A power-supply designer performs many chores. And in today's environment, the development time and cost are major constraints. Additionally, the loads have become very demanding in both power and response. The power-supply designer must judiciously study the various components that make up the power stage to meet the desired power transfer and cost target. After the designer finds the best compromise of components, the engineer must make them work in the intended application. To this end, many days are spent tuning the circuit across various operating conditions to arrive at, once again, the best compromise for load response.
With the advent of digitally controlled power, the engineer now has the opportunity to optimize loop compensation. The mechanism that compensates the loop is also available for analysis. The loop data converter is used to determine the disturbance in the power-supply output by slightly modifying the duty cycle that is controlled by the digital pulse width modulator (PWM). The loop control components provide both the stimulus and response needed to determine the power-supply frequency characteristics. The digital system helps in the analysis and also remembers the optimal compensation at various conditions of line and load. These compensation parameters are used based on the digital power solution either recognizing this same condition or when the system informs the power supply that a particular operating mode is pending.
To perform the power-supply loop analysis, an injection signal is constructed from a sine-wave sequence with a specified amplitude and frequency. This sine-wave sequence is injected into the feedback loop by adding the sequence to one of the control loop variables. At a different place in the loop, the response to the injected sequence is measured by performing a discrete fourier transform (DFT) at the injection frequency. If the DFT operation includes a sine and cosine component, the magnitude and phase of the response can be calculated from the orthogonal results of the DFT operation.
Loop Analysis Using Digital Power Controller
The techniques for measuring the frequency response for a feedback-controlled system are essentially the same, whether the system to be measured is a continuous-time analog system or a discrete-time digital system. Fig. 1 shows two possible locations to inject the excitation sine-wave signal, labeled as d1 and d2. The figure also shows the possible locations to measure the response to the excitation signal. They are labeled y, u, c, x and e.
In the case of d1, the sine wave is added digitally to the results of the error calculation. The error calculation is simply the difference between the digitized voltage output and the digital equivalent of the preferred voltage output. In a likewise manner, d2 is added to the digital value used to generate the pulse width in the digital PWM.
A more complete analysis of the transfer gains can be found in the many complete references listed at the end of this article.[1-6] However, the table lists the various gains for each of the designated measurement locations shown in Fig. 1.
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Digital Controller Design
Let's define a buck converter stage as shown in Fig. 2. The equation for the small signal ac transfer function for this converter is shown in Eq. 1:
For representative values, use
KDUTY = 24/100; V/% duty
R = 1800 Ω
C = 3 × 22 µF
RC = 0.001 Ω
L = 0.65 µH
RL = 0.058 Ω.
This produces a power stage with a zero at 2.4 MHz, and a complex second-order pole at 24.3 kHz with a Q of 1.68. The gain is defined for purposes of simulating the differences between measurement methods such that the input to the power stage is in percent duty cycle and the output is in volts.
The controller consists of a two-pole, two-zero digital compensator. In this example, the compensator zeros are both set to 30 kHz, and the poles are set to zero (to form an integrator) and 300 kHz. The gain of the compensator, defined at 1 kHz, is 43 dB. The digital sample rate is set to the PWM switching frequency of 700 kHz.
Fig. 3 shows the simulated open-loop transfer function for the representative system. So now let's use this system to evaluate the four transfer gains defined in Fig. 4. The G/(1+GH) trace has the lowest gain. At best its gain is -20 dB. This means that you would have to inject a large signal to get a small amplitude at the measurement location — not good. The 1/(1+GH) trace has a low gain at low frequencies, but a gain at or above 1.0 for high frequencies. Likewise, the GH/(1+GH) trace has good gain at low frequencies and low gain at high frequencies. Finally, the H/(1+GH) trace is the transfer gain seen when injecting the excitation signal at the input to the compensator and measuring it at the output of the compensator. In this case, we get to use the gain in the compensator, and it has the highest measurement gain.
Bode Analysis Design Tool
An in-circuit loop analysis was developed based on a TMS320F2808 and a PC-based design tool for a digital telecom rectifier reference design. The PC communicates to the power supply through an RS-232 interface. The telecom rectifier has three loops that can be analyzed with the system, the power factor correction (PFC) voltage loop, the PFC current loop and the dc-dc voltage loop. Commands are defined to select one of three loops in the power-supply system.
To characterize the system, an injection node and a response-measurement node are selected. The analysis start frequency, stop frequency, number of frequency steps, injection amplitude, number of dwell samples and number of measurement samples are specified. The PC test program sends commands to the digital controller to make a frequency response measurement for each frequency step. At the end of each measurement, the digital controller returns the two accumulated sine and cosine coefficients for that frequency. The PC program calculates the complex open-loop transfer function, and then plots the magnitude and phase for that frequency.
Because the power-stage compensator is digital, the test program queries the digital controller for the compensator coefficients, then calculates the exact frequency response of the compensator. Once the compensator frequency response is known, it is factored out of the open-loop transfer function to calculate the transfer function for the power stage.
Once these measurements and calculations are made, the user selects the display of frequency response of the power stage, the frequency response of the digital compensator, the frequency response of the open-loop system or the frequency response of the closed-loop system.
After the loop analysis measurement has been made and the frequency response of the analog power stage is determined, the Bode tool can be used to quickly explore the effect of changing the compensator coefficients, since the compensator is deterministic due to the digital nature.
Digitally Controlled Power Analytical Benefits
The digitally controlled power-supply loop analysis is beneficial during the power-supply design, in manufacturing and during system operation. The power-supply designer determines the compensation for the desired operating conditions as they do today for analog supplies. Where today the designer uses a network analyzer for this analysis and adjusts the resistors and capacitors of the compensation network, the digital power designer can work in virtual space adjusting the compensation terms to achieve the best results. At the same time, the designer can be assured that the compensator will be very deterministic and without the circuit tolerances of the analog components.
During manufacturing, each supply can be optimized for frequency response based on the characteristics on the power-stage components rather than compromised by the variances that are expected. This allows for a broader range of acceptability of the power-stage component values without compromising the supply's frequency response.
One of the major nemeses of the power-supply designer is the system designer. The system designer may place large amounts of capacitors around various components to help bypass or provide local energy storage. In many cases, the actual results of such liberal use of capacitance can actually reduce the power-supply frequency performance.
Fig. 5 shows examples of adding various capacitances to the output of a 1-kW telecom rectifier without adjusting the compensation. As the capacitance is added, the gain of the system is reduced and, therefore, the frequency response may not meet the system needs. With the ability of in-system frequency analysis, the power supply can be recompensated to adjust for this unexpected capacitance. If this operation is not done, then at a minimum, the host can be notified that the power-supply frequency response may not meet the requirement.
The technique described for measuring the transfer function of the power stage is efficient in its use of memory and mixed-integer programming. This technique also has very good signal-to-noise ratio characteristics if the right nodes are chosen for injecting the excitation signal and for measuring the response. Ultimately, this measurement capability opens up the possibility of moving the measurement and calculation of loop compensation from the designer's lab bench to the factory floor, or to the end customer's application.
Miftakhutdinov, Rais. “Compensating DC/DC Converters with Ceramic Output Capacitors.” Texas Instruments Power-Supply Design Seminar (SEM1600), 2004.
Figoli, David. “Creating a Two Channel Sine Wave Generator,” Texas Instruments application note (spra412.htm, 1 KB), January 1999.
TMS320x280x DSP Boot ROM Reference Guide Rev. B (spru722b.htm, 9 KB), Aug. 26, 2005.
Zwillinger, Daniel (editor), Standard Mathematical Tables and Formula, 30th Edition, CRC Press, 1996.
Hagen, Mark. “In-situ Bode Measurement in a Digitally Controlled Power Supply,” Darnell Digital Power Forum, 2006.