π⋅Figure 1: Undamped LC filter very large, and amplify t - PDF document

π⋅Figure 1: Undamped LC filter very large, and amplify the noise at that frequency. of the problem it is necessary to analyze the transfer function of the filter: =11sLRload ⋅⋅ Cutoff frequency [Hz] (resonance frequenc y Second Order Input filter 100110311041105 40 30 20 1001020 Magnitude, dB30.120.707 © 2010 National Semiconductor Corporationwww.national.com 1LC⋅−Z⋅ ⋅ 1j2 ⋅ Z⋅= Cutoff frequency in radiantL2RLC Damping factor (zeta) The transfer function presents two negative poles at: 01 −The damping factor the two poles are complex, and the imaginAs the damping factor becomes smaller, the gain at the corner frequency becomes larger, the ideal limit for zero damping wresistance of the real components limits the maximum gain. With a damping factor equal to one the imaginary component isdamping factor on the input filter design could have other side effects on the final performance of the system. It can influencontrol loop, and cause some oscillatiThe Middlebrook’s extra element theorem (pap (pap)does not significantly modify the converter loop gain if the output impedance curve of words to avoid oscillations it is important to keep the peak output impedance of the filter below the input impedance From a design point of view, a good compromise between size of the filter and performance is obtained with a minimum damping factor of 1/final control system. © 2010 National Semiconductor Corporationwww.national.com impedance. The output impedance and the transfer function of the filter can be calculated the same way as the parallel damped filter: rom the approximated transfer function of the series damped filter, the damping The optimal damped resistance is: Figure 7 : Series damped filter factor can be calcuThe peaking is minimi ⋅ ⋅ Zfilter3s()11Z1s() 1Z2s() 1Z3s() ⋅ ⋅⋅⋅ = 321 Rdn1() C L n34n⋅12n⋅214n Ffilter3s()Z2Z2Zeq1.3 sLL ⋅⋅⋅⋅ ⋅= 1n1() ⋅sL ⋅ © 2010 National Semiconductor Corporationwww.national.com REFERENCES Rudolf P. Severns, Gordon E. Bloom “Modern DC to DC Switchmode Power Converter Circuits” R.D. Middlebrook, “Design Techniques for Preventing Input Filter Oscillations in Robert W. Erickson “Optimal Single ReH. Dean Venable “Minimizing Input Filter” Jim Riche “Feedback Loop StabilizBruce W. Carsten “Design Techniques for Examples of filters using a basic step down simple switcher power supply Downloads: Mathcad example EXE files PTC® Mathcad website (links to PTC website) Basic step-down simple switcher power supply: Input parameters Results Maximum input voltage: Vinput40VOutput current: Iout1A⋅=Output voltage: Vout5VOutput inductor: Lo66DC resistance: © 2010 National Semiconductor Corporationwww.national.com c3.14210Maximum input impedance of the power supply: Rin25ohm⋅=Input Capacitance of the power supply: C15UNDAMPED LC FILTER Inductance calculated: L0.068mHDamping factor: 2Rin Inductor used: Lf33Rf0.030:⋅=Capacitor used: Cf47ESRci0.150:=Cut off frequency of the filter: π⋅LfCf 4.041kHz Transfer function: Rfs⋅= = © 2010 National Semiconductor Corporationwww.national.com 20log ⋅= 100110311041105 40 30 20 1001020 Filter output impedance: 1001103110411050.010.1110100 Filter output impedance In order to avoid oscillations it is important to keep the peak output impedance of the filter below the input impedance of the converter. The two curves should not overlap. In most of the cases a parallel damped filter easily meets the damping and impedance requirements. © 2010 National Semiconductor Corporationwww.national.com 10011031104110511060.010.1110100 Undumped filter Parallel damped filter SERIES DAMPED FILTER π⋅LfCf Series inductor: LdLfn⋅= Ld4.4 Series damping resistance: Cf Rds0.838:= Rdss⋅= Transfer function: © 2010 National Semiconductor Corporationwww.national.com 20log ⋅= 100110311041105 40 30 20 1001020 Undumped Filter Series damped filter With the series damped filter the gain at high frequency is attenuated. Filter output impedance: 1001103110411050.010.1110100 Undumped filter Series damped filter Power supply input impedanceOutput impedanceFrequency, HzOutput Impedance, Ohm © 2010 National Semiconductor Corporationwww.national.com MULTIPLE FILTER SECTIONS First LC filter: L18.25 RL10.1:= C111.75 ESRc10.120:=π⋅L1C1 fm116.165kHzSecond LC filter: L27L1⋅= L257.75 RL20.1:=C24C1⋅= C247 ESRc20.120:=π⋅L2C2 fm23.055kHz Rd40.419:= Ld4L18 = =Rd4s⋅Rd4s⋅⋅ Zm4i1siC2 = © 2010 National Semiconductor Corporationwww.national.com Transfer function: © 2010 National Semiconductor Corporationwww.national.com20log Zm3iZm4i Zm2iZm1iZm2i ⋅= 100110311041105 60 50 40 30 20 1001020 Two stage filter Series damped filter Filter output impedance: Zm3iZm1iZm2iZm1iZm2i Zm3iZm4i 10011031104110511060.010.1110100 Two stage filter Series damped filter Power supply input impedanceOutput impedanceFrequency, HzOutput Impedance, Ohm IMPORTANTNOTICE TexasInstrumentsIncorporatedanditssubsidiaries(TI)reservetherighttomakecorrections,modifications,enhancements,improvements, andotherchangestoitsproductsandservicesatanytimeandtodiscontinueanyproductorservicewithoutnotice.Customersshould obtainthelatestrelevantinformationbeforeplacingordersandshouldverifythatsuchinformationiscurrentandcomplete.Allproductsare soldsubjecttoTI¶stermsandconditionsofsalesuppliedatthetimeoforderacknowledgment. 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