Multiple methods are available to monitor the health of a power supply, ultimately leading to improved reliability of the power subsystem and, subsequently, the total system. These improvements can come from adjusting system operating parameters based on these real-time diagnostics or by alerting the host system that the power subsystem performance is degraded, allowing the system to adjust or schedule maintenance.

Because discrete values of the powersystem states already exist, digital control makes it convenient for this monitoring and evaluation to occur within the power supply itself. It also simplifies monitoring parameters that otherwise might require additional circuitry to sense.

One important advantage of a digitally controlled power solution is that it’s possible to monitor complex parameters. In addition to simple parameters (like switching frequency, duty cycle, input and output voltage, input and output current, and the temperature of various components), complex parameters (such as power dissipation, efficiency, stability margin, output ripple voltage, input ripple voltage, phase-current mismatch, pulsewidth jitter, and fault history) can be captured and reported to the host system.

Traditionally, things like current, voltage, and temperature have been easy to measure. However, we need the embedded intelligence of the digital controller to determine parameters such as stability margin or pulse-width jitter. Access to such information and the controller’s embedded intelligence can allow for complex operations; for example, adjusting its own compensation if it senses the stability margin is unacceptable.

The Bode characteristics of the loop gain can provide considerable insight into component values, efficiency, and stability margins. The ability of a digital controller to make this measurement while the power supply is deployed in an actual product offers a unique opportunity to improve the reliability of the overall system.

Once the Bode characteristics are determined, classical stability metrics like phase margin, gain margin, and loop bandwidth can be extracted from the resulting data. In addition, the output filter’s resonant frequency and quality factor (Q) also can be extracted. This data then can be compared to expected values. If the observed changes are statistically significant, conclusions about the component values or efficiency can be made and, if deemed necessary, a maintenance request can be sent to the system.

Figure 1 shows a typical power-supply
application. The transfer function from the
switching node to the output has the form
of Equation 1 with passive loss elements
shown in Equation 2.^{1}

The Q of the output filter is related to
the loss elements connected to the energy
storage components L and C; ?_{Z} is related
to the output capacitance and its associated
equivalent series resistance (ESR);
and ?_{0} is primarily determined by the
resonance of the inductor and capacitor.

While in this example the resonant
frequency is a function of R, ESR, and
DCR (Fig. 1, again), efficiency requirements
demand that R be much larger than
either ESR or DCR. The result is that ?_{0} is
approximately a function of L and C only.
Because Q is linked to the losses, a large
change in its value means either a passive
component value has changed, or a large
change occurred in the MOSFET’s losses.
Either way, it’s possible to alert the system
that maintenance is needed. A history of
the Bode metrics can be stored in memory
for later statistical analysis.

In addition to making measurements, the controller must be able to interpret an appropriate time to take the measurement. Bode characteristics are only relevant during steady-state conditions with known input voltage, load characteristics, and temperatures. A digital controller can monitor these items before, after, and during the measurement. If any of these parameters are unacceptable, delay the measurement until such time as they are acceptable.

As an application, the controller can
measure and record critical loop gain
characteristics right before the product is
deployed into the field. If the system can
record bandwidth, Q, and ?_{0} at a known
load and temperature when the product
is new, the power supply can periodically
monitor these parameters to see if a statistically
significant change has occurred and
alert the host system as appropriate.

System Identification

Measuring the power system’s transfer
function and creating the Bode plot of the
loop gain is called system identification.
The classical way a network analyzer measures
a system is to inject an excitation signal
at a summing junction at one location
around the loop and measure the response
at another point. If we chose locations
within the controller where the control signals
are discrete samples, we can use digital
techniques to apply the excitation and measurement.
The power system can be excited
by injecting a signal at x_{1} or x_{2} (Fig. 2). The
response to the excitation can be measured
at e, c, d, or u. Reference 2 describes the
associated math for each case.

Continued on page 2

Because the compensating filter for a digital controller operates numerically, minimal offset and gain tolerances are associated with its transfer function. It will also have little drift with time and temperature. This means that the only variation in the transfer function of the compensator will be due to tolerance in the clock frequency for the digital logic.

Therefore, any variation in the measured loop transfer function should be due to changes in the analog power stage and not the controller. If the compensator transfer function is divided out of the measured open-loop response, an accurate picture of the power stage, and any variation, can be observed.

In addition to monitoring the small signal ac transfer function, a digital controller has ready access to the instantaneous and average duty cycle. In a digital pulsewidth modulation (PWM) controller, a digital filter performs compensation (Fig. 3). The filter’s output is proportional to the control effort necessary to regulate the output voltage.

Since the filter is digital, the filter’s output can be easily sampled by the supervising microcontroller. In fact, the authors of the PMBus Command Standard for digital power supplies anticipated this and defined the standard command: READ_ DUTY_CYCLE.

Using Multiple Parameters

For a buck regulator, it’s well known
that the duty cycle must increase as losses
grow in the system. This concept can be
used to estimate the series resistive losses
in the power stage. In a simplified buck
power stage, we can see that the series
resistive losses are lumped together as R_{S} (Fig. 4). At dc, we can write the expression
for the output voltage as:

Solving for the average duty cycle D and
replacing R_{LOAD} with V_{OUT}/i_{L} yields:

Then we can solve for R_{S}:

This says that, if we monitor the duty
cycle, V_{IN}, and inductor current (all things
that the controller already monitors), we
can estimate the series resistance in the
power stage. A change in this parameter
would indicate that the health of the power
stage has been compromised.

Any real-world power supply has some
associated switching losses. In part, they
will affect the value of R_{S} measured by this
method. However, when making a health
assessment of the power supply in situ, the
principal item of interest isn’t the absolute
value of R_{S}, but the relative change in R_{S}. As such, this method of R_{S} measurement
also provides a figure of merit on the
switching losses in the regulator.

Statistical Process Control

A digital controller’s embedded processing
power can be utilized to interpret
measured and calculated data through statistics.
Manufacturers use statistical process
control (SPC) techniques to maintain
control of their manufacturing process.
An electronic system can use the same
technique to measure critical parameters
relating to the power supply.

The general approach is to first estimate the expected mean and standard deviation for a measurement. This is usually done during product development. Then periodic measurements are made, and the measured value is compared against limits based on a confidence interval.

To determine the deviation that represents a problem, define some interval \[µ – k, µ + k\] such that, if the averaged measurement values fall outside of this interval, we can state with some percent confidence that the mean has changed. Here, k is calculated as:

where s is the expected population standard
deviation, n is the sample size, and
z_{a}/2 is the double-sided probability that
the sample mean is within the confidence
interval. Some typical values for z_{a}/2 are
1.96 for 95%, 2.58 for 99%, and 6.0 for 2
parts per billion.^{3}

Continued on page 3

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*