Electronic Design

Digital PWM Controllers Augment System Reliability

Intelligent digital controllers can simplify power-supply design as well as enhance overall reliability through monitoring and statistical interpretation of key performance metrics.

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 x1 or x2 (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.

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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 RS (Fig. 4). At dc, we can write the expression for the output voltage as:

Solving for the average duty cycle D and replacing RLOAD with VOUT/iL yields:

Then we can solve for RS:

This says that, if we monitor the duty cycle, VIN, 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 RS 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 RS, but the relative change in RS. As such, this method of RS 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 za/2 is the double-sided probability that the sample mean is within the confidence interval. Some typical values for za/2 are 1.96 for 95%, 2.58 for 99%, and 6.0 for 2 parts per billion.3

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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

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