Sallen-Key Filtresi / Sallen and Key Filter

Sallen and switch filter design is a second-order active filter topology that we can use as basic building blocks to implement higher-order filter circuits such as low-pass (LPF), high-pass (HPF), and band-pass (BPF) filter circuits.

As we see in this filters section, passive or active electronic filters are used in circuits where signal amplitude is required only in a limited frequency range. The advantage of using Sallen-Key filter designs is that they are easy to apply and understand.

Sallen and Switch topology evolve operational amplifiers and two resistances, thus creating a voltage-controlled voltage in design with welding (VCVS), high input impedance and low output impedance and good stability filter features- an active filter design based around a single cigarette and such individual Sallen-key filter sections together allow cascading to produce much higher grade filters.

However, before looking at the design and operation of the Sallen-key filter, let's first remember the characteristics of a single resistance capacitor or RC network when exposed to a series of input frequencies.

Voltage Divider

When two (or more) resistors are connected along a DC supply voltage, different voltage values will be developed in each resistance and will basically be called a voltage divider or potential divider network.

Recissive Voltage Divider

Sallen-key Filter
Recissive Voltage Divider

The basic circuit shown consists of two resistances in series connected to vin, which is an input voltage.

The Ohm Act tells us that the voltage that falls along a resistance is multiplied by V = I*r, the reactive value of the current flowing through it, so if the two resistors are equal, the voltage falling along both resistances, R1 and R2, will be equal and divided evenly between them.

Recissive Voltage Divider Transfer Function

Sallen-key Filter
Recissive Voltage Divider Transfer Function

If we change the input voltage to an AC source or signal and change the frequency range, what happens to the output voltage? Since resistors are usually not affected by changes in frequency (except wired Cables), there is nothing, so frequency responses are zero, which allows AC, Irms2*R voltages to develop or fall along the resistors.

RC voltage divider

If we change the R1 resistance above to a capacitor C, as shown, how does this affect our previous transmission function. From our tutorials about capacitors, we know that when a capacitor is connected to a DC voltage source, it acts like an open circuit after charging.

Sallen-key Filter
RC voltage divider

When a fixed DC source is connected to the VIN, the capacitor fully charges after 5 time constants (5T = 5RC). During this time, it does not draw current from the feed. Therefore, there is no current flowing from the resistance. Voltage Drop on R and above has not developed, so there is no output voltage. In other words, after the capacitors are discharged, the constant state blocks DC voltages.

Now if we change the input feed to an AC sinusoidal voltage, the properties of this simple RC circuit completely change when the DC or fixed part of the signal is blocked. Now we're analyzing the RC circuit in the frequency field, part of the signal that's connected to this time.

In an AC circuit, a capacitor has capacitive reactance feature, XC, but we can analyze the RC circuit only as we do with resistance circuits, the difference is that the impedance of the capacitor now depends on the frequency.

For AC circuits and signals, capacitive reactandency (XC) is against alternating current flow from a capacitor measured in Ohm. capacitive reacques are connected to the frequency, that is, at low frequencies (ε ≤ 0), the capacitor acts as an open circuit and blocks them

At very high frequencies (ε ε), the capacitor acts as a short circuit and transmits signals directly to the output as vout = Vin. However, somewhere between these two frequency endpoints, the capacitor has an impedance given by XC:

Sallen-key Filter

RC filter circuit

Sallen-key Filter
RC filter circuit

The graph shows the frequency response of this simple 1st degree RC circuit. At low frequencies, the voltage gain is extremely low as the input signal is blocked by the capacitor's reactance. At high frequencies, the voltage gain is high because it is reactax (unity), causing the capacitor to effectively short-circuit these high frequencies, so VOUT = VIN

However, there is a frequency point where the recessor of the capacitor equals the resistance of resistance, that is: XC = R, and this is called the "critical frequency" point, or more commonly the cutting frequency or corner frequency εC.

Cutting frequency equation

Sallen-key Filter
Cutting frequency equation

The cutting frequency, εC, in this example defines where the circuit changes from weakening or blocking all frequencies below εc and begins to exceed all frequencies above this εC point. Thus, the circuit is called a "high pass filter".

The cutting frequency is where the ratio of the input-output signal has a magnitude of 0.707 and is equal to -3db when converted to decibels. This is often referred to as the 3db down point of filters.

Since the recessor of the capacitor depends on the frequency, that is, capacitive reacquence (XC) varies inversely to the applied frequency, we can change the voltage divider equation above to achieve the transfer function of this simple RC high-pass filter circuit.

RC filter circuit

Sallen-key Filter
RC filter circuit

One of the main drawbacks of an RC filter is that the output amplitude will always be less than the input. That's why it's never bigger than a union. In addition, external loading of the output by more RC stages or circuits will affect the properties of the filters. One way to overcome this problem is to add a process amplifier to the basic RC configuration and convert the passive RC filter to an "active RC filter".

By adding an operational amplifier, the basic RC filter can be designed to provide the required amount of voltage gain at the output. Thus, it changes the filter from a debilitator to an amplifier. In addition, due to its high input impedance and low output impedance, the operational amplifier prevents external loading of the filter and allows it to be easily adjusted in a wide frequency range without changing the designed frequency response.

Active High Pass Filter

Sallen-key Filter
Active High Pass Filter

The RC filter part of the circuit responds in the same way as above, that is, it passes high frequencies, but blocks low frequencies, the cutting frequency is adjusted with R and C values. The operational amplifier, or op-amp for short, is configured as an inverted amplifier whose voltage gain is adjusted by two resistance, R1 and R2 ratios.

Then the closed loop voltage gain is given as follows: AV on the transition band of an inverted operational amplifier:

Interrupt Frequency Equation

Sallen-key Filter
Interrupt Frequency Equation

RC Filter Example

From a simple 1st order, the active high-pass filter must have a cutting frequency of 500hz and a transition band gain of 9db. Calculate the required components assuming that a standard 741 operational amplifier is used.

We found that the cutting frequency from above was determined by the R and C values in the frequency selector RC circuit of εc. Assuming a value for 5kΩ R (which makes any reasonable value), then the C value is calculated as follows:

Sallen-key Filter

The calculated C value is 63.65 nf, so the nearest preferred value used is 62nf.

The gain of the high transition filter in the transition belt zone should be +9db, which equates to 2.83 AV voltage gain. Suppose an arbitrary value for feedback resistance, R1 15kΩ, this resistance gives a value for R1:

Sallen-key Filter

Again, the calculated value of R2 is 8197Ω. The nearest preferred value will be 8200Ω or 8.2 kΩ. This then gives us the final circuit for our active high-pass filter sample:

High Pass Filter Circuit

Sallen-key Filter
High Pass Filter Circuit

We found that filters that pass higher than a simple first order can be made using a cutting frequency, a single resistance and capacitor that produces a point of εC, where the output amplitude is -3dB below the input amplitude. By adding a second RC filter stage to the first, we can convert the circuit from a second degree to a high-pass filter.

Quadrored RC filter

The second simplest RC filter consists of two RC parts cascading together, as shown. However, for this basic configuration to work correctly, the input and output impedances of the two RC stages should not affect each other's work, that is, they should be non-interactive.

High Pass Filter Circuit

Sallen-key Filter
High Pass Filter Circuit

Cascading one RC filter stage with another (the same or different RC values) does not work very well. Because each consecutive stage loads the previous one, and when more RC stages are added, the cutting frequency point moves further away from the designed or required frequency.

One way to overcome this problem for a passive filter design is that the input impedance of the second RC stage is at least 10 times larger than the output impedance of the first RC stage. This cutting frequency is RB = 10*R1 and CB = CA/10.

The advantage of increasing component values by 10 times is that the second order obtained produces a rounding 40dB/decade steeper than the cascading RC stages of the filter. However, if you want to design a 4th or 6th degree filter, calculating ten times the value of previous components can be time consuming and complex.

A simple way to combine RC filter stages that don't interact or load with each other to create higher-level filters (individual filter sections don't have to be the same) that can be easily adjusted and designed to provide the required voltage gain is to use the Sallen-Key filter stages.

Sallen-Key Cutting Frequency Equation

Sallen-key Filter
Sallen-Key Cutting Frequency Equation

If the two serial capacitors CA and CB are made equal (CA = CB = C) and the two resistance ra and RB are made equal (RA = RB = R), the above equation simplifies the original cutting frequency equation:

Sallen-key Filter

Since the transactional amplifier is configured as a union gain buffer, that is, A = 1, cutting frequency, εC and Q are completely independent of each other. It provides a simpler filter design. The magnification factor is then calculated as Q:

Sallen-key Filter

For union-gain buffer configuration, the voltage gain (AV) of the filter circuit is equal to 0.5 or-6db (over-damped) at the cutting frequency point. We expect to see this because a circumstrial filter response is 0.7071*0.7071 = 0.5. So-3dB * – 3dB = – 6dB.

However, since the Q value determines the response properties of the filter, the two feedback resistances of operational amplifiers, the correct selection of R1 and R2, allow us to choose the required transition band gain for the selected magnification factor.

Note that selecting a Sallen-key filter topology too close to the maximum value of 3 will result in high Q values. A high Q filter will make the design susceptible to tolerance changes in the values of R1 and R2 feedback resistors. For example, setting the voltage gain to 2.9 (A = 2.9) is 10 (1/(3-2.9)) of the Q value, making the filter extremely sensitive around εC.

Sallen-Key Filter Response

Sallen-key Filter
Sallen-Key Filter Response

Next, we can see that the lower the Q value, the more stable the Sallen-Key filter design will be. While high Q values can make the design unstable, very high gains producing negative Q will lead to oscillations.

Sallen-Key high pass filter

Sallen-key Filter

Then, with a cutting or corner frequency of 200hz, a transition band gain of 2,667 and maximum voltage gain at the cutting frequency of 8 (2,667*3) due to Q = 3, we can show the properties of this second-degree high-pass Sallen-Key filter in the Bode chart below.

Sallen-key Filter
Sallen-Key Filter Bode Diagram