JP2848997B2 - Method for determining equivalent circuit element constants of electric circuits - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for determining an equivalent circuit element constant of an electric circuit.

In recent years, in designing an electric circuit including a GaAs FET, such as a microwave amplifier, which is well known in satellite broadcasting and the like, it has become extremely important to determine the equivalent circuit element constant of the FET.

[0003]

2. Description of the Related Art It has been proposed that the configuration of an equivalent circuit such as an FET is shown as shown in FIG. 1, and this configuration has been almost solidified. This is described, for example, in “IEEE Transaction Microwave Theory”, Vol. 32, No. 12, pp. 1573-1578, December 1984.
(IEEE Trans. Microwave Theory Tech. Vol.32 No.12 D
ec. 1984 pp1573-1578), Curtis and Kamisa
(Curtice & Camisa), "Consistent GaAsFET Model (Self-Consistent GaA) for Amplifier Design and Device Diagnosis.
sFET Models for Amplifier Design and Device Diagno
stics).

In such an equivalent circuit, Ri, Rd
s, Rg, Rs, and Rd indicate resistance components, and Lg, L
s and Ld indicate inductance components, and Cgs and Cg
d and Cds indicate capacitance components, and gm indicates a current source. Further, resistance components Ri and Rds and capacitance components Cgs and Cg surrounded by dotted lines in the drawing are shown.
The impedance part ZF composed of d, Cds and the current source gm is called an elementary element part, and the impedance parts Zg (= Lg + Rg) and Zs (= Ls +
Rs) and Zd (= Ld + Rd) are parasitic elements (Ext
rinsic element) part.

As a method for determining the element constants of such an equivalent circuit, a method is adopted in which the S parameters obtained by the measurement are taken into CAD software for determination.

In order to obtain the S parameter of the equivalent circuit of FIG. 1, first, the Y ( admittance) parameter of the basic element portion ZF and the Z (impedance) parameter of the parasitic element portions Zg, Zs, Zd are obtained as follows. become.

[0007]

(Equation 1)

[0008]

Further, the Y parameter of the basic element portion ZF and all the Z parameters including the parasitic element portions Zg, Zs, Zd are as follows.

[0010]

(Equation 2)

Here, Δ = det (Y (basic part)) (5)

When all the Z parameters are converted into S parameters, the following equation is obtained.

[0013]

(Equation 3)

Zo is a characteristic impedance (50Ω)
It is.

[0015]

However, in the above-described method, since each element constant is determined only by the successive approximation method, an initial value determined empirically is used. This is because the parasitic element Z shown in FIG.
This is because g, Zs, and Zd must be determined in the high-frequency range, and a successive approximation method applicable to the previous frequency range is required. Therefore, convergence depends on the method of selecting the initial values and the method of the successive approximation. However, there are problems that the final device constants are different or have no physical validity, and that the verification requires a certain amount of experience.

In view of the above, a method has been proposed which eliminates such a problem when only the successive approximation method is used. This method is described in "EEE Transaction Microwave Theory Technology", 1988. July 36, No. 7, pp. 1151-1159 (IEEE Trans. Microwave Theory Tec
h. vol.32 No.12 Dec.1984 pp1151-1159), a new method for determining an FET small-signal equivalent circuit (A New Method for Determinator) by Dambrine et al.
ining the FET Small-Signal Equivalent Circuit). In this paper, the basic element portion ZF shown in FIG.
g, Zs, and Zd. First, the parasitic element portions Zg, Z
s and Zd are determined by measurement, and thereafter, each element constant in the basic element portion ZF is determined using the measured value of the S parameter.

According to such a determination method, the problem when only the successive approximation method is used can be eliminated, but on the other hand, there is a regret that the number of types of measurement increases.

Accordingly, an object of the present invention is to determine the element constants of an equivalent circuit using the four-terminal parameters of an electric circuit by using an optimal approximation method with few types of measurement.

[0019]

In the method of determining the equivalent circuit element constant of an electric circuit according to the present invention, the measured values of the four-terminal parameters of the electric circuit are expressed by a rational expression of angular frequency , and the minimum engagement of the organic physical expression by applying the square approximation method
It is characterized in that an equivalent circuit element constant of the electric circuit is determined by calculating the number .

As a result, the measured values of the four-terminal parameters of the electric circuit are expressed by rational expressions, and can be applied to the four-terminal parameters of all electric circuits. And
Furthermore, by applying the least-squares approximation method to the measured values of the four-terminal parameters of the electric circuit expressed in this way, the rational coefficients of the measured values of the four-terminal parameters can be determined. If the rational coefficient can be determined in this way, the equivalent circuit element constant of the electric circuit can be obtained.

Since the rational formula is used without using the successive approximation method in this way, there is uniqueness, and the obtained initial value can be obtained as a reasonable equivalent circuit element constant. Is obtained without being influenced by the method of selecting the initial value, and is physically appropriate.

Further, in the present invention, the value of the parasitic element of the electric circuit can be more accurately determined by further combining the least squares approximation method applied to the above rational expression with the successive approximation method. This is because the measured values of the four-terminal parameters include an error, and thus the coefficients of the obtained rational expression are compressed but include errors. This is because a parameter corresponding to a high-frequency band that must be determined cannot be directly derived from the coefficient.

Therefore, in this case, in the least-squares approximation method applied to the rational expression, the initial value of the successive approximation method is uniquely determined by the measured values of the four-terminal parameters. All equivalent circuit element constants can be obtained by the method.

[0024]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of a method for determining an equivalent circuit element constant of an electric circuit according to the present invention will be described below.

(1) Least-squares (fractional formula) approximation method : In the above equation (6), if it is considered that g _m exp (jωτ) is Taylor-expanded by ω, the S parameter is a fractional formula of ω. You can see it. Also, the denominator is considered to be common.

Therefore, as described later, the least square approximation of the measured value of the S parameter by giving a constraint condition that the denominator is common is first considered as a general theory apart from the S parameter.

That is, a physical quantity F whose property is known from physical considerations to be given by a rational (fractional) expression of ω
_P (ω _i ) can be expressed as in the following equation.

[0028]

(Equation 4)

The denominator and numerator constants a _k , b of the equation (8)
A general method for determining _k to best approximate the measured value F (ω _i ) (ω _i : sample) is described below.

[0030] In order to have the above rational fractional expression (8) makes sense a _0, a _1, ......, at least one non-zero of a _n. When this is the a _l ≠ 0, the denominator in this a _l, even by dividing the molecular function F (ω _i) is does not change, a _l =
1 is good.

Accordingly, the physical quantity F _P (ω _i ) and the measured value F (ω
The difference ε _{i from i} ) is expressed by the following equation.

[0032]

(Equation 5)

When both sides of the equation (9) are multiplied by the denominator and the square of the absolute value is added to the entire measured value, a least square equation is obtained as in the following equation.

[0034]

(Equation 6)

The constants a _k and b that minimize the equation (10)
_k is obtained by setting the coefficient obtained when the error function ε is partially differentiated by a _k and b _k to zero as in the following equation.

[0036]

(Equation 7)

Where p = 0, 1,..., N, p ≠ l q = 0, 1,.

The above equation (11) is a simultaneous linear equation, which can be written as the following equation.

[0039]

(Equation 8)

Each element in this equation is obtained as follows.

[0041]

(Equation 9)

By solving these linear forms, constants a _k and b _k that minimize the error function ε are uniquely obtained.

[0043](2) S-parameter rational (fractional) expression
Similar: Next, the S parameter expressed by the above equation (6) is calculated by the equation
Considering the case expressed by the rational (fractional) expression of (8), ω = 0
Definite denominator constant a₀≠ 0. Also a fraction
Since the denominator of the term is common, make all the denominators the same as
I do.

[0044]

(Equation 10)

Here, when the error function shown in the equation (10) is applied to the above equations (14) to (17), the following equation is obtained.

[0046]

[Equation 11]

Therefore, from equation (18), similarly,
A simultaneous equation giving a _k and b _k ^pq that minimizes the error function T when T / ∂a _k = 0 and ∂T / ∂b _k ^pq = 0 is obtained. The simultaneous equations are as follows.

[0048]

(Equation 12)

However, the partial matrix in the above equation (19) is as follows.

[0050]

(Equation 13)

Such a partial matrix (20.a),
By solving the simultaneous equations (19) including (20.b), the values of the matrices a, b ¹¹ , b ¹² , b ²¹ , and b ²² can be obtained. (1)
The element constants shown in FIG. 1 can be obtained by substituting and performing calculations.

(2) Determination of Constant of Equivalent Circuit Element : Extraction of Initial Value As described above, the fractional expression (8) should include all information of the equivalent circuit element, and at least in principle, All of the equivalent circuit elements should be able to be determined from the fractional expressions.

However, since the measured value of the S parameter always contains an error, which is also included in the coefficient of the fractional expression, many operations are performed on the coefficient to obtain the element values (particularly, the parasitic element portions Zg and Zs). , Zd) becomes more difficult as the frequency becomes higher, and it becomes necessary to use a successive approximation method. Therefore, an initial value of the successive approximation method is obtained using only relatively low-order coefficients.

First, the expression will be started by using the element values of the equation (6). Using equation (4), the denominator of equation (6), P, is as follows. _{P = (Z 11 + Z 0} ) (Z 22 + Z 0) -Z 12 Z 21 = (Z g + Z 0) (Z d + Z 0) + Z S (Z g + Z d + 2Z 0) + [1+ (Z ₀ + Z _d + Z _s ) Y ₂₂ + (Z ₀ + Z _g + Z _s ) Y ₁₁ + Z _S (Y ₁₂ + Y ₂₁ )] / Δ

When this is used, the left side of the equations (14) to (17) is represented by the following equation when represented by S parameters.

[0056]

[Equation 14]

Further, when the equation (1) is used, the following equation is obtained.

[0058]

(Equation 15)

The lowest order coefficient of the denominator of the above equations (25) to (28) is expressed by the following equation.

[0060]

(Equation 16)

Therefore, all coefficients are normalized with this value.

Now, let the coefficients of the numerator of 1-S ₁₁ -S _{12 in} equation (25) be a ₀ , a ₁ , a ₂ , a _3, and S _{12 in} equation (26)
B ₀ , b ₁ , b ₂ , and b ₃ , the numerator coefficient of S _{21 in} equation (27) is c ₀ , c ₁ , c ₂ , and c ₃ , and the 1-S the coefficients of the numerator of _{22 -S 12 d 0, d 1} ,
Assuming that d ₂ and d ₃ , low-order components of each coefficient (basic part ZF
) Is as follows.

[0063]

[Equation 17]

From the above coefficients c ₀ , c ₁ and c ₂ , the following equation is obtained.

[0065]

(Equation 18)

It should be noted, however, that although there is no root sign due to a measurement error, it may be negative. Therefore, the following equation is set.

[0067]

[Equation 19]

In addition, c ₀ / d ₀ = g _m R _ds , and when Z ₀ (= 50Ω) >> R _d , R _s is considered, the following expression is obtained.

[0069]

(Equation 20)

Also,

[0071]

(Equation 21)

Therefore, the relationship between element constants is obtained as shown in the following equation.

[0073]

(Equation 22)

In the above calculation, since the resistance of the parasitic portion such as R _s , R _d , R _g is smaller than Z ₀ , R _ds, etc., it is considered that it is difficult to determine from the coefficients, so the initial value is 0. can be, since the processing for the coefficient becomes larger the inductance, the initial value is set to 0.

Determination of Element Values by Successive Approximation Since the initial values of the respective element constants of the basic element portion ZF shown in FIG. 1 have been given by the above calculations, first, the values of the elements in the parasitic portion are updated.

That is, the element constants R _s , R _g , R _d , L _s ,
Slope of S parameter with respect to L _g and L _d ∂S _ij / ∂R
_s ... ∂S _ij / ∂L _s ... is calculated by Rafson-Newton method, and the following equation is obtained.

[0077]

(Equation 23)

In order to minimize the error ε as an objective function,
Perform successive approximation. However, [S_pq] (Ω _i) Is the measured value
You. This minimum value becomes unstable when a large value is changed.
The damping coefficient α (= 0.5)
You.

Next, the slope of the S parameter with respect to the basic element portion ZF is calculated, and the value of the basic element portion ZF is updated.

This is repeated alternately until convergence. It should be noted that when the equation (44) is calculated with respect to the first fractional formula approximation, the degree of the experimental error can be determined, and this value can be used as a guide for convergence determination.

[0081]

As described above, according to the method for determining the equivalent circuit element constant of the electric circuit according to the present invention, the measured values of the four-terminal parameters of the electric circuit are expressed by the rational expression of the angular frequency, Apply the approximation method to find the coefficient of the rational expression.
And then, it is determined by the following effects can be obtained. (1) Since a rational expression can be applied, equivalent circuit constants of any electric circuit element such as a filter in addition to the above-described FET can be obtained. (2) Since the calculation is performed uniquely, the objectivity of the evaluation is improved, and the reliability in constructing the database is improved. (3) Since there is no need to verify the evaluation results, even unskilled persons who have not performed characterization can perform the work. (4) The process of obtaining information for stabilizing the process can be performed almost in parallel with the measurement of the sample, which is useful for eliminating bottlenecks. (5) Since the improvement of the evaluation accuracy can be dealt with by increasing the order of the rational expression, there is no need to consider floating / parasitic elements.

[Brief description of the drawings]

FIG. 1 is a diagram showing an equivalent circuit of a GaAs FET used in a method for determining an equivalent circuit element constant of an electric circuit according to the present invention.

[Explanation of symbols]

 ZF Basic element part Zg, Zs, Zd Parasitic element part

────────────────────────────────────────────────── ─── Continuation of front page (56) References JP-A-3-105268 (JP, A) JP-A-2-1577674 (JP, A) (58) Fields investigated (Int.Cl. ⁶ , DB name) G01R 31/26 G01R 31/28 H01L 21/66

Claims (2)

(57) [Claims] 1. Measured values of four-terminal parameters of an electric circuit
A method characterized in that the equivalent circuit element constant of the electric circuit is determined by expressing the angular frequency as a rational expression and applying a least-squares approximation method thereto to obtain the coefficient of the rational expression. 2. The method for determining an equivalent circuit element constant of an electric circuit according to claim 1, wherein the value of a parasitic element of the electric circuit is determined by a successive approximation method.