CN107167747A - The monitoring device and method of CCM buck converter inductance and output capacitance - Google Patents

The monitoring device and method of CCM buck converter inductance and output capacitance Download PDF

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CN107167747A
CN107167747A CN201710508163.4A CN201710508163A CN107167747A CN 107167747 A CN107167747 A CN 107167747A CN 201710508163 A CN201710508163 A CN 201710508163A CN 107167747 A CN107167747 A CN 107167747A
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msub
mrow
centerdot
inductance
output
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CN107167747B (en
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杨思文
姚凯
韩旭芝
周世林
曹诚
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Nanjing University of Science and Technology
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Nanjing University of Science and Technology
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/40Testing power supplies
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R27/00Arrangements for measuring resistance, reactance, impedance, or electric characteristics derived therefrom
    • G01R27/02Measuring real or complex resistance, reactance, impedance, or other two-pole characteristics derived therefrom, e.g. time constant
    • G01R27/26Measuring inductance or capacitance; Measuring quality factor, e.g. by using the resonance method; Measuring loss factor; Measuring dielectric constants ; Measuring impedance or related variables
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R27/00Arrangements for measuring resistance, reactance, impedance, or electric characteristics derived therefrom
    • G01R27/02Measuring real or complex resistance, reactance, impedance, or other two-pole characteristics derived therefrom, e.g. time constant
    • G01R27/26Measuring inductance or capacitance; Measuring quality factor, e.g. by using the resonance method; Measuring loss factor; Measuring dielectric constants ; Measuring impedance or related variables
    • G01R27/2605Measuring capacitance

Abstract

The invention discloses a kind of CCM buck converter inductance and the monitoring device and method of output capacitance, the device includes Buck converters main power circuit, drive circuit, display unit, electric capacity and current transformer known to one parameter, and signal processing module, wherein signal processing module include power circuit control unit, switching frequency fsComputing unit, dutycycle D computing units, output voltage sampling unit, capacitance current triggering sampling unit, inductance L and electric capacity ESR and C computing unit.The parameter L and the parameter ESR and C of electric capacity that the present invention can be in the case where not influenceing circuit normal work to inductance are monitored, and the life prediction for electric capacity and power supply provides foundation, without additional parameter, convenient to realize.

Description

The monitoring device and method of CCM buck converter inductance and output capacitance
Technical field
The invention belongs to the monitoring technical field in electrical energy changer, more particularly to a kind of CCM buck converter electricity Sense and the monitoring device and method of output capacitance.
Background technology
Switching Power Supply has the advantages that efficiency high, small volume, the extensive use in daily production and living.Decompression (Buck), Boost (Boost), buck (Buck-Boost) converter is three kinds of most basic switching power converters, other converters It can be developed by these three converters.Wherein, CCM (Continuous Current Mode, continuous current mode) Buck converters are widely used in fields such as computer power supply, communication power supply, Aero-Space.In Buck converter circuits in order to Relatively stable output voltage is obtained, generally requires using electric capacity to filter out high-frequency noise.Converter worked after a period of time, The capacitance (Capacitance, C) and equivalent series resistance (Equivalent Series Resistance, ESR) of electric capacity can be sent out Changing, when the variable quantity is larger, that is, thinks that the electric capacity has failed, it will cause the operation troubles of power supply and system, therefore The ESR and C of the output filter capacitor of CCM Buck converters are monitored, predicts that its life-span is extremely important.Existing technology mainly may be used It is divided into off-line monitoring technology and on-line monitoring technique, offline inspection technology application method is simple, and cost is low, but general only monitoring is only The electric capacity stood on outside circuit, and online measuring technique can detect electric capacity in running order in circuit, but application method Complexity, and need to know many other parameters in circuit.
The content of the invention
It is an object of the invention to provide a kind of CCM buck converter inductance and the monitoring device and method of output capacitance, The inductance value L and equivalent series resistance ESR of inductance and the capacitance C of electric capacity change can be monitored in real time, to electrochemical capacitor and electricity The life-span in source carries out Accurate Prediction.
The technical solution for realizing the object of the invention is:A kind of CCM buck converter output capacitance ESR and C monitoring Device includes Buck converters main power circuit, drive circuit, display unit, electric capacity and Current Mutual Inductance known to a parameter Device, and signal processing module, the signal processing module include power circuit control unit, switching frequency fsComputing unit, Dutycycle D computing units, output voltage sampling unit, capacitance current triggering sampling unit, inductance L and electric capacity ESR and C are calculated Unit;
The Buck converters main power circuit includes input voltage source Vin, switching tube Qb, sustained diodeb, filtered electrical Feel L, output filter capacitor and load RL, the output filter capacitor includes equivalent series resistance ESR and electric capacity C, wherein switching Pipe QbDrain electrode and voltage source VinPositive pole connection, sustained diodebNegative electrode and switching tube QbDrain electrode connection, the pole of afterflow two Pipe DbAnode and voltage source VinNegative pole connection, filter inductance L one end and sustained diodebNegative electrode connection, filtered electrical Feel the L other end respectively with equivalent series resistance ESR one end and load RLOne end connection, equivalent series resistance ESR's is another One end is connected with electric capacity C one end, electric capacity the C other end and load RLThe other end with voltage source VinNegative pole connection, bear Carry RLTwo ends are output average voltage Vo
The voltage source V of the input of the power circuit control unit respectively with Buck converter main power circuitsinWith it is defeated Go out average voltage VoConnection, the pwm signal of power circuit control unit output end is respectively connected to switching frequency fsComputing unit and account for Sky is than D computing unit, the output average voltage V of Buck converter main power circuitsoOutput voltage sampling unit is accessed, electric current is mutual Sense isolation amplifying unit and the pwm signal of power circuit control unit output end access capacitance current triggering sampling unit, open Close frequency fsComputing unit, dutycycle D computing units, output voltage sampling unit and capacitance current trigger the output of sampling unit The output end access display unit of electric capacity ESR and C computing unit, inductance L and electric capacity ESR and C computing unit is accessed at end;
The input of the drive circuit is connected with the pwm signal of power circuit control unit output end, drive circuit Output end access switching tube QbGate pole.
A kind of CCM buck converter output capacitance ESR and C monitoring method, comprises the following steps:
Step 1, the electric capacity known to a parameter in output end parallel connection, creates power circuit control in signal processing module Unit processed, switching frequency fsComputing unit, dutycycle D computing units, output voltage sampling unit, capacitance current triggering sampling are single Member, inductance L and electric capacity ESR and C computing unit;
Step 2, the power circuit control unit of signal processing module is flat according to the output of Buck converter main power circuits Equal voltage Vo, obtain pwm signal and through drive circuit driving switch pipe Qb
Step 3, the pwm signal feeding switching frequency f of power circuit control unit outputsComputing unit and dutycycle D meters Unit is calculated, through switching frequency fsComputing unit processing draws the current switching frequency f of converters, at dutycycle D computing units Reason draws the current dutycycle D of converter;
Step 4, the output average voltage V of Buck converters main power circuitoOutput voltage sampling unit is sent into, obtains defeated Go out the average value of voltage;
Step 5, the electric capacity electricity of the pwm signal of power circuit control unit output and Current Mutual Inductance isolation amplifying unit (8) Flow ixCapacitance current triggering sampling unit is sent into, by delay procedure to DT such as capacitance currentss/ 10 interval samplings, obtain ix(0)、 ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix(3DTs/5)、ix(7DTs/10)、ix (4DTs/5)、ix(9DTs/10)、ix(DTs) totally 11 value;
Step 6, by obtained switching frequency fs, dutycycle D, the average value V of output voltageoAnd capacitance current is instantaneous Value ix(0)、ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix(3DTs/5)、ix(7DTs/ 10)、ix(4DTs/5)、ix(9DTs/10)、ix(DTs) feeding inductance L and electric capacity ESR and C computing unit carry out curve fitting and Integrated treatment, obtains value and the output filter capacitor current equivalent series resistance ESR and electric capacity C of inductance L in Buck converters Value;
Step 7, inductance L and electric capacity ESR and C computing unit are by the inductance L and equivalent series resistance ESR and electricity of gained The value feeding display unit for holding C is shown in real time.
Compared with prior art, remarkable advantage of the invention is:The present invention is for the inductance of CCM Buck converters and defeated Go out filter capacitor, design the inductance L and output filter capacitor equivalent series resistance ESR and electric capacity C of a kind of efficient stable On-Line Monitor Device and method, parameter L and electricity that this method can be in the case where not influenceing circuit normal work to inductance The parameter ESR and C of appearance are monitored, and the life prediction for electric capacity and power supply provides foundation, without additional parameter, convenient to realize.
Brief description of the drawings
Fig. 1 is the working waveform figure in the CCM Buck converter switches cycles.
Fig. 2 is the structural representation of CCM buck converter inductance L of the present invention and output capacitance ESR and C monitoring device Figure.
Wherein:Vin- input voltage, Iin- input current, iL- inductive current, iC- capacitance current, iCx- shunt capacitance electric current, Io- output current, Vo- output voltage average value, Qb- switching tube, Db- diode, L- inductance, C- output filter capacitor values, ESR- Equivalent series impedance, the capacitance of Cx- shunt capacitances, the equivalent series impedance of ESRx- shunt capacitances, RL- load, Vgs- Switching tube QbDriving voltage, D- dutycycles, t- times, fs- converter switches frequency, Δ IL- inductive current ripple peak-to-peak value, vESRVoltage on-equivalent series resistance, vCVoltage on-electric capacity.
Embodiment
Below in conjunction with the accompanying drawings and specific embodiment makes further description to the present invention.
1st, theory deduction:
Fig. 1 is the working waveform figure in the CCM Buck converter switches cycles.As switching tube QbDuring conducting, diode DbCut Only, the voltage at inductance L two ends is Vo/D-Vo, its inductive current iLWith (Vo/D-Vo)/L slope linear rise.As diode Db During shut-off, inductive current iLPass through diode DbAfterflow, now the voltage at inductance L two ends is-Vo, inductive current iLWith Vo/ L's is oblique Rate declines.Because Buck converters are operated in CCM patterns, therefore before switch periods terminate, inductive current iLDo not drop to zero. Inductive current iLAverage value in a switch periods is output current Io
Inductive current iLExpression formula in one cycle is as follows:
Wherein VoFor output voltage average value, L is inductance value, fsFor the switching frequency of Buck converters, D is switching tube Dutycycle, t is the time.
It assume that
It can obtain, the electric current and i of two electric capacityC+iCxExpression formula be:
The voltage expression of two electric capacity is respectively
Because two electric capacity are in parallel, then two capacitance voltages are equal
vC(t)=vCx(t) (9)
WhenWhen,
By formula (10), derivation can be obtained
Respectively formula (11) equal sign both sides are done with Laplace conversion to obtain
Abbreviation can be obtained
Doing Laplace inverse transformations to formula (13) equal sign can obtain
Wherein
Matched curve can be made according to 11 shunt capacitance current values that sampling is obtained, X is obtained1、X2、X3And iCx(0)。
Wushu (2), (3) substitution (15), (16), (17) can be obtained:
In formula, L is the inductance value of inductance, and ESR is the resistance of equivalent series resistance, and C is the capacitance of electric capacity, fsOpened for converter Close frequency, VoFor output voltage average value, D is the dutycycle of converter, and ESRx is the equivalent series resistance of institute's shunt capacitance Resistance, Cx is the capacitance of the electric capacity of institute's shunt capacitance, X1、X2、X3For the parameter of matched curve.
Based on formula (18), (19), (20), can obtain CCM Buck converter inductance L and output filter capacitor ESR and C monitoring method.
2nd, the monitoring device and method of CCM buck converter inductance of the present invention and output capacitance
With reference to Fig. 2, the monitoring device of CCM buck converter inductance of the present invention and output capacitance, including Buck converter masters Power circuit 1, drive circuit 3, known to parameter at electric capacity 7, Current Mutual Inductance isolation amplifying unit 8, display unit 11 and signal Module is managed, the signal processing module includes power circuit control unit 2, switching frequency fsComputing unit 4, dutycycle D are calculated Unit 5, output voltage sampling unit 6, capacitance current triggering sampling unit 9, inductance L and electric capacity ESR and C computing unit 10;
The Buck converters main power circuit 1 includes input voltage source Vin, switching tube Qb, sustained diodeb, filtering Inductance L, output filter capacitor and load RL, the output filter capacitor includes equivalent series resistance ESR and electric capacity C, wherein opening Close pipe QbDrain electrode and voltage source VinPositive pole connection, sustained diodebNegative electrode and switching tube QbDrain electrode connection, afterflow two Pole pipe DbAnode and voltage source VinNegative pole connection, filter inductance L one end and sustained diodebNegative electrode connection, filtering The inductance L other end respectively with equivalent series resistance ESR one end and load RLOne end connection, equivalent series resistance ESR's The other end is connected with electric capacity C one end, electric capacity the C other end and load RLThe other end with voltage source VinNegative pole connection, Load RLIn parallel with electric capacity known to parameter (7), its two ends is output average voltage Vo
The voltage source V of the input of the power circuit control unit 2 respectively with Buck converters main power circuit 1inWith Export average voltage VoConnection, the pwm signal of the output end of power circuit control unit 2 is respectively connected to switching frequency fsComputing unit 4 With dutycycle D computing units 5, the output average voltage V of Buck converters main power circuit 1oAccess output voltage sampling unit 6, Current Mutual Inductance isolation amplifying unit 8 and the output end of power circuit control unit 2 pwm signal access capacitance current triggering adopt Sample unit 9, switching frequency fsComputing unit 4, dutycycle D computing units 5, output voltage sampling unit 6 and capacitance current triggering The output end of sampling unit 9 accesses electric capacity ESR and C computing unit 7, inductance L and electric capacity ESR and C computing unit 10 it is defeated Go out and terminate into display unit 11;The input of the drive circuit 3 connects with the pwm signal of the output end of power circuit control unit 2 Connect, the output end access switching tube Q of drive circuit 3bGate pole.The signal processing module is dsp chip TMS320F28335; The display unit 11 is 1602 LCDs.
The monitoring method of monitoring device based on CCM buck converter inductance of the present invention and output capacitance, including following step Suddenly:
Step 1, the electric capacity 7 known to a parameter in output end parallel connection, creates power circuit in signal processing module Control unit 2, switching frequency fsComputing unit 4, dutycycle D computing units 5, output voltage sampling unit 6, capacitance current triggering Sampling unit 9, inductance L and electric capacity ESR and C computing unit 10;
Step 2, the power circuit control unit 2 of signal processing module is according to the output of Buck converters main power circuit 1 Average voltage Vo, obtain pwm signal and through the driving switch pipe Q of drive circuit 3b
Step 3, the pwm signal feeding switching frequency f that power circuit control unit 2 is exportedsComputing unit 4 and dutycycle D Computing unit 5, through switching frequency fsThe processing of computing unit 4 draws the current switching frequency f of converters, calculate single through dutycycle D The processing of member 5 draws the current dutycycle D of converter;
Step 4, the output average voltage V of Buck converters main power circuit 1oOutput voltage sampling unit 6 is sent into, is obtained The average value of output voltage;
Step 5, the pwm signal and Current Mutual Inductance that power circuit control unit 2 is exported isolate the capacitance current of amplifying unit 8 ixCapacitance current triggering sampling unit 9 is sent into, is handled by delay procedure to DT such as capacitance currentss/ 10 interval samplings, obtain electricity The instantaneous value i of capacitance currentx(0)、ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix (3DTs/5)、ix(7DTs/10)、ix(4DTs/5)、ix(9DTs/10)、ix(DTs) totally 11 value;
Step 6, by obtained switching frequency fs, dutycycle D, the average value V of output voltageoAnd capacitance current is instantaneous Value ix(0)、ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix(3DTs/5)、ix(7DTs/ 10)、ix(4DTs/5)、ix(9DTs/10)、ix(DTs) feeding inductance L and electric capacity ESR and C computing unit 10 carry out curve fitting And integrated treatment, obtain the current equivalent series resistance ESR of value and output filter capacitor and electric capacity of inductance L in Buck converters C value, specifically:
L, ESR and C computing unit described in step 6 (10) Fitting curve equation is as follows:
Try to achieve X1、X2、X3And iCx(0) after, L, ESR and C computing unit (10) carries out integrated treatment to matched curve, Value and the output filter capacitor current equivalent series resistance ESR and electric capacity C of inductance L in Buck converters value is obtained, specifically Formula is as follows:
In formula, L is the inductance value of inductance, and ESR is the resistance of equivalent series resistance, and C is the capacitance of electric capacity, fsOpened for converter Close frequency, VoFor output voltage average value, D is the dutycycle of converter, and ESRx is the equivalent series resistance of institute's shunt capacitance Resistance, Cx is the capacitance of the electric capacity of institute's shunt capacitance, X1、X2、X3For the parameter of matched curve.
Step 7, inductance L and electric capacity ESR and C computing unit 10 by the inductance L and equivalent series resistance ESR of gained and Electric capacity C value feeding display unit 11 is shown in real time.

Claims (5)

1. the monitoring device of a kind of CCM buck converter inductance and output capacitance, it is characterised in that including the main work(of Buck converters Rate circuit (1), drive circuit (3), electric capacity (7), Current Mutual Inductance isolation amplifying unit (8), display unit (11) known to parameter And signal processing module, the signal processing module include power circuit control unit (2), switching frequency fsComputing unit (4), Dutycycle D computing units (5), output voltage sampling unit (6), capacitance current triggering sampling unit (9), inductance L and electric capacity ESR and C computing units (10);
The Buck converters main power circuit (1) includes input voltage source Vin, switching tube Qb, sustained diodeb, filter inductance L, output filter capacitor and load RL, the output filter capacitor include equivalent series resistance ESR and electric capacity C, wherein switching tube Qb Drain electrode and voltage source VinPositive pole connection, sustained diodebNegative electrode and switching tube QbDrain electrode connection, sustained diodeb Anode and voltage source VinNegative pole connection, filter inductance L one end and sustained diodebNegative electrode connection, filter inductance L The other end respectively with equivalent series resistance ESR one end and load RLOne end connection, the equivalent series resistance ESR other end It is connected with electric capacity C one end, electric capacity the C other end and load RLThe other end with voltage source VinNegative pole connection, load RL In parallel with electric capacity known to parameter (7), its two ends is output average voltage Vo
The voltage source V of the input of the power circuit control unit (2) respectively with Buck converters main power circuit (1)inWith Export average voltage VoConnection, the pwm signal of power circuit control unit (2) output end is respectively connected to switching frequency fsCalculate single First (4) and dutycycle D computing units (5), the output average voltage V of Buck converters main power circuit (1)oAccess output voltage Sampling unit (6), the pwm signal of Current Mutual Inductance isolation amplifying unit (8) and power circuit control unit (2) output end is accessed Capacitance current triggering sampling unit (9), switching frequency fsComputing unit (4), dutycycle D computing units (5), output voltage sampling The output end of unit (6) and capacitance current triggering sampling unit (9) accesses electric capacity ESR and C computing unit (7), inductance L and The output end access display unit (11) of electric capacity ESR and C computing unit (10);
The input of the drive circuit (3) is connected with the pwm signal of power circuit control unit (2) output end, drive circuit (3) output end access switching tube QbGate pole.
2. the monitoring device of CCM buck converter inductance according to claim 1 and output capacitance, it is characterised in that institute Signal processing module is stated for dsp chip TMS320F28335.
3. the monitoring device of CCM buck converter inductance according to claim 1 and output capacitance, it is characterised in that institute Display unit (11) is stated for 1602 LCDs.
4. a kind of CCM buck converter inductance L and output capacitance ESR and C monitoring method, it is characterised in that including following Step:
Step 1, the electric capacity (7) known to a parameter in output end parallel connection, creates power circuit control in signal processing module Unit (2) processed, switching frequency fsComputing unit (4), dutycycle D computing units (5), output voltage sampling unit (6), electric capacity electricity Stream triggering sampling unit (9), inductance L and electric capacity ESR and C computing unit (10);
Step 2, the power circuit control unit (2) of signal processing module is according to the output of Buck converters main power circuit (1) Average voltage Vo, obtain pwm signal and through drive circuit (3) driving switch pipe Qb
Step 3, the pwm signal feeding switching frequency f of power circuit control unit (2) outputsComputing unit (4) and dutycycle D meters Unit (5) is calculated, through switching frequency fsComputing unit (4) processing draws the current switching frequency f of converters, calculated through dutycycle D Unit (5) processing draws the current dutycycle D of converter;
Step 4, the output average voltage V of Buck converters main power circuit (1)oOutput voltage sampling unit (6) is sent into, is obtained The average value of output voltage;
Step 5, the capacitance current of the pwm signal of power circuit control unit (2) output and Current Mutual Inductance isolation amplifying unit (8) ixCapacitance current triggering sampling unit (9) is sent into, by delay procedure to DT such as capacitance currentss/ 10 interval samplings, obtain ix (0)、ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix(3DTs/5)、ix(7DTs/10)、 ix(4DTs/5)、ix(9DTs/10)、ix(DTs) totally 11 capacitance current instantaneous values;
Step 6, by obtained switching frequency fs, dutycycle D, the average value V of output voltageoAnd the instantaneous value i of capacitance currentx (0)、ix(DTs/10)、ix(DTs/5)、ix(3DTs/10)、ix(2DTs/5)、ix(DTs/2)、ix(3DTs/5)、ix(7DTs/10)、 ix(4DTs/5)、ix(9DTs/10)、ix(DTs) feeding inductance L and electric capacity ESR and C computing unit (10) carry out curve fitting and Integrated treatment, obtains value and the output filter capacitor current equivalent series resistance ESR and electric capacity C of inductance L in Buck converters Value;
Step 7, inductance L and electric capacity ESR and C computing unit (10) are by the inductance L and equivalent series resistance ESR and electricity of gained The value feeding display unit (11) for holding C is shown in real time.
5. CCM buck converter inductance L according to claim 4 and output capacitance ESR and C monitoring method, it is special Levy and be, L, ESR and C computing unit described in step 6 (10) Fitting curve equation is as follows:
<mrow> <msub> <mi>i</mi> <mrow> <mi>C</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>t</mi> <mo>+</mo> <mo>&amp;lsqb;</mo> <msub> <mi>i</mi> <mrow> <mi>C</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>X</mi> <mn>3</mn> </msub> </mrow>
Try to achieve X1、X2、X3And iCx(0) after, L, ESR and C computing unit (10) carries out integrated treatment to matched curve, obtains Inductance L value and output filter capacitor current equivalent series resistance ESR and electric capacity C value, specific formula in Buck converters It is as follows:
<mrow> <mi>L</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>V</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>D</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>S</mi> <mi>R</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>D</mi> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>+</mo> <mi>D</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>S</mi> <mi>R</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow>
<mrow> <mi>E</mi> <mi>S</mi> <mi>R</mi> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>S</mi> <mi>R</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>(</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>+</mo> <mi>D</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mrow> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>D</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow>
<mrow> <mi>C</mi> <mo>=</mo> <mfrac> <mrow> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mi>D</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>C</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>S</mi> <mi>R</mi> <mi>x</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow>
In formula, L is the inductance value of inductance, and ESR is the resistance of equivalent series resistance, and C is the capacitance of electric capacity, fsFor converter switches frequency Rate, VoFor output voltage average value, D is the dutycycle of converter, ESRx for the equivalent series resistance of institute's shunt capacitance resistance, Cx is the capacitance of the electric capacity of institute's shunt capacitance, X1、X2、X3For the parameter of matched curve.
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