JP2009033483A - Calculation method of oscillation frequency deviation and calculation program of oscillation frequency deviation - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a calculation method of oscillation frequency deviation of a piezoelectric oscillator capable of reflecting an influence of a resistive component in the piezoelectric oscillator on oscillation frequency deviation highly accurately and so of analyzing the oscillation frequency deviation more accurately. <P>SOLUTION: A highly accurate oscillation frequency deviation analysis can be attained by using a recurrence equation that reflects on oscillation frequency deviation highly accurately an influence of equivalent series resistance R1 and equivalent load resistance RL of a piezoelectric oscillator to repeatedly calculating the oscillation frequency for obtaining a converged constant value. More specifically, calculations of determining an arbitrary initial substitution value f0 and substituting n order fn for a predetermined recurrence equation are repeated to determine n+1 order oscillation frequency fn+1, and the constant value to which the fn converges is obtained by repeating the calculation to calculate the frequency deviation. This enables highly accurate calculation of the oscillation frequency deviation of the piezoelectric oscillator. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a method for analyzing an oscillation frequency deviation of a piezoelectric oscillator including a piezoelectric vibrator, and more particularly to an analysis method in which the analysis is performed with high accuracy.

  At the design stage of the piezoelectric oscillator, it is necessary to grasp the performance of the piezoelectric oscillator by simulation before making an actual prototype, and to reflect it in the design of a higher performance oscillator in order to reduce useless cost and time. Connection is very important. Among various parameters indicating the performance of the piezoelectric oscillator, the oscillation frequency deviation Δf / f is a parameter that is very important in the design stage, and is extremely important for measuring the performance of the piezoelectric oscillator often used as a frequency generation source. Is one of the most important parameters.

  FIG. 1 is a diagram showing an equivalent circuit 100 of a piezoelectric oscillator. The piezoelectric vibrator 10 includes an equivalent series inductance L1, an equivalent series capacitance C1, an equivalent series resistance R1, and an interelectrode capacitance C0. An equivalent series inductance L1, an equivalent series capacitance C1, and an equivalent series resistance R1 are connected in series, and these are further connected in parallel to the interelectrode capacitance C0. An equivalent circuit of the oscillation circuit 20 is indicated by an equivalent load capacitance CL connected in series with the equivalent load resistor RL.

滝貞男、人工水晶とその電気的応用、日刊工業新聞社、（1974） ｐ122〜126 The oscillation frequency deviation Δf / f is also called a resonance frequency offset, and is expressed by an approximate expression given by Expression (1). The capacity ratio γ is given by the equation (2) (see Non-Patent Document 1).

Sadao Taki, artificial quartz and its electrical application, Nikkan Kogyo Shimbun, (1974) p122 ～ 126

  However, the equation (1) does not include the equivalent series resistance R1 of the piezoelectric vibrator 10 and the equivalent load resistance RL of the oscillation circuit 20, and does not consider the influence of the resistance included in the piezoelectric oscillator. Therefore, in the equation (1), it is impossible to obtain the influence of the resistance on the oscillation frequency. Since it is practically impossible that the actual piezoelectric oscillator does not include a resistor, it is impossible to analyze the oscillation frequency deviation with high accuracy even if the equation (1) is used.

  The present invention has been made to eliminate the drawbacks of the prior art, and it is possible to consider the influence of the equivalent series resistance R1 and the equivalent load resistance RL, which are resistances included in the piezoelectric oscillator, on the oscillation frequency. Thus, an oscillation frequency deviation analysis method that is very highly accurate compared to the prior art is provided.

  In order to achieve the above object, a method for calculating an oscillation frequency deviation according to the present invention is a calculation method for an oscillation frequency deviation for calculating an oscillation frequency deviation of a piezoelectric oscillator with high accuracy. An initial value for determining an arbitrary initial substitution value f0 with respect to a predetermined recurrence formula used to determine the oscillation frequency of the piezoelectric oscillator, reflecting the influence of the equivalent series resistance of the child and the negative resistance of the piezoelectric oscillator. Using a step, an iterative step of substituting the n-th order fn into the predetermined recurrence formula and repeating the calculation for determining the n + 1-order oscillation frequency fn + 1, and a constant value where fn has converged by the repetition of the calculation, And a calculating step for calculating a frequency deviation.

  Also, in the calculation method of the oscillation frequency deviation according to the present invention, the predetermined recurrence formula is such that a resistance value of an equivalent series resistance of a piezoelectric vibrator included in the piezoelectric oscillator, a capacitance value of an equivalent series capacitance of the piezoelectric vibrator, an equivalent series It is characterized by further including an inductance value of inductance, a capacitance value of interelectrode capacitance, and a capacitance value of an equivalent load capacitance of the piezoelectric oscillator.

Also, in the calculation method of the oscillation frequency deviation of the present invention, the predetermined recurrence formula is such that the resistance value of the equivalent series resistance of the piezoelectric vibrator included in the piezoelectric oscillator is R1, the capacitance value of the equivalent series capacitance is C1, and the equivalent series. When the inductance value of the inductance is represented by L1, the capacitance value of the interelectrode capacitance is represented by C0, and the capacitance value of the equivalent load capacitance of the piezoelectric oscillator is represented by CL, it is represented by the following formula.

In addition, the calculation method of the oscillation frequency deviation according to the present invention is characterized in that, in the calculation step, the frequency deviation is calculated by the following formula when a constant value where fn converges is expressed as fc.

  The oscillation frequency deviation calculation program of the present invention is an oscillation frequency deviation calculation program for calculating the oscillation frequency deviation of the piezoelectric oscillator with high accuracy, and the equivalent series resistance of the piezoelectric vibrator of the piezoelectric oscillator to be calculated is calculated. An initial process for determining an arbitrary initial substitution value f0 for a predetermined recurrence formula including a resistance value and a resistance value of a negative resistance of the piezoelectric oscillator, and n in the predetermined recurrence formula Substituting the next fn and repeating the calculation for determining the (n + 1) th order oscillation frequency fn + 1, and the calculation process for calculating the frequency deviation using a constant value where fn has converged by the repetition of the calculation It is characterized by that.

  According to the present invention, it is possible to analyze the frequency deviation with high accuracy in consideration of the influence of the resistance included in the piezoelectric oscillator.

  As described above, the oscillation frequency deviation analyzing method of the present invention can reflect the influence of the equivalent series resistance R1 and equivalent load resistance RL of the piezoelectric oscillator on the oscillation frequency with high accuracy. And

  Further, the oscillation frequency deviation analysis method of the present invention is characterized in that the oscillation frequency deviation can be analyzed with higher accuracy by repeatedly calculating the oscillation frequency and making it coincide with the actual oscillation frequency.

Means for solving the drawbacks of the prior art using the present invention will be described in detail below. FIG. 2 is a diagram showing an equivalent circuit 200 after the interelectrode capacitance C0 is synthesized on the oscillation circuit side. The equivalent circuit 200 includes a motion arm 30, a combined equivalent resistance -Rcci, and a combined equivalent capacitance Ccci. -Rcci and Ccci are expressed by equations (3) and (4), respectively.

In the equations (3) and (4), ω is an angular frequency of the oscillation frequency f of the piezoelectric oscillator, and is represented by the equation (5).

The oscillation conditions and frequency conditions of the equivalent circuit 200 after the interelectrode capacitance C0 is synthesized on the oscillation circuit side are shown in equations (6) and (7), respectively. The frequency condition of the equivalent circuit 200 is a conditional expression for determining the resonance frequency of the equivalent circuit 200.

Ω1 in the equation (7) is an angular frequency calculated from the equivalent series inductance L1 and the equivalent series capacitance C1 of the piezoelectric vibrator 10, and is represented by the equation (8).

Here, it is assumed that the following equation (9) is established at the time of steady oscillation, and the equation (10) is obtained.

M in the equation (10) is a figure of merit and is represented by the equation (11).

Substituting RL in the equation (10) into Ccci in the equation (4) to obtain Cocci in the equation (12). Here, Cocci means Ccci when R1 = Rcci.

Here, the left side of the equation (7) is determined from the angular frequency ω of the piezoelectric oscillator, but Ccci included in the right side of the equation (7) is also a function of the angular frequency ω as is apparent from the equation (4). Therefore, if the left side is not determined, the value on the right side cannot be determined, and vice versa. Therefore, considering that Ccci is a function Ccci (ωn) of the angular frequency ω, Equation (7) is expressed as Equation (13) to solve this problem. That is, Ccci is determined by setting the angular frequency ωn first, and the oscillation angular frequency ωn + 1 is determined. ωn is shown in equation (14).

  FIG. 3 is a flowchart 300 relating to the determination of the oscillation frequency.

A method for calculating the oscillation frequency deviation will be described below in accordance with the flowchart 300 of FIG. When fn in the initial state (n = 0), that is, f0 is determined, ω0 is determined from the equation (14). Subsequently, Ccci is determined from ω0, R1, C0, and CL, and ω1 is determined from equation (13). From the equation (14), f1 corresponding to ω1 is determined. Next, returning to the first procedure, ω2 is calculated from ω1 at fn + 1, that is, f1, and thereafter this routine is continued until the oscillation frequency fn converges. The oscillation frequency can be determined by this routine, and the frequency deviation Δf / fpt can be determined by equation (15).

  The above calculation method of the oscillation frequency deviation can be executed by a calculation device by a program. An example of the embodiment will be described below. FIG. 4 is a block diagram illustrating a functional configuration of the calculation apparatus 500.

  As the calculation device 500, for example, a PC can be used. The input unit 501 is used for a user to input a value necessary for calculation, and is configured by a pointing device such as a keyboard or a mouse. The values necessary for the calculation are, for example, an equivalent series inductance L1, an equivalent series capacitance C1, an equivalent series resistance R1, an interelectrode capacitance C0, an equivalent load capacitance CL, and the like of the piezoelectric oscillator.

  The initial value determination unit 502 determines an initial value to be substituted into the recurrence formula. For example, a value input by the user may be determined as an initial value as it is, or an optimal value may be automatically determined under a predetermined condition. The iterative calculation unit 503 performs iterative calculation by the above recurrence formula based on the determined initial value, and calculates the oscillation frequency fn.

  The determination unit 504 determines whether or not the difference between the oscillation frequencies fn and fn + 1 calculated by the iterative calculation unit 503 is within a predetermined numerical range. The predetermined numerical range is a numerical range preset by the user. When the difference between the oscillation frequencies fn and fn + 1 is not within a predetermined numerical range, the determination unit 504 further causes the iterative calculation unit 503 to repeat the calculation. When the difference between the oscillation frequencies fn and fn + 1 is within a predetermined numerical range, it is determined that the oscillation frequency has converged, and the calculation unit 505 calculates the oscillation frequency deviation.

  The calculation unit 505 calculates the oscillation frequency deviation by substituting the converged oscillation frequency into the equation (15). Then, the display unit 506 displays the obtained value as a result. The display unit 506 is a display such as an LCD, for example, and displays confirmation to the user even when a numerical value is input. Based on the oscillation frequency deviation obtained in this way, the user can adjust the variable capacitance Cx of the circuit. Each unit that performs the above calculation is configured by a storage device such as a CPU or a memory. In the calculation apparatus 500, each unit operates according to the calculation method shown in the flowchart of FIG.

  FIG. 5 is a diagram showing an equivalent circuit 400 of a general transistor Colpitts oscillation circuit. Cx is a variable capacitor, RB is a base resistor, C2 and C3 are capacitors, R3 is a resistor, 40 is an equivalent circuit of an npn transistor, B is a base terminal of the transistor, C is a collector terminal of the transistor, E is an emitter terminal of the transistor is there.

The combined equivalent resistance Rπ between the base and emitter of the equivalent circuit 400 and the combined equivalent capacitance Cπ between the base and emitter are given by the equations (16) and (17), respectively, and Rπ is expressed by a function of β.

In the equations (16) and (17), β is a current amplification factor, τF is a base running time, and gm is a mutual conductance. Here, it is assumed that gm and τF are constant (collector current is constant). Circuit analysis is performed on the equivalent circuit 400 to obtain equations (18), (19), and (20). The equivalent load capacitance CL is given by equation (18), the equivalent circuit capacitance Cci included in equation (18) is given by equation (19), and the equivalent load resistance RL is given by equation (20).

Rc and Cc included in equations (19) and (20) are given by equations (21) and (22).

R2, c2, r3, and c3 included in the equations (21) and (22) are given by equations (23) to (26), respectively. Here, r2 and c2 are functions of ω and β, and r3 and c3 are functions of ω.

  The combined equivalent resistance Rcci and the combined equivalent capacitance Ccci of the equivalent circuit of the oscillation circuit shown in FIG. 2 are expressed by equations (3) and (4). Further, the combined equivalent capacity Cocci when the equation (9) in the steady state is established is represented by the equation (12). The oscillation frequency fn is determined by the relationship between the Cocci, the equation (13), and the vibrator. This frequency is substituted into the equation (23) to the equation (26), and f (n + 1) is determined by the calculation method described so far. By repeating this, f converges to a constant value, that is, converges to a frequency at which the oscillation frequency coincides with the frequency that determines the equivalent capacitance of the circuit. This frequency becomes the oscillation frequency.

  In the case of the equivalent circuit 400, CL is a combined capacity of the equivalent circuit capacities Cci and Cx, and Cci is an equivalent circuit capacity determined by a proportionality constant K and an equivalent circuit resistance Rci (a function of f and β). After substituting CL in the equation (18) into the equation (12), the oscillation frequency is calculated from the routine, and the frequency deviation is obtained from the equation (15) to obtain FIG. FIG. 6 shows the relationship of the oscillation frequency deviation Δf / fpt with high accuracy to the change of the variable capacitor Cx obtained from the equivalent circuit 400. The circuit parameters are fpt = 12.995 MHz, gm = 38 mA / V, Cπ = 4 pF, C2 = C3 = 30 pF, R3 = 2 kΩ, RB = 20 kΩ.

  L1 = 30.8 mH, C1 = 4.87 fF, CO = 1.35 pF, R1 = 17.8Ω. FIG. 6 characteristically shows that a large current amplification factor is required in order to maintain steady oscillation in a range where Cx is small.

FIG. 7 shows an equivalent circuit 700 of the CMOS inverter oscillation circuit. The output resistance of each MOSFET of the CMOS inverter is indicated by r0, and the mutual conductance of FET (field-effect transistor) in the CMOS inverter is indicated by gm. Cocci is the combined equivalent capacitance Ccci when R1 = Rcci. PchMOS is a P channel metal-oxide semiconductor. NchMOS is an N channel metal-oxide semiconductor. Rc, Cc, r3, and r3 in FIG. 7 are given by equations (27) to (30).

  Rc, Cc, r3, and r3 are introduced into the equations (18), (19), and (20), and the convergence frequency is obtained by the equations (12), (13), and (15) and the flowchart.

This is an equivalent circuit of a general piezoelectric oscillator. In an equivalent circuit of a general piezoelectric oscillator, this is an equivalent circuit after the interelectrode capacitance C0 is synthesized on the oscillation circuit side. 3 is a flowchart illustrating a method for calculating an oscillation frequency deviation according to the present invention. It is a block diagram which shows the functional structure of the calculation apparatus which concerns on this invention. This is an equivalent circuit of a general transistor Colpitts oscillation circuit. It is a figure which shows the relationship of highly accurate frequency deviation (DELTA) f / fpt with respect to the change of the variable capacitance Cx of a transistor Colpitts oscillation circuit. It is an equivalent circuit of a CMOS inverter oscillation circuit.

Explanation of symbols

DESCRIPTION OF SYMBOLS 10 Piezoelectric vibrator 20 Oscillation circuit 30 Motion arm 40 Equivalent circuit 100, 200 Equivalent circuit 300 of piezoelectric oscillator 300 Flowchart diagram 400 Equivalent circuit 500 of transistor Colpitts oscillation circuit Calculation device 501 Input unit 502 Initial value determination unit 503 Iterative calculation unit 504 Determination unit 505 Calculation unit 506 Display unit 700 Inverter oscillation circuit equivalent circuit C2, C3 Capacitor R3 Resistance Cx Variable capacitance RB Base resistance Rπ Base-emitter combined equivalent resistance Cπ Base-emitter combined equivalent capacitance β Current gain τF Base travel time gm Mutual conductance L1 Equivalent series inductance C1 Equivalent series capacitance R1 Equivalent series resistance C0 Electrode capacitance -RL Equivalent load resistance CL Equivalent load capacity Rci Equivalent circuit resistance Cci Equivalent circuit capacity -Rcci Composite equivalent resistance Ccc Synthetic equivalent capacitance f Oscillation frequency fn of the piezoelectric oscillator Oscillation frequency fc of the piezoelectric oscillator A convergence value fpt of the calculated oscillation frequency fpt A frequency calculated from L1 and C1 An angular frequency ωn of the piezoelectric oscillator An angular frequency Δf / fpt of the piezoelectric oscillator Frequency deviation, resonance frequency offset γ Capacity ratio M Figure of Merit
Combined equivalent capacitance Ccci when Cocci R1 = Rcci
PchMOS P channel metal-oxide semiconductor
NchMOS Nchannel metal-oxide semiconductor
Output resistance of r0 MOSFET

Claims (5)

An oscillation frequency deviation calculation method for calculating an oscillation frequency deviation of a piezoelectric oscillator with high accuracy,
Reflects the influence of the equivalent series resistance of the piezoelectric vibrator of the piezoelectric oscillator to be calculated and the negative resistance of the piezoelectric oscillator, and is arbitrary for the predetermined recurrence formula used to determine the oscillation frequency of the piezoelectric oscillator. An initial step for determining an initial substitution value f0 of
An iterative step of substituting the nth order fn into the predetermined recurrence formula and repeating the calculation to determine the n + 1st order oscillation frequency fn + 1;
A calculation step of calculating a frequency deviation by using a constant value in which fn is converged by repetition of the calculation.   The predetermined recurrence formula is: a resistance value of an equivalent series resistance of a piezoelectric vibrator included in the piezoelectric oscillator, a capacitance value of an equivalent series capacitance of the piezoelectric vibrator, an inductance value of an equivalent series inductance, and a capacitance value of an interelectrode capacitance, and 2. The oscillation frequency deviation calculation method according to claim 1, further comprising a capacitance value of an equivalent load capacitance of the piezoelectric oscillator. In the predetermined recurrence formula, the resistance value of the equivalent series resistance of the piezoelectric vibrator included in the piezoelectric oscillator is R1, the capacitance value of the equivalent series capacitance is C1, the inductance value of the equivalent series inductance is L1, and the capacitance value of the interelectrode capacitance. 3. The oscillation frequency deviation calculation method according to claim 2, wherein C0 is represented by C0 and the capacitance value of the equivalent load capacitance of the piezoelectric oscillator is represented by CL.

4. The oscillation frequency deviation calculation method according to claim 1, wherein, in the calculation step, when a constant value where fn converges is expressed as fc, the frequency deviation is calculated by the following formula.

An oscillation frequency deviation calculation program for calculating the oscillation frequency deviation of a piezoelectric oscillator with high accuracy,
Arbitrary initial substitution value f0 with respect to a predetermined recurrence formula including the resistance value of the equivalent series resistance of the piezoelectric vibrator of the piezoelectric oscillator to be calculated and the resistance value of the negative resistance of the piezoelectric oscillator. Initial processing to determine,
An iterative process for substituting the nth order fn into the predetermined recurrence formula and repeating the calculation for determining the n + 1st order oscillation frequency fn + 1;
And a calculation process for calculating a frequency deviation using a constant value where fn has converged by the repetition of the calculation.

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