Active High Pass Filter

Active High Pass Filters

The basic operation of an Active High Pass Filter (HPF) is exactly the same as that for its equivalent RC passive filter circuit, except that this type of circuit has an operational amplifier or op-amp included within its design for amplification and gain control. Like the active Low Pass filter circuit, the simplest form of an active high pass filter is to connect a standard inverting or non-inverting operational amplifier to the basic RC high pass passive filter circuit.

However, unlike Passive High Pass Filters which have an "infinite" frequency response, the maximum pass band frequency response of an active High Pass Filter is limited to the characteristics or bandwidth of the op-amp being used within the circuit design. In the Operational Amplifier tutorial we saw that the maximum frequency response of an op-amp is limited to the Gain/Bandwidth product or open loop voltage gain ( A _V ) of the operational amplifier being used. A commonly available operational amplifier such as the uA741 has a typical "Open-loop" (without any feedback) DC voltage gain of about 100dB maximum reducing at a roll off rate of -20dB/Decade (-6db/Octave) as the input frequency increases. The gain of the uA741 reduces until it reaches unity gain, (0dB) or its "Transition Frequency" ( Ft ) which is about 1MHz. This causes the op-amp to have a frequency response curve very similar to that of a 1st-Order Low Pass filter and this is shown below.

Frequency Response Curve of a typical Operational Amplifier.

This unity gain crossover frequency determines the overall bandwidth of the open-loop amplifier and starts at about 100kHz for small signal amplifiers up to about 1GHz for high-speed digital video amplifiers. Under normal circumstances the maximum pass band required for a closed loop active high pass or band pass filter is well below that of the maximum open-loop transition frequency. However, when designing active filter circuits it is important to choose the correct op-amp for the circuit as the loss of high frequency signals may result in distortion.

Active High Pass Filter

A 1st-Order (single-pole) Active High Pass Filter as its name implies, attenuates low frequencies and passes high frequency signals. It consists simply of a passive filter section followed by a non-inverting operational amplifier. The frequency response of the circuit is the same as that of the passive filter, except that the amplitude of the signal is increased by the gain of the amplifier and for a non-inverting amplifier the value of the pass band voltage gain is given as 1 + R2/R1.

First-order High Pass Butterworth Filter

This 1st-Order high pass Butterworth type filter consists simply of a passive filter followed by a non-inverting amplifier. The frequency response of the circuit is the same as that of the passive filter, except that the amplitude of the signal is increased by the gain of the amplifier.

For a non-inverting amplifier circuit, the magnitude of the voltage gain for the filter is given as a function of the feedback resistor (R2) divided by its corresponding input resistor (R1) value and is given as:

Voltage Gain for a First-order High Pass Filter

• Where:
•   A_F = the Pass band Gain of the filter, (1 + R2/R1)
•   ƒ = the Frequency of the Input Signal in Hertz, (Hz)
•   ƒc = the Cut-off Frequency in Hertz, (Hz)

When dealing with filter circuits the magnitude of the pass band gain of the circuit is generally expressed in Decibels or dB as a function of the voltage gain, and this is given as:

Magnitude of Voltage Gain in (dB)

For a 1st-Order filter the frequency response curve of the filter increases by 20dB/Decade or 6dB/Octave up to the determined cut-off frequency point which is always at -3dB below the maximum gain value. As with the previous filter circuits, the cut-off or corner frequency (ƒc) can be found by using the same formula:

The corresponding phase angle or phase shift of the output signal is the same as that given for the passive RC filter and leads that of the input signal. It is equal to +45^o at the cut-off frequency ƒc value and is given as:

A simple 1st-Order active high pass filter can also be made using an inverting operational amplifier configuration as well, and an example of this circuit design is given along with its corresponding frequency response curve. A gain of 40dB has been assumed for the circuit.

Inverting Operational Amplifier Circuit

Frequency Response Curve

Example No1

A first order High Pass filter has a pass band gain of 2 and a cut-off corner frequency of 1kHz. If the input capacitor has a value of 10nF, calculate the value of the cut-off frequency determining resistor and the gain resistors. Also, plot the frequency response of the filter.

With a cut-off corner frequency of 1kHz and a capacitor of 10nF, the value of R will be:

The pass band gain of the filter, A_F is given as being, 2.

As the value of resistor, R₂ divided by resistor, R₁ gives a value of 1. Then, resistor R₁ must be equal to resistor R₂, since the pass band gain, A_F = 2. We can therefore select a suitable value for the two resistors of say, 10kΩ's each for both feedback resistors.

So for a high pass filter with a cut-off corner frequency of 1kHz, the values of R and C will be, 10kΩ's and 10nF respectively. The values of the two feedback resistors to produce a pass band gain of 2 are given as: R₁ = R₂ = 10kΩ's

The data for the frequency response bode plot can be obtained by substituting the values obtained above over a frequency range from 100Hz to 100kHz into the equation for voltage gain:

This then will give us the following table of data.

The frequency response data from the table above can now be plotted as shown below. In the stop band (from 100Hz to 1kHz), the gain increases at a rate of 20dB/decade. However, in the pass band after the cut-off frequency, ƒ_C = 1kHz, the gain remains constant at 6.02dB. The upper-frequency limit of the pass band is determined by the open loop bandwidth of the operational amplifier used as we discussed earlier. Then the bode plot of the filter circuit will look like this.

The Frequency Response Bode-plot for our example.

Applications of Active High Pass Filters are in audio amplifiers, equalizers or speaker systems to direct the high frequency signals to the smaller tweeter speakers or to reduce any low frequency noise or "rumble" type distortion. When used like this in audio applications the active high pass filter is sometimes called a "Treble Boost" filter.

As discussed previously in the Passive High Pass Filter section this type of filter circuit can also be used as a differentiator circuit when the input signal is of a non-sinusoidal or step type response.