How does a Sallen-Key filter provide second-order filtering using operational amplifiers?
1 Answer
A Sallen-Key filter is a type of active filter circuit that uses operational amplifiers (op-amps) to provide second-order filtering. It is a popular choice for implementing low-pass, high-pass, and band-pass filters due to its simplicity and effectiveness. The second-order filtering capability means that the filter's frequency response has a roll-off rate of -40 dB/decade after the cutoff frequency, providing steeper attenuation compared to first-order filters.
The basic Sallen-Key filter consists of two cascaded RC (resistor-capacitor) filter stages, with an op-amp in between. The op-amp is configured as a non-inverting amplifier, and its gain and frequency response characteristics determine the filter's behavior. The two stages of the Sallen-Key filter work together to achieve the second-order filtering.
Here's a step-by-step explanation of how the Sallen-Key filter provides second-order filtering:
First RC Stage:
The first RC stage consists of a resistor (R1) and a capacitor (C1). The input signal is applied to the non-inverting input (+) of the op-amp. The op-amp amplifies the input signal, and the voltage across C1 is fed back to the inverting input (-) of the op-amp. This feedback network, together with C1, forms a high-pass filter with a cutoff frequency determined by the values of R1 and C1.
Second RC Stage:
The second RC stage consists of another resistor (R2) and capacitor (C2). The output of the first stage is connected to the non-inverting input (+) of the op-amp in the second stage. The voltage across C2 is fed back to the inverting input (-) of the op-amp. This feedback network, together with C2, forms a low-pass filter with a cutoff frequency determined by the values of R2 and C2.
Overall Transfer Function:
The cascaded combination of the first and second stages results in a second-order filter with a transfer function that combines the characteristics of the high-pass and low-pass filters. The overall transfer function H(s) of the Sallen-Key filter is a function of the complex frequency 's' and is given by:
H(s) = K / (s^2 + (ω0/Q)s + ω0^2)
where:
K is the overall gain of the filter (usually set by resistor ratios).
ω0 is the angular cutoff frequency (in radians per second).
Q is the quality factor of the filter, which determines the shape of the frequency response.
Frequency Response:
The frequency response of the Sallen-Key filter depends on the value of Q. For Q values greater than 0.5, the filter behaves as a band-pass filter with a peak at the cutoff frequency. For Q values less than 0.5, the filter behaves as a low-pass filter. As Q approaches 0.5, the filter's response becomes flat, resembling a Butterworth filter.
By appropriately selecting the resistor and capacitor values, as well as the op-amp characteristics, the Sallen-Key filter can be designed to achieve the desired frequency response for various filtering applications.
The basic Sallen-Key filter consists of two cascaded RC (resistor-capacitor) filter stages, with an op-amp in between. The op-amp is configured as a non-inverting amplifier, and its gain and frequency response characteristics determine the filter's behavior. The two stages of the Sallen-Key filter work together to achieve the second-order filtering.
Here's a step-by-step explanation of how the Sallen-Key filter provides second-order filtering:
First RC Stage:
The first RC stage consists of a resistor (R1) and a capacitor (C1). The input signal is applied to the non-inverting input (+) of the op-amp. The op-amp amplifies the input signal, and the voltage across C1 is fed back to the inverting input (-) of the op-amp. This feedback network, together with C1, forms a high-pass filter with a cutoff frequency determined by the values of R1 and C1.
Second RC Stage:
The second RC stage consists of another resistor (R2) and capacitor (C2). The output of the first stage is connected to the non-inverting input (+) of the op-amp in the second stage. The voltage across C2 is fed back to the inverting input (-) of the op-amp. This feedback network, together with C2, forms a low-pass filter with a cutoff frequency determined by the values of R2 and C2.
Overall Transfer Function:
The cascaded combination of the first and second stages results in a second-order filter with a transfer function that combines the characteristics of the high-pass and low-pass filters. The overall transfer function H(s) of the Sallen-Key filter is a function of the complex frequency 's' and is given by:
H(s) = K / (s^2 + (ω0/Q)s + ω0^2)
where:
K is the overall gain of the filter (usually set by resistor ratios).
ω0 is the angular cutoff frequency (in radians per second).
Q is the quality factor of the filter, which determines the shape of the frequency response.
Frequency Response:
The frequency response of the Sallen-Key filter depends on the value of Q. For Q values greater than 0.5, the filter behaves as a band-pass filter with a peak at the cutoff frequency. For Q values less than 0.5, the filter behaves as a low-pass filter. As Q approaches 0.5, the filter's response becomes flat, resembling a Butterworth filter.
By appropriately selecting the resistor and capacitor values, as well as the op-amp characteristics, the Sallen-Key filter can be designed to achieve the desired frequency response for various filtering applications.