The Basic Low Pass Filter Circuit of Fig. 15.7 is commonly used for low pass active filters. The filtering is done by the use of an RC network. The opamp is used as a unity gain amplifier. The resistor RF is equal to R1.
(At dc, the capacitive reactance is infinite and the dc resistance path to ground for both input terminals must be equal.) The difference voltage between inverting and non-inverting inputs is essentially 0 V. Hence, the voltage across the capacitor C equals the output voltage. Since this circuit is a voltage follower, Vin divides between R and C. The capacitor voltage Vo is given by
where ω is the frequency of Vin in radians per second (ω = 2 πf and j is the imaginary term. To obtain the closed loop voltage gain ACL, we have,
Consider the Eq. (15.12). At very low frequencies, as ω approaches 0, | ACL | = 1, and at very high frequencies, as ω approaches infinity, | ACL | = 0. Hence this filter is a Basic Low Pass Filter Circuit.
Figure 15.8 shows a frequency response of 0) versus | ACL |. For frequencies greater than the cutoff frequency ωc, | ACL | decreases at a rate of 20 db/decade. This is the same as saying that the voltage gain is divided by 10 when the frequency of co is increased by 10.
The cutoff frequency is defined as that frequency of Vin where | ACL | is reduced to 0.707 times its low frequency value. The cutoff frequency is calculated from
fc = is the cutoff frequency in Hz
R = resistance in Ω
C = capacitance in Farad.
The Eq. (15.13) can be rearranged to solve R, ignore to give
Figure 15.9 shows a first order low pass Butterworth filter that uses an RC network for filtering. The opamp is used in the non-inverting configuration, which does not load the RC network. R1 and RF determine the gain of the filter (in this case unity).
Using the voltage divider rule, the voltage across the capacitor, i.e. at the non-inverting input is
Simplifying, we get
As output voltage
Vo/Vin = Gain of the filter as a function of frequency
AF = 1 + RF/R1= pass band gain of the filter
f= frequency of the input signal
fH = 1/2 π RC = high cutoff frequency
The gain magnitude and phase angle can be obtained by applying modulus to Eq. (15.14).
where Φ is the phase angle in degrees.
The operation of the Basic Low Pass Filter Circuit can be verified from the gain magnitude Eq. (15.15).
Hence the Basic Low Pass Filter Circuit has a constant gain, AF, from 0 Hz to the high cutoff frequency fH. At fH, the gain is 0.707 AF and after fH the gain decreases at a constant rate with increase in frequency; when the frequency is increased 10 times (one decade), the voltage gain is divided by 10. In other words, the gain decreases by 20 db (20 log 10) each time the frequency is increased by 10. Hence the rate at which the gain rolls off after fH is 20 db/decade or 6 db/octave, where octave signifies a two fold increase in frequency. The frequency f = fH is called the cutoff frequency.
The procedure of converting a cutoff frequency to a new cutoff frequency is called frequency scaling.
To obtain a new cutoff frequency, R or C (but not both) is multiplied by the ratio of the original cutoff frequency to the new cutoff frequency.
In filter design, the values required for R and C are often not standard, and a variable capacitor C is not commonly used. Hence, we choose a standard value of the capacitor and then calculate the value of the resistor required for a desired cutoff frequency. This is because for a non-standing value of a resistor, a potentiometer can be used.