An active filters means using amplifiers to improve the filter. An acive second-order RC low-pass filter still has two RC components in series.

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1 Active Filters An active filters means using amplifiers to improve the filter. An acive second-order low-pass filter still has two components in series. Hjω ( ) = = jω ω B 2jω ω B ( ω ω B ) 2 G( ω) = = 2( ω ω B ) 2 ( ω ω B ) 4 The amplifier buffer changes the impedance. This filter is sharper than the non-buffered filter ( ω ω B ) 2 of 2

2 Gyrator Active feedback alters the effective impedance of an element. The gyrator alters the apparent impedance of the load element. The op-amp rules are used to find the effective input impedance. On the non-inverting input: On the inverting input: i in Z load i in = i in Z load = Z load v = Z load ( out Z load v i ) in in i in -- Z load = v Z in = v load Z in load So and the impedance is the negative of the load. Z in = = Z i load in 2 of 2

3 Artificial Inductor If the load on a gyrator is a capacitor the phase behaves like an inductor. Z in = = jω j ω This would not have the frequency dependence of an inductor. With two gyrators a capacitor can act as an inductor in phase and frequency. Z in L = 2 GY GY Z in ( jω) = = jω 2 jω Inductors are difficult to miniaturize and have intrinsic resistance that may be undesirable. The inductor can be replaced by a gyrator circuit. 3 of 2

4 Active Bandpass A single pole filter uses a parallel L circuit as the feedback network. L 2 Use a notch filter in a inverting amplifier also makes a bandpass filter. Twin-T 2 At the notch frequency the Twin-T has infinite impedance, A = - 2 /. If >> Z TT, for other frequencies A = - TT / = 0. 4 of 2

5 Bootstrapping Bootstrapping refers to the use of feedback to make a very large input impedance by setting i in = 0 through a coupling capacitor for A frequencies of interest. 2 = The circuit is set for an A input signal with capacitive coupling to the non-inverting input. The feedback circuit will give unity gain of the A part of the signal. If the A part of the signal can pass through the coupling capacitor, it can also pass through the capacitor at the inverting input. The signal is the same on both sides of resristor. Since the voltage drop across = 0 and no current enters the opamp, i in = 0. The input impedance Z in = /i in, and should be very large. i in 5 of 2

6 Sallen-Key Filter The Sallen-Key filter looks like two filters (two pole) and a x amplifier (buffer). There is a bootstrap to create a large input impedance. For example, a high-pass Sallen-Key filter uses a resistor as the bootstrap. 2 i in 2 = The breakpoint frequency is ω b = The roll-off for very low frequencies is 40 db/decade or 2 db/octave. 6 of 2

7 Low Pass Sallen-Key The low-pass Sallen-Key filter swaps the resistors and capacitors. v 2 in 2 The circuit behavior is equivalent to a damped driven mechanical oscillator. The driving force is. The damping factor is d 0 = --- = ( Q 2 ) 2 ω b As a passive network, the oscillator can be overdamped, underdamped or critically damped: overdamped (d 2 0 > 2), underdamped (d 2 0 < 2), critically damped (d 2 0 = 2). 7 of 2

8 Voltage-ontrolled Voltage Source A voltage-controlled voltage source (VVS) replaces the Sallen-Key unity gain buffer with a noninverting amplifier. The high-pass VVS has two additional resistors to create a gain A A = 3 / 4 The gain is usually expressed as a factor K. 3 K = A = of 2

9 VVS Damping As with the Sallen-Key, the low-pass VVS swaps pairs of resistors and capacitors A = 3 / 4 With a variable gain for negative feedback and matching = 2 =, = 2 =, the gain and damping are independent of the break frequency = The gain remains A 0 = ω B The damping is 3 d 0 = 3 A 0 = of 2

10 Butterworth Low-Pass The VVS can be used to make an active filter version of the Butterworth filter. 4 3 K = 3 / 4 The desired damping is d 0 =.44, so K = 3 d 0 =.586. The ratio 3 / 4 = 2 d 0 = For a cutoff frequency at KHz, ω B = 6.28 x 0 3 s -. esistors are best in range from KΩ to 0 KΩ So typical = /ω B ; about 0. μf at.5 KΩ. High pass just inverts and in the circuit. Higher order filters requires multiple stages, with K-values set for each stage. 0 of 2

11 Bessel and hebyshev VVS The VVS can also be used to make other special purpose active filters. The Bessel filter needs a normalizing factor f n for the frequency: = /f n ω B. The gain for a Bessel filter requires K =.268, and the frequency factor is f n =.272. The hebyshev filter has an additional variable that controls the ripple band size. For a ripple band of 0.5 db: K =.842, f n =.23. For a ripple band of 2.0 db: K = 2.4, f n = Users of specific values and higher order filters rely on tanles of K and f n values for circuit design. of 2

12 Bandpass VVS Filter The VVS circuit can also be used to create a bandpass filter. Use one-pole for each cutoff frequency. 4 3 v 5 in 2 2 The cutoff frequencies are at / and / of 2