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Sallen-Key filters

What are Sallen-Key filters?

The Sallen–Key topology is used to implement second-order active filters that easily enhance the Q of the filter using controlled positive feedback. The topology is more simple in comparison to other active filter topologies. In 1955, R. P. Sallen and E. L. Key described these filter circuits, and hence they are generally known as Sallen-Key filters.

Sallen Key Filter Calculator

While Sallen-Key might be more beneficial in terms of component count and ease of use, Multiple-Feedback (MFB) Filters are better in terms of distortion and stability.

Second order passive low-pass filter

Figure 1 shows a RC-RC network that forms a second-order low-pass filter. With R1=R2=R and C1=C2=C the quality factor Q becomes 1/3. Q approaches the maximum value of 1/2 if the impedance of the second RC stage is made much larger than the first. Most filters require Q larger than 1/2.

second order passive low pass filter

Fig 1 : Passive second order low pass filter

The transfer function of a passive second-order low-pass filter is :

$$\cfrac{V_{out}}{V_{in}}=\cfrac{1}{s^2R^2C^2+3sRC+1}$$

$$Q=\cfrac{1}{3}$$

$$\omega{}_o=\cfrac{1}{RC}$$

Sallen-Key low pass filter circuit

Larger Qs are attainable by using a positive feedback. If the positive feedback is controlled, and localized to the cut-off frequency of the filter, almost any Q can be realized. Obtaining higher Q is limited mainly by the component tolerances and opamps gain-bandwidth product. The filter cutoff frequency should be much less than opamps's gain-bandwidth product. Figure 2 shows an opamp in non-inverting gain used in this manner. Capacitor C1, which is not connected to the ground, provides a positive feedback path.

sallen key low pass filter

Fig 2 : A Sallen-Key second order active low pass filter

If R1=R2=R and C1=C2=C, the Q factor is :

From the above equation, it can be concluded that Q > 1/2 can be achieved if Gain (G) > 1. This achievable Q is higher than any passive filter.

Also, keeping G greater than or equal to 3 will make the system unstable. This is because it will cause at least one pole to appear on the right-hand side of s-plane. This results in either perpetual oscillation or exponential growth in the output amplitude.

active_filter_sallen_key_lpf_explaination_near_fo-1

Fig 3 : How Sallen-Key gets high Q

Intuitively, we can understand the operation in the following steps:

  1. At low frequencies, where C1 and C2 appear as open circuits, the system behaves as an opamp in a non-inverting configuration providing a gain of G (=1+Rf/Rg).
  2. At high frequencies, where C1 and C2 appear as short circuits, the signal is attenuated at the amplifier’s input, the amplifier amplifies this input to its output, and the signal does not appear at Vo.
  3. Near the cut-off frequency, if gain G is set close to but less than 3 (e.g., 2.5), Vo becomes very high which is basically feeding current through C1 from Vo to vx. This increases the input at the non-inverting pin (vy) resulting in high Q of this filter.

Second order passive high-pass filter

Figure 4 shows a two-stage RC network that forms a second-order high-pass filter. For simplicity, if R1=R2=R and C1=C2=C. This results in Q=1/3. This low Q results in attenuation in the passband and not enough attenuation in the stop band. A higher Q is desired. A higher Q is achievable using a Sallen-Key topology .

second_order_passive_hpf-1

Fig 4 : A second order passive low pass filter

The transfer function of a passive second-order high-pass filter is :

Sallen-Key high pass filter circuit

sallen_key_hpf-1

Fig 5 : Sallen-Key second order active high pass filter

Figure 5 mentions a Sallen-Key high pass filter. Instead of R1 connected to the ground, it is connected to the output of the amplifier to provide positive feedback. Intuitively, we can understand the operation in the following steps:

  1. At low frequencies, where C1 and C2 appear as open circuits, R2 is pulling down the non-inverting node to the ground. This means no signal is getting amplified.
  2. At high frequencies, where C1 and C2 appear as short circuits, vi completely appears at the non-inverting pin of the opamp because the impedance of C2 is lower than R2.
  3. Near the cut-off frequency, similar to the Sallen-Key low pass filter, a current is injected through R1 into C2 from the Vo node. This leads to higher signal strength at the non-inverting pin which results in higher Q.

Generalized Sallen-Key filter

Generalized Sallen Key

The circuit illustrated above represents a generalized version of the Sallen-Key configuration, where the passive filter elements are expressed as generalized impedances (Z), and the resistors Rf and Rg determine the passband gain.

Derivation

KCL at note Vf: $$V_f\left(\cfrac{1}{Z_1}+\cfrac{1}{Z_2}+\cfrac{1}{Z_4}+\cfrac{1}{Z_5}\right)=V_i\left(\cfrac{1}{Z_1}\right)+V_p\left(\cfrac{1}{Z_2}\right)+V_o\left(\cfrac{1}{Z_4}\right)$$ KCL at note Vp: $$V_p\left(\cfrac{1}{Z_2}+\cfrac{1}{Z_3}\right)=V_f\left(\cfrac{1}{Z_2}\right)$$ $$V_f=V_p\left(1+\cfrac{Z_2}{Z_3}\right)$$ KCL at node Vn: $$V_n\left(\cfrac{1}{R_3}+\cfrac{1}{R_4}\right)=V_o\left(\cfrac{1}{R_4}\right)$$ $$V_n=V_o\left(\cfrac{R_3}{R_3+R_4}\right)$$ Using the virtual short property of opamp : $$V_p=V_n$$ Using above relations, we get : $$\cfrac{V_o}{V_i}=\cfrac{1}{Z_1}\left(\cfrac{1}{\cfrac{1}{K}\left(1+\cfrac{Z_2}{Z_3}\right)\left(\cfrac{1}{Z_1}+\cfrac{1}{Z_2}+\cfrac{1}{Z_4}+\cfrac{1}{Z_5}\right)-\cfrac{1}{Z_2K}-\cfrac{1}{Z_4}}\right)$$ Where K: $$K=1+\cfrac{R_f}{R_g}$$

$$\cfrac{V_o}{V_i}=\cfrac{1}{Z_1}\left(\cfrac{1}{\cfrac{1}{K}\left(1+\cfrac{Z_2}{Z_3}\right)\left(\cfrac{1}{Z_1}+\cfrac{1}{Z_2}+\cfrac{1}{Z_4}+\cfrac{1}{Z_5}\right)-\cfrac{1}{Z_2K}-\cfrac{1}{Z_4}}\right)$$ Where K: $$K=1+\cfrac{R_f}{R_g}$$

Low pass High pass Band pass
$$Z_1$$ $$R_1$$ $$\cfrac{1}{sC_1}$$ $$R_1$$
$$Z_2$$ $$R_2$$ $$\cfrac{1}{sC_2}$$ $$\cfrac{1}{sC_2}$$
$$Z_3$$ $$\cfrac{1}{sC_1}$$ $$R_2$$ $$R_3$$
$$Z_4$$ $$\cfrac{1}{sC_2}$$ $$R_1$$ $$R_2$$
$$Z_5$$ $$\infty{}$$ $$\infty{}$$ $$\cfrac{1}{sC_1}$$

Sallen Key vs Multi-Feedback Filter

Sallen Key LPF
Sallen Key LPF
Multifeedback LPF
MultiFeedback LPF

Sallen Key Vs MultiFeedback Filter

In the AC response of Sallen Key low pass filter, the gain starts increasing after certain frequency due to finite output impedance of the operational amplifier. This is not desired. However, if we observe the multi-feedback filter's response, the gain continue decreasing with frequency which is desired. However, the rate of decrease has reduced.

Equivalent model circuit

The equivalent model which explains the high frequency behaviour of the Sallen Key is shown.

Equivalent model frequency response

At high frequency, the response merges indicating operational-amplifier's non-zero output impedance and finite gain-bandwidth product.

References

  1. Sallen Key low pass filter