Basic Electronics on the Go - Active Band Pass Filter

From http://www.electronicshub.org/active-band-pass-filter/
From http://www.electronics-tutorials.ws/filter/filter_7.html

Introduction

A Band Pass Filter is a circuit which allows only particular band of frequencies to pass through it. This Pass band is mainly between the cut-off frequencies and they are fL and fH.  Where fL is the lower cut-off frequency and fH is higher cut-off frequency. The centre frequency is denoted by ‘fC’ and it is also called as resonant frequency or peak frequency.

The fL value must always be less than the value of fH. The pass band of the filter is nothing but the bandwidth. The gain of the filter is maximum at resonant or centre frequency and this is referred as total pass band gain. This pass band gain is denoted by ‘Amax’.

For low pass filter this pass band starts from 0 Hz and continues until it reaches the resonant frequency value at -3 dB down from a maximum pass band gain.

Where as in the case of high pass filter this pass band begins from the -3 dB resonant frequency and ends at the value of the maximum loop gain for active filter. Combination of low pass and high pass responses gives us band pass response as shown below:

Active Band Pass Filter

Depending on the quality factor the band pass filter is classified into Wide band pass filter and Narrow band pass filter. The quality factor is also referred as ‘figure of merit’. By cascading High Pass Filter and Low Pass Filter with an amplifying component we obtain band pass filter.

The cut-off or corner frequency of the low pass filter (LPF) is higher than the cut-off frequency of the high pass filter (HPF) and the difference between the frequencies at the -3dB point will determine the “bandwidth” of the band pass filter while attenuating any signals outside of these points. Obviously, a reasonable separation is required between the two cut-off points to prevent any interaction between the low pass and high pass stages. The amplifier circuit between these high pass and low pass filter will provide isolation and  voltage gain for the circuit.

The circuit diagram for active band pass filter is shown below:

Wide Band Pass Filter

If the value of quality factor is less than ten, then the pass band is wide, which gives us the larger bandwidth. This band pass filter is called Wide Band Pass Filter. In this filter the high cut-off frequency must be greater than the lower cut-off frequency. It uses two amplifying elements (Op-amps) in design.

First the signal will pass through the high pass filter, the output signal of this high pass filter will tends to infinity and thus the signal which tends to infinity is given to the low pass filter at the end. 

By cascading one first order low pass and high pass gives us the second order band pass filter and by cascading two first order low pass filters with two high pass filters forms a fourth order band pass filter. Due to this cascading the circuit produces a low value quality factor. The capacitor in the first order high pass filter will block any DC biasing from the input signal.

The gain rolls off at both the stop bands is ± 20 dB /decade in the case of second order filter (High + Low). The high Pass and low pass filters must be in first order only. Similarly, when the high pass and low pass filters are at second order, then the gain roll off at both the stop bands is ± 40dB/Decade.

The voltage gain expression for band pass filter is given as:

| Vout / Vin | = [Amax * (f/fL)] / √{[1+(f/fL)²][1+(f/fH)²]}

It is obtained by the individual gains of both high pass and low pass filters, the individual gains of both high pass and low pass filter are given below.

Voltage Gain for High Pass filter:

| Vout / Vin | = [Amax1 * (f/fL)] / √[1+(f/fL)²]

Voltage Gain for Low pass filter:

| Vout / Vin | = Amax2 / √[1+(f/fH)²]

Amax = Amax1 * Amax2

Where Amax1 is the gain of the high pass stage and Amax2 is the gain of the low pass stage.

The response of the wide band filter is shown below

Narrow Band Pass Filter

If the value of quality factor is greater than ten then the pass band is narrow and bandwidth of the pass band is also less. This band pass filter is called as Narrow Band Pass Filter. It uses only one active component (op-amp) rather than two and this op-amp is in inverting configuration. In this filter the gain of the op-amp is maximum at centre frequency fc.

Narrow Band Pass Filter Circuit

The input is applied to the inverting input terminal. This shows that the Op-amp is in inverting configuration. This filter circuit produces narrow band pass filter response.

The voltage gain of the above filter circuit is AV = – R2 / R1 

The cut-off frequencies of the filter circuit are

fC1 = 1 / (2πR1C1) and fC2 = 1 / (2πR2C2)

Multi Feedback Active Band Pass Filter

This filter circuit produces a tuned circuit based on the negative feedback of the filter. The important advantage of this multiple feedback is that without any change in the maximum gain at the centre frequency we can change the value of the cut-off frequency. This change in the cut-off frequency can be done by the resistance ‘R3’.

By considering the  active filter circuit below, let us consider the changed resistor value as R3′and the changed cut-off frequency value as fc′, then we can equate for the new resistor value as follows:

R3′ = R3(fc /fc′)²

t consists of two feedback paths, because of this multiple feedback paths it is also referred as ‘Multiple feedback band pass circuit’. This circuit produce an infinity gain multiple feedback band pass filter. Due to this circuit the quality factor value increases maximum up to 20.

fc = 1/√(R1R2C1C2)

Q = fc/Bandwidth = (½){√[R2/R1]}

Amax = -R2/2R1

R1 = Q/{2πfcCAmax}

R2 = Q/πfcC

R3 = Q/{2πfcC(2Q² – Amax)}

The gain at the centre frequency ‘Amax’ must be less than 2Q². That is,

Amax < 2Q²

Where,

fc = cut off frequency in Hz
C = Capacitance, (C1 = C2 = C)
Q = Quality factor
Amax = Maximum gain

The Frequency Response of Active Band Filter

 

It has two centre frequencies, one is from the  high pass filter and the other is from the  low pass filter. The centre frequency of the high pass filter must be lower than the centre frequency of the low pass filter. The centre frequency of band pass filter is the geometric mean of lower and upper cut-off frequencies f_r² = f_H * f_L. 

The gain of the filter is 20 log (V_out/V_in) dB/Decade. The amplitude response is similar to the responses of the low pass and high pass filter. Let us consider two cut-off frequencies as 300 Hz and 900Hz, then, the bandwidth of the filter is 300 Hz -900 Hz = 600 Hz.

The Quality Factor

The quality factor depends on the bandwidth of the pass band. Quality factor is inversely proportional to the Bandwidth. That means if band width increases the quality factor decrease and if band width decreases the quality factor increases.

Q = f_c/Bandwidth

For wide band pass filter the quality factor is low because the pass band width is high. For the narrow band pass filter the quality factor is high. Selectivity and un-selectivity depends on the width of the pass band. This quality factor is also relates to the damping factor. If damping co-efficient value is more, the flatness of the output response is also more. This is equated as follows:

ε = 2/Q


 For different quality factor values the normalized gain response of a second order band pass filter is given as:

 

By this graph, it is clear that the selectivity is more for higher quality factor.

Basic Electronics on the Go - Active High Pass Filter

From http://www.electronicshub.org/active-high-pass-filter/
From http://www.electronics-tutorials.ws/filter/filter_6.html

Introduction

A high pass filter will allow the frequencies which are higher than the cut-off frequency and attenuate the frequencies lower than the cut off frequency. In some cases this filter is also termed as ‘Low-Cut’ filter or ‘Base-cut’ filter. The amount of attenuation or the pass band range will depend on the designing parameters of the filter.

The pass band gain of an active filter is more than unity gain. The operation of the active high pass filter is same as passive high pass filter, but the main difference is that the active high pass filter uses an  operational  amplifier which provides amplification of the output signals and controls gain.

The ideal characteristics of the high pass filter are shown below

Technically, there is no such thing as an active high pass filter. 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 by the open-loop characteristics or bandwidth of the operational amplifier being used, making them appear as if they are band pass filters with a high frequency cut-off determined by the selection of op-amp and gain.

Active High Pass Filter

By connecting a passive RC high pass filter circuit to the inverting or non-inverting terminal of the op-amp gives us first order active high pass filter. The passive RC high pass filter circuit connected to the non-inverting terminal of the unity gain operational amplifier is shown below.

The gain Amax = 1 and cutoff frequency fc = 1/2πRC

Active High Pass Filter With High Voltage Gain

The operation is the same as that of the passive high pass filter, but the input signal is amplified by the amplifier at the output. The amount of amplification depends on the gain of the amplifier. The magnitude of the pass band gain is equal to 1 + (R3/R2). Where R3 is the feedback resistor in Ω (ohms) and R2 is the input resistor. The circuit of  an active high pass filter with amplification is given below

Voltage Gain Of An Active High Pass Filter

Voltage Gain Av = Amax (f/fc) / √{1 + (f/fc)²}

Where f = operating frequency

fc = cut-off frequency

Amax = pass band gain of the filter = 1 + (R3/R2)

At low frequencies means when the operating frequency is less than the cut-off frequency, the voltage gain is less than the pass band gain Amax. At high frequencies means when the operating frequency is greater than the cut-off frequency, the voltage gain of the filter is equal to pass band gain. If operating frequency is equal to the cut-off frequency,then the voltage gain of the filter is equal to 0.707 Amax.

Voltage Gain in (dB)

The magnitude of the  voltage gain is generally taken in decibels (dB):

Av(dB) = 20 log10 (Vout/Vin)

-3 dB = 20 log10 (0.707 * Vout/Vin)

The cut-off frequency which separates both pass band and stop band can be calculated using the  formula
fc = 1 / (2πRC)

The phase shift of the active high pass filter is equal to that of the passive filter. It is equal to +45° at the cut-off frequency fc and this phase shift value is equated as

Ø = tan-1(1/2πfcRC)

Frequency Response Of Active High Pass Filter

The frequency response curve with respect to the amplifiers open loop gain is shown below.

In frequency response of the active high pass filter the maximum pass band frequency is limited by the bandwidth or the open loop characteristics of the operational amplifier. Due to this limitation the active high pass filter response will appears like a wide band filter response. By using this op-amp based active high pass filter we can achieve high accuracy with the use of low tolerance resistors and capacitors.

Active High Pass Filter using Inverting Operational Amplifier

We know that the active high pass filter can be designed by using either inverting terminal or the non-inverting terminal of an operational amplifier. Till now we saw the high pass filter circuit and response curves of the non-inverting active high pass filter. Now let us see the active high pass filter using inverting op-amp.

Gain derivation in Laplace form

Let us consider the inverting amplifier as shown below

The input impedance Z1 = 1/sC1

Where s = Laplace Variable

C1 = Capacitance

The currents flowing in the circuit are I1, I2 and Iin,

Where I1 = I2 and Iin = 0

Vin / Z1 = -Vout / R1

Vout / Vin = – R1 / Z1

Vout / Vin = – R1 / (1/sC1)

Vout / Vin = -sR1C1 = Gain

Bode-plot

To analyse the circuit frequency response this bode plot is used. It is nothing but a graph of the transfer function of linear, time variant versus frequency. This is plotted with the log frequency axis. It consists of mainly two plots; one is magnitude plot and the other is phase plot.

The magnitude plot will express the magnitude of the frequency response i.e., gain and the phase plot is used to express the response of the frequency shift.

The frequency response bode-plot according to the values which are tabulated above is given below:

According to the values calculated, at frequency 10 Hz the gain of the filter obtained in dB is -56.48. If we increase the value of frequency to 100 Hz the obtained gain is -36.48 dB and at frequency 500 Hz the gain of the filter is -22.51 dB, at frequency 1000 Hz gain in dB is -16.52.By this we can say that if frequency increases the gain of the filter increases at the rate of 20dB/decade.

Till the cut-off frequency 10 KHz the gain of the filter increases but after the cut-off frequency the gain reaches maximum value and it is constant.

Second Order High Pass Filter

Second order active high pass filter frequency response is exactly opposite to the second order active low pass filter response because this filter will attenuate the voltages below the cut-off frequency. The transfer function of the second order filter is given below

Vout(s) / Vin(s) = -Ks² / s² + (ω0/Q)s + ω0²

Where K = R1/R2 and ω0 = 1/CR

This is the general form of the second order high pass filter.

Second Order Active High Pass Filter Circuit

The designing procedure for the second order active filter is same as that of the first order filter because the only variation is in the roll-off. If the roll-off of the first order active high pass filter is 20dB/decade, then roll-off of the second order filter is 40 dB/ decade. It means the twice of the value of the first order filter. The circuit of second order filter is shown below

The gain of the filter is 1+ R1/R2 and the equation of the cut-off frequency is fc = 1/ 2π√R3R4C1C2

Higher Order High Pass Filters

By cascading first order filter with second order filter , we can obtain the third order filter. When we cascade two second order filters we can get the fourth order filter. Like this with the help of first order and second order filters we get the higher order filters.

With the increase in the order of the filter,the difference between actual stop band and theoretical stop band increases. But the overall gain of the higher order filter is equal because we already saw that the resistors and capacitors which determine the frequency response values will be the same.

Applications of active high pass filters

• These are used in the loud speakers to reduce the low level noise.
• Eliminates rumble distortions in audio applications so these are also called are treble boost filters.
• These are used in audio amplifiers to amplify the higher frequency signals.
• These are also used in equalisers.