CN104601115A - Sensitivity function based wide-temperature-range filter design method - Google Patents

Abstract

The invention discloses a sensitivity function based wide-temperature-range filter design method. The method comprises performing deduction to obtain a filter transfer function, performing deduction by comparing with a filter standard transfer function to obtain a filter parameter expression, obtaining sensitivity functions of all element parameters to the filter parameter according to the filter parameter expression, and the temperature coefficients of two elements are set to be opposite numbers if the sensitivity functions of two elements are constants and are identical in value; the temperature coefficients of two elements are set to be identical if the sensitivity functions of two elements are constants and are opposite in value; if the sensitivity function of certain element is not a constant and if an element parameter enables the sensitivity expression value to be 0, the element parameter is adjusted, otherwise, no operation is performed. By the aid of the method, the sensitivity function is analyzed, the element parameters are set reasonably, and the filter normal working temperature range is increased.

Description

Based on the wide temperature range filter design method of sensitivity function

Technical field

The invention belongs to wave filter technology field, more specifically say, relate to a kind of wide temperature range filter design method based on sensitivity function.

Background technology

The effect of filter makes the signal of a certain frequency range pass through and suppressed out-of-band signal or decay.For the filter designed, due to the impact of component value tolerance and operational amplifier imperfection, filter real response likely departs from the response of its theory expectation, make real response close to theoretical response although fine setting can be carried out to element, but be subject to the impact of component ageing and thermal drift, depart from or there will be.Usual employing sensitivity function evaluates the sensitivity that filter changes for component value at present.Typical sensitivity function is:

$S_x^y=\frac{\∂y/y}{\∂x/x}=\frac{x}{y}\frac{\∂y}{\∂x}---\left(1\right)$

Wherein, y represents filter parameter, and x represents component value.

As can be seen from formula (1), sensitivity function partial differential illustrates that filter parameter depends on component value situation of change, and for minor variations, formula (1) can be approximately parameters variation Δ y/y caused when can estimate component value changes delta x/x thus.Needing by rationally arranging component parameters in design of filter, making the filter parameter changes delta y/y caused by Δ x/x minimum.

The major parameter of filter has passband gain, center angular frequency ω ₀and quality factor Q, wherein passband gain refers to the signal amplification factor in filter passband, center angular frequency is the frequency that passband (stopband) interior signal gain maximum (minimum) is put, quality factor is for band logical (band resistance) filter, size equals the ratio of center angular frequency and passband (stopband) width B (difference of-3dB cut-off angular frequency), i.e. Q=ω ₀/ B, can find out, Q value is higher, and bandwidth is narrower, the selective power of filter to signal is stronger.For the feature of parameters, filter is also not quite similar to its requirement according to the difference of type.

Component value is produced to two factors departed from: component ageing and thermal drift, wherein component ageing affects the long-time stability of filter, the short-term stability of thermal drift effects filter.Be at present design from element itself or production technology for the improvement mode major part of component ageing performance and thermal drift performance, improve effect and not obvious.

Summary of the invention

The object of the invention is to overcome the deficiencies in the prior art, the wide temperature range filter design method based on sensitivity function is provided, by the analysis of sensitivity function, the parameter value of element is rationally arranged, increase the normal working temperature scope of filter.

For achieving the above object, the present invention is based on the wide temperature range filter design method of sensitivity function, comprise the following steps:

S1: derive and obtain the transfer function of filter;

S2: the transfer function obtained by step S1 and the standard transfer function of filter contrast, derives and obtains the expression formula of filter parameter;

S3: according to filter parameter expression formula, asks for the sensitivity function of each device parameter values to filter parameter;

S4: analyze each sensitivity function, arrange component parameters, comprises following situation:

1) sensitivity function of two elements is constant, and is worth identical, then the temperature coefficient arranging two elements is opposite number;

2) sensitivity function of two elements is constant, and is worth opposite number each other, then the temperature coefficient arranging two elements is identical;

3) the sensitivity function expression formula of certain element is not constant, if there is device parameter values, the value making sensitivity expression formula is 0, so adjusts device parameter values, otherwise does not do any operation.

The present invention is based on the wide temperature range filter design method of sensitivity function, first derive and obtain the transfer function of filter, then the standard transfer function derivation contrasting filter obtains the expression formula of filter parameter, the sensitivity function of each device parameter values to filter parameter is tried to achieve according to filter parameter expression formula, if the sensitivity function of two elements is constant, and being worth identical, then the temperature coefficient arranging two elements is opposite number; If the sensitivity function of two elements is constant, and be worth opposite number each other, then the temperature coefficient arranging two elements is identical; If the sensitivity function expression formula of certain element is not constant, if there is device parameter values, the value making sensitivity expression formula is 0, so adjusts device parameter values, otherwise does not do any operation.

The present invention, by analyzing sensitivity function, is rationally arranged the parameter value of element, and the change that component parameters is produced when thermal migration can not bring impact to filter parameter, thus increases the normal working temperature scope of filter.

Accompanying drawing explanation

Fig. 1 is the embodiment flow chart of the wide temperature range filter design method that the present invention is based on sensitivity function;

Fig. 2 is multiple feedback low-pass filter circuit figure;

Fig. 3 is multiple feedback band pass filter circuit figure.

Embodiment

Below in conjunction with accompanying drawing, the specific embodiment of the present invention is described, so that those skilled in the art understands the present invention better.Requiring particular attention is that, in the following description, when perhaps the detailed description of known function and design can desalinate main contents of the present invention, these are described in and will be left in the basket here.

Fig. 1 is the embodiment flow chart of the wide temperature range filter design method that the present invention is based on sensitivity function.As shown in Figure 1, the wide temperature range filter design method that the present invention is based on sensitivity function comprises the following steps:

S101: the transfer function of derivation filter:

According to correlation theorem, namely " empty short void is broken " principle and Kirchhoff's law, derive and obtain the transfer function of filter.

S102: derivation filter parameter expression formula:

The transfer function obtained by step S101 and the standard transfer function of filter contrast, and derive and obtain the expression formula of filter parameter.

Need the filter parameter considered to be one or more of passband gain, center angular frequency and quality factor under normal circumstances, can determine according to actual needs.

S103: ask for sensitivity function:

According to filter parameter expression formula, ask for the sensitivity function of each device parameter values to filter parameter.

S104: sensitivity for analysis function, arranges component parameters:

Each sensitivity function is analyzed, component parameters is arranged.The invention provides the set-up mode in three kinds of situations, comprising:

1) sensitivity function of two elements is constant, and is worth identical, then the temperature coefficient arranging two elements is opposite number;

2) sensitivity function of two elements is constant, and is worth opposite number each other, then the temperature coefficient arranging two elements is identical;

3) the sensitivity function expression formula of certain element is not constant, if there is device parameter values, the value making sensitivity expression formula is 0, so adjusts device parameter values, otherwise does not do any operation.

If design time consider various filters parameter simultaneously, then determine the priority of parameter as required, when when carrying out optimum configurations, to the optimum configurations of certain element can not reach all filter parameters require time, meet the parameter that priority is high.

Embodiment 1

For low pass and high pass filter, be described for multiple feedback low pass filter in the present embodiment.Fig. 2 is multiple feedback low-pass filter circuit figure.

The transfer function utilizing " empty short void is broken " principle and Kirchhoff's law can obtain embodiment 1 median filter is:

$H_1\left(\mathrm{j\ω}\right)=-\frac{R_{13}}{R_{11}-{\mathrm{\ω}}^2{R}_{11}{R}_{12}{R}_{13}{C}_{11}{C}_{12}+\mathrm{j\ω}(R_{12}{R}_{13}{C}_{12}+R_{11}{R}_{13}{C}_{12}+R_{11}{R}_{12}{C}_{12})}---\left(2\right)$

Low pass filter standard transfer function expression formula is:

$H\left(\mathrm{j\ω}\right)=H_{0\mathrm{LP}}{H}_{\mathrm{LP}}\left(\mathrm{j\ω}\right)=\frac{H_{0\mathrm{LP}}}{1-{(\mathrm{\ω}/{\mathrm{\ω}}_0)}^2+(\mathrm{j\ω}/{\mathrm{\ω}}_0)/Q}---\left(3\right)$

Contrast equation (2) and formula (3), can obtain the parameter passband gain in embodiment 1 center angular frequency and quality factor Q ¹expression formula be:

$H_{0\mathrm{LP}}^1=-\frac{R_{13}}{R_{11}}$

${\mathrm{\ω}}_0^1=\frac{1}{\sqrt{R_{12}{R}_{13}{C}_{11}{C}_{12}}}---\left(4\right)$

$Q^1=\frac{\sqrt{C_{11}/C_{12}}}{\sqrt{R_{12}{R}_{13}}/R_{11}+\sqrt{R_{13}/R_{12}}+\sqrt{R_{12}/R_{13}}}$

Try to achieve each device parameter values pair and Q ¹sensitivity function:

$S_{R_{13}}^{H_{0\mathrm{LP}}^1}=-S_{R_{11}}^{H_{0\mathrm{LP}}^1}=1;$

$S_{R_{11}}^{{\mathrm{\ω}}_0^1}=S_{R_{12}}^{{\mathrm{\ω}}_0^1}=S_{C_{11}}^{{\mathrm{\ω}}_0^1}=S_{C_{12}}^{{\mathrm{\ω}}_0^1}=-1/2;$

$S_{C_{11}}^{Q^1}=-S_{C_{12}}^{Q^1}=1/2;$

$S_{R_{11}}^{Q^1}=-\frac{1}{1+R_{11}/R_{12}+R_{11}/R_{13}};---\left(5\right)$

$S_{R_{12}}^{Q^1}=\frac{1}{2}-\frac{R_{11}/R_{12}}{1+R_{11}/R_{12}+R_{11}/R_{13}};$

$S_{R_{13}}^{Q^1}=\frac{1}{2}-\frac{R_{11}/R_{13}}{1+R_{11}/R_{12}+R_{11}/R_{13}}$

In formula (5), on the right of equation, in result, negative sign represents negative correlation, with for example, this formula represents when resistance R1 increases 1%, and passband gain reduces 1%.

The present embodiment considers three filter parameters when carrying out component parameters configuration simultaneously, therefore needs the priority determining parameter.For low pass filter, usually more pay close attention to passband gain then be center angular frequency successively quality factor Q ¹, therefore first the present embodiment is analyzed when carrying out component parameters and selecting from the sensitivity function result of passband gain.

From passband gain sensitivity function result can find out, resistance R ₁₃change and passband gain change positive correlation, resistance R ₁₁change and passband gain change negative correlation, and degree of correlation is equal, i.e. resistance R ₁₃and R ₁₁be constant to the sensitivity function of passband gain, and be opposite number, as long as therefore the two gets identical temperature coefficient, be namely all positive temperature coefficient or negative temperature coefficient and equal and opposite in direction, can ensure when the temperature is changed, resistance R ₁₃and R ₁₁passband gain variable quantity caused by resistance change is positive and negative to offset, and reaches the optimum stabilization in wide temperature range.

From quality factor sensitivity function result can find out, electric capacity C ₁₁change and quality factor Q ¹change positive correlation, electric capacity C ₁₂change and quality factor Q ¹change negative correlation, and degree of correlation is equal, and thus, the two also gets identical temperature coefficient.

From center angular frequency sensitivity function result can find out, element R ₁₁, R ₁₂, C ₁₁, C ₁₂change all with center angular frequency ω ₀change negative correlation, and degree of correlation is equal, therefore, in order to ensure the stability of center angular frequency at wide temperature range, should have two positive temperature coefficients in four parameters, two negative temperature coefficients, and equal and opposite in direction.Analyze from quality factor sensitivity function, electric capacity C ₁₁with C ₁₂identical temperature coefficient should be got, so, resistance R ₁₁with R ₁₂the temperature coefficient contrary with electric capacity should be got, due to R ₁₃and R ₁₁temperature coefficient identical, thus can obtain electric capacity C ₁₁, C ₁₂get identical positive temperature coefficient, resistance R ₁₁, R ₁₂, R ₁₃get the conclusion of identical negative temperature coefficient, or electric capacity C ₁₁, C ₁₂get identical negative temperature coefficient, resistance R ₁₁, R ₁₂, R ₁₃get the conclusion of identical positive temperature coefficient.The needs of center angular frequency and quality factor can be met so simultaneously.

And for remaining quality factor sensitivity function result because it is not constant, there is not device parameter values yet and be 0, the inventive method therefore cannot be adopted to be configured.

Embodiment 2

For band, logical and band stop filter, is described for multiple feedback band pass filter in the present embodiment.Fig. 3 is multiple feedback band pass filter circuit figure.

Similarly, the transfer function utilizing " empty short void is broken " principle and Kirchhoff's law can obtain embodiment 2 median filter is:

$H_2\left(\mathrm{j\ω}\right)=-\frac{\mathrm{j\ω}{R}_{22}{C}_{22}}{1-{\mathrm{\ω}}^2{R}_{21}{R}_{22}{C}_{21}{C}_{22}+\mathrm{j\ω}{R}_{21}(C_{21}+C_{22})}---\left(6\right)$

Band pass filter standard transfer function expression formula is:

$H\left(\mathrm{j\ω}\right)=H_{0\mathrm{BP}}{H}_{\mathrm{BP}}\left(\mathrm{j\ω}\right)=H_{0\mathrm{BP}}\·\frac{(\mathrm{j\ω}/{\mathrm{\ω}}_0)/Q}{1-{(\mathrm{\ω}/{\mathrm{\ω}}_0)}^2+(\mathrm{j\ω}/{\mathrm{\ω}}_0)/Q}---\left(7\right)$

Contrast equation (6) and formula (7), can obtain the parameter passband gain in embodiment 2 center angular frequency and quality factor Q ²expression formula be:

$H_{0\mathrm{BP}}^2=-\frac{R_{22}/R_{21}}{1+C_{21}/C_{22}}$

${\mathrm{\ω}}_0^2=\frac{1}{\sqrt{R_{22}{R}_{23}{C}_{21}{C}_{22}}}---\left(8\right)$

$Q^2=\frac{\sqrt{R_{22}/R_{21}}}{\sqrt{C_{21}/C_{22}}+\sqrt{C_{22}/C_{21}}}$

Try to achieve each device parameter values pair and Q ²sensitivity function:

$S_{R_{21}}^{H_{0\mathrm{BP}}^2}=-S_{R_{22}}^{H_{0\mathrm{BP}}^2}=-S_{C_{21}}^{H_{0\mathrm{BP}}^2}=S_{C_{22}}^{H_{0\mathrm{BP}}^2}=-1;$

$S_{R_{21}}^{{\mathrm{\ω}}_0^2}=S_{R_{22}}^{{\mathrm{\ω}}_0^2}=S_{C_{21}}^{{\mathrm{\ω}}_0^2}=S_{C_{22}}^{{\mathrm{\ω}}_0^2}=-1/2;$

$S_{R_{21}}^{Q^2}=-S_{R_{22}}^{Q^2}=-1/2;---\left(9\right)$

$S_{C_{21}}^{Q^2}=-S_{C_{22}}^{Q^2}=\frac{1}{2}\·\frac{C_{22}-C_{21}}{C_{22}+C_{21}};$

Owing to considering three filter parameters in the present embodiment simultaneously, need the priority determining parameter.For band pass filter, more the center of interest angular frequency usually then be channel gain successively quality factor Q ², therefore first the present embodiment is analyzed when carrying out component parameters and selecting from the sensitivity function result of center angular frequency.

From the result of center angular frequency sensitivity function , element R ₂₁, R ₂₂, C ₂₁, C ₂₂change all and center angular frequency change negative correlation, and degree of correlation is identical, therefore, in order to ensure the stability of center angular frequency at wide temperature range, should have two positive temperature coefficients in four parameters, two negative temperature coefficients, and equal and opposite in direction.This conclusion is in conjunction with passband gain sensitivity function result , be obviously be unfavorable for the stability of passband gain in wide temperature range, but consider band pass filter this factor of concern degree of priority to parameter, still adopt two positive temperature coefficients, two these conclusions of negative temperature coefficient here.

From the result of quality factor sensitivity function , resistance R ₂₁change and quality factor Q ²change negative correlation, resistance R ₂₂change and quality factor Q ²change positive correlation, and degree of correlation is equal, in order to ensure the stability of quality factor in wide temperature range, resistance R ₂₁with R ₂₂identical temperature coefficient should be got, in conjunction with analysis result above, electric capacity C ₂₁with C ₂₂should get in contrast and equal-sized temperature coefficient.Therefore resistance R can be obtained ₂₁, R ₂₂get identical positive temperature coefficient, electric capacity C ₂₁, C ₂₂get the conclusion of identical negative temperature coefficient, or resistance R ₂₁, R ₂₂get identical negative temperature coefficient, electric capacity C ₂₁, C ₂₂get the conclusion of identical positive temperature coefficient.

And for the result of quality factor sensitivity function this result is not constant, but known as electric capacity C ₂₁=C ₂₂time, now electric capacity C ₂₁with C ₂₂change to quality factor Q ²impact, therefore, should not make C during design of filter ₂₁=C ₂₂.

Although be described the illustrative embodiment of the present invention above; so that those skilled in the art understand the present invention; but should be clear; the invention is not restricted to the scope of embodiment; to those skilled in the art; as long as various change to limit and in the spirit and scope of the present invention determined, these changes are apparent, and all innovation and creation utilizing the present invention to conceive are all at the row of protection in appended claim.

Claims (3)

1., based on a wide temperature range filter design method for sensitivity function, it is characterized in that, comprise the following steps:

S1: derive and obtain the transfer function of filter;

S2: the transfer function obtained by step S1 and the standard transfer function of filter contrast, derives and obtains the expression formula of filter parameter;

S3: according to filter parameter expression formula, asks for the sensitivity function of each device parameter values to filter parameter;

S4: analyze each sensitivity function, arrange component parameters, comprises following situation:

1) sensitivity function of two elements is constant, and is worth identical, then the temperature coefficient arranging two elements is opposite number;

2) sensitivity function of two elements is constant, and is worth opposite number each other, then the temperature coefficient arranging two elements is identical;

3) the sensitivity function expression formula of certain element is not constant, if there is device parameter values, the value making sensitivity function expression formula is 0, so adjusts device parameter values, otherwise does not do any operation.

2. wide temperature filter design method according to claim 1, is characterized in that, described filter parameter is one or more of passband gain, center angular frequency and quality factor.

3. wide temperature range filter design method according to claim 1, it is characterized in that, if consider various filters parameter when designing simultaneously, determine the priority of parameter as required, when in step S3 to the optimum configurations of certain element can not reach all filter parameters require time, meet the parameter that priority is high.