CN103268377A - Radio link control (RLC) series second-order circuit model analysis method based on matrix laboratory (MATLAB) - Google Patents

Abstract

The invention discloses a radio link control (RLC) series second-order circuit model analysis method based on a matrix laboratory (MATLAB). The RLC series second-order circuit model analysis method includes: setting up an RLC series circuit under the effect of square wave excitation signals; building and analyzing a mathematical model of a two-order complex frequency domain algebraic equation, and obtaining a complex frequency domain algebraic equation model with zero state input response and a complex frequency domain algebraic equation model with nonzero state input response through a differential equation of he circuit; building an MATLAB program module, and setting up a SIMULINK simulation model; and achieving RLC series circuit two-order system transient state response analysis of different states of overdamping, critical damping, underdamping and the like. The RLC series second-order circuit model analysis method based on the MATLAB analyzes a transient state response rule wholly, discloses dynamic properties of an RLC two-order circuit, and combines advantages of theoretical analysis, simulation operation and experimental demonstration.

Description

A kind of RLC series connection second-order circuit model analysis method based on MATLAB

Technical field

The invention belongs to the circuit engineering field, relate in particular to a kind of RLC series connection second-order circuit model analysis method based on MATLAB.

Background technology

Computer science and technology has improved research tool and research method, and computing machine combines with relevant experimental observation instrument, can carry out field notes, arrangement, processing, analyze and draw a diagram experimental data, improves quality and the efficient of experimental work significantly.Computer simulation technique is to use electronic simulation software to carry out circuit design, emulation, debugging at computer platform, finishes the experiment that just can finish in corresponding hardware experiments chamber usually

The going deep into of the application in each subject along with computer science and technology calculated and simulation has become the 3rd approach of research work, becomes a kind of important research, experiment and design tool.Effectively promote people in depth to containing the research of dynamic element circuit, though this kind circuit is very of short duration from the process that a kind of stable state arrives another kind of stable state, but it not only has high researching value in theory, have more application more widely in the engineering field, RLC series connection second-order circuit particularly, many periodical literatures are studied this.Rlc circuit in recent years, has become the important experimental circuit that research contains the dynamic element circuit characteristic because having advantages such as theoretical analysis, simulation calculating and experimental demonstration three are consistent.

Specific analytical method and result have following a few class: the transient state process of the RLC series connection second-order circuit under the excitation of (1) offset of sinusoidal is studied, provide circuit directly enters steady-state response when connecting sinusoidal voltage condition, analyzed the superpotential in the transient state process, excess current phenomenon.(2) calculated value in cycle of RLC series circuit transient state process is revised, reduced the relative error with experiment value; (3) use the Matlab operational software to the RLC second-order circuit zero input response study; (4) measuring method of RLC series circuit critical resistance and influence the reason of accuracy of measurement has been discussed.Mainly be provided the RC circuit time constant greater than, be less than or equal to solution and the corresponding experimental result of the circuit differential equation under several situations of square wave semiperiod.(5) investigate by means of the RL integration/differentiating circuit of circuit analysis software Multisim8; (6) use the VisualBasic language that RLC series resonant circuit experimental data is carried out process of fitting treatment research; (7) investigate by means of the RLC series connection of Proteus second-order circuit; (8) utilize the second-order circuit of Simulink to carry out the discussion of emulation and encapsulation; (9) the underdamping situation of using the second-order circuit of Matlab is observed in conjunction with example.

At present, MATLAB has obtained using widely in the characteristic research that contains the dynamic element circuit.MATLAB is used for advanced techniques computational language and the interactive environment of algorithm development, data visualization, data analysis and numerical evaluation, and it has friendly workbench and programmed environment, is simple and easy to the program language of usefulness, powerful science is calculated and graphics capability.Also lack at present the RLC series connection second-order circuit model analysis method based on MATLAB.

Summary of the invention

The purpose of the embodiment of the invention is to provide a kind of RLC series connection second-order circuit model analysis method based on MATLAB, is intended to solve the problem of utilizing the RLC series connection of MATLAB second-order circuit model to analyze.

The embodiment of the invention is achieved in that a kind of RLC series connection second-order circuit model analysis method based on MATLAB, and this analytical approach can be divided into 4 steps:

Build the RLC series circuit under the effect of square wave excitation signal;

Set up the mathematical model of second order complex frequency domain algebraic equation and analyzed, obtain the complex frequency domain algebraic equation model of zero condition input response and the complex frequency domain algebraic equation model of non-zero status input response by the differential equation of circuit;

Set up the MATLAB procedural model and build the SIMULINK realistic model;

Realize the RLC series circuit second-order system instantaneous response analysis of different conditions such as overdamping, critical damping, underdamping.

Further, described RLC series connection second-order circuit figure is connected to: power supply U (t), resistance R, inductance L and capacitor C are connected into the loop.

Further, SIMULINK realistic model structure is: square-wave generator Step, gain module Gain, summer Add, integrator Integrator, integrator Integrator1, oscillograph Scope head and the tail are connected successively; Simultaneously, the SIMULINK realistic model also comprises two negative feedback loops, and concrete is connected to:

The outer shroud negative feedback, integrator Integrator1 is connected with gain G ain2, and gain G ain2 is connected with summer Add;

The negative feedback of interior ring, integrator Integrator is connected with gain G ain1, and gain G ain1 is connected with summer Add.

The complex frequency domain algebraic equation model of setting up is:

According to Kirchhoff's law, the differential equation of circuit is

$\mathrm{LC}\frac{d^2{u}_C\left(t\right)}{d{t}^2}+\mathrm{RC}\frac{d{u}_C\left(t\right)}{\mathrm{dt}}+u_C\left(t\right)=u\left(t\right)---\left(1\right)$

Wherein, $u\left(t\right)=\left\{\begin{array}{cc}E& \mathrm{nT}\≤t\≤T/2+\mathrm{nT}\\ 0& \mathrm{nT}+T/2\≤t\≤(n+1)T\end{array}\right.(n=\mathrm{0,1,2},\·\·\·)$

The initialization condition of equation (1) is

$\frac{d{u}_C\left(t\right)}{\mathrm{dt}}{|}_{t=0}=0,$ Wherein, make τ=2L/R,

ξ=1/ τ=R/2L, condition discussed here be T＞＞τ;

The complex frequency domain algebraic equation model of zero condition input response is:

$G\left(s\right)=\frac{U_C\left(s\right)}{U\left(s\right)}=\frac{1}{\mathrm{LC}{s}^2+\mathrm{RCs}+1}$

$=\frac{\frac{1}{\mathrm{LC}}}{s^2+\frac{R}{L}s+\frac{1}{\mathrm{LC}}}$

$=\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}$

That is, rlc circuit charges under the zero state response condition;

The complex frequency domain algebraic equation model of non-zero status input response is:

$U_c\left(s\right)=\frac{(s+2\mathrm{\ξ})u_c(0\_)+u_c^{\′}(0\_)}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}+\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}U\left(s\right).$

Further, the MATLAB programming step that comprises of this method is:

Remove the MATLAB memory headroom;

Parameter, the initial value on the energy-storage travelling wave tube, decay factor and circuit resonance angular frequency, the time step of setting the RLC element are 0.01 second, and simulation time is 3s;

Obtain by U _S(S) array that numerator coefficients constitutes, the array that denominator coefficients constitutes;

Obtain in the time domain $u_c\left(t\right)=k_1{e}^{p_{1}t}+k_2{e}^{p_{2}t};$

Obtain by the Voltammetric Relation on the electric capacity $i_L\left(t\right)=C{u}_c^{\′}\left(t\right)=C\frac{d{u}_C\left(t\right)}{\mathrm{dt}};$

Draw.

The present invention has carried out detailed mathematical modeling analysis to the transient response of RLC series connection second-order circuit.By building the RLC series circuit under the effect of square wave excitation signal, the mathematical model of setting up second order complex frequency domain algebraic equation obtains the MATLAB procedural model and builds the instantaneous response analysis that the SIMULINK realistic model has been realized RLC series circuit second-order system different damping state.Not only analyzed the transient response rule of RLC series connection second-order circuit all sidedly, disclosed the dynamic property of RLC second-order circuit, designed complete research method and the analysis thinking of a cover simultaneously, not only that theoretical analysis, simulation calculating and experimental demonstration three's advantage is complete combination is used, and also the similar application of researching and analysing that contains the important experimental circuit of dynamic element circuit characteristic is had directive significance.

Description of drawings

Fig. 1 is the process flow diagram of RLC series connection second-order circuit analytical approach;

Fig. 2 is RLC series connection second-order circuit figure connection diagram provided by the invention;

Fig. 3 is SIMULINK realistic model connection layout provided by the invention;

Fig. 4 is MATLAB programming step figure provided by the invention.

Embodiment

In order to make purpose of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that specific embodiment described herein only in order to explaining the present invention, and be not used in restriction the present invention.

Fig. 1 is the process flow diagram of RLC series connection second-order circuit analytical approach, and as shown in the figure, analytical approach can be divided into 4 steps:

STEP1: build the RLC series circuit under the effect of square wave excitation signal;

STEP2: set up the mathematical model of second order complex frequency domain algebraic equation and analyzed, obtain the complex frequency domain algebraic equation model of zero condition input response and the complex frequency domain algebraic equation model of non-zero status input response by the differential equation of circuit;

STEP3: set up the MATLAB procedural model and build the SIMULINK realistic model;

STEP4: realize the RLC series circuit second-order system instantaneous response analysis of different conditions such as overdamping, critical damping, underdamping.

Fig. 2 is RLC series connection second-order circuit figure connection diagram of the present invention.As shown in the figure, power supply U (t), resistance R, inductance L and capacitor C are connected into the loop.

Fig. 3 is SIMULINK realistic model connection layout.As shown in the figure, square-wave generator Step, gain module Gain, summer Add, integrator Integrator, integrator Integrator1, oscillograph Scope head and the tail are connected successively.Simultaneously, this analogous diagram also comprises two negative feedback loops, and concrete is connected to:

The outer shroud negative feedback, integrator Integrator1 is connected with gain G ain2, and gain G ain2 is connected with summer Add;

The negative feedback of interior ring, integrator Integrator is connected with gain G ain1, and gain G ain1 is connected with summer Add.

Fig. 4 is MATLAB programming step figure, comprises 6 steps altogether:

STEP1: remove the MATLAB memory headroom;

STEP2: parameter, the initial value on the energy-storage travelling wave tube, decay factor and circuit resonance angular frequency, the time step of setting the RLC element are 0.01 second, and simulation time is 3s;

STEP3: obtain by U _S(S) array that numerator coefficients constitutes, the array that denominator coefficients constitutes;

STEP4: obtain in the time domain $u_c\left(t\right)=k_1{e}^{p_{1}t}+k_2{e}^{p_{2}t};$

STEP5: obtain by the Voltammetric Relation on the electric capacity $i_L\left(t\right)=C{u}_c^{\′}\left(t\right)=C\frac{d{u}_C\left(t\right)}{\mathrm{dt}};$

STEP6: draw.

The present invention has carried out detail analysis to the mathematical model of the second order complex frequency domain algebraic equation of RLC series circuit under the effect of square wave excitation signal, and utilizes the MATLA graphics processing function that the dynamic property of the second-order system of realization different damping state is carried out emulation.Concrete step comprises following four steps: build the RLC series circuit under the effect of square wave excitation signal; Set up the mathematical model of second order complex frequency domain algebraic equation and analyzed, obtain the complex frequency domain algebraic equation model of zero condition input response and the complex frequency domain algebraic equation model of non-zero status input response by the differential equation of circuit; Set up the MATLAB procedural model and build the SIMULINK realistic model; Realize the RLC series circuit second-order system instantaneous response analysis of different conditions such as overdamping, critical damping, underdamping.The specific implementation in per step thes contents are as follows:

(1) builds RLC series circuit under the effect of square wave excitation signal

The passive two terminal circuit network that RLC series connection second-order circuit system as shown in Figure 2 is made up of basic electronic component such as resistance, inductance and electric capacity, wherein, u (t) is extraneous input voltage, u _c(t) be output voltage.

(2) set up complex frequency domain algebraic equation model

According to Kirchhoff's law, the differential equation of circuit shown in Figure 1 is

$\mathrm{LC}\frac{d^2{u}_C\left(t\right)}{d{t}^2}+\mathrm{RC}\frac{d{u}_C\left(t\right)}{\mathrm{dt}}+u_C\left(t\right)=u\left(t\right)---\left(1\right)$

Wherein, $u\left(t\right)=\left\{\begin{array}{cc}E& \mathrm{nT}\≤t\≤T/2+\mathrm{nT}\\ 0& \mathrm{nT}+T/2\≤t\≤(n+1)T\end{array}\right.(n=\mathrm{0,1,2},\·\·\·)$

The initialization condition of equation (1) is

$\frac{d{u}_C\left(t\right)}{\mathrm{dt}}{|}_{t=0}=0,$ Wherein, make τ=2L/R,

ξ=1/ τ=R/2L, condition discussed here be T＞＞τ.

The complex frequency domain algebraic equation model of zero condition input response is:

$G\left(s\right)=\frac{U_C\left(s\right)}{U\left(s\right)}=\frac{1}{\mathrm{LC}{s}^2+\mathrm{RCs}+1}$

$=\frac{\frac{1}{\mathrm{LC}}}{s^2+\frac{R}{L}s+\frac{1}{\mathrm{LC}}}$

$=\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}$

That is, rlc circuit charges under the zero state response condition;

The complex frequency domain algebraic equation model of non-zero status input response is:

$U_c\left(s\right)=\frac{(s+2\mathrm{\ξ})u_c(0\_)+u_c^{\′}(0\_)}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}+\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}U\left(s\right).$

Use the Laplace transform rule, equation (1) is carried out conversion, the complex frequency domain algebraic equation that then gets the second order differential equation (1) of its zero state response under charged state is

(LCs ²+RCs+1)U _c(s)＝U(s) (2)

Then the function model of system is that transport function is

$G\left(s\right)=\frac{U_C\left(s\right)}{U\left(s\right)}=\frac{1}{\mathrm{LC}{s}^2+\mathrm{RCs}+1}$

$=\frac{\frac{1}{\mathrm{LC}}}{s^2+\frac{R}{L}s+\frac{1}{\mathrm{LC}}}---\left(3\right)$

$=\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}$

The complex frequency domain algebraic equation model of non-zero status input response is:

As u (t)=0, u _c| _T=0=E, $i_L\left(0\right)=C{u}_c^{\′}\left(0\right)=C\frac{d{u}_C\left(t\right)}{\mathrm{dt}}{|}_{t=0}=0,$ The T of nT+T/2≤t≤(n+1), (n=0,1,2 ...), that is, rlc circuit is being put in fact, also i.e. zero input state response.Then the complex frequency domain algebraic equation that gets the second order differential equation (1) of non-zero status response by the Laplace transform rule is

$s^2{U}_c\left(s\right)-s{u}_c(0\_)-u_c^{\′}(0\_)+2\mathrm{\ξ}[s{U}_c\left(s\right)-u_c(0\_)]+{\mathrm{\ω}}_0^2{U}_c\left(s\right)={\mathrm{\ω}}_0^{2}U\left(s\right)$

That is, $U_c\left(s\right)=\frac{(s+2\mathrm{\ξ})u_c(0\_)+u_c^{\′}(0\_)}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}+\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}U\left(s\right)---\left(4\right)$

U then _c(s) inverse Laplace transform gets its time domain differential equation u _c(t) be

$u_c\left(t\right)=L^{-1}\left[U_c\left(s\right)\right]=L^{-1}\left[\frac{s{u}_c(0\_)+2\mathrm{\ξ}{u}_c(0\_)+u_c^{\′}(0\_)}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}\right]+L^{-1}\left[\frac{{\mathrm{\ω}}_0^2}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}U\left(s\right)\right]---\left(5\right)$

By first known conditions u (t)=0 as can be known, (4) turn to

$U_c\left(s\right)=\frac{(s+2\mathrm{\ξ})u_c(0\_)+u_c^{\′}\left(0\right)}{s^2+2\mathrm{\ξs}+{\mathrm{\ω}}_0^2}---\left(6\right)$

(6) formula is expanded into partial fraction

$U_c\left(s\right)=\frac{k_1}{s-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HSA00000895984900021.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}+\frac{k_2}{s-p_2}$

P wherein ₁And p ₂It is equation Two roots, i.e. polynomial expression U _c(s) limit of fraction, and k ₁And k ₂It then is the residual of its correspondence.

Determine k with Heaviside expansion theorem method ₁, k ₂, ${\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HSA00000895984900021.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=(s-p_1)U_c\left(s\right){|}_{s={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HSA00000895984900021.png"\; state="\{\{state\}\}">p</figure-callout>}}_1},$ $k_2=(s-p_2)U_c\left(s\right){|}_{s=p_2},$

Then have $u_c\left(t\right)=L^{-1}\left[U_c\left(s\right)\right]=k_1{e}^{p_{1}t}+k_2{e}^{p_{2}t}.$

ω wherein ₀Be called system's undamped oscillation frequency, ξ is the system damping ratio.

(3) set up the MATLAB procedural model, set up the SIMULINK realistic model

In MATLAB software, can directly come design factor k with the residue function ₁, k ₂, p ₁And p ₂Its form be [k, p, n]=residue (num, den), num wherein, den is respectively polynomial expression U _c(s) molecule, denominator polynomial expression are by falling the vectorial array that every coefficient that power arranges is formed.K is the array of residual phasor; P is the array of limit phasor; N is the polynomial expression array of s, under normal conditions, because U _c(s) has regularity, so n=0.U then _c(s) corresponding to the time domain expression formula u of MATLAB language _c(t) be

u _c(t)＝k(1)exp(p(1)*t)+k(2)exp(p(2)*t)

Its main MATLAB program description is as follows:

% filename RLC.M

Clear, % removes the MATLAB memory headroom

R＝input(‘Resistance is：’)；

L＝input(‘Iductance is：’)；

C=input (' Capacitance is: '); % sets the parameter of RLC element, R=126; L=0.001; C=0.0000025;

Uc0=2; IL0=0; % sets the initial value on the energy-storage travelling wave tube

Xi=R/ (2*L); W0=sqrt (1/ (L*C)); % definition decay factor and circuit resonance angular frequency

s1＝-xi+sqrt(xi^2-w0^2)；

S2=-xi-sqrt (xi^2-w0^2); Two roots of % secular equation

Dt=0.01; T=0:dt:3; It is 0.01 second that % arranges time step, and simulation time is 3 seconds

Num=[uc0,2*xi*uc0+iL0/C]; % is by U _c(S) array that numerator coefficients constitutes

Den=[1,2*xi, w0^2]; % is by U _c(S) array that denominator coefficients constitutes

[k, p, n]=residue (num, den); The limit of % and corresponding undetermined coefficient

Uc2=k (1) * exp (p (1) * t)+k (2) * exp (p (2) * t); % obtains in the time domain

IL2=C*diff (uc2)/dt; % obtains by the Voltammetric Relation on the electric capacity

figure(2)，subplot(2，1，1)；

Plot (t, uc2), grid, xlabel (' t '), ylabel (' uc (t) '); The draw X-Y scheme window of Figue1 of %, the waveform of first subgraph of 2 row, 1 row has mesh lines

subplot(2，1，2)；

Plot (t (1:end-1), iL2), grid, xlabel (' t '), ylabel (' il (t) '); The draw X-Y scheme window of Figue1 of %, the waveform of second subgraph of row 1 row

According to the formula differential expressions, its circuit exciter response relation is still the second order linear differential equation with constant coefficients.Can describe with following mathematical expression

$\frac{d^{2}y\left(t\right)}{d{t}^2}+a\frac{\mathrm{dy}\left(t\right)}{\mathrm{dt}}+\mathrm{by}\left(t\right)=\mathrm{mf}\left(t\right)$

Wherein, a, b, m are normal real number, adopt basic integration module to set up model, as Fig. 3.Module is set to be worth accordingly.Next just can carry out instantaneous response analysis to the RLC series circuit.

(4) RLC series circuit instantaneous response analysis

At

With Three kinds of situation meter opinions, namely the value of the selected R of this article, L and C is for satisfying

With Three kinds of situations are namely carried out simulation analysis research to the duty that RLC series connection second-order system is in the second order dynamic circuit under overdamping, underdamping and three kinds of situations of critical damping.Wherein: R=126 Ω, L=1mH, C can change ^[8]

The overdamping situation: get R=126 Ω, L=1mH, C=2500nF, namely

At this moment

Two unequal negative real number roots are arranged, and in the case, the transient response of circuit is the overdamping state of non-oscillatory, the simulation result of its Changing Pattern as shown in Figure 3, its observations conforms to theoretical analysis.

The critical damping situation: get R=126 Ω, L=1mH, C=250nF, namely

At this moment

Two equal negative real number roots are arranged, and in the case, the transient response of circuit is the critical damping state of non-oscillatory.

The underdamping situation: get R=126 Ω, L=1mH, C=25nF, namely

At this moment

It is that the conjugate complex of negative is several that a pair of real part is arranged, and in the case, the transient response of circuit is the underdamping state of oscillatory.

From above research as can be known, when

Namely

The time, the resistance R of comparing is bigger, and energy has little time to exchange between L and C and has just consumed in resistance, so circuit is overdamped non-oscillatory charge and discharge process from the transient process that a kind of stable state arrives another kind of stable state; When

Namely

The time, comparatively speaking, resistance R is very little, L, and energy exchange accounts for leading role between the C, the energy of resistance consumption is less, in whole process, waveform will present the state of damped oscillation, with changing direction periodically, energy-storage travelling wave tube is also incited somebody to action periodically positive energy exchange, so the character of transitional circuit is underdamped vibration charge and discharge process.In the zero input response, electric capacity discharges the electric energy that stores always in whole process, and electric current does not change direction all the time, when t=0, and i=0; When t → ∞, i=0, so electric current must experience the variation that goes to zero again from small to large in discharge process, electric current reaches before the maximal value, inductance absorbs energy, set up magnetic field after inductance release energy, decay gradually in magnetic field, trend disappears.When

Namely

The time, the character of transitional circuit is the non-oscillatory charge and discharge process of critical damping, and in electromagnetic oscillation, critical damping is compared with overdamping with underdamping, and system is the shortest from motion required time of tending to balance.When R=0, circuit is continuous oscillation, and the oscillation amplitude of voltage or electric current remains unchanged in the circuit, consumed energy not in the oscillatory process.This shows that the course of work of electric capacity is continuous charge and discharge process.Between action period (being equivalent to DC power supply), be charging process at the square wave high level, in square wave low level action time (being equivalent to power supply short circuit), be discharge process.Charge and discharge process is controlled by R, L, C parameter.Therefore can determine the undamped oscillation frequencies omega of second order series circuit system according to the numerical value of circuit parameter R, L, C ₀With damping ratio ξ, by Theory of Automatic Control as can be known, the response characteristic of second-order system fully can be by damping ratio ξ and undamped oscillation frequencies omega ₀Two parameters are determined, make second-order system have satisfied dynamic performance index, must select suitable damping ratio ξ and undamped oscillation frequencies omega ₀

The above only is preferred embodiment of the present invention, not in order to limiting the present invention, all any modifications of doing within the spirit and principles in the present invention, is equal to and replaces and improvement etc., all should be included within protection scope of the present invention.

Claims (5)

1. one kind based on the RLC of MATLAB series connection second-order circuit model analysis method, it is characterized in that this analytical approach may further comprise the steps:

Build the RLC series circuit under the effect of square wave excitation signal;

Set up the mathematical model of second order complex frequency domain algebraic equation and analyzed, obtain the complex frequency domain algebraic equation model of zero condition input response and the complex frequency domain algebraic equation model of non-zero status input response by the differential equation of circuit;

Set up the MATLAB procedural model and build the SIMULINK realistic model;

Realize the RLC series circuit second-order system instantaneous response analysis of overdamping, critical damping, underdamping different conditions.

2. the RLC series connection second-order circuit model analysis method based on MATLAB as claimed in claim 1 is characterized in that described RLC series connection a kind of of second-order circuit is connected to: power supply U (t), resistance R, inductance L and capacitor C are connected into the loop.

3. the RLC series connection second-order circuit model analysis method based on MATLAB as claimed in claim 1, it is characterized in that a kind of structure of SIMULINK realistic model is: square-wave generator Step, gain module Gain, summer Add, integrator Integrator, integrator Integrator1, oscillograph Scope head and the tail are connected successively; Simultaneously, the SIMULINK realistic model also comprises two negative feedback loops, and concrete is connected to:

The outer shroud negative feedback, integrator Integrator1 is connected with gain G ain2, and gain G ain2 is connected with summer Add;

The negative feedback of interior ring, integrator Integrator is connected with gain G ain1, and gain G ain1 is connected with summer Add.

4. the RLC series connection second-order circuit model analysis method based on MATLAB as claimed in claim 1 is characterized in that the complex frequency domain algebraic equation model of foundation is:

According to Kirchhoff's law, the complex frequency domain algebraic equation is

$\mathrm{LC}\frac{d^2{u}_C\left(t\right)}{d{t}^2}+\mathrm{RC}\frac{d{u}_C\left(t\right)}{\mathrm{dt}}+u_C\left(t\right)=u\left(t\right)---\left(1\right)$

Wherein,

$u\left(t\right)=\left\{\begin{array}{cc}E& \mathrm{nT}\&le;t\&le;T/2+\mathrm{nT}\\ 0& \mathrm{nT}+T/2\&le;t\&le;(n+1)T\end{array}\right.(n=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;)$

The initialization condition of equation (1) is

$\frac{d{u}_C\left(t\right)}{\mathrm{dt}}{|}_{t=0}=0,$

Wherein, make τ=2L/R,

ξ=1/ τ=R/2L, condition discussed here be T＞＞τ;

The complex frequency domain algebraic equation model of zero condition input response is:

$G\left(s\right)=\frac{U_C\left(s\right)}{U\left(s\right)}=\frac{1}{\mathrm{LC}{s}^2+\mathrm{RCs}+1}$

$=\frac{\frac{1}{\mathrm{LC}}}{s^2+\frac{R}{L}s+\frac{1}{\mathrm{LC}}}$

$=\frac{{\mathrm{\&omega;}}_0^2}{s^2+2\mathrm{\&xi;s}+{\mathrm{\&omega;}}_0^2}$

That is, rlc circuit charges under the zero state response condition;

The complex frequency domain algebraic equation model of non-zero status input response is:

$U_c\left(s\right)=\frac{(s+2\mathrm{\&xi;})u_c(0\_)+u_c^{\&prime;}(0\_)}{s^2+2\mathrm{\&xi;s}+{\mathrm{\&omega;}}_0^2}+\frac{{\mathrm{\&omega;}}_0^2}{s^2+2\mathrm{\&xi;s}+{\mathrm{\&omega;}}_0^2}U\left(s\right).$

5. the RLC series connection second-order circuit model analysis method based on MATLAB as claimed in claim 1 is characterized in that the MATLAB programming step that this method comprises is:

Remove the MATLAB memory headroom;

Parameter, the initial value on the energy-storage travelling wave tube, decay factor and circuit resonance angular frequency, the time step of setting the RLC element are 0.01 second, and simulation time is 3s;

Obtain by U _S(S) array that numerator coefficients constitutes, the array that denominator coefficients constitutes;

Obtain in the time domain $u_c\left(t\right)=k_1{e}^{p_{1}t}+k_2{e}^{p_{2}t};$

Obtain by the Voltammetric Relation on the electric capacity $i_L\left(t\right)=C{u}_c^{\&prime;}\left(t\right)=C\frac{d{u}_C\left(t\right)}{\mathrm{dt}};$

Image draws.

Images