# High pass

Circuit symbol of a high-pass filter

As a high-pass (also low barrier , English low-cut filter , high-pass filter ) is known filters , the frequencies above its cut-off frequency can be approximately unattenuated pass and attenuate lower frequencies.

Such filters are common in electronics , but corresponding filter functions can also be found in other areas, such as mechanics , acoustics , hydraulics or electrical engineering, although they are usually not called that there.

## Applications

Are high-pass filter in the low frequency technology application specific as depth lock , bass filter , low-cut filter , bass-cut filter , low-cut filter , respectively. These terms are common in sound engineering; They indicate that such a filter, for example in an equalizer, attenuates the "depths" of the signal or corresponding hum interferences which mainly contain low frequencies; see also equalization (sound engineering) . Furthermore, high-pass filters are connected upstream of the tweeters.

High- pass filters are also used for coupling in and out high-frequency signals , e.g. B. used in antenna switches , ADSL or RF signal transmission over power lines.

With the help of filter transformations , a low pass or a bandstop filter can be formed from the high pass .

## 1st order high pass

Simple RC high pass
Amplitude responses of 1st and 2nd order high-pass filters

The function of an electrical filter circuit is given below as an example of a high pass . The image shows the basic structure of a capacitor C and a resistor R . At a low frequency, the reactance ( ) of the capacitor largely blocks the current. ${\ displaystyle X _ {\ mathrm {C}}}$

According to the voltage divider formula , only the portion of the input voltage appears at the output  : ${\ displaystyle {\ hat {U}} _ {\ mathrm {e}}}$${\ displaystyle {\ hat {U}} _ {\ mathrm {a}}}$

${\ displaystyle {\ hat {U}} _ {\ mathrm {a}} = {\ hat {U}} _ {\ mathrm {e}} \ cdot {\ frac {R} {\ sqrt {X _ {\ mathrm {C}} ^ {2} + R ^ {2}}}} = {\ hat {U}} _ {\ mathrm {e}} \ cdot {\ frac {\ omega CR} {\ sqrt {1+ ( \ omega CR) ^ {2}}}}}$           (For derivation see low-pass formula derivation )

Phase response :${\ displaystyle {\ varphi} (\ omega)}$ ${\ displaystyle = \ arctan \ left ({\ frac {1} {{\ omega} {C} {R}}} \ right)}$

where and denote the amounts of the input and output voltage. ${\ displaystyle U _ {\ mathrm {e}}}$${\ displaystyle U _ {\ mathrm {a}}}$

Locus of a passive first-order high pass

The cut-off frequency (ger .: cutoff frequency ) of such a high-pass filter is . The cutoff frequency is the frequency at which is, i.e. i.e., is weakened by 3 decibels compared to this . The attenuation increases below the cutoff frequency by 20 decibels per decade . With a logarithmic representation on both axes, this results in a straight line. Since it becomes smaller with increasing frequency, the division ratio tends towards 1 with increasing frequency, and becomes for high frequencies . ${\ displaystyle f _ {\ mathrm {c}}}$${\ displaystyle f _ {\ mathrm {c}} = {\ tfrac {1} {2 \ pi RC}}}$${\ displaystyle U _ {\ mathrm {a}} = U _ {\ mathrm {e}} / {\ sqrt {2}}}$${\ displaystyle U _ {\ mathrm {a}}}$${\ displaystyle U _ {\ mathrm {e}}}$${\ displaystyle X _ {\ mathrm {C}}}$${\ displaystyle U _ {\ mathrm {a}} = U _ {\ mathrm {e}}}$

${\ displaystyle X _ {\ mathrm {C}} = {\ frac {1} {\ omega C}}}$ With ${\ displaystyle \ omega = 2 \ pi f}$

The attenuation is then 0 dB.

The frequency response of the circuit is also often represented by a locus curve in the complex plane . It turns A the tension in complex notation is:

${\ displaystyle {\ underline {A}} = {\ frac {{\ underline {u_ {a}}} (t)} {{\ underline {u_ {e}}} (t)}}}$ .

The length of the time-independent pointer A represents the amplitude ratio as it changes with frequency; the angle to the positive real axis stands for  φ .

## 2nd order high pass

Second order passive high pass

A second order high-pass filter is obtained by R by an inductance L is replaced, since this in turn is a - although in opposite direction and to the capacitor - has frequency dependence, and a resistor R in series with the capacitor C on. Here, R is selected so large that no or only a small resonance step-up occurs the frequency response.

The frequency response of such a high pass is

${\ displaystyle H (\ omega) = {\ frac {j \, X_ {L}} {R + j (X_ {L} + X_ {C})}}}$
with .${\ displaystyle X_ {L} = \ omega L, \ quad X_ {C} = {\ frac {-1} {\ omega C}} \ ,, \ quad \ omega = 2 \ pi f}$

The magnitude of the transfer function is

${\ displaystyle {\ frac {U_ {a}} {U_ {e}}} = \ vert H (\ omega) \ vert = {\ frac {X_ {L}} {\ sqrt {R ^ {2} + ( X_ {L} + X_ {C}) ^ {2}}}}}$

This reduces the output voltage below f G to a greater extent (by 40 dB / decade), since now not only | X C | larger, but at the same time X L becomes smaller.

With static frequency response change, emphasis and deemphasis , the time constant is usually specified instead of the cutoff frequency .

Nowadays, high-pass filters of the second and higher order are usually implemented using operational amplifier circuits. These filters are known as active high-pass filters (or active filters) and are also known as Sallen-Key filters after their inventors .

## Higher order high pass

By connecting several high passes in series, their order is increased. Two second-order high-pass filters connected in series accordingly form a fourth-order high-pass filter. The damping changes below the cutoff frequency with:

${\ displaystyle 4 \ cdot 20 \, {\ text {dB / decade}} = 80 \, {\ text {dB / decade}}}$,

which corresponds to a slope of 24 dB / octave. 6 dB per octave is equal to 20 dB per decade: a change by one octave (change by a factor of 2) corresponds to the -fold change by one decade: ${\ displaystyle {\ tfrac {6} {20}}}$

${\ displaystyle \ log _ {10} (2) \ approx 0 {,} 30 = {\ frac {6} {20}}}$.