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Bewise Inc. www.tool-tool.com Reference source from the internet.

Un filtre passe-bas est un filtre qui laisse passer les basses fréquences et qui atténue les hautes fréquences, c'est-à-dire les fréquences supérieures à la fréquence de coupure. Il pourrait également être appelé filtre coupe-haut. Le filtre passe-bas est l'inverse du filtre passe-haut et ces deux filtres combinés forment un filtre passe-bande.

Le concept de filtre passe-bas est une transformation mathématique appliquée à des données (un signal). L'implémentation d'un filtre passe-bas peut se faire numériquement ou avec des composantes électroniques. Cette transformation a pour fonction d'atténuer les fréquences supérieures à sa fréquence de coupure fc et ce, dans le but de conserver uniquement les basses fréquences. La fréquence de coupure du filtre est la fréquence séparant les deux modes de fonctionnement idéaux du filtre : passant ou bloquant.

Sommaire

[masquer]

Filtre idéal [modifier]

Un filtre passe-bas idéal a un gain constant dans sa bande passante et un gain nul dans la bande coupée. La transition entre les deux états est instantanée. Mathématiquement, il peut être réalisé en multipliant le signal par une fenêtre rectangulaire dans le domaine fréquentiel ou par une convolution avec un sinus cardinal (sinc) dans le domaine temporel. Ce type de filtre est appelé « mur de brique » dans le jargon des ingénieurs.

Naturellement, un filtre idéal n'est pratiquement pas réalisable, car un sinus cardinal est une fonction infinie. Ainsi, le filtre devrait prédire le futur et avoir une connaissance infinie du passé pour effectuer la convolution et obtenir l'effet désiré. Il est possible d'approximer très fidèlement ce filtre de manière numérique lorsqu'on dispose d'un signal pré-enregistré (en ajoutant des zéros aux deux extrémités de la série d'échantillons) ou pour un signal périodique.

En temps réel, les filtres numériques peuvent approximer ce filtre en insérant un délai volontaire dans le signal, ce qui permet de « connaître le futur du signal ». Cette opération crée un déphasage entre la sortie et l'entrée et naturellement, plus le délai inséré est court, plus le filtre se rapprochera du filtre idéal.

Filtre passe-bas analogique [modifier]

Un filtre passe-bas peut être implémenté de façon analogique avec des composantes électroniques. Ainsi, ce genre de filtre s'applique sur des signaux continus en temps réel. Les composantes et la configuration du circuit fixeront les différentes caractéristiques du filtre, telles que l'ordre, la fréquence de coupure et son diagramme de Bode. Les filtres analogiques classiques sont du premier ou du second ordre. Il existe plusieurs familles de filtres analogiques : Butterworth, Tchebychev, Bessel, elliptique, etc. L'implémentation des filtres de même famille se fait généralement en utilisant la même configuration de circuit, et ceux-ci possèdent la même forme de fonction de transfert, mais ce sont les paramètres de celle-ci qui changent, donc la valeur des composantes du circuit électrique.

Filtre passe-bas du premier ordre [modifier]

Un filtre passe-bas du premier ordre est caractérisé par sa fréquence de coupure fc. La fonction de transfert du filtre est obtenue en dénormalisant le filtre passe-bas normalisé en substituant ωn par ω / ωc, ce qui donne la fonction de transfert suivante :

H(j \omega) = \frac{v_o}{v_i} = \frac {K}{1+j\frac{\omega}{\omega_c}}

où

\displaystyle \omega = 2 \pi f

\displaystyle \omega_c =2 \pi f_c

Le module et la phase de la fonction de transfert égalent à :

|H(\omega)| = \left | \frac{v_o}{v_i} \right |=\frac{K}{\sqrt{1+\big(\frac{\omega}{\omega_c}\big)^2}}

\phi(\omega) = \arg H(j \omega) = - \arg \left ( 1+j\frac{\omega}{\omega_c} \right) = - \arctan \left( \frac{\omega}{\omega_c} \right)

Il y a plusieurs méthodes pour implémenter ce filtre. Une réalisation active et une réalisation passive sont ici présentées. K est le gain du filtre.

Circuit passif [modifier]

La manière la plus simple de réaliser physiquement ce filtre est d'utiliser un circuit RC. Comme son nom l'indique, ce circuit est constitué d'une résistance R et d'un condensateur de capacité C. Ces deux éléments sont placés en série avec la source vi du signal. Le signal de sortie vo est récupéré aux bornes du condensateur. Pour retrouver la fonction de transfert de ce filtre, il faut travailler dans le domaine de Laplace en utilisant les impédances des éléments. Avec cette technique, le circuit devient un simple diviseur de tension, et on obtient :

H(j \omega) = \frac{v_o}{v_i} = \frac {1}{1+jRC\omega}

Dans cette équation, j est un nombre complexe (j tel que j²=-1) et ω est la pulsation du circuit ou fréquence radiale, exprimée en rad/s. Comme la fréquence de coupure d'un circuit RC est :

Un filtre passe-bas analogique d'ordre 1 réalisé avec un circuit RC

f_c = \frac {1}{2\pi RC} ou \omega_c= \frac {1}{RC}

Ici ωc, la pulsation de coupure, est également la pulsation propre ωo du circuit, elle est également l'inverse de la constante de temps τ du circuit (majorée de la constante 2π). Ainsi, on obtient bel et bien la fonction de transfert typique du filtre passe-bas du premier ordre.

Avec cette fonction de transfert, on peut obtenir les diagrammes de Bode :

G_{dB}(\omega) = 20 \cdot \log |H(\omega)| = -10 \cdot \log \left(1+ \big (\omega RC)^2\right)

\phi(\omega) = -\arctan \big(\omega RC)

On distingue alors deux situations idéales :

 

Lieux de Bode du filtre passe-bas passif d'ordre 1

  • Lorsque \omega \ll \omega_c, on a :

G_{dB} \simeq 0 et \phi \simeq 0

(le filtre est passant)

  • Lorsque \omega \gg \omega_c, on a :

G_{dB} \sim -20 \cdot \log \left(\frac{\omega}{\omega_c}\right) et \phi \simeq -90

(le signal est alors filtré)

On remarque que pour ω = ωc, on a GdB = -3 dB.

Circuit actif [modifier]

Il est également possible de réaliser un filtre passe-bas avec un circuit actif. Cette option permet d'ajouter du gain au signal de sortie, c'est-à-dire d'obtenir une amplitude supérieure à 0 dB dans la bande passante. Plusieurs configurations permettent d'implémenter ce genre de filtre.

 

Un filtre passe-bas actif

Dans la configuration présentée ici, la fréquence de coupure se définit comme suit :

f_c = {1 \over 2 \pi R_2 C } ou \omega_\mathrm{c} = \frac{1}{R_2 C}

En utilisant les propriétés des amplificateurs opérationnels, et les impédances des éléments, on obtient la fonction de transfert suivante :

H(j \omega) = \frac{v_o}{v_i} = \frac{-R_2}{R_1} \cdot \frac {1}{1+jR_2C\omega}

En basse fréquence, le condensateur agit comme un circuit ouvert, ce qui est confirmé par le fait que le terme de droite de l'équation précédente tend vers 1. La formule simplifiée ainsi obtenue nous donne le gain dans la bande passante :

H(\omega)_{\omega \ll \omega_c} = \frac{v_o}{v_i} = \frac{-R_2}{R_1}

En haute fréquence, le condensateur agit comme un circuit fermé et le terme de droite tend vers 0, ce qui fait tendre la formule vers zéro.

H(\omega)_{\omega \gg \omega_c} = \frac{v_o}{v_i} \simeq 0

Avec la fonction de transfert, on peut démontrer que l'atténuation dans la bande rejetée est de 20 dB/décade ou de 6 dB par octave telle qu'attendu pour un filtre d'ordre 1.

Il est fréquent de voir un circuit d'amplification ou d'atténuation transformé en filtre passe-bas en ajoutant un condensateur C. Ceci diminue la réponse du circuit à haute fréquence et aide à diminuer les oscillations dans l'amplificateur. Par exemple, un amplificateur audio peut être un filtre passe-bas actif avec une fréquence de coupure de l'ordre de 100 kHz pour réduire le gain à des fréquences qui autrement oscilleraient. Cette modification du signal n'altère pas les informations « utiles » du signal, car la bande audio (bande de fréquence audible par l'humain) s'étend jusqu’à environ 20 kHz, ce qui est largement inclus dans la bande passante du circuit.

Filtre passe-bas du second ordre [modifier]

Un filtre passe-bas du second ordre est caractérisé par sa fréquence de résonance fo et par le facteur de qualité Q. Il est représenté par la fonction de transfert suivante :

H(j \omega) = \frac{v_o}{v_i} = \frac {K}{1-(\frac{\omega}{\omega_0})^2+j\frac{(\frac{\omega}{\omega_0})}{Q}}

où
\omega = 2 \pi f\,
\omega_o =2 \pi f_o\,

Le module et la phase de la fonction de transfert sont donc égaux à :

|H(\omega)| = |\frac{v_o}{v_i}|= \frac{K}{\sqrt{(1-(\frac{\omega}{\omega_0})^2)^2+(\frac{(\frac{\omega}{\omega_0})}{Q})^2}}

\phi(\omega) = - \arctan\left (\frac{\frac{(\frac{\omega}{\omega_0})}{Q}}{1 - (\frac{\omega}{\omega_0})^2}\right )

Circuit passif [modifier]

La manière la plus simple de réaliser physiquement ce filtre est d'utiliser un circuit RLC. Comme son nom l'indique, ce circuit est constitué d'une résistance R, d'un condensateur de capacité C et d'une inductance L. Ces trois éléments sont placés en série avec la source vi du signal. Le signal de sortie vo est récupéré aux bornes du troisième et dernier élément, le condensateur. Pour retrouver la fonction de transfert de ce filtre, il faut travailler dans le domaine de Laplace en utilisant les impédances des éléments. Avec cette technique, le circuit devient un simple diviseur de tension, et on obtient :

H(j \omega) = \frac{v_o}{v_i} = \frac {\frac{-1}{LC}}{\omega^2-j \omega \frac{R}{L}- \frac{1}{LC}}

Avec :

\omega_o = \frac{1}{\sqrt{LC}}

Q = \frac{1}{R} \sqrt{\frac{L}{C}}

Le module et la phase de ce circuit sont :

|H(\omega)| = \left|\frac{v_o}{v_i}\right|=\frac{1}{\sqrt{R^2 C^2 {\omega}^2 + \big(1 - LC{\omega}^2)^2}}

\phi(\omega) = - \arctan\left(-\frac{RC\omega}{1 - LC{\omega}^2}\right)

Circuit actif [modifier]

 

Un filtre passe-bas actif du second ordre.

Plusieurs types de filtres existent pour réaliser un filtre actif du deuxième ordre. Les plus populaires sont les structures MFB et VCVS.

Filtre d'ordre supérieur [modifier]

Les filtres d'ordre supérieur sont généralement composés de filtres d'ordre 1 et 2 en cascade. La réalisation d'un filtre d'ordre 5, par exemple, se fait en plaçant deux filtres d'ordre 2 et un filtre d'ordre 1. Il serait possible de réaliser directement un filtre d'ordre 5, mais la difficulté de conception en serait grandement augmentée.

Filtre passe-bas numérique [modifier]

Voir filtre numérique.

Voir aussi [modifier]

Liens internes [modifier]

Liens externes [modifier]

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Bewise Inc. www.tool-tool.com Reference source from the internet.

Un filtro paso bajo corresponde a un filtro caracterizado por permitir el paso de las frecuencias más bajas y atenuar las frecuencias más altas. El filtro requiere de dos terminales de entrada y dos de salida, de una caja negra, también denominada cuadripolo o bipuerto, así todas las frecuencias se pueden presentar a la entrada, pero a la salida solo estarán presentes las que permita pasar el filtro. De la teoría se obtiene que los filtros están caracterizados por sus funciones de transferencia, así cualquier configuración de elementos activos o pasivos que consigan cierta función de transferencia serán considerados un filtro de cierto tipo.

En particular la función de transferencia de un filtro paso bajo de primer orden corresponde a H(s)=k\frac{1}{1+\frac{s}{\omega_c}} \,\!, donde la constante k \,\! es sólo una ponderación correspondiente a la ganacia del filtro, y la real importancia reside en la forma de la función de transferencia \frac{1}{1+\frac{s}{\omega_c}} \,\!, la cual determina el comportamiento del filtro. En la función de transferencia anterior \omega_c \,\! corresponde a la frecuencia de corte propia del filtro, es decir la frecuencia a partir de la cual el se empieza a atenuar la señal de entrada.

De forma análoga al caso de primer orden, los filtros de pasa bajo de mayor orden también se caracterízan por su función de transferencia, por ejemplo la de un filtro paso bajo de segundo orden corresponde a H(s)=K\frac{\omega_o^2}{s^2+2\xi\omega_os+\omega_o^2} \,\!, donde \omega_o \,\! es la frecuencia natural del filtro y \xi \,\! es el factor de amortiguamiento de este.

Contenido

[ocultar]

Filtro analógico [editar]

Cualquier filtro, tiene una entrada y una salida, así que si hablamos de un filtro pasabajo, lo podemos ver, como una caja negra con dos terminales de entrada y dos de salida. Si un terminal de entrada es común a la salida tendremos un sistema desbalanceado (Unbalance, en inglés), así las cosas, si llamamos e1 y e2 a los terminales de entrada y s1 y s2 a los de salida, un filtro pasabajo sencillo, sería, colocar una resistencia entre e1 y s1 y un condensador, entre s1 y s2, uniendo e2 con s2, tenemos un filtro pasabajo desbalanceado. Ahora veamos como trabaja: las diferentes frecuencias ingresan por e1-e2 y salen por s1-s2, las altas frecuencias verán en el condensador una baja impedancia (cortocircuito) mientras que las bajas frecuencias seguirán de largo por las salidas s1-s2 hacia el circuito siguiente, cumpliendo con la función de dejar pasar las bajas frecuencias y atenuar las altas. Esto a grandes rasgos. También como se describe abajo, se puede usar una bobina, entre e1 y s1 y los terminales e2 y s2 se unen, teniendo así un filtro pasabajo desbalanceado, el cual se rige por XL=WL,donde XL es la reactancia inductiva y w la frecuencia angular y L la inductancia, como se ve abajo del escrito.

El más sencillo está armado en una resistencia y un condensador (o bobina). Pero podría ser mejor. Un filtro analógico elemental compuesto por un capacitor se denomina, "Filtro pasa altos" (debido a que la Reactancia Capacitiva Xc = 1/WC). Mientras que el compuesto por una inductancia (bobina, o choque) es un "filtro pasa bajos" (debido a que la Reactancia Inductiva Xl = WL).

Filtro digital [editar]

La ecuación de un filtro paso bajo digital de primer orden es: y[n]=y[n-1]+\frac {x[n]-y[n-1]}{A} \,\!

Donde A ha de ser mayor que uno. También es llamado filtro promediador, debido a que promedia las muestras de la entrada y por lo tanto suprime variaciones rápidas, característica que le otorga el carácter de paso bajo.

Su transformada Z es: H(Z)=\frac{1}{A}(1+Z^{-1}) \,\!

Ancho de Banda [editar]

Un filtro pasa bandas ideal posee dos espectros uno ubicado en wo y otro en -wo, siendo wo la frecuencia central del filtro, si el mismo posee un ancho de banda b los espectros seria:

-wo-b/2\,\! y -wo+b/2\,\!
wo-b/2\,\! y wo+b/2\,\!

Véase también [editar]

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A low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications.

The concept of a low-pass filter exists in many different forms, including electronic circuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data, acoustic barriers, blurring of images, and so on. Low-pass filters play the same role in signal processing that moving averages do in some other fields, such as finance; both tools provide a smoother form of a signal which removes the short-term oscillations, leaving only the long-term trend.

[edit] Examples of low-pass filters

Figure 1: A low-pass electronic filter realized by an RC circuit

Figure 1 shows a low-pass RC filter for voltage signals, discussed in more detail below. Signal Vout contains frequencies from the input signal, with high frequencies attenuated, but with little attentuation below the cutoff frequency of the filter determined by its RC time constant. For current signals, a similar circuit using a resistor and capacitor in parallel works the same way. See current divider.

[edit] Acoustic

A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated.

[edit] Electronic

Electronic low-pass filters are used to drive subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast.

Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications.

An integrator is another example of a low-pass filter.

DSL splitters use low-pass and high-pass filters to separate DSL and POTS signals sharing the same pair of wires.

Low-pass filters also play a significant role in the sculpting of sound for electronic music as created by analogue synthesisers. See subtractive synthesis.

[edit] Ideal and real filters

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged. The transition region present in practical filters does not exist in an ideal filter. An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with a sinc function in the time domain.

However, the ideal filter is impossible to realize without also having signals of infinite extent, and so generally needs to be approximated for real ongoing signals, because the sinc function's support region extends to all past and future times. The filter would therefore need to have infinite delay, or knowledge of the infinite future and past, in order to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, but even that is not typically practical.

Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a finite impulse response; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future. This delay is manifested as phase shift. Greater accuracy in approximation requires a longer delay.

The Whittaker–Shannon interpolation formula describes how to use a perfect low-pass filter to reconstruct a continuous signal from a sampled digital signal. Real digital-to-analog converters use real filter approximations.

[edit] Continuous-time low-pass filters

 

The gain-magnitude frequency response of a first-order (one-pole) low-pass filter. Power gain is shown in decibels (i.e., a -3 dB decline reflects an additional half-power attenuation). Angular frequency is shown on a logarithmic scale in units of radians per second.

There are a great many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot, and the filter is characterized by its cutoff frequency and rate of frequency rolloff. In all cases, at the cutoff frequency, the filter attenuates the input power by half or -3 dB. So the order of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.

  • A first-order filter, for example, will reduce the signal amplitude by half (so power reduces by –6 dB) every time the frequency doubles (goes up one octave); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this one-pole–one-zero filter is still a first-order low-pass. See Pole–zero plot and RC circuit.
  • A second-order filter attenuates higher frequencies more steeply. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles (so power decreases by –12 dB per octave, or –40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of –12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See RLC circuit.
  • Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-n all-pole filter is 6n dB per octave (i.e., 20n dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they will intersect at exactly the "cutoff frequency". The frequency response at the cutoff frequency in a first-order filter is –3 dB below the horizontal line. The various types of filters — Butterworth filter, Chebyshev filter, Bessel filter, etc. — all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. See electronic filter for other types.

The meanings of 'low' and 'high' — that is, the cutoff frequency — depend on the characteristics of the filter. The term "low-pass filter" merely refers to the shape of the filter's response; a high-pass filter could be built that cuts off at a lower frequency than any low-pass filter – it is their responses that set them apart. Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1000 MHz) and higher.

[edit] Laplace notation

Continuous-time filters can also be described in terms of the Laplace transform of their impulse response in a way that allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of the Laplace transform in the complex plane (in discrete time, one can similarly consider the Z-transform of the impulse response).

For example, a first-order low-pass filter can be described in Laplace notation as

\frac{\text{Output}}{\text{Input}} = K \frac{1}{1 + s \tau}

where s is the Laplace transform variable, τ is the filter time constant, and K is the filter passband gain.

[edit] Electronic low-pass filters

[edit] Passive electronic realization

 

Passive, first order low-pass RC filter

One simple electrical circuit that will serve as a low-pass filter consists of a resistor in series with a load, and a capacitor in parallel with the load. The capacitor exhibits reactance, and blocks low-frequency signals, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The combination of resistance and capacitance gives you the time constant of the filter τ = RC (represented by the Greek letter tau). The break frequency, also called the turnover frequency or cutoff frequency (in hertz), is determined by the time constant:

f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}

or equivalently (in radians per second):

\omega_\mathrm{c} = {1 \over \tau} = { 1 \over R C}.

One way to understand this circuit is to focus on the time the capacitor takes to charge. It takes time to charge or discharge the capacitor through that resistor:

  • At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.
  • At high frequencies, the capacitor only has time to charge up a small amount before the input switches direction. The output goes up and down only a small fraction of the amount the input goes up and down. At double the frequency, there's only time for it to charge up half the amount.

Another way to understand this circuit is with the idea of reactance at a particular frequency:

  • Since DC cannot flow through the capacitor, DC input must "flow out" the path marked Vout (analogous to removing the capacitor).
  • Since AC flows very well through the capacitor — almost as well as it flows through solid wire — AC input "flows out" through the capacitor, effectively short circuiting to ground (analogous to replacing the capacitor with just a wire).

It should be noted that the capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the Bode plot and frequency response that show this variability.

[edit] Active electronic realization

 

An active low-pass filter

Another type of electrical circuit is an active low-pass filter.

In the operational amplifier circuit shown in the figure, the cutoff frequency (in hertz) is defined as:

f_{\text{c}} = \frac{1}{2 \pi R_2 C}

or equivalently (in radians per second):

\omega_{\text{c}} = \frac{1}{R_2 C}

The gain in the passband is \frac{-R_2}{R_1}, and the stopband drops off at −6 dB per octave as it is a first-order filter.

Sometimes, a simple gain amplifier (as opposed to the very-high-gain operation amplifier) is turned into a low-pass filter by simply adding a feedback capacitor C. This feedback decreases the frequency response at high frequencies via the Miller effect, and helps to avoid oscillation in the amplifier. For example, an audio amplifier can be made into a low-pass filter with cutoff frequency 100 kHz to reduce gain at frequencies which would otherwise oscillate. Since the audio band (what we can hear) only goes up to 20 kHz or so, the frequencies of interest fall entirely in the passband, and the amplifier behaves the same way as far as audio is concerned.

[edit] Discrete-time realization

For another method of conversion from continuous- to discrete-time, see Bilinear transform.

The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model.

A simple low-pass RC filter

From the circuit diagram to the right, according to Kirchoff's Laws and the definition of capacitance:

\begin{cases} v_{\text{in}}(t) - v_{\text{out}}(t) = R \, i(t)&\text{(V)}\ Q_c(t) = C \, v_{\text{out}}(t)&\text{(Q)}\ i(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t} &\text{(I)} \end{cases}

where Qc(t) is the charge stored in the capacitor at time t. Substituting Equation (Q) into Equation (I) gives i(t) = C \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}, which can be substituted into Equation (V) so that:

v_{\text{in}}(t) - v_{\text{out}}(t) = RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}\,

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by ΔT time. Let the samples of vin be represented by the sequence (x_1, x_2, \ldots, x_n), and let vout be represented by the sequence (y_1, y_2, \ldots, y_n) which correspond to the same points in time. Making these substitutions:

x_i - y_i = RC \, \frac{y_{i}-y_{i-1}}{\Delta_T}\,

And rearranging terms gives the recurrence relation

y_i = \overbrace{x_i \left( \frac{\Delta_T}{RC + \Delta_T} \right)}^{\text{Input contribution}} + \overbrace{y_{i-1} \left( \frac{RC}{RC + \Delta_T} \right)}^{\text{Inertia from previous output}}

That is, this discrete-time implementation of a simple RC low-pass filter is the exponentially-weighted moving average

y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha \triangleq \frac{\Delta_T}{RC + \Delta_T}\,

By definition, the smoothing factor 0 \leq \alpha \leq 1. The expression for α yields the equivalent time constant RC in terms of the sampling period ΔT and smoothing factor α:

RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right)

If α = 0.5, then the RC time constant equal to the sampling period. If \alpha \ll 0.5, then RC is significantly larger than the sampling interval, and \Delta_T \approx \alpha RC.

[edit] Algorithmic implementation

The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following pseudocode algorithm will simulate the effect of a low-pass filter on a series of digital samples:

 // Return RC low-pass filter output samples, given input samples,
 // time interval dt, and time constant RC
 function lowpass(real[0..n] x, real dt, real RC)
   var real[0..n] y
   var real α := dt / (RC + dt)
       y[0] := x[0]
   for i from 1 to n
       y[i] := α * x[i] + (1-α) * y[i-1]    
   return y

The loop which calculates each of the n outputs can be refactored into the equivalent:

   for i from 1 to n
       y[i] := y[i-1] + α * (x[i] - y[i-1])

That is, the change from one filter output to the next is proportional to the difference between the previous output and the next input. This exponential smoothing property matches the exponential decay seen in the continuous-time system. As expected, as the time constant RC increases, the discrete-time smoothing parameter α decreases, and the output samples (y_1,y_2,\ldots,y_n) respond more slowly to a change in the input samples (x_1,x_2,\ldots,x_n) – the system will have more inertia.

[edit] See also


Electronics portal

[edit] External links

歡迎來到Bewise Inc.的世界，首先恭喜您來到這接受新的資訊讓產業更有競爭力，我們是提供專業刀具製造商，應對客戶高品質的刀具需求，我們可以協助客戶滿足您對產業的不同要求，我們有能力達到非常卓越的客戶需求品質，這是現有相關技術無法比擬的，我們成功的滿足了各行各業的要求，包括：精密HSS DIN切削刀具、協助客戶設計刀具流程、DIN or JIS 鎢鋼切削刀具設計、NAS986 NAS965 NAS897 NAS937orNAS907 航太切削刀具,NAS航太刀具設計、超高硬度的切削刀具、醫療配件刀具設計、複合式再研磨機、PCD地板專用企口鑽石組合刀具、NSK高數主軸與馬達、專業模具修補工具-氣動與電動、粉末造粒成型機、主機版專用頂級電桿、PCD V-Cut刀、捨棄式圓鋸片組、粉末成型機、主機版專用頂級電感、’汽車業刀具設計、電子產業鑽石刀具、木工產業鑽石刀具、銑刀與切斷複合再研磨機、銑刀與鑽頭複合再研磨機、銑刀與螺絲攻複合再研磨機等等。我們的產品涵蓋了從民生刀具到工業級的刀具設計；從微細刀具到大型刀具；從小型生產到大型量產；全自動整合；我們的技術可提供您連續生產的效能，我們整體的服務及卓越的技術，恭迎您親自體驗！！

BW Bewise Inc. Willy Chen willy@tool-tool.com bw@tool-tool.com www.tool-tool.com skype:willy_chen_bw mobile:0937-618-190 Head &Administration Office No.13,Shiang Shang 2nd St., West Chiu Taichung,Taiwan 40356 http://www.tool-tool.com / FAX:+886 4 2471 4839 N.Branch 5F,No.460,Fu Shin North Rd.,Taipei,Taiwan S.Branch No.24,Sec.1,Chia Pu East Rd.,Taipao City,Chiayi Hsien,Taiwan

Welcome to BW tool world! We are an experienced tool maker specialized in cutting tools. We focus on what you need and endeavor to research the best cutter to satisfy users’ demand. Our customers involve wide range of industries, like mold & die, aerospace, electronic, machinery, etc. We are professional expert in cutting field. We would like to solve every problem from you. Please feel free to contact us, its our pleasure to serve for you. BW product including: cutting tool、aerospace tool .HSS  DIN Cutting tool、Carbide end mills、Carbide cutting tool、NAS Cutting tool、NAS986 NAS965 NAS897 NAS937orNAS907 Cutting Tools,Carbide end mill、disc milling cutter,Aerospace cutting tool、hss drill’Фрезеры’Carbide drill、High speed steel、Compound Sharpener’Milling cutter、INDUCTORS FOR PCD’CVDD(Chemical Vapor Deposition Diamond )’PCBN (Polycrystalline Cubic Boron Nitride) ’Core drill、Tapered end mills、CVD Diamond Tools Inserts’PCD Edge-Beveling Cutter(Golden Finger’PCD V-Cutter’PCD Wood tools’PCD Cutting tools’PCD Circular Saw Blade’PVDD End Mills’diamond tool. INDUCTORS FOR PCD . POWDER FORMING MACHINE ‘Single Crystal Diamond ‘Metric end mills、Miniature end mills、Специальные режущие инструменты ‘Пустотелое сверло ‘Pilot reamer、Fraises’Fresas con mango’ PCD (Polycrystalline diamond) ‘Frese’POWDER FORMING MACHINE’Electronics cutter、Step drill、Metal cutting saw、Double margin drill、Gun barrel、Angle milling cutter、Carbide burrs、Carbide tipped cutter、Chamfering tool、IC card engraving cutter、Side cutter、Staple Cutter’PCD diamond cutter specialized in grooving floors’V-Cut PCD Circular Diamond Tipped Saw Blade with Indexable Insert’ PCD Diamond Tool’ Saw Blade with Indexable Insert’NAS tool、DIN or JIS tool、Special tool、Metal slitting saws、Shell end mills、Side and face milling cutters、Side chip clearance saws、Long end mills’end mill grinder’drill grinder’sharpener、Stub roughing end mills、Dovetail milling cutters、Carbide slot drills、Carbide torus cutters、Angel carbide end mills、Carbide torus cutters、Carbide ball-nosed slot drills、Mould cutter、Tool manufacturer.

Bewise Inc. www.tool-tool.com

ようこそＢｅｗｉｓｅ　Ｉｎｃ.の世界へお越し下さいませ、先ず御目出度たいのは新たな

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BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web  www.tool-tool.com  for more info.

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Als Tiefpass bezeichnet man in der Elektronik Filter, die Signalanteile mit Frequenzen unterhalb ihrer Grenzfrequenz annähernd ungeschwächt passieren lassen, Anteile mit höheren Frequenzen dagegen abschwächen.

Inhaltsverzeichnis

[Verbergen]

Tiefpass 1. Ordnung [Bearbeiten]

Im einfachsten Fall besteht ein Tiefpass aus einer Widerstand-Kondensator-Kombination (RC-Glied). Ein solcher passiver Tiefpass 1. Ordnung sieht folgendermaßen aus:

 

Amplitudenverlauf Tiefpass 1. Ordnung

 

Spannung VC an der Kapazität als Funktion der Zeit

Einfacher RC-Tiefpass (Tiefpass 1.Ordnung)

Von der Eingangsspannung U_e\ erscheint am Ausgang gemäß der Spannungsteilerformel nur der Anteil U_a\ :

U_a = U_e \cdot \frac{\vert X_C \vert}{\sqrt{X_C^2 + R^2}} = U_e \cdot \frac {1} {\sqrt{ 1 + (\omega CR)^2}}

wobei U_a\ und U_e\ die Beträge der Aus- bzw. Eingangsspannung bezeichnen.

Unter der Grenzfrequenz fc (cutoff frequency) versteht man diejenige Frequenz, bei der U_{a} = U_{e}/\sqrt{2}\approx U_{e} \cdot 0{,}707
(d. h. Ua gegenüber Ue um etwa 3 Dezibel abgeschwächt) ist. Die Grenzfrequenz berechnet sich zu f_c=\frac{1}{2\,\pi\,R\cdot C}

Da XC, der Blindwiderstand des Kondensators mit steigender Frequenz kleiner wird,

X_C = \frac{-1}{\omega\,C} mit \omega = 2\,\pi\,f,

geht das Teilungsverhältnis mit sinkender Frequenz gegen 1, für Gleichspannung (Frequenz f = 0) wird U_a = U_e\ .

In logarithmischer Darstellung nimmt die Dämpfung oberhalb der Grenzfrequenz um 6 dB/Oktave bzw. 20 dB/Dekade zu.

 

Aktiver Tiefpass

Mit Operationsverstärkern können aktive Tiefpässe realisiert werden. Diese haben den Vorteil, dass der Frequenzgang unabhängig von der am Ausgang angeschlossenen Last ist. Der Betrag der Ausgangsspannung dieses Tiefpasses ist

U_a = -U_e \cdot \frac {R_2} {R_1} \cdot \frac{\vert X_C \vert}{\sqrt{X_C^2 + R_2^2}}\ .

Die Grenzfrequenz ist hier die Frequenz, bei der die Verstärkung auf das 1/\sqrt{2}-fache der Gleichspannungsverstärkung (R2 / R1) abgefallen ist.

Herleitung der Formel [Bearbeiten]

laut Spannungsteiler gilt:

\frac {\underline {u_a}(t)}{\underline {u_e}(t)} = \frac{\underline {Z_c}}{\underline {Z_c} + R} \ = \frac{\frac{1}{{\mathrm j}{\omega}{C}}}{\frac{1}{{\mathrm j}{\omega}{C}} + R}= \frac{1}{1 + {\mathrm j}{\omega}{C}{R}} \qquad \text{mit} \qquad j^2=-1

mit \underline {Z_c} = \frac{1}{{\mathrm j}{\omega}{C}} = Widerstandsoperator bzw. Impedanz des Kondensators

komplexe Umformung [Bearbeiten]
z = x + \mathrm j y = 1 + {\mathrm j}{\omega}{C}{R} \,
z = r \mathrm e^{{\mathrm j}{\varphi}} \Rightarrow r = \sqrt{x^2 + y^2} \Rightarrow {\varphi} = \arctan\left(\frac yx\right)
z = \sqrt{1 + ({\omega}{C}{R})^2} \cdot {\mathrm e^{{\mathrm j} \cdot {\arctan({\omega}{C}{R})}}}
\frac {\underline {u_a}(t)}{\underline {u_e}(t)} = \frac {1} {\sqrt{ 1 + (\omega CR)^2}} \cdot {\mathrm e^{-{\mathrm j} \cdot {\arctan({\omega}{C}{R})}}}

diese Gleichung stellt die normierte Spannungsortskurve dar (Bildbereich)

Rücktransformation [Bearbeiten]

(nach den Transformationsregeln für harmonische Schwingungen)

a = {\hat a} \sin({\omega}t + {\varphi}) \Leftrightarrow \underline {a} = {\hat a} \cdot {e^{{\mathrm j} \cdot ({{\omega}{t} + {\varphi}})}}
\frac {u_a(t)}{u_e(t)} = \frac {1} {\sqrt{ 1 + (\omega CR)^2}} \cdot \sin(- \arctan({ \omega}{C}{R}))

Gleichungen [Bearbeiten]

Amplitudengang

H(\omega) = \frac {\hat u_a}{\hat u_e} = \frac {1} {\sqrt{ 1 + (\omega CR)^2}}

Phasengang

{\varphi}(\omega) = {-\arctan({\omega}{C}{R})}

Gleichung zur Darstellung im Bode-Diagramm [Bearbeiten]

(Amplitudenverlauf Tiefpass 1. Ordnung)

H_\mathrm{dB}(\omega) = 20 \lg \frac {1} {\sqrt{ 1 + (\omega CR)^2}}

Umformung zur hier verwendeten Formel [Bearbeiten]

(mit Blindwiderstand)

\frac {1} {\sqrt{ 1 + (\omega CR)^2}}=\frac {1} {\sqrt{({\omega C})^2 ((\frac {1}{\omega C})^2 + R^2)}}=\frac {(\frac {1}{\omega C})} {\sqrt{ ((\frac {1}{\omega C})^2 + R^2)}}=\frac{\vert X_C \vert}{\sqrt{X_C^2 + R^2}}

Tiefpass 2. Ordnung [Bearbeiten]

Einen Tiefpass zweiter Ordnung erhält man, indem man R durch eine Reihenschaltung von R mit einer Induktivität L ersetzt, da diese ihrerseits eine – und zwar zum Kondensator gegenläufige – Frequenzabhängigkeit besitzt. Dabei wird R so groß gewählt, dass keine oder nur eine geringe Resonanzüberhöhung des Frequenzgangs entsteht.

 

Passiver Tiefpass 2. Ordnung

 

Aktiver Tiefpass 2. Ordnung

Die Übertragungsfunktion eines solchen Tiefpasses ist

H(\omega)=\frac{\mathrm j\, X_C}{R+\mathrm j(X_L+X_C)}
mit X_L=\omega \, L, \quad X_C=\frac{-1}{\omega \, C}\, , \quad \omega = 2\,\pi\,f.

Der Betrag der Übertragungsfunktion ist

\begin{align} \frac{U_a}{U_e} &= \vert H(\omega) \vert = \frac{\vert X_C \vert}{\sqrt{R^2+(X_L+X_C)^2}}\ &=\sqrt{\frac{1}{\omega^4 \, L^2 \, C^2 + \omega^2 \, R^2 \, C^2 - 2 \, \omega^2 \, L \, C + 1 }} \end{align}

Damit fällt die Ausgangsspannung Ua oberhalb von fG schneller (mit 12 dB/Oktave bzw. 40 dB/Dekade) ab, da nun nicht nur |XC| kleiner sondern zugleich |XL| größer wird.

In dieser Variante werden im Niederfrequenzbereich große Induktivitäten gebraucht (bis zu mehreren Henry). Diese haben schlechte elektrische Eigenschaften und besitzen recht große Dimensionen. Deshalb werden Tiefpässe zweiter und höherer Ordnung heutzutage üblicherweise durch Operationsverstärker-Schaltungen realisiert. Diese Filter werden als aktive Tiefpässe (bzw. aktive Filter) bezeichnet und sind nach ihren Erfindern auch als Sallen-Key-Filter bekannt.

Im Hochfrequenzbereich, beispielsweise beim Bau von Sendeanlagen ist R immer Null, um Wärmeverluste zu vermeiden. Diese Schaltung wird aus zwei Gründen verwendet:

Tiefpass n-ter Ordnung [Bearbeiten]

Durch das Hintereinanderschalten von mehreren Tiefpässen kann man dessen Ordnung erhöhen, beispielsweise bilden zwei hintereinandergeschaltete Tiefpässe 2. Ordnung einen Tiefpass 4. Ordnung. Die Dämpfung ändert sich hierbei oberhalb der Grenzfrequenz mit 4•20 dB/Dekade = 80 dB/Dekade, was einer Flankensteilheit von 24 dB/Oktave entspricht.

Zwei zusammengeschaltete Tiefpässe mit gleicher Grenzfrequenz ergeben aber keinen Tiefpass höherer Ordnung derselben Grenzfrequenz. Für die Dimensionierung eines Tiefpasses mit gewünschter Grenzfrequenz stehen spezielle Formeln und Tabellen zur Verfügung.

Zusätzlich tritt das Problem auf, dass ein Tiefpass in einer Kette vom Ausgangswiderstand des vorgeschaltenen und dem Eingangswiderstand des nachgeschalteten Tiefpasses beeinflusst wird. Diesem Effekt kann mit Impedanzwandlern entgegengewirkt werden.

Allgemein werden für ein Filter n-ter Ordnung n speichernde Elemente (also Kondensatoren oder Spulen) benötigt.

Die Dämpfung eines Tiefpasses n-ter Ordnung nimmt oberhalb der Grenzfrequenz mit n·20 dB/Dekade zu.

Emphasis und Deemphasis [Bearbeiten]

Bei der statischen Frequenzgangveränderung, der Emphasis und der Deemphasis wird anstatt der Grenzfrequenz üblicherweise die Zeitkonstante angegeben [1].

Anwendung [Bearbeiten]

Tiefpässe für schwache Signale können entweder in analoger Technik aus Widerständen, Spulen und Kondensatoren aufgebaut sein und zur Verbesserung ihrer Filterfunktion aktive Bauelemente wie Transistoren enthalten. Sie können aber auch – wie andere Filterfunktionen – durch Computerprogramme nachgebildet werden und spielen in der digitalen Signalverarbeitung und in der Computergrafik (Anti-Aliasing) eine Rolle.

Tiefpässe für starke Signale und für Leistungsübertragung können sinnvoll nur in analoger Technik aus Kondensatoren und Spulen aufgebaut werden. Hauptanwendung ist die Hochfrequenztechnik, man findet sie auch an den Lastausgängen von Frequenzumrichtern, Klasse-D-Verstärkern, Schaltnetzteilen und in Netzfiltern.

Tiefpass-Filter in der Niederfrequenztechnik werden anwendungsbezogen auch als Höhensperre, Höhenfilter, Treble-Cut-Filter, High-Cut-Filter, oder Rauschfilter bezeichnet. Diese Begriffe sind in der Tontechnik gebräuchlich; sie weisen darauf hin, dass ein solches Filter, zum Beispiel in einem Equalizer, die „Höhen“ des Signals bzw. das Rauschen abschwächt, das vorwiegend hohe Frequenzen enthält; siehe auch Entzerrung (Tontechnik).

Tiefpassfunktionen kommen auch in der Mechanik (Schwingungsdämpfung), Akustik (die Schallausbreitung tiefer Frequenzen ist verlustärmer), Optik (Kantenfilter), Hydraulik oder der Lichtausbreitung in der Atmosphäre vor, werden dort jedoch nicht so genannt.

Eine Sonderstellung eines Tiefpassfilters nimmt der ideale Tiefpass ein. Dieser Tiefpass weist eine nicht kausale Übertragungsfunktion auf und kann daher in Praxis nicht realisiert werden. Er dient wegen seiner einfachen Übertragungsfunktion in der Filtertheorie als vereinfachtes Modell. Reale Tiefpässe können sich nur möglichst gut der Eigenschaft des idealen Tiefpasses annähern.

Literatur [Bearbeiten]

  • Ulrich Tietze, Christoph Schenk und Eberhard Gamm: Halbleiter-Schaltungstechnik. Springer-Verlag, 2002, 12. Auflage, ISBN 3-540-42849-6

Siehe auch [Bearbeiten]

Weblinks [Bearbeiten]

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情報を受け取って頂き、もっと各産業に競争力プラス展開。

弊社は専門なエンド・ミルの製造メーカーで、客先に色んな分野のニーズ、

豊富なパリエーションを満足させ、特にハイテク品質要求にサポート致します。

弊社は各領域に供給できる内容は：

（１）精密ＨＳＳエンド・ミルのＲ＆Ｄ

（２）Carbide Cutting tools設計

（３）鎢鋼エンド・ミル設計

（４）航空エンド・ミル設計

（５）超高硬度エンド・ミル

（６）ダイヤモンド・エンド・ミル

（７）医療用品エンド・ミル設計

（８）自動車部品＆材料加工向けエンド・ミル設計

弊社の製品の供給調達機能は：

（１）生活産業～ハイテク工業までのエンド・ミル設計

（２）ミクロ・エンド・ミル～大型エンド・ミル供給

（３）小Ｌｏｔ生産～大量発注対応供給

（４）オートメーション整備調達

（５）スポット対応～流れ生産対応

弊社の全般供給体制及び技術自慢の総合専門製造メーカーに貴方のご体験を御待ちしております。

Bewise Inc. talaşlı imalat sanayinde en fazla kullanılan ve üç eksende (x,y,z) talaş kaldırabilen freze takımlarından olan Parmak Freze imalatçısıdır. Çok geniş ürün yelpazesine sahip olan firmanın başlıca ürünlerini Karbür Parmak Frezeler, Kalıpçı Frezeleri, Kaba Talaş Frezeleri, Konik Alın Frezeler, Köşe Radyüs Frezeler, İki Ağızlı Kısa ve Uzun Küresel Frezeler, İç Bükey Frezeler vb. şeklinde sıralayabiliriz.

BW специализируется в научных исследованиях и разработках, и снабжаем самым высокотехнологичным карбидовым материалом для поставки режущих / фрезеровочных инструментов для почвы, воздушного пространства и электронной индустрии. В нашу основную продукцию входит твердый карбид / быстрорежущая сталь, а также двигатели, микроэлектрические дрели, IC картонорезальные машины, фрезы для гравирования, режущие пилы, фрезеры-расширители, фрезеры-расширители с резцом, дрели, резаки форм для шлицевого вала / звездочки роликовой цепи, и специальные нано инструменты. Пожалуйста, посетите сайт  www.tool-tool.com  для получения большей информации.

BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web  www.tool-tool.com  for more info.

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Et lavpasled er en sammenstilling af en kondensator og en modstand, som lader vekselspændinger med lave frekvenser passere næsten uhindret, mens spændinger med højere frekvenser bliver dæmpet - dette kaldes også for efterbetoning. Lavpasleddet kan sammenlignes med en spændingsdeler sammensat af en modstand og en kondensator; da kondensatorens impedans varierer med frekvensen, vil spændingsforholdet afhænge af signalets frekvens.

Indholdsfortegnelse

[skjul]

[redigér] Sådan virker lavpasleddet

 

Diagram, kurveformer, Bode-diagram og phasordiagram for et lavpasled

I feltet øverst til venstre på illustrationen til højre ses diagrammet for et lavpasled: Det består af modstanden R og kondensatoren C, koblet i serie. Indgangssignalet, i form af spændingen u indføres på terminalerne til venstre i diagrammet, og udgangssignalet "tappes" over kondensatoren i diagrammets højre side.

Øverst til højre på illustrationen ses hvad der "sker" med et sinusformet signal i lavpasleddet: Omkring de tidspunkter hvor indgangssignalet nærmer sig en "top" eller en "bund", søger strømmen i i kredsløbet at op- eller aflade kondensatoren. Hvis signalet har en høj frekvens, dvs. skifter retning hurtigt, "når" kondensatoren ikke at blive ladet til særlige store spændinger. Ved lavere frekvenser får kondensatoren derimod bedre "tid" til at nå op i nærheden af indgangssignalets spænding. Den lavere spænding over kondensatoren ved højere frekvenser skyldes, at kondensatorens reaktans (vekselstrømsmodstand) bliver mindre ved stigende frekvens. Spændingsfalder falder derfor over C, men bliver tilsvarende større over R, så der til sidst ligger hele indgangssignalet over R.

Det ses at der opstår en vis "forsinkelse", eller fasedrejning, benævnt θ, mellem ind- og udgangssignalet: Denne fasevinkel kan være alt mellem en anelse over 0, til lige knap 90 grader, og er størst for høje frekvenser, dvs. når leddet dæmper signalet kraftigt.
Da udgangssignalet er "bagud" i forhold til indgangssignalet, ser man ofte denne vinkel angivet med negativt fortegn, altså som et tal mellem −90 og 0.

Lavpass ledet giver 90 grader pr pol. Det viste lavpass led er af 1 orden og vil derfor giver en fasedrejning på -90 grader. Ved 3db knækfrekvensen vil fasedrejningen være -45 grader,og 1 dekade før vil den være 0grader,og en dekade efter være -90 grader.

[redigér] Frekvensgang og overgangsspænding

Nederst til venstre på illustrationen ses et såkaldt Bode-diagram, som viser hvor stor udgangssignalets amplitude er i forhold til indgangssignalets ditto: Indtil en vis frekvens f0, kaldet overgangsfrekvensen eller grænsefrekvensen, "slipper" signalet igennem ved næsten fuld styrke. Ved frekvenser over overgangsfrekvensen dæmpes signalet gradvist mere og mere efterhånden som frekvensen stiger. For frekvenser et godt stykke over overgangsfrekvensen gælder mere præcist, at for hver gang frekvensen fordobles, "taber" udgangssignalet yderligere 6 decibel i styrke, svarende til 20 dB hvis frekvensen stiger til det 10-dobbelte.

Overgangsfrekvensen defineres som det sted hvor signalet dæmpes til \sqrt{\frac{1}{2}} af sin oprindelige styrke, svarende til ca. 3 dB. Hvis modstandens værdi er R og kondensatorens C, kan man beregne overgangsfrekvensen med denne formel:

f_o = \frac{1}{2 \cdot \pi\ \cdot R \cdot C}

Når leddet arbejder ved lige netop overgangsfrekvensen, er impedansen i kondensatoren netop lige så stor som den rent ohmske modstand i modstanden: I den situation svarer leddet jo til en spændingsdeler med to lige store "modstande".

Man kan beregne føromtalte fasevinkel ved en given frekvens fi forhold til overgangsfrekvensen f0, idét trigonometrien giver det simple forhold mellem en vinkel i en retvinklet trekant og de to kateter:

\tan \theta\ = \frac{f}{f_0}

[redigér] Phasordiagram og faseforhold

Nederst til højre på illustrationen ses et phasordiagram for lavpasleddet: Da modstandens størrelse er et reelt tal og kondensatorens impedans et imaginært, bliver summen af spændingerne over komponenterne et komplekst tal. Et sted i den komplekse plan findes phasoren for indgangsspændingen, som ifølge Kirchhoffs spændingslov skal være summen af de to seriekoblede komponenter i leddet: Afhængigt af frekvensen, og dermed kondensatorens reaktans, danner denne phasor en vis vinkel &theta i forhold til phasoren for spændingen over kondensatoren, som jo samtidig er udgangsspændingen fra lavpasleddet.

Da phasorerne for spændingerne over kondensatoren og modstanden står vinkelret på hinanden, giver den pythagoræiske læresætning forholdet:

u^2 = {u'}^2+u_R^2

Ved hjælp af trigonometri kan man desuden bl.a. finde følgende formel for fasedrejningen i leddet:

\cos \theta\ = \frac{u^\prime}{u}

[redigér] Se også

歡迎來到Bewise Inc.的世界，首先恭喜您來到這接受新的資訊讓產業更有競爭力，我們是提供專業刀具製造商，應對客戶高品質的刀具需求，我們可以協助客戶滿足您對產業的不同要求，我們有能力達到非常卓越的客戶需求品質，這是現有相關技術無法比擬的，我們成功的滿足了各行各業的要求，包括：精密HSS DIN切削刀具、協助客戶設計刀具流程、DIN or JIS 鎢鋼切削刀具設計、NAS986 NAS965 NAS897 NAS937orNAS907 航太切削刀具,NAS航太刀具設計、超高硬度的切削刀具、醫療配件刀具設計、複合式再研磨機、PCD地板專用企口鑽石組合刀具、NSK高數主軸與馬達、專業模具修補工具-氣動與電動、粉末造粒成型機、主機版專用頂級電桿、PCD V-Cut刀、捨棄式圓鋸片組、粉末成型機、主機版專用頂級電感、’汽車業刀具設計、電子產業鑽石刀具、木工產業鑽石刀具、銑刀與切斷複合再研磨機、銑刀與鑽頭複合再研磨機、銑刀與螺絲攻複合再研磨機等等。我們的產品涵蓋了從民生刀具到工業級的刀具設計；從微細刀具到大型刀具；從小型生產到大型量產；全自動整合；我們的技術可提供您連續生產的效能，我們整體的服務及卓越的技術，恭迎您親自體驗！！

BW Bewise Inc. Willy Chen willy@tool-tool.com bw@tool-tool.com www.tool-tool.com skype:willy_chen_bw mobile:0937-618-190 Head &Administration Office No.13,Shiang Shang 2nd St., West Chiu Taichung,Taiwan 40356 http://www.tool-tool.com / FAX:+886 4 2471 4839 N.Branch 5F,No.460,Fu Shin North Rd.,Taipei,Taiwan S.Branch No.24,Sec.1,Chia Pu East Rd.,Taipao City,Chiayi Hsien,Taiwan

Welcome to BW tool world! We are an experienced tool maker specialized in cutting tools. We focus on what you need and endeavor to research the best cutter to satisfy users’ demand. Our customers involve wide range of industries, like mold & die, aerospace, electronic, machinery, etc. We are professional expert in cutting field. We would like to solve every problem from you. Please feel free to contact us, its our pleasure to serve for you. BW product including: cutting tool、aerospace tool .HSS  DIN Cutting tool、Carbide end mills、Carbide cutting tool、NAS Cutting tool、NAS986 NAS965 NAS897 NAS937orNAS907 Cutting Tools,Carbide end mill、disc milling cutter,Aerospace cutting tool、hss drill’Фрезеры’Carbide drill、High speed steel、Compound Sharpener’Milling cutter、INDUCTORS FOR PCD’CVDD(Chemical Vapor Deposition Diamond )’PCBN (Polycrystalline Cubic Boron Nitride) ’Core drill、Tapered end mills、CVD Diamond Tools Inserts’PCD Edge-Beveling Cutter(Golden Finger’PCD V-Cutter’PCD Wood tools’PCD Cutting tools’PCD Circular Saw Blade’PVDD End Mills’diamond tool. INDUCTORS FOR PCD . POWDER FORMING MACHINE ‘Single Crystal Diamond ‘Metric end mills、Miniature end mills、Специальные режущие инструменты ‘Пустотелое сверло ‘Pilot reamer、Fraises’Fresas con mango’ PCD (Polycrystalline diamond) ‘Frese’POWDER FORMING MACHINE’Electronics cutter、Step drill、Metal cutting saw、Double margin drill、Gun barrel、Angle milling cutter、Carbide burrs、Carbide tipped cutter、Chamfering tool、IC card engraving cutter、Side cutter、Staple Cutter’PCD diamond cutter specialized in grooving floors’V-Cut PCD Circular Diamond Tipped Saw Blade with Indexable Insert’ PCD Diamond Tool’ Saw Blade with Indexable Insert’NAS tool、DIN or JIS tool、Special tool、Metal slitting saws、Shell end mills、Side and face milling cutters、Side chip clearance saws、Long end mills’end mill grinder’drill grinder’sharpener、Stub roughing end mills、Dovetail milling cutters、Carbide slot drills、Carbide torus cutters、Angel carbide end mills、Carbide torus cutters、Carbide ball-nosed slot drills、Mould cutter、Tool manufacturer.

Bewise Inc. www.tool-tool.com

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情報を受け取って頂き、もっと各産業に競争力プラス展開。

弊社は専門なエンド・ミルの製造メーカーで、客先に色んな分野のニーズ、

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弊社は各領域に供給できる内容は：

（１）精密ＨＳＳエンド・ミルのＲ＆Ｄ

（２）Carbide Cutting tools設計

（３）鎢鋼エンド・ミル設計

（４）航空エンド・ミル設計

（５）超高硬度エンド・ミル

（６）ダイヤモンド・エンド・ミル

（７）医療用品エンド・ミル設計

（８）自動車部品＆材料加工向けエンド・ミル設計

弊社の製品の供給調達機能は：

（１）生活産業～ハイテク工業までのエンド・ミル設計

（２）ミクロ・エンド・ミル～大型エンド・ミル供給

（３）小Ｌｏｔ生産～大量発注対応供給

（４）オートメーション整備調達

（５）スポット対応～流れ生産対応

弊社の全般供給体制及び技術自慢の総合専門製造メーカーに貴方のご体験を御待ちしております。

Bewise Inc. talaşlı imalat sanayinde en fazla kullanılan ve üç eksende (x,y,z) talaş kaldırabilen freze takımlarından olan Parmak Freze imalatçısıdır. Çok geniş ürün yelpazesine sahip olan firmanın başlıca ürünlerini Karbür Parmak Frezeler, Kalıpçı Frezeleri, Kaba Talaş Frezeleri, Konik Alın Frezeler, Köşe Radyüs Frezeler, İki Ağızlı Kısa ve Uzun Küresel Frezeler, İç Bükey Frezeler vb. şeklinde sıralayabiliriz.

BW специализируется в научных исследованиях и разработках, и снабжаем самым высокотехнологичным карбидовым материалом для поставки режущих / фрезеровочных инструментов для почвы, воздушного пространства и электронной индустрии. В нашу основную продукцию входит твердый карбид / быстрорежущая сталь, а также двигатели, микроэлектрические дрели, IC картонорезальные машины, фрезы для гравирования, режущие пилы, фрезеры-расширители, фрезеры-расширители с резцом, дрели, резаки форм для шлицевого вала / звездочки роликовой цепи, и специальные нано инструменты. Пожалуйста, посетите сайт  www.tool-tool.com  для получения большей информации.

BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web  www.tool-tool.com  for more info.

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Dolní propust' v elektronice označuje frekvenční filtr, který nepropouští signál vyšších frekvencí.

Nejjednodušší je zapojení s paralelním kondenzátorem a seriovým rezistorem, pak pro nejvyšší propustnou frekvenci fm platí vztah:

f_m = \frac{1}{2 \pi R C}

歡迎來到Bewise Inc.的世界，首先恭喜您來到這接受新的資訊讓產業更有競爭力，我們是提供專業刀具製造商，應對客戶高品質的刀具需求，我們可以協助客戶滿足您對產業的不同要求，我們有能力達到非常卓越的客戶需求品質，這是現有相關技術無法比擬的，我們成功的滿足了各行各業的要求，包括：精密HSS DIN切削刀具、協助客戶設計刀具流程、DIN or JIS 鎢鋼切削刀具設計、NAS986 NAS965 NAS897 NAS937orNAS907 航太切削刀具,NAS航太刀具設計、超高硬度的切削刀具、醫療配件刀具設計、複合式再研磨機、PCD地板專用企口鑽石組合刀具、NSK高數主軸與馬達、專業模具修補工具-氣動與電動、粉末造粒成型機、主機版專用頂級電桿、PCD V-Cut刀、捨棄式圓鋸片組、粉末成型機、主機版專用頂級電感、’汽車業刀具設計、電子產業鑽石刀具、木工產業鑽石刀具、銑刀與切斷複合再研磨機、銑刀與鑽頭複合再研磨機、銑刀與螺絲攻複合再研磨機等等。我們的產品涵蓋了從民生刀具到工業級的刀具設計；從微細刀具到大型刀具；從小型生產到大型量產；全自動整合；我們的技術可提供您連續生產的效能，我們整體的服務及卓越的技術，恭迎您親自體驗！！

BW Bewise Inc. Willy Chen willy@tool-tool.com bw@tool-tool.com www.tool-tool.com skype:willy_chen_bw mobile:0937-618-190 Head &Administration Office No.13,Shiang Shang 2nd St., West Chiu Taichung,Taiwan 40356 http://www.tool-tool.com / FAX:+886 4 2471 4839 N.Branch 5F,No.460,Fu Shin North Rd.,Taipei,Taiwan S.Branch No.24,Sec.1,Chia Pu East Rd.,Taipao City,Chiayi Hsien,Taiwan

Welcome to BW tool world! We are an experienced tool maker specialized in cutting tools. We focus on what you need and endeavor to research the best cutter to satisfy users’ demand. Our customers involve wide range of industries, like mold & die, aerospace, electronic, machinery, etc. We are professional expert in cutting field. We would like to solve every problem from you. Please feel free to contact us, its our pleasure to serve for you. BW product including: cutting tool、aerospace tool .HSS  DIN Cutting tool、Carbide end mills、Carbide cutting tool、NAS Cutting tool、NAS986 NAS965 NAS897 NAS937orNAS907 Cutting Tools,Carbide end mill、disc milling cutter,Aerospace cutting tool、hss drill’Фрезеры’Carbide drill、High speed steel、Compound Sharpener’Milling cutter、INDUCTORS FOR PCD’CVDD(Chemical Vapor Deposition Diamond )’PCBN (Polycrystalline Cubic Boron Nitride) ’Core drill、Tapered end mills、CVD Diamond Tools Inserts’PCD Edge-Beveling Cutter(Golden Finger’PCD V-Cutter’PCD Wood tools’PCD Cutting tools’PCD Circular Saw Blade’PVDD End Mills’diamond tool. INDUCTORS FOR PCD . POWDER FORMING MACHINE ‘Single Crystal Diamond ‘Metric end mills、Miniature end mills、Специальные режущие инструменты ‘Пустотелое сверло ‘Pilot reamer、Fraises’Fresas con mango’ PCD (Polycrystalline diamond) ‘Frese’POWDER FORMING MACHINE’Electronics cutter、Step drill、Metal cutting saw、Double margin drill、Gun barrel、Angle milling cutter、Carbide burrs、Carbide tipped cutter、Chamfering tool、IC card engraving cutter、Side cutter、Staple Cutter’PCD diamond cutter specialized in grooving floors’V-Cut PCD Circular Diamond Tipped Saw Blade with Indexable Insert’ PCD Diamond Tool’ Saw Blade with Indexable Insert’NAS tool、DIN or JIS tool、Special tool、Metal slitting saws、Shell end mills、Side and face milling cutters、Side chip clearance saws、Long end mills’end mill grinder’drill grinder’sharpener、Stub roughing end mills、Dovetail milling cutters、Carbide slot drills、Carbide torus cutters、Angel carbide end mills、Carbide torus cutters、Carbide ball-nosed slot drills、Mould cutter、Tool manufacturer.

Bewise Inc. www.tool-tool.com

ようこそＢｅｗｉｓｅ　Ｉｎｃ.の世界へお越し下さいませ、先ず御目出度たいのは新たな

情報を受け取って頂き、もっと各産業に競争力プラス展開。

弊社は専門なエンド・ミルの製造メーカーで、客先に色んな分野のニーズ、

豊富なパリエーションを満足させ、特にハイテク品質要求にサポート致します。

弊社は各領域に供給できる内容は：

（１）精密ＨＳＳエンド・ミルのＲ＆Ｄ

（２）Carbide Cutting tools設計

（３）鎢鋼エンド・ミル設計

（４）航空エンド・ミル設計

（５）超高硬度エンド・ミル

（６）ダイヤモンド・エンド・ミル

（７）医療用品エンド・ミル設計

（８）自動車部品＆材料加工向けエンド・ミル設計

弊社の製品の供給調達機能は：

（１）生活産業～ハイテク工業までのエンド・ミル設計

（２）ミクロ・エンド・ミル～大型エンド・ミル供給

（３）小Ｌｏｔ生産～大量発注対応供給

（４）オートメーション整備調達

（５）スポット対応～流れ生産対応

弊社の全般供給体制及び技術自慢の総合専門製造メーカーに貴方のご体験を御待ちしております。

Bewise Inc. talaşlı imalat sanayinde en fazla kullanılan ve üç eksende (x,y,z) talaş kaldırabilen freze takımlarından olan Parmak Freze imalatçısıdır. Çok geniş ürün yelpazesine sahip olan firmanın başlıca ürünlerini Karbür Parmak Frezeler, Kalıpçı Frezeleri, Kaba Talaş Frezeleri, Konik Alın Frezeler, Köşe Radyüs Frezeler, İki Ağızlı Kısa ve Uzun Küresel Frezeler, İç Bükey Frezeler vb. şeklinde sıralayabiliriz.

BW специализируется в научных исследованиях и разработках, и снабжаем самым высокотехнологичным карбидовым материалом для поставки режущих / фрезеровочных инструментов для почвы, воздушного пространства и электронной индустрии. В нашу основную продукцию входит твердый карбид / быстрорежущая сталь, а также двигатели, микроэлектрические дрели, IC картонорезальные машины, фрезы для гравирования, режущие пилы, фрезеры-расширители, фрезеры-расширители с резцом, дрели, резаки форм для шлицевого вала / звездочки роликовой цепи, и специальные нано инструменты. Пожалуйста, посетите сайт  www.tool-tool.com  для получения большей информации.

BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web  www.tool-tool.com  for more info.

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مرشح الترددات المنخفضة هو مرشح يقوم بتمرير الأشارات ذات الترددات المنخفضة و يقوم باضعاف (تقليل شدة) الأشارات ذات الترددات الأعلى من تردد الايقاف.

نظرية مرشح الترددات المنخفضة تتمثل في الكثير من التطبيقات مثل الدوائر الاليكترونية و الخواريزميات الرقمية و الموانع الصوتية... الخ.

مرشحات الترددات المنخفضة تلعب ايضا دورا هاما في مجال معالجة الاشارات.

歡迎來到Bewise Inc.的世界，首先恭喜您來到這接受新的資訊讓產業更有競爭力，我們是提供專業刀具製造商，應對客戶高品質的刀具需求，我們可以協助客戶滿足您對產業的不同要求，我們有能力達到非常卓越的客戶需求品質，這是現有相關技術無法比擬的，我們成功的滿足了各行各業的要求，包括：精密HSS DIN切削刀具、協助客戶設計刀具流程、DIN or JIS 鎢鋼切削刀具設計、NAS986 NAS965 NAS897 NAS937orNAS907 航太切削刀具,NAS航太刀具設計、超高硬度的切削刀具、醫療配件刀具設計、複合式再研磨機、PCD地板專用企口鑽石組合刀具、NSK高數主軸與馬達、專業模具修補工具-氣動與電動、粉末造粒成型機、主機版專用頂級電桿、PCD V-Cut刀、捨棄式圓鋸片組、粉末成型機、主機版專用頂級電感、’汽車業刀具設計、電子產業鑽石刀具、木工產業鑽石刀具、銑刀與切斷複合再研磨機、銑刀與鑽頭複合再研磨機、銑刀與螺絲攻複合再研磨機等等。我們的產品涵蓋了從民生刀具到工業級的刀具設計；從微細刀具到大型刀具；從小型生產到大型量產；全自動整合；我們的技術可提供您連續生產的效能，我們整體的服務及卓越的技術，恭迎您親自體驗！！

BW Bewise Inc. Willy Chen willy@tool-tool.com bw@tool-tool.com www.tool-tool.com skype:willy_chen_bw mobile:0937-618-190 Head &Administration Office No.13,Shiang Shang 2nd St., West Chiu Taichung,Taiwan 40356 http://www.tool-tool.com / FAX:+886 4 2471 4839 N.Branch 5F,No.460,Fu Shin North Rd.,Taipei,Taiwan S.Branch No.24,Sec.1,Chia Pu East Rd.,Taipao City,Chiayi Hsien,Taiwan

Welcome to BW tool world! We are an experienced tool maker specialized in cutting tools. We focus on what you need and endeavor to research the best cutter to satisfy users’ demand. Our customers involve wide range of industries, like mold & die, aerospace, electronic, machinery, etc. We are professional expert in cutting field. We would like to solve every problem from you. Please feel free to contact us, its our pleasure to serve for you. BW product including: cutting tool、aerospace tool .HSS  DIN Cutting tool、Carbide end mills、Carbide cutting tool、NAS Cutting tool、NAS986 NAS965 NAS897 NAS937orNAS907 Cutting Tools,Carbide end mill、disc milling cutter,Aerospace cutting tool、hss drill’Фрезеры’Carbide drill、High speed steel、Compound Sharpener’Milling cutter、INDUCTORS FOR PCD’CVDD(Chemical Vapor Deposition Diamond )’PCBN (Polycrystalline Cubic Boron Nitride) ’Core drill、Tapered end mills、CVD Diamond Tools Inserts’PCD Edge-Beveling Cutter(Golden Finger’PCD V-Cutter’PCD Wood tools’PCD Cutting tools’PCD Circular Saw Blade’PVDD End Mills’diamond tool. INDUCTORS FOR PCD . POWDER FORMING MACHINE ‘Single Crystal Diamond ‘Metric end mills、Miniature end mills、Специальные режущие инструменты ‘Пустотелое сверло ‘Pilot reamer、Fraises’Fresas con mango’ PCD (Polycrystalline diamond) ‘Frese’POWDER FORMING MACHINE’Electronics cutter、Step drill、Metal cutting saw、Double margin drill、Gun barrel、Angle milling cutter、Carbide burrs、Carbide tipped cutter、Chamfering tool、IC card engraving cutter、Side cutter、Staple Cutter’PCD diamond cutter specialized in grooving floors’V-Cut PCD Circular Diamond Tipped Saw Blade with Indexable Insert’ PCD Diamond Tool’ Saw Blade with Indexable Insert’NAS tool、DIN or JIS tool、Special tool、Metal slitting saws、Shell end mills、Side and face milling cutters、Side chip clearance saws、Long end mills’end mill grinder’drill grinder’sharpener、Stub roughing end mills、Dovetail milling cutters、Carbide slot drills、Carbide torus cutters、Angel carbide end mills、Carbide torus cutters、Carbide ball-nosed slot drills、Mould cutter、Tool manufacturer.

Bewise Inc. www.tool-tool.com

ようこそＢｅｗｉｓｅ　Ｉｎｃ.の世界へお越し下さいませ、先ず御目出度たいのは新たな

情報を受け取って頂き、もっと各産業に競争力プラス展開。

弊社は専門なエンド・ミルの製造メーカーで、客先に色んな分野のニーズ、

豊富なパリエーションを満足させ、特にハイテク品質要求にサポート致します。

弊社は各領域に供給できる内容は：

（１）精密ＨＳＳエンド・ミルのＲ＆Ｄ

（２）Carbide Cutting tools設計

（３）鎢鋼エンド・ミル設計

（４）航空エンド・ミル設計

（５）超高硬度エンド・ミル

（６）ダイヤモンド・エンド・ミル

（７）医療用品エンド・ミル設計

（８）自動車部品＆材料加工向けエンド・ミル設計

弊社の製品の供給調達機能は：

（１）生活産業～ハイテク工業までのエンド・ミル設計

（２）ミクロ・エンド・ミル～大型エンド・ミル供給

（３）小Ｌｏｔ生産～大量発注対応供給

（４）オートメーション整備調達

（５）スポット対応～流れ生産対応

弊社の全般供給体制及び技術自慢の総合専門製造メーカーに貴方のご体験を御待ちしております。

Bewise Inc. talaşlı imalat sanayinde en fazla kullanılan ve üç eksende (x,y,z) talaş kaldırabilen freze takımlarından olan Parmak Freze imalatçısıdır. Çok geniş ürün yelpazesine sahip olan firmanın başlıca ürünlerini Karbür Parmak Frezeler, Kalıpçı Frezeleri, Kaba Talaş Frezeleri, Konik Alın Frezeler, Köşe Radyüs Frezeler, İki Ağızlı Kısa ve Uzun Küresel Frezeler, İç Bükey Frezeler vb. şeklinde sıralayabiliriz.

BW специализируется в научных исследованиях и разработках, и снабжаем самым высокотехнологичным карбидовым материалом для поставки режущих / фрезеровочных инструментов для почвы, воздушного пространства и электронной индустрии. В нашу основную продукцию входит твердый карбид / быстрорежущая сталь, а также двигатели, микроэлектрические дрели, IC картонорезальные машины, фрезы для гравирования, режущие пилы, фрезеры-расширители, фрезеры-расширители с резцом, дрели, резаки форм для шлицевого вала / звездочки роликовой цепи, и специальные нано инструменты. Пожалуйста, посетите сайт  www.tool-tool.com  для получения большей информации.

BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web  www.tool-tool.com  for more info.

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