Low-pass filter

As low pass is referred to in electronics such filters, which can happen almost unattenuated signal components with frequencies below the crossover frequency, attenuating components with higher frequencies, however. Corresponding filter functions can also occur in other areas, such as mechanics, acoustics or hydraulics, they are there but usually not called that. Also, any kind of mechanical inertia affects low-pass -forming. Connected to the slowdown is a time delay, by the shifts for sinusoidal waveform of the phase angle.

Application

Low-pass filter can be realized in different ways. At the electronic passive analog low-pass filters are usually made ​​up of resistors, capacitors and inductors. Through circuitry extensions to active devices such as operational amplifiers or transistors, active analog low-pass filters can be realized.

Another variation is made in the context of digital signal processing as a time-discrete low-pass filter in filter structures, such as FIR or IIR filter. The implementation can be done in digital circuits such as FPGAs or by means of sequential computer programs.

Low-pass filters for high performance in the field of high frequency technology and electrical engineering are built in analogue technology of capacitors and coils. The main application is the high- frequency technology, they are also found in the load outputs of frequency converters, class D amplifiers, switching power supplies and line filters.

Low-pass filter in the low- frequency technology are referred to as application specific height barrier height filter Treble -cut filter, high-cut filter, or noise filter. These terms are commonly used in audio engineering; they point out that such a filter, for example in an equalizer, the " heights " of the signal or the noise attenuates containing predominantly high frequencies; see also equalization (audio engineering ). Furthermore, low-pass filters are preceded by the low-frequency loudspeakers (woofer ) in speakers.

Low-pass functions are also used in mechanics ( vibration damping ), acoustic ( sound propagation of low frequencies is less loss ), optics before ( edge filters ), hydraulic or the propagation of light in the atmosphere, but there are not so called. In the measurement of the low-pass filter is also referred to as an arithmetic mean value generator and applied, for example, in the moving coil instrument or the generation of a variable DC voltage by means of pulse width modulation.

A special case of a low-pass filter takes the ideal low-pass. This has a low-pass transfer function of non-causal and can not therefore in practice be realized. It is used because of its simple transfer function of the filter theory as a simplified model. Reliable low-pass filters can only approximate as closely as possible to the property of the ideal low-pass filter.

Using filter transforms may be formed from the low-pass, a high pass or a band-pass.

Representation

The general mathematical approach for a filter leads to a differential equation. Especially for sinusoidal quantities can be the often tedious solution to deal with the use of complex-valued variables, see complex AC circuit analysis.

Also the frequency response of full value describes the behavior of the filter. This is prepared by the complex voltage ratio H = Ua / Ue (or by the amount of gain A ( w) = 20 log10 | H (w ) | ) and the phase shift angle φ between Ua and Ue dar. vividly with Bode diagram or locus

Low pass 1st order

Description

In the simplest case, a low pass from a resistor-capacitor combination ( RC element) and provides a Butterworth filter with first -order in the following arrangement is:

A sudden change in the input voltage Ue follows the output voltage Vout to the same jump height, but delayed in the course of an exponential function with a time constant τ = RC.

A sinusoidal input voltage of the frequency f at the output follows the voltage divider in accordance with the formula again a sinusoidal, but a frequency-dependent voltage attenuated

Wherein the amounts and the output and input voltage denote the amount of the reactance of the capacitor and the angular frequency.

In a logarithmic representation of frequency ( Bode plot ), the division ratio has two asymptotes. It is at low frequencies to 1 and for DC voltage ( frequency f = 0). At high frequencies it decreases with 6 dB / octave or 20 dB / decade from. Under the cut-off frequency fc (cutoff frequency ) is defined as the frequency at which the asymptotes intersect. Here is

(that is, Ua is over Ue attenuated by 3 dB). The cutoff frequency is

The frequency deviates by more than an order of magnitude of the limit frequency ( up or down ), then the curve may be replaced by the respective asymptotic with a relative deviation of less than ½%.

With operational amplifiers active low-pass filters can be realized. These have the advantage that the response is independent of the load connected to the output. The magnitude of the output voltage of this low-pass filter

At the cut-off frequency the gain has dropped in accordance to the times the DC gain, is (apart from the sign inversion ), the.

Derivation of the formula

In the representation of alternating quantities by complex quantities valid for the stress ratio according to voltage divider rule:

= operator with resistance or impedance of the capacitor.

With an auxiliary variable

Obtained

This equation represents the locus of the complex voltage transfer function

Conclusions

From this are derived from:

On transition to amounts and reactance ( real quantities ) yields the formula given above

The time function of the sinusoidal vibration is obtained from the imaginary part of the complex vector of the rotating trigonometric form:

Then follows for the timing function:

With

Low-pass 2nd order

A second-order low-pass filter obtained by an inductance L is switched to R in series, since the reactance XL also a - has frequency dependency - and the capacitor reactance XC opposite. Where R is chosen so large that no or only a slight increase in voltage is produced the frequency response.

Is the transfer function of such a low-pass filter

Is the magnitude of the transfer function

Thus, the output voltage drops Ua above fc faster ( with 12 dB / octave or 40 dB / decade ), because now not only | XC | smaller but at the same time | XL | becomes larger.

In this variant, large inductors are used ( up to several Henry) in the low frequency range. These have poor electrical properties, and have quite large geometric proportions. Nowadays, low-pass second and higher order only come in power converter technology used. In communication engineering, however, filters are now implemented by operational amplifier circuits. These filters are referred to as active low-pass filters (or active filters ), and, after its inventors also known as a Sallen-Key filter.

In the high frequency range, for example in the construction of transmission facilities R is always zero in order to avoid heat losses. This circuit is used for two reasons:

• You attenuates harmonics generated by the C- operation of electron tubes to an acceptable level.
• The values ​​of the components can be chosen so that the circuit operates as a resonant transformer and an impedance matching between the transmitter and antenna allowed.

Low-pass n- th order

By cascading several low passes you can increase its order. For example, form two cascaded low-pass 2nd order low-pass 4th order. The attenuation change at this time above the cutoff frequency of 4 • 20 dB / decade = 80 dB / decade, corresponding to a slope of 24 dB / octave.

However, two interconnected low-pass filters with the same cut-off frequency low-pass yield no higher order the same cutoff frequency. For the dimensioning of a low-pass filter with desired cutoff frequency, special formulas and tables.

In addition, there arises a problem that a low-pass filter is affected in a chain from the output resistance of the upstream and downstream of the input resistor of the low-pass filter. This effect can be counteracted with impedance transducers.

N- th order are generally for a filter requires n storage elements (ie capacitors or coils).

The attenuation of a low-pass filter of order n increases above the cutoff frequency with n • 20 dB / decade to.

Emphasis and de-emphasis

The static frequency response change, the emphasis and the de-emphasis time constant is given instead of the cut-off frequency typically.