CN105553330B - Non-linear piezoelectric energy recovery interface circuit inductor design and method of controlling switch - Google Patents

Non-linear piezoelectric energy recovery interface circuit inductor design and method of controlling switch Download PDF

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CN105553330B
CN105553330B CN201510902004.3A CN201510902004A CN105553330B CN 105553330 B CN105553330 B CN 105553330B CN 201510902004 A CN201510902004 A CN 201510902004A CN 105553330 B CN105553330 B CN 105553330B
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CN105553330A (en
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黄以华
曾泽丰
望庆磊
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Sun Yat Sen University
SYSU CMU Shunde International Joint Research Institute
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Sun Yat Sen University
SYSU CMU Shunde International Joint Research Institute
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02N—ELECTRIC MACHINES NOT OTHERWISE PROVIDED FOR
    • H02N2/00—Electric machines in general using piezoelectric effect, electrostriction or magnetostriction
    • H02N2/18—Electric machines in general using piezoelectric effect, electrostriction or magnetostriction producing electrical output from mechanical input, e.g. generators
    • H02N2/181—Circuits; Control arrangements or methods
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02N—ELECTRIC MACHINES NOT OTHERWISE PROVIDED FOR
    • H02N2/00—Electric machines in general using piezoelectric effect, electrostriction or magnetostriction
    • H02N2/18—Electric machines in general using piezoelectric effect, electrostriction or magnetostriction producing electrical output from mechanical input, e.g. generators
    • H02N2/186—Vibration harvesters

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Abstract

Non-linear piezoelectric energy recovery interface circuit inductor design provided by the invention and method of controlling switch, which consider inductance and switch under truth, has the characteristics of non-ideal characteristic：All there is conducting resistance so that interface circuit not simply carries out analysis design according to LC vibrating circuits；When designing SSHI circuits, design method proposed by the present invention is used, it is possible to achieve the optimized design of system, this has great significance to actual piezoelectric energy recovery interface circuit design, is with a wide range of applications in piezoelectric energy collection.

Description

Nonlinear piezoelectric energy recovery interface circuit inductance design and switch control method
Technical Field
The invention relates to the technical field of energy conservation, in particular to a nonlinear piezoelectric energy recovery interface circuit inductance design and switch control method.
Background
Devices such as microelectronic systems and wireless network nodes generally rely on chemical batteries for power supply, but the limited service life of the chemical batteries can cause frequent replacement and environmental pollution caused by discarded batteries. Therefore, the acquisition and utilization of natural energy is a big solution to these problems. Wherein piezoelectric energy recovery by natural vibrations offers possibilities for this.
In the piezoelectric energy recovery, the interface circuit plays a crucial role for the whole piezoelectric energy acquisition system, and is related to the efficiency of the system and whether the target load can be provided with effective voltage and current.
The standard interface circuit consists of a diode full-wave rectifier bridge and a filter capacitor, but the recovery power of the standard interface circuit is low. To provide power, synchronous switched inductor recovery (SSHI) circuits were proposed and later developed to extend to AC-P-SSHI, DC-P-SSHI, AC-S-SSHI, DC-S-SSHI, and the like.
When mechanical amplitude reaches an extreme value, namely the maximum charge amount is accumulated on the piezoelectric element, the traditional synchronous switch inductance recovery circuit closes the switch, so that the subsequent energy storage and collection circuit extracts all charges on the piezoelectric element to the energy storage and collection circuit. And after half LC vibration period, the switch is switched off, the electric charge is accumulated on the piezoelectric element again, and the cycle is performed sequentially.
However, in the design of nonlinear piezoelectric energy recovery interface circuits (including AC-P-SSHI, DC-P-SSHI, AC-S-SSHI, and DC-S-SSHI), people often ignore the on-resistance of connected inductors and switches, and still analyze and design the piezoelectric energy recovery interface circuits according to the standard LC vibration circuits, so that the piezoelectric energy recovery efficiency has deviation. Even if some documents introduce the quality factor Q of the inductor, the influence of the on-resistance on the circuit oscillation is not considered.
Disclosure of Invention
The invention aims to provide an inductance design and switch control method for a nonlinear piezoelectric energy recovery interface circuit, which brings the non-ideal characteristics of the switch and the inductance with on-resistance into analysis under the real condition and has great significance for the design of the actual piezoelectric energy recovery interface circuit.
In order to realize the purpose, the technical scheme is as follows:
a nonlinear piezoelectric energy recovery interface circuit inductance design and switch control method is disclosed, wherein the nonlinear piezoelectric energy recovery interface circuit is used for extracting piezoelectric energy stored by a piezoelectric element; the nonlinear piezoelectric energy recovery interface circuit comprises two working stages:
the first stage is as follows: the piezoelectric element is equivalent to a current source which is connected in parallel with a piezoelectric sheet clamped capacitor, the switch is switched off, the current source charges the piezoelectric sheet clamped capacitor, and the piezoelectric sheet clamped capacitor obtains output voltage;
and a second stage: when the voltage of the clamped capacitor of the piezoelectric sheet reaches an extreme value, the switch is closed to supply power to the load;
the inductance design and switch control method enables the nonlinear piezoelectric energy recovery interface circuit to be in an underdamped state in the second stage by controlling the inductive reactance value of the inductance, and controls the switch to be closed continuously until the oscillation output voltage in the underdamped state is at the moment of the first extreme value.
In the second stage, under the non-ideal characteristic, the nonlinear piezoelectric energy recovery interface circuit is not a pure LC oscillating circuit, and the switch has an on-resistance and the inductor has a loss resistance. And the voltage slice clamped capacitor has a certain voltage value in the first stage, according to the RLC second-order circuit principle with an excitation source, under the condition that the piezoelectric element clamped capacitor is fixed, the different designs of the inductor can make the circuit have 3 different damping states: critical damping state, over-damping state, under-damping state. In these three cases, the voltage reversal situation is different.
For critical damping and over-damping conditions, the second-order circuit voltage cannot oscillate and only exponentially attenuates, and under-damping conditions, oscillation attenuation occurs. Under the condition of the same initial voltage, the voltage after the reversal is close to 0 in the states of over damping and critical damping. However, the polarity of the voltage after the under-damped state is reversed, and the absolute value is maximum. Because the larger the output voltage of the piezoelectric internal clamp capacitor in the first stage is, the more energy can be collected by the piezoelectric energy recovery interface circuit, the larger the absolute value of the voltage after overturning is, the better the absolute value is, and the requirement can be kept consistent with the polarity of a current source, and under-damping obviously meets the requirement most under the three conditions.
Summarizing the design scheme, the circuit at the second stage of the system works in an underdamped state by controlling the inductive reactance value of the inductor, and the switch is controlled to be closed continuously until the oscillation output voltage in the underdamped state has the first extreme value, so that the output voltage of the whole system is higher when the whole system reaches a steady state, and the energy obtained on the load is the maximum.
Preferably, the nonlinear piezoelectric energy recovery interface circuit comprises an AC-P-SSHI circuit, a DC-P-SSHI circuit, an AC-S-SSHI circuit and a DC-S-SSHI circuit.
Preferably, for an AC-P-SSHI circuit, the inductive reactance L of the inductor should be designed as:
wherein, CpRepresenting the clamped capacitance of the piezoelectric plate, L representing the inductive reactance, RsRepresenting the sum of the on-resistance of the inductor and the on-resistance of the switch, RLRepresenting a load;
and switch on time tauonThe design is as follows:
wherein,mu is a circuit RLC parameter, α and β are the real part and the imaginary part of the characteristic root of a circuit differential equation respectively, and the characteristic root can be obtained by solving the circuit differential equation.
Preferably, for a DC-P-SSHI circuit, the inductive reactance L of the inductor should be designed as:
wherein, CpRepresenting the clamped capacitance of the piezoelectric patch, RsThe sum of the on-resistance of the inductor and the on-resistance of the switch is represented, and L represents the inductive reactance of the inductor;
and switch on time tauonThe design is as follows:
wherein,mu is a circuit RLC parameter, α and β are the real part and the imaginary part of the characteristic root of a circuit differential equation respectively, and the characteristic root can be obtained by solving the circuit differential equation.
Preferably, for an AC-S-SSHI circuit, the inductive reactance L of the inductor should be designed as:
Cprepresenting the clamped capacitance of the piezoelectric patch, RsRepresenting the sum of the inductor on-resistance and the switch on-resistance, L representing the inductor inductance, RLRepresenting a load;
and switch on time tauonThe design is as follows:
wherein,mu is a circuit RLC parameter, α and β are the real part and the imaginary part of the characteristic root of a circuit differential equation respectively, and the characteristic root can be obtained by solving the circuit differential equation.
Preferably, for a DC-S-SSHI circuit, the inductive reactance L of the inductor is designed to be:
Cprepresenting the clamped capacitance of the piezoelectric plate, L representing the inductive reactance, RsRepresenting the sum of the on-resistance of the inductor and the on-resistance of the switch, RdRepresenting the on-resistance of the diode;
and switch on time tauonThe design is as follows:
wherein,mu is a circuit RLC parameter, α and β are the real part and the imaginary part of the characteristic root of a circuit differential equation respectively, and the characteristic root can be obtained by solving the circuit differential equation.
Compared with the prior art, the invention has the beneficial effects that:
the inductance design and switch control method of the nonlinear piezoelectric energy recovery interface circuit provided by the invention considers the characteristics that the inductance and the switch have non-ideal characteristics under the real condition: the on-resistance exists, so that the interface circuit is not simply designed according to the analysis of the LC vibration circuit; when the SSHI circuit is designed, the design method provided by the invention can realize the optimal design of the system, which has great significance for the design of the practical piezoelectric energy recovery interface circuit and has wide application prospect in piezoelectric energy collection.
Drawings
FIG. 1(a), FIG. 1(b), FIG. 1(c) and FIG. 1(d) are schematic design diagrams of the AC-P-SSHI circuit, the DC-P-SSHI circuit, the AC-S-SSHI circuit and the DC-S-SSHI circuit, respectively.
FIG. 2 is an equivalent circuit diagram of the AC-P-SSHI circuit at the first stage.
FIG. 3 is an equivalent circuit diagram of the AC-P-SSHI circuit in the second stage.
Fig. 4 is a voltage reversal waveform diagram of the critical damping state, the over-damping state and the under-damping state at the second stage.
Fig. 5 is a voltage waveform diagram of the output voltage in the critical damping state, the over-damping state and the under-damping state.
Fig. 6 is a waveform diagram of the output voltage in the underdamped state.
Detailed Description
The drawings are for illustrative purposes only and are not to be construed as limiting the patent;
the invention is further illustrated below with reference to the figures and examples.
Example 1
FIG. 1 is a schematic diagram of the design of an AC-P-SSHI circuit, a DC-P-SSHI circuit, an AC-S-SSHI circuit and a DC-S-SSHI circuit, respectively, in consideration of the fact that both the inductor and the switch have non-ideal characteristics and have on-resistance. The piezoelectric element is equivalent to a current source which is connected in parallel with a piezoelectric sheet clamped capacitor. The inductor and the switch with the on-resistance in the actual situation are connected in series, the design of the control strategy of the inductor and the switch is optimized, the circuit is in an underdamping state, and therefore the energy extraction power of the whole circuit is improved.
FIG. 2 is an equivalent circuit diagram of the AC-P-SSHI circuit at a first stage when the switch S is turned off and the equivalent current source inside the piezoelectric patch is applied to the clamped capacitance C of the piezoelectric patchpCharging and pressingUnder the excitation of the electric equivalent current source, the output voltage of the clamped capacitor of the piezoelectric sheet in the first stage is obtained.
Fig. 3 is an equivalent circuit diagram of the AC-P-SSHI circuit in the second stage, when the voltage across the clamped capacitor of the piezoelectric patch reaches an extreme value, i.e. the amplitude of the mechanical vibration reaches an extreme value, the switch S is closed, and under the non-ideal characteristic, the nonlinear piezoelectric energy recovery interface circuit is not a pure LC oscillating circuit, and the switch has an on-resistance and the inductor has a loss resistance. And the voltage slice clamped capacitor has a certain voltage value in the first stage, according to the RLC second-order circuit principle with an excitation source, under the condition that the piezoelectric element clamped capacitor is fixed, the different designs of the inductor can make the circuit have 3 different damping states: critical damping state, over-damping state, under-damping state. In these three cases, the voltage reversal situation is different.
Fig. 4 shows the voltage inversion waveform diagram of the second stage of the piezoelectric energy recovery interface circuit under three conditions of critical damping, over-damping and under-damping. For critical damping and over-damping conditions, the second-order circuit voltage cannot oscillate and only exponentially attenuates, and under-damping conditions, oscillation attenuation occurs. Under the condition of the same initial voltage, the voltage after the reversal is close to 0 in the states of over damping and critical damping. However, the polarity of the voltage after the under-damped state is reversed, and the absolute value is maximum. Because the larger the output voltage of the piezoelectric internal clamp capacitor in the first stage is, the more energy can be collected by the piezoelectric energy recovery interface circuit, the larger the absolute value of the voltage after overturning is, the better the absolute value is, and the requirement can be kept consistent with the polarity of a current source, and under-damping obviously meets the requirement most under the three conditions.
By controlling the inductive reactance value of the inductor, the circuit in the second stage of the system works in an underdamped state, and the switch is controlled to be closed continuously until the oscillation output voltage in the underdamped state has the first extreme value, so that the output voltage of the whole system is higher when the whole system reaches a steady state, and the energy obtained by the load is the largest.
For the following four different circuitsAll can be set with the clamped capacitance of the piezoelectric sheet as CpThe inductance of the inductor is L, and the current on the inductor is iLThe sum of the on-resistance of the inductor and the on-resistance of the switch is RsThe load resistance is RLThe on-resistance of the diode is RdThe angular frequency of the vibration of the piezoelectric sheet is omega, and the initial phase of the equivalent current source in the piezoelectric sheet isThe maximum amplitude of the current is IoTherefore, the instantaneous equivalent current of the piezoelectric plate can be expressed as:
the differential equation for an AC-P-SSHI circuit can be derived as:
the equation is given as iLFor an unknown RLC circuit differential equation, solving the equation, can set iL＝AeptP is then the feature root, α and β are the real and imaginary parts of the feature root, respectively.
The discriminant of the characteristic equation is as follows:
fig. 5 shows the output voltage waveform of the system under three damping conditions. In three cases, after the circuit reaches steady state, the underdamped output voltage is higher and the energy available on the load is maximum. Therefore, when the inductance L is actually selected, it should be satisfied that the circuit operates in an underdamped state, i.e., Δ < 0, so that it is possible to obtain
Fig. 6 shows the voltage output waveform in the underdamped state, and in order to obtain a better voltage flipping effect, the switch is controlled to be closed continuously until the moment when the oscillation output voltage in the underdamped state has the first extreme value, i.e. the lowest point F is taken as the time point when the switch is opened, and the switch closing duration is:
wherein,mu is a circuit RLC parameter, α and β are the real part and the imaginary part of the characteristic root of a circuit differential equation respectively, and the characteristic root can be obtained by solving the circuit differential equation.
Fig. 1(b) shows a block diagram of a DC-P-SSHI circuit, which, unlike an AC-P-SSHI circuit,
the DC-P-SSHI circuit adds a diode rectifier bridge circuit and an output filter capacitor C between a nonlinear circuit and a loadoThe system can provide DC voltage to the load due to the filter capacitor CoLarge value and output voltage VoutRemains almost constant and can be set to VoutIs a constant.
Let second stage circuit work in the underdamping district just can make the voltage absolute value after the upset the biggest, the electric induction design is:
FIG. 1(c) shows a block diagram of an AC-S-SSHI circuit, for which the differential equation is:
the discriminant of the characteristic equation is as follows: Δ ═ R (R)s+RL)2Cp 2-4CpL, to make it reach an underdamped state, i.e. Δ < 0, so that the electric induction design can be obtained as:
FIG. 1(d) shows a structure diagram of a DC-S-SSHI circuit, considering the non-ideal characteristics of the diode with the on-voltage and on-resistance, and for the DC-S-SSHI circuit, the differential equation is
The discriminant of the characteristic equation is as follows: Δ ═ 2Rd+Rs)2Cp 2-4CpL, to make it reach an underdamped state, i.e. Δ < 0, so that the electric induction design can be obtained as:
similar to the AC-P-SSHI circuit, for the DC-P-SSHI circuit, the AC-S-SSHI circuit and the DC-S-SSHI circuit, the closing time of the switches is controlled to be kept until the moment that the oscillation output voltage in the underdamped state has the first extreme value, and the closing time is designed as follows:
wherein,mu is circuit RLC parameter, α and β are real part and imaginary part of characteristic root of second order differential equation, respectively, and can be obtained by solving characteristic root of circuit differential equation。
It should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.

Claims (1)

1. A nonlinear piezoelectric energy recovery interface circuit inductance design and switch control method is disclosed, wherein the nonlinear piezoelectric energy recovery interface circuit is used for extracting piezoelectric energy stored by a piezoelectric element; the nonlinear piezoelectric energy recovery interface circuit comprises two working stages:
the first stage is as follows: the piezoelectric element is equivalent to a current source which is connected in parallel with a piezoelectric sheet clamped capacitor, the switch is switched off, the current source charges the piezoelectric sheet clamped capacitor, and the piezoelectric sheet clamped capacitor obtains output voltage;
and a second stage: when the voltage of the clamped capacitor of the piezoelectric sheet reaches an extreme value, the switch is closed to supply power to the load;
the method is characterized in that: the inductance design and switch control method controls the inductive reactance value of the inductance to enable the nonlinear piezoelectric energy recovery interface circuit to be in an underdamped state in the second stage, and controls the switch to be closed continuously until the oscillation output voltage in the underdamped state has the first extreme value;
the nonlinear piezoelectric energy recovery interface circuit comprises an AC-P-SSHI circuit, a DC-P-SSHI circuit, an AC-S-SSHI circuit and a DC-S-SSHI circuit;
for an AC-P-SSHI circuit, the inductive reactance L of the inductor should be designed as follows:
<mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>L</mi> </msub> <mo>-</mo> <mn>2</mn> <msqrt> <mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <msub> <mi>R</mi> <mi>L</mi> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mi>L</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> <msub> <mi>R</mi> <mi>L</mi> </msub> <msub> <mi>C</mi> <mi>p</mi> </msub> <mo>&lt;</mo> <mi>L</mi> <mo>&lt;</mo> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>L</mi> </msub> <mo>+</mo> <mn>2</mn> <msqrt> <mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <msub> <mi>R</mi> <mi>L</mi> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mi>L</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> <msub> <mi>R</mi> <mi>L</mi> </msub> <msub> <mi>C</mi> <mi>p</mi> </msub> </mrow>
wherein, CpRepresenting the clamped capacitance of the piezoelectric plate, L representing the inductive reactance, RsRepresenting the sum of the on-resistance of the inductor and the on-resistance of the switch, RLRepresenting a load;
and switch on time tauonThe design is as follows:
<mrow> <msub> <mi>&amp;tau;</mi> <mrow> <mi>o</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mo>&amp;prime;</mo> </msup> </mrow> <mi>&amp;beta;</mi> </mfrac> <mo>;</mo> </mrow>
wherein,α and β are respectively the real part and the imaginary part of the characteristic root of the circuit differential equation of the AC-P-SSHI circuit, and can be obtained by solving the characteristic root through the circuit differential equation of the AC-P-SSHI circuit;
for a DC-P-SSHI circuit, the inductive reactance L of the inductor is designed as follows:
<mrow> <mi>L</mi> <mo>&gt;</mo> <mfrac> <mrow> <msup> <msub> <mi>R</mi> <mi>s</mi> </msub> <mn>2</mn> </msup> <msub> <mi>C</mi> <mi>p</mi> </msub> </mrow> <mn>4</mn> </mfrac> </mrow>
wherein, CpRepresenting the clamped capacitance of the piezoelectric patch, RsThe sum of the on-resistance of the inductor and the on-resistance of the switch is represented, and L represents the inductive reactance of the inductor;
and switch on time tauonThe design is as follows:
<mrow> <msub> <mi>&amp;tau;</mi> <mrow> <mi>o</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mo>&amp;prime;</mo> </msup> </mrow> <mi>&amp;beta;</mi> </mfrac> <mo>;</mo> </mrow>
wherein,α and β are respectively the real part and the imaginary part of the characteristic root of the circuit differential equation of the DC-P-SSHI circuit, and can be obtained by solving the characteristic root through the circuit differential equation of the DC-P-SSHI circuit;
for an AC-S-SSHI circuit, the inductive reactance L of the inductor should be designed as follows:
<mrow> <mi>L</mi> <mo>&gt;</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>L</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>C</mi> <mi>p</mi> </msub> </mrow>
Cprepresenting the clamped capacitance of the piezoelectric patch, RsRepresenting the sum of the inductor on-resistance and the switch on-resistance, L representing the inductor inductance, RLRepresenting a load;
and switch on time tauonThe design is as follows:
<mrow> <msub> <mi>&amp;tau;</mi> <mrow> <mi>o</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mo>&amp;prime;</mo> </msup> </mrow> <mi>&amp;beta;</mi> </mfrac> <mo>;</mo> </mrow>
wherein,α and β are respectively the real part and the imaginary part of the characteristic root of the circuit differential equation of the AC-S-SSHI circuit, and can be obtained by solving the characteristic root through the circuit differential equation of the AC-S-SSHI circuit;
for a DC-S-SSHI circuit, the inductive reactance L of the inductor is designed as follows:
<mrow> <mi>L</mi> <mo>&gt;</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>C</mi> <mi>p</mi> </msub> <mo>;</mo> </mrow>
Cprepresenting the clamped capacitance of the piezoelectric plate, L representing the inductive reactance, RsRepresenting the sum of the on-resistance of the inductor and the on-resistance of the switch, RdRepresenting the on-resistance of the diode;
and switch on time tauonThe design is as follows:
<mrow> <msub> <mi>&amp;tau;</mi> <mrow> <mi>o</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mo>&amp;prime;</mo> </msup> </mrow> <mi>&amp;beta;</mi> </mfrac> <mo>;</mo> </mrow>
wherein,mu is a circuit RLC parameter, α and β are respectively a real part and an imaginary part of a characteristic root of a circuit differential equation of the DC-S-SSHI circuit, and the characteristic root can be obtained by solving the circuit differential equation of the DC-S-SSHI circuit.
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