CN101425107A - Linearized model establishing method for non-linear radio frequency microwave circuit - Google Patents

Abstract

The invention discloses a linear model establishment method for a nonlinear radiofrequency microwave circuit, which comprises the steps: firstly, selecting a quality factor Q value of a no-load resonance loop according to a double parameter bifurcation diagram; secondly, carrying out linearization on a normalization state equation to ensure open loop gain g <*> of an oscillator; thirdly, calculating an inductance L value; fourthly, calculating a value of resistance R; fifthly, utilizing the parameter values to establish a Colpitts circuit simulation model. A nonlinear exponential function n (X2) equal to e<X>2 -1is converted to a linear function by taylor series expansion, thus the linearization conversion process of the nonlinear radiofrequency microwave circuit is realized. The establishment of the traditional Colpitts circuit linearization model needs to respectively select five parameter values which are L, C1, C2, IL and R, and the invention can ensure all the parameter values of the circuit model only by selecting the values of the quality factor Q, a current source I0 and capacitance C2.

Description

A kind of linearized model establishing method of non-linear radio frequency microwave circuit

Technical field

The present invention relates to a kind of method for building up of circuit model, specifically, be meant a kind of linearized model establishing method of non-linear radio frequency microwave circuit.

Background technology

The Colpitts circuit is the known pierce circuits of people, as typical capacitance three-point type feedback oscillation circuit, the Colpitts circuit is widely used in the signal source from extremely low frequency to the millimeter wave frequency band very much, and is to realize the most promising scheme of microwave chaos at this stage.The Colpitts circuit mainly comprises booster element that is made of triode and the resonant tank that is made of two electric capacity of an inductance.The basic structure of Colpitts circuit such as Fig. 1 wherein, are connected between power Vcc and the triode T collector C after resistance R and the inductance L series connection, and the emitter E of triode T is connected with current source, current source I ₀Ground connection; The base stage B of triode T and capacitor C ₂Negative pole connect capacitor C ₁And capacitor C ₂Series connection back and triode T and current source I ₀In parallel.

According to following normalization state equation:

$\left(\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\end{array}\right)=\left(\begin{array}{c}\frac{g^{*}}{Q(1-k)}[-\left(\frac{I_0-I_s}{I_0}\right)n\left(x_2\right)+x_3]\\ \frac{g^{*}}{\mathrm{Qk}}{x}_3\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}(x_1+x_2)-\frac{1}{Q}{x}_3\end{array}\right)---\left(1\right)$

Wherein, $n\left(x_2\right)=e^{x_2}-1,$ Parameter g ^*Be the open-loop gain of oscillator, Q is the quality factor of no-load resonant tank, and k is a nondimensional ratio, I _sBe the triode reverse saturation current, and g ^*, Q, k be defined as follows:

g ^*＝I ₀L/V _TR(C ₁+C ₂) (2)

Q＝ω ₀L/R (3)

k＝C ₂/(C ₁+C ₂) (4)

In the formula: I ₀Be current source, V _TBe thermal voltage, C ₁, C ₂Be the electric capacity in the circuit resonance network, ω ₀Be basic angular frequency, L is the inductance in the circuit resonance network, and R is the resistance that is equivalent to the voltage source correspondence of current source.Angular frequency ₀Expression formula is as follows:

${\mathrm{\&omega;}}_0=2\mathrm{\&pi;}{\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="A200810227076E00132.png"\; state="\{\{state\}\}">f</figure-callout>}}_0=\sqrt{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)/\left({\mathrm{LC}}_1{C}_2\right)}---\left(5\right)$

In the formula: f ₀Be the circuit basic frequency.

Under the room temperature condition, thermal voltage V _T≈ 26mV.Triode reverse saturation current I _sValue be far smaller than current source I in the Colpitts circuit usually ₀Value, so in the formula (1) $\frac{I_0-I_s}{{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="A200810227076E00132.png"\; state="\{\{state\}\}">I</figure-callout>}}_0}\&ap;1.$

Traditional Colpitts circuit linearized model establishing method need be chosen L, C respectively ₁, C ₂, I _L, five parameter values of R because the sensitivity to parameter of Colpitts circuit, parameter variation range is all very big, the variation of each parameter has nothing in common with each other to the influence of output characteristics, makes when determining component parameters to have no way of doing it.And, because the mathematical model of Colpitts circuit relates to exponential function, be a kind of nonlinear function, unavoidably can be much complicated when handling various application problem.Simultaneously, in communication system, oscillatory circuit is often required certain frequecy characteristic usually,, can just become very important from the method that specific output characteristics is set up model so find a kind of simple and feasible more.

Summary of the invention

The present invention changes responsively in order to solve the Colpitts circuit to parameter, and parameter variation range is big, and the problem that is not easy to determine proposes a kind of linearized model establishing method of non-linear radio frequency microwave circuit, and is main by to the normalization state equation $\left(\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\end{array}\right)=\left(\begin{array}{c}\frac{g^{*}}{Q(1-k)}[-\left(\frac{I_0-I_s}{I_0}\right)n\left(x_2\right)+x_3]\\ \frac{g^{*}}{\mathrm{Qk}}{x}_3\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}(x_1+x_2)-\frac{1}{Q}{x}_3\end{array}\right)$ Linearization, determine quality factor q, open-loop gain g in the equation ^*, select dimensionless ratio k, current source I ₀, capacitor C ₂Thereby, derive other parameter capacitor C of model ₁, inductance L, resistance R, greatly reduce the difficulty of setting up the non-linear radio frequency microwave circuit model.

The linearized model establishing method of a kind of non-linear radio frequency microwave circuit of the present invention may further comprise the steps:

Step 1: the quality factor q value of choosing the no-load resonant tank according to two-parameter bifurcation graphs;

In dividing represented chaos state district, two-parameter bifurcation graphs black part chooses log ₁₀(Q) scope is in interval [0.2,1.5], and then definite Q value interval [0.6310,31.6228].

Step 2:, determine the open-loop gain g of oscillator to the linearization of normalization state equation ^*

In formula (1), because x ₂Magnitude generally 10 ^-3Below, will $n\left(x_2\right)=e^{x_2}-1$ Carry out Taylor series expansion, as shown in the formula

$e^{x_2}=1+x_2---\left(6\right)$

According to formula (6), then normalization equation of state (1) is deformed into:

$\left(\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{g^{*}}{Q(1-k)}& 1\\ 0& 0& \frac{g^{*}}{\mathrm{Qk}}\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{1}{Q}\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)---\left(7\right)$

In the chaos turnoff, the secular equation of normalization state equation is:

${\mathrm{\&lambda;}}^3+\frac{1}{Q}{\mathrm{\&lambda;}}^2+[1-k+\frac{\mathrm{Qk}(1-k]}{g^{*}})\mathrm{\&lambda;}-\frac{g^{*}}{Q}=0--\left(8\right)$

According to the Liapunov stability principle, in the chaos turnoff, state equation has a real root and a pair of pure imaginary root.If real root is γ, pure imaginary root be i ω ,-i ω, bring these three roots into formula (8) respectively, can get:

$Q=\frac{-{\left(g^{*}\right)}^2-g^{*}+k{g}^{*}}{k(1-k)}---\left(9\right)$

With formula (9) substitution formula (8):

$(\mathrm{\&lambda;}-\frac{k(1-k)}{{\left(g^{*}\right)}^2+g^{*}-{\mathrm{kg}}^{*}})({\mathrm{\&lambda;}}^2-g^{*})=0---\left(10\right)$

Get by formula (10):

$\mathrm{\&gamma;}=\frac{k(1-k)}{{\left(g^{*}\right)}^2+g^{*}-{\mathrm{kg}}^{*}},$ $\mathrm{\&omega;}=\sqrt{g^{*}}---\left(11\right)$

The frequency f of normalization Colpitts circuit correspondence then _HCan represent by formula (11):

$f_H=\frac{\mathrm{\&omega;}}{2\mathrm{\&pi;}}=\frac{\sqrt{g^{*}}}{2\mathrm{\&pi;}}---\left(12\right)$

The frequency F of non-normalized Colpitts circuit correspondence _HCan be expressed as

$F_H=\frac{I_0\mathrm{Qk}}{V_T{C}_2{g}^{*}}{f}_H---\left(13\right)$

Draw by formula (12) substitution formula (13) and to ask g ^*Formula:

$g^{*}={\left(\frac{{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="A200810227076E00132.png"\; state="\{\{state\}\}">I</figure-callout>}}_0\mathrm{<figure-callout\; id="2"\; label="Qk"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">Qk</figure-callout>}}{2\mathrm{\&pi;}{F}_H{V}_T{C}_2}\right)}^2---\left(14\right)$

Current source I ₀Generally get 20mA, capacitor C ₂, the value order of magnitude is pF, the capacitor C of choosing among the present invention ₂Scope is [1pF, 70pF].The dimensionless ratio k can only cause the variation on the state variable yardstick, and can not cause the variation of system's state of living in, gets 0.5 usually.Thermal voltage V _T, under the normal temperature state, get 26mV, F _HBe the known device frequency.Q chooses in step 1.With all known parameters substitution formulas (14), obtain g ^*Value.

Step 3: calculate the inductance L value;

Formula (2) is got than formula (3):

$\frac{g^{*}}{Q}=\frac{I_{0}L}{{\mathrm{\&omega;}}_{0}L{V}_T(C_1+C_2)}---\left(15\right)$

Formula (5) substitution formula (15) is got:

$\frac{g^{*}}{Q}=\frac{I_{0}L\sqrt{{\mathrm{LC}}_1{C}_2}}{L{V}_{\mathrm{<figure-callout\; id="1"\; label="T\; C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">T</figure-callout>}}\sqrt{C_{}}\sqrt{{}_1+C_2}({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)}---\left(16\right)$

Formula (4) substitution formula (16) is got:

$L=8C_2\left(\frac{1-k}{k}\right){\left(\frac{g^{*}{V}_T}{{\mathrm{QI}}_0}\right)}^2---\left(17\right)$

All parameters are known in the formula (17), and substitution can obtain the inductance L value.

Step 4: calculated resistance R value;

Formula (5) substitution formula (3) is got:

$R=\sqrt{\frac{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)L}{Q^2{C}_1{C}_2}}---\left(18\right)$

Get by formula (4);

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1=\frac{1-k}{k}{C}_2---\left(19\right)$

Capacitor C ₁Through type (19) calculates.Inductance L is obtained by step 3, and the quality factor q of no-load resonant tank is determined by step 1, with known parameters substitution formula (18), obtained the resistance R value.

Step 5: utilize above-mentioned gained parameter value, set up the Colpitts circuit simulation model.

The invention has the advantages that:

(1) traditional Colpitts circuit parameter selection method need be chosen L, C respectively ₁, C ₂, I _L, five parameter values of R, the present invention only need choose quality factor q, current source I ₀, capacitor C ₂Numerical value just can be determined the device parameter values that circuit is every other;

(2) in choosing the process of quality factor, adopted clear and definite bifurcation graphs, in a limited scope, determine parameter value, the problem of not clear scope when having solved selection of parameter;

(3) modeling process is changed to simple computation by separating complicated equation;

(4) solved describing method, when finding the solution the secular equation of normalization state equation, used Taylor series expansion, thereby solved the problem that the nonlinear function exponential function is deformed into linear function nonlinear characteristic;

(5) mathematical model with the Colpitts circuit is deformed into the approximately linear model, therefore can utilize the model of more simplifying to analyze problems and solve them in the types of applications of this model;

Description of drawings

Fig. 1 is the basic structure of Colpitts circuit;

Fig. 2 is the process flow diagram of linearized model establishing method of the present invention;

Fig. 3 is two-parameter bifurcation graphs;

Fig. 4 is the amplitude spectrum of the emulation output among the embodiment;

Fig. 5 is the chaotic attractor of emulation output of the present invention.

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples.

The present invention is a kind of linearized model establishing method of non-linear radio frequency microwave circuit, and this method is passed through the normalization state equation $\left(\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\end{array}\right)=\left(\begin{array}{c}\frac{g^{*}}{Q(1-k)}[-\left(\frac{I_0-I_s}{I_0}\right)n\left(x_2\right)+x_3]\\ \frac{g^{*}}{\mathrm{Qk}}{x}_3\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}(x_1+x_2)-\frac{1}{Q}{x}_3\end{array}\right)$ Linearization, determine quality factor q, open-loop gain g ^*, select dimensionless ratio k, current source I ₀, capacitor C ₂Thereby, derive other parameter capacitor C of model ₁, inductance L, resistance R parameter value, set up the inearized model of non-linear radio frequency microwave circuit.The process flow diagram of described method for establishing model as shown in Figure 2, concrete implementation step is as follows:

Step 1: the quality factor q value of choosing the no-load resonant tank according to two-parameter bifurcation graphs.

Two-parameter bifurcation graphs such as Fig. 3, wherein white portion is represented the state of one-period, the ash color part is represented the multicycle state, black part is divided the expression chaos state, The present invention be directed to chaos circuit and carry out modeling, so should in black part is divided represented chaos state district, choose the Q value, choose log according to Fig. 3 ₁₀(Q) numerical value in interval [0.2,1.5], and then definite Q value interval [0.6310,31.6228].

Step 2:, determine the open-loop gain g of oscillator to the linearization of normalization state equation ^*

In formula (1), because x ₂Magnitude generally 10 ^-3Below, will $n\left(x_2\right)=e^{x_2}-1$ Carry out Taylor series expansion, as shown in the formula:

$e^{x_2}=1+x_2---\left(6\right)$

According to formula (6), then normalization equation of state (1) is deformed into:

$\left(\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{g^{*}}{Q(1-k)}& 1\\ 0& 0& \frac{g^{*}}{\mathrm{Qk}}\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{1}{Q}\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)---\left(7\right)$

In the chaos turnoff, the secular equation of normalization state equation is:

${\mathrm{\&lambda;}}^3+\frac{1}{Q}{\mathrm{\&lambda;}}^2+[1-k+\frac{\mathrm{Qk}(1-k]}{g^{*}})\mathrm{\&lambda;}-\frac{g^{*}}{Q}=0--\left(8\right)$

According to the Liapunov stability principle, in the chaos turnoff, above-mentioned state equation (8) has a real root and a pair of pure imaginary root.If real root is γ, pure imaginary root be i ω ,-i ω, bring these three roots into formula (8) respectively, can draw

$Q=\frac{-{\left(g^{*}\right)}^2-g^{*}+k{g}^{*}}{k(1-k)}---\left(9\right)$

With formula (9) substitution formula (8), can draw formula (10):

$(\mathrm{\&lambda;}-\frac{k(1-k)}{{\left(g^{*}\right)}^2+g^{*}-{\mathrm{kg}}^{*}})({\mathrm{\&lambda;}}^2-g^{*})=0---\left(10\right)$

Solving equation (10) gets:

$\mathrm{\&gamma;}=\frac{k(1-k)}{{\left(g^{*}\right)}^2+g^{*}-{\mathrm{kg}}^{*}},$ $\mathrm{\&omega;}=\sqrt{g^{*}}---\left(11\right)$

The frequency f of normalization Colpitts circuit correspondence then _HBe expressed as:

$f_H=\frac{\mathrm{\&omega;}}{2\mathrm{\&pi;}}=\frac{\sqrt{g^{*}}}{2\mathrm{\&pi;}}---\left(12\right)$

The frequency F of non-normalized Colpitts circuit correspondence _HCan be expressed as:

$F_H=\frac{I_0\mathrm{Qk}}{V_T{C}_2{g}^{*}}{f}_H---\left(13\right)$

Formula (12) substitution formula (13) is got:

$g^{*}={\left(\frac{I_0\mathrm{<figure-callout\; id="2"\; label="Qk"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">Qk</figure-callout>}}{2\mathrm{\&pi;}{F}_H{V}_T{C}_2}\right)}^2---\left(14\right)$

Current source I in the formula (14) ₀Generally get 20mA; Capacitor C ₂The value order of magnitude is pF, the capacitor C of choosing among the present invention ₂Scope is [1pF, 70pF]; Thermal voltage V _TGet 26mV at normal temperatures; The dimensionless ratio k can only cause the variation on the state variable yardstick, and can not cause the variation of system's state of living in, gets k=0.5 usually; F _HBe the known device frequency.

With current source I ₀, capacitor C ₂, dimensionless ratio k, thermal voltage V _T, known device frequency F _H, the no-load resonant tank quality factor q substitution formula (14), just can calculate the open-loop gain g of oscillator ^*Numerical value.

Step 3: calculate the inductance L value;

Get according to formula (2) and formula (3):

$\frac{g^{*}}{Q}=\frac{I_{0}L}{{\mathrm{\&omega;}}_{0}L{V}_T({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)}---\left(15\right)$

Formula (5) substitution formula (15) is drawn:

$\frac{g^{*}}{Q}=\frac{I_{0}L\sqrt{{\mathrm{LC}}_1{C}_2}}{L{\mathrm{<figure-callout\; id="1"\; label="V\; T\; C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">V</figure-callout>}}_T\sqrt{C_{}}\sqrt{{}_1+C_2}({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)}---\left(16\right)$

Formula (4) substitution formula (16) is drawn:

$L=8C_2\left(\frac{1-k}{k}\right){\left(\frac{g^{*}{V}_T}{{\mathrm{QI}}_0}\right)}^2---\left(17\right)$

With dimensionless ratio k, capacitor C ₂, oscillator open-loop gain g ^*, thermal voltage V _T, the quality factor q of no-load resonant tank, current source I ₀Substitution formula (17) promptly gets the inductance L value.

The numerical value of step 4: calculated resistance R.

Can get by formula (5) substitution formula (3):

$R=\sqrt{\frac{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)L}{Q^2{C}_1{C}_2}}---\left(18\right)$

Known capacitance C ₂Value is obtained capacitor C according to formula (19) ₁Value.

With capacitor C ₁, capacitor C ₂, inductance L, no-load resonant tank quality factor q substitution formula (18), draw the resistance R value.

Step 5: utilize above-mentioned steps gained parameter value, set up the Colpitts circuit simulation model.

The linearized model establishing method of a kind of non-linear radio frequency microwave circuit of the present invention is described with an embodiment below, carries out according to following steps:

Step 1: the quality factor q value of choosing the no-load resonant tank according to two-parameter bifurcation graphs.

This example has adopted a frequency microwave frequency 800MHz to carry out algorithm and has implemented and emulation, i.e. F _H=800MHz is according to the chaos bifurcation graphs of Fig. 3, log of the present invention ₁₀(Q) value is got in the interval [0.2,1.5], gets lg Q=0.3250 herein, gets Q=2.1135.

Step 2:, determine the open-loop gain g of oscillator to the linearization of normalization state equation ^*

Choose current source I among the embodiment ₀=20mA, the capacitor C of choosing ₂Scope is [1pF, 70pF], gets C in the present embodiment ₂=4pF, the dimensionless ratio k only to the variation on the output generation yardstick, is got k=0.5, under the room temperature condition, thermal voltage V _T≈ 26mV, the quality factor q of no-load resonant tank=2.1135, known device frequency microwave frequency F _H=800MHz, the parameter value substitution formula (14) that all are known draws the open-loop gain of oscillator:

$g^{*}={\left(\frac{I_0\mathrm{<figure-callout\; id="2"\; label="Qk"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">Qk</figure-callout>}}{2\mathrm{\&pi;}{F}_H{V}_T{C}_2}\right)}^2={\left(\frac{20\&times;2.1135\&times;0.5}{2\mathrm{\&pi;}\&times;800\&times;26\&times;4}\right)}^2=25.4026$

So, lgg ^*=1.4049.

Step 3: calculate the inductance L value.

With capacitor C ₂=4pF, thermal voltage V _TQuality factor q=2.1135 of=26mV, dimensionless ratio k=0.5, no-load resonant tank, current source I ₀The open-loop gain g of=20mA, oscillator ^*=25.4026 substitution formula (17) draw:

$L=8C_2\left(\frac{1-k}{k}\right){\left(\frac{g^{*}{V}_T}{{\mathrm{QI}}_0}\right)}^2=8\&times;4\&times;\left(\frac{1-0.5}{0.5}\right)\&times;\left(\frac{25.4026\&times;26}{2.1135\&times;20}\right)=7.81\left(\mathrm{nH}\right).$

The numerical value of step 4: calculated resistance R.

Known capacitance C ₂=4pF, substitution formula (19) obtains capacitor C ₁=4pF.With capacitor C ₁=4pF, capacitor C ₂The quality factor q of=4pF, inductance L=7.81nH, no-load resonant tank=2.1135 substitution formulas (18):

$R=\sqrt{\frac{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_2)L}{Q^2{C}_1{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="A200810227076E00142.png"\; state="\{\{state\}\}">C</figure-callout>}}_2}}=\sqrt{\frac{(4+4)\&times;7.81}{{2.1135}^2\&times;4\&times;4}}=29.6\left(\mathrm{\&Omega;}\right)$

Step 5: utilize above-mentioned gained parameter value, set up the Colpitts circuit simulation model.

Adopt ADS software to set up the Colpitts circuit model.Triode amplifier utilizes the existing model in the ADS software model storehouse, with step 1 to 4 all parameter values of choosing and this realistic model of parameter value substitution of calculating, amplitude spectrum that draws and phase-plane diagram are respectively as Fig. 4 and Fig. 5, point shown in the vernier is the channel frequency point among Fig. 4, the result is 816.7MHz, meets substantially with expected frequence, and Fig. 5 shows that this circuit working is at the chaotic oscillation state, and have the attractor form, so the chaotic oscillation that meets design requirement of result.

Claims (4)

1, a kind of linearized model establishing method of non-linear radio frequency microwave circuit is characterized in that: main passing through the normalization state equation $\left(\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\\ {\stackrel{.}{x}}_3\end{array}\right)=\left(\begin{array}{c}\frac{g^{*}}{Q(1-k)}[-\left(\frac{I_0-I_s}{I_0}\right)n\left(x_2\right)+x_3]\\ \frac{g^{*}}{\mathrm{Qk}}{x}_3\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}(x_1+x_2)-\frac{1}{Q}{x}_3\end{array}\right)$ Linearization, determine quality factor q, open-loop gain g ^*, select dimensionless ratio k, current source I ₀, capacitor C ₂Thereby, derive other parameter capacitor C of model ₁, inductance L, resistance R, comprise following steps:

1) chooses the quality factor q value of no-load resonant tank according to two-parameter bifurcation graphs;

In dividing represented chaos state district, two-parameter bifurcation graphs black part chooses log ₁₀(Q) scope is in interval [0.2,1.5], and then definite Q value scope [0.6310,31.6228];

2), determine the open-loop gain g of oscillator to the linearization of normalization state equation ^*

With the normalization state equation $\left(\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\\ {\stackrel{.}{x}}_3\end{array}\right)=\left(\begin{array}{c}\frac{g^{*}}{Q(1-k)}[-\left(\frac{I_0-I_s}{I_0}\right)n\left(x_2\right)+x_3]\\ \frac{g^{*}}{\mathrm{Qk}}{x}_3\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}(x_1+x_2)-\frac{1}{Q}{x}_3\end{array}\right)$ In $n\left(x_2\right)=e^{x_2}-1$ Carry out Taylor series expansion,

$e^{x_2}=1+x_2$

Normalization equation of state (1) is deformed into: $\left(\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\\ {\stackrel{.}{x}}_3\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{g^{*}}{Q(1-k)}& 1\\ 0& 0& \frac{g^{*}}{\mathrm{Qk}}\\ -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{\mathrm{Qk}(1-k)}{g^{*}}& -\frac{1}{Q}\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right),$ In the chaos turnoff, the secular equation of normalization state equation is:

${\mathrm{\&lambda;}}^3+\frac{1}{Q}{\mathrm{\&lambda;}}^2+[1-k+\frac{\mathrm{Qk}(1-k)}{g^{*}}]\mathrm{\&lambda;}-\frac{g^{*}}{Q}=0,$

The secular equation of finding the solution state equation gets: $g^{*}={\left(\frac{I_0\mathrm{Qk}}{2\mathrm{\&pi;}{F}_H{V}_T{C}_2}\right)}^2,$

In the formula: V _T--thermal voltage, F _H--the known device frequency;

3) calculate the inductance L value;

According to formula $L=8C_2\left(\frac{1-k}{k}\right){\left(\frac{g^{*}{V}_T}{{\mathrm{QI}}_0}\right)}^2$ Calculate the inductance L value;

4) calculated resistance R value;

According to formula $R=\sqrt{\frac{(C_1+C_2)L}{Q^2{C}_1{C}_2}}$ Calculated resistance R value, C in the formula ₁Be the electric capacity in the resonant network;

5) utilize above-mentioned gained parameter value, set up the Colpitts circuit simulation model.

2, the linearized model establishing method of a kind of non-linear radio frequency microwave circuit according to claim 1 is characterized in that: in step 2) in, current source I ₀Get 20mA, capacitor C ₂The value order of magnitude is pF, and scope is [1pF, 70pF], dimensionless ratio k=0.5, and under the normal temperature, thermal voltage V _T=26mV;

3, the linearized model establishing method of a kind of non-linear radio frequency microwave circuit according to claim 1 is characterized in that: in step 3), and the determining of inductance L through following method:

According to g ^*=I ₀L/V _TR (G ₁+ C ₂) and Q=ω ₀L/R gets:

$\frac{g^{*}}{Q}=\frac{I_{0}L}{{\mathrm{\&omega;}}_{0}L{V}_T(C_1+C_2)}---\left(15\right)$

In the formula: ω ₀--basic angular frequency; f ₀--basic frequency;

Will ${\mathrm{\&omega;}}_0=2\mathrm{\&pi;}{f}_0=\sqrt{(C_1+C_2)/\left({\mathrm{LC}}_1{C}_2\right)}$ Substitution formula (15):

$\frac{g^{*}}{Q}=\frac{I_{0}L\sqrt{{\mathrm{LC}}_1{C}_2}}{{\mathrm{LV}}_T\sqrt{C_1+C_2}(C_1+C_2)}---\left(16\right)$

With k=C ₂/ (C ₁+ C ₂) substitution formula (16):

$L=8C_2\left(\frac{1-k}{k}\right){\left(\frac{g^{*}{V}_T}{{\mathrm{QI}}_0}\right)}^2---\left(17\right)$

Current source I in the formula (17) ₀, capacitor C ₂, dimensionless ratio k, thermal voltage V _T, open-loop gain g ^*, quality factor q is known, substitution can obtain the inductance L value.

4, according to the linearized model establishing method of the described non-linear radio frequency microwave circuit of claim 1, it is characterized in that: in step 4), resistance $R=\sqrt{\frac{(C_1+C_2)L}{Q^2{C}_1{C}_2}}$ By inciting somebody to action ${\mathrm{\&omega;}}_0=2\mathrm{\&pi;}{f}_0=\sqrt{(C_1+C_2)/\left({\mathrm{LC}}_1{C}_2\right)}$ Bring Q=ω into ₀L/R draws; Capacitor C ₁By $C_1=\frac{1-k}{k}{C}_2$ Calculate.

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