The design of a switching power supply has always been considered a blend of magic and art by every engineer that designs one for the first time. Fortunately, today the market offers different tools to help designers. For example, National Semiconductor offers the “Simple Switcher” software and an on-line simulation tool for designing and simulating a switching power supply. New ultra-fast MOSFETs and synchronous high-switching-frequency PWM controllers result in high efficiency and smaller switching power supplies.
All these advantages can be lost if the input filter is not properly designed. An oversized input filter unnecessarily adds cost and volume to the design and compromises system performance. This paper explains how to choose and design the optimal input filter for a switching power supply application.
The input filter on a switching power supply has two primary functions. One is to prevent electromagnetic interference, generated by the switching source, from reaching the power line and affecting other equipment. The second is to prevent high-frequency voltage on the power line from passing through the output of the power supply.
A passive L-C filter solution can achieve both filtering requirements. The goal for the input filter design should be to achieve the best balance between filter performance versus size and cost.
The first simple passive-filter solution is the undamped L-C passive filter shown in Figure 1 . Ideally, a second order filter provides 12dB per octave of attenuation after the cutoff frequency f_{0} , has no gain before f_{0} , and has peak response at the resonant frequency.
Cutoff frequency [Hz] (resonance frequency)
Figure 1: Â Schematic of an undamped LC filter |
Figure 2: Â Transfer function of an LC filter with different damping factors |
One of the critical factors involved in designing a second-order filter is the attenuation characteristic at the corner frequency f_{0} . The gain near the cutoff frequency could be very large, and amplify the noise at that frequency. To have a better understanding of the nature of the problem, it is necessary to analyze the transfer function of the filter:
The transfer function can be rewritten with the frequency expressed in radians:
Cutoff frequency in radiant
Damping factor (z )
The transfer function presents two negative poles at:
The damping factor z describes the gain at the corner frequency.
For z > 1 the two poles are complex and the imaginary part gives the peak response at the resonant frequency.
As the damping factor becomes smaller, the gain at the corner frequency becomes larger. The ideal limit for zero damping would be infinite gain, but the internal resistance of the real components limits the maximum gain. With a damping factor equal to one, the imaginary component is zero and there is no peaking.
A poor damping factor on the input-filter could have other side effects on the final performance of the system. The damping factor can influence the transfer function of the feedback control loop and cause oscillation at the output of the power supply.
Middlebrook's extra element theorem explains that the input filter does not significantly modify the converter loop gain if the output impedance curve of the input filter is far below the input impedance curve of the converter. In other words, to avoid oscillation it is important to keep the peak output impedance of the filter below the input impedance of the converter. (Figure 3 ). From a design point of view, a good compromise between filter size and performance is obtained with a minimum damping factor of 1/square root 2, which provides a 3 dB attenuation at the corner frequency and favorable control over the stability of the final control system.
Figure 3: Â The dip in input impedance of the power supply should be well separated from the peak in output impedance of the input filter |
In most cases an undamped second-order filter, such as the one shown in Figure 1 , does not easily meet the damping requirements; thus, a damped version is preferred:
Figure 4: Â Parallel damped filter |
Figure 4 shows a damped filter made with a resistor R_{d} in series with a capacitor C_{d} , both connected in parallel with the filter's capacitor C_{f} . The purpose of R_{d} is to reduce the output peak impedance of the filter at the cutoff frequency. C_{d} blocks the dc component of the input voltage, and avoids power dissipation in R_{d} . C_{d} should have lower impedance than R_{d} at the resonant frequency and have a bigger value than the filter capacitor, so as not to influence the cutoff point of the main RL filter.
The filter's output impedance comes from the parallel connection of the three block impedances: Z_{1} , Z_{2} , and Z_{3} .
The transfer function is:
where Z_{eq2.3} is Z_{2} in parallel with Z_{3} .
The transfer function presents a zero and three poles, where the zero and the first pole fall close to each other at frequency w almost equal to 1/R_{d} C_{d} . The other two dominant poles occur at the cutoff frequency, w _{o} = 1/square root LC. Without compromising the results, the first pole and the zero can be ignored, and the formula can be approximated to a second-order equation:
(for frequencies higher than w almost equal to 1/R_{d} C_{d} , [1+R_{d} C_{d} * s] almost is equal to R_{d} C_{d} * s)
The approximated formula for the parallel damped filter is identical to the transfer function of the undamped filter; the only difference being that the damping factor z is calculated with the R_{d} resistance.
For a parallel damped filter, peaking is minimized with a damping factor equal to:
Combining the last two equations, the optimum damping resistance value R_{d} is equal to:
With the blocking capacitor C_{d} equal to four times the filter capacitor C.
Figures 5 and 6 show the output impedance and the transfer function of the parallel damped filter, respectively.
Figure 5: Â Output impedance of the parallel damped filter |
Figure 6: Â Transfer function of the parallel damped filter |
Another way to obtain a damped filter is with a resistance Rd in series with an inductor L_{d} , all connected in parallel with the filter inductor L (Figure 7 ). At the cutoff frequency, the resistance R_{d} has to be larger than L_{d} 's impedance.
Figure 7: Â Schematic of a series damped filter |
The output impedance and the transfer function of the filter are obtained the same way as they were for the parallel damped filter:
From the approximated transfer function of the series damped filter, the damping factor is:
The peaking is minimized with a damping factor:
The optimal damped resistance is:
The disadvantage of this damped filter is that the high-frequency attenuation is degraded (Figure 10 ).
Most of the time, a multiple section filter allows higher attenuation at high frequencies with less volume and cost, because if the number of single components is increased, it allows the use of smaller inductances and capacitances (Figure 8 ).
Figure 8: Â Schematic of a two-section input filter |
The output impedance and transfer function can be calculated from the combination of each block impedance:
Figures 9 and 10 show the output impedance and the transfer function of the series damped filter compared with the undamped filter.
The two-stage filter has been optimized with the following ratios:
The filter provides an attenuation of 80 dB with a peak filter output impedance lower than 2 Ohms.
Figure 9: Â Output impedance of the series damped filter and two-stage damped filter |
Figure 10: Â Transfer function of the series damped filter and two-stage damped filter |
The switching power supply rejects noise for frequencies below the crossover frequency of the feedback control loopâ€”the input filter should reject higher frequencies. To be able to meet the forward filtering with a small solution, the input filter has to have the corner frequency around one decade below the bandwidth of the feedback loop.
Another important issue affecting the final performance of the filter is the right selection of capacitors and inductors. For high-frequency attenuation, you need capacitors with low ESL and low ESR for ripple-current performance. The most common capacitors used are aluminum electrolytic types. To achieve low ESR and ESL the output capacitor could be split into separate smaller capacitors in parallel to achieve the a larger total value. Design filter inductors to reduce parasitic capacitance as much as possible. The input and output leads should be kept as far apart as possible, and single layer or banked windings are preferred.
At the National Semiconductor Web site, you can find all the information and tools needed to design a complete switching power supply. On the Web site are datasheets, application notes, selection guides, and WEBENCH power supply design software.
About the Author
Michele Sclocchi is a staff application engineer for National Semiconductor, Europe. Prior to joining National Semiconductor, Mr. Sclocchi was a hardware analog design engineer for Baker Atlas Logging Services in Houston, TX. He has a Master Degree in Electrical Engineering from Politecnico University of Milan, Italy. |
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