 and Q: By exciting the
system over a range of frequencies that
include the expected resonant frequency
of the power stage, we can construct a
health metric for power-supply components
that otherwise would be difficult
to measure. A health metric based on ?0 can be an indicator of a change in output
capacitance or inductor value. This could
be due to damage to the capacitor dielectric
or a cracked inductor.
A health metric based on the quality
factor of the output filter can be used to
identify changes in the series resistance
of the filter components. At low load currents,
the load resistance is larger than
the ESR of the capacitors and DCR of the inductor and MOSFETs. In this case, the
Q of the power stage response is:
Therefore, Q will decrease with increasing
series resistance.
Average duty cycle: Separate from the
dynamic measurements used to estimate
?0 and Q of the plant, we can estimate the
series losses by comparing the average
duty cycle to the measured voltage and
supply current. This is an indicator of efficiency,
a performance metric that’s become
increasingly important in today’s world.
In addition to steady-state duty cycle,
the digital controller can collect statistics
on the duty-cycle jitter. This jitter can be
used as an extra input when determining
the optimal loop compensation. For example,
if the controller determines a given
Bode response from the TFA algorithm
and then implements what it believes to be
an appropriate compensation, the results
of that compensation on the system duty
cycle can be checked. Variations in duty
cycle can be interpreted as having a direct
impact on the system noise and output
voltage ripple. If the jitter is deemed
excessive, an alternate compensation can
be chosen with a larger gain margin to
quiet the duty-cycle jitter.
Experimental Results
Figure 5 shows the measured plant
response for a single-phase power stage
driven by a UCD9240 digital PWM controller.4 The controller accepts commands
over a serial interface to excite the loop at a
given frequency and returns a complex (real
and imaginary) response for that frequency.
In this case, a host computer was used
to issue the commands and collect the
complex data. From the closed-loop
response to the excitation, the open-loop
gain was calculated. Then the gains of the
error voltage analog-to-digital converter
(ADC), compensation filter, and PWM
modulator were divided out of the openloop
response, yielding the transfer function
for the power stage.
To simulate a fault, the DCR of the inductor
was increased from 2 mO to 42 mO. As
you can see, the Q of the power-stage
response dropped substantially.
References:
1. R.W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, Second
Edition, Springer Science + Business
Media Inc., 2001
2. Mark Hagen, “In Situ Transfer Function
Analysis,” 2006 Digital Power Forum
3. Daniel Zwillinger (editor), Standard
Mathematical Tables and Formula, 30th
Edition, CRC Press, 1996
4. UCD9240 Digital Point-of-Load System
Controller, Rev. C, Texas Instruments,
2008: http://focus.ti.com/docs/prod/folders/print/ucd9240.html