Active Filters

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

R R − vout vin + C C

1 2 1 Hj()ω ==⎛⎞------⎝⎠ωω⁄ 2 1 + j B ωω⁄ωω()⁄ 12+ j B – B 1 1 G()ω ==------2 ()ωω⁄ 2 ()ωω⁄ 4 1 + ()ωω⁄ 12++B B B • The buffer changes the impedance. • This filter is sharper than the non-buffered filter.

LABORATORY ELECTRONICS II 1 of 12

• Active feedback alters the effective impedance of an element. • The gyrator alters the apparent impedance of the load element. vin Zload R iin − vout +

iin v in R • The op-amp rules are used to find the effective input impedance. On the non-inverting input:

vin = vout + iinR On the inverting input: Z Z load load () vin ==------vout ------vin – iinR Zload + R Zload + R

1 Zload + R 1 iin ==--- 1–------vin –------vin R Zload Zload

vin So and the impedance is the negative of the load. Zin ==------–Zload iin

LABORATORY ELECTRONICS II 2 of 12 Artificial

• If the load on a gyrator is a capacitor the phase behaves like an inductor. 1 j Z = –------= ------in jωC ωC • This would not have the frequency dependence of an inductor.

• With two a capacitor can act as an inductor in phase and frequency.

Zin GYR R R C 2 GYR LR= C R

–RR()+ 1 ⁄ jωC 2 Z ==– R + ------jωR C in 1 ⁄ jωC

• are difficult to miniaturize and have intrinsic resistance that may be undesirable. • The inductor can be replaced by a gyrator circuit.

LABORATORY ELECTRONICS II 3 of 12 Active Bandpass

• A single pole filter uses a parallel RLC circuit as the feedback network. C

L

R vin 2 − vout + R1

• Use a notch filter in a inverting amplifier also makes a bandpass filter.

Twin-T

vin R2 − vout + R1

At the notch frequency the Twin-T has infinite impedance, A = -R2/R1.

If R1 >> ZTT, for other frequencies A = -RTT/R1 = 0.

LABORATORY ELECTRONICS II 4 of 12 Bootstrapping

• Bootstrapping refers to the use of feedback to make a very large input impedance by setting iin = 0 through a coupling capacitor for AC frequencies of interest.

vin vin

R2 C − vout = vin R1 + C vin vin

iin • The circuit is set for an AC input signal with capacitive coupling to the non-inverting input. • The feedback circuit will give unity gain of the AC part of the signal. • If the AC 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 R1.

• Since the voltage drop across R1 = 0 and no current enters the opamp, iin = 0.

• The input impedance Zin = vin/iin, and should be very large.

LABORATORY ELECTRONICS II 5 of 12 Sallen-Key Filter

• The Sallen-Key filter looks like two RC filters (two pole) and a x1 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. R1 vin

− C1 C2 vout = vin + vin vin vin R2 iin

• The breakpoint frequency is

ω 1 b = ------R1C1R2C2

• The roll-off for very low frequencies is 40 dB/decade or 12 dB/octave.

LABORATORY ELECTRONICS II 6 of 12 Low Pass Sallen-Key

• The low-pass Sallen-Key filter swaps the resistors and capacitors. C1

− R1 R2 vout vin +

C2

• The circuit behavior is equivalent to a damped driven mechanical oscillator.

• The driving force is vin. • The damping factor is 1 d ==---- ()R + R C ω 0 Q 1 2 2 b • As a passive network, the oscillator can be overdamped, underdamped or critically damped: 2 overdamped (d0 > 2), 2 underdamped (d0 < 2), 2 critically damped (d0 = 2).

LABORATORY ELECTRONICS II 7 of 12 Voltage-Controlled Voltage Source

• A voltage-controlled voltage source (VCVS) replaces the Sallen-Key unity gain buffer with a non- inverting amplifier. • The high-pass VCVS has two additional resistors to create a gain A.

R4 R3 R1

C1 − vout + vin C2 R2 A = 1 + R3/R4

• The gain is usually expressed as a factor K. R KA==1 + -----3- R4

LABORATORY ELECTRONICS II 8 of 12 VCVS Damping

• As with the Sallen-Key, the low-pass VCVS swaps pairs of resistors and capacitors.

R4 R3 C1

R1 − vout + vin R2 C2 A = 1 + R3/R4

• With a variable gain for negative feedback and matching R1 = R2 = R, C1 = C2 = C, the gain and 1 damping are independent of the break frequency ω = ------. B RC R ------3 • The gain remains A0 = 1 + . R4 R -----3- • The damping is d0 ==3 –2A0 – R4

LABORATORY ELECTRONICS II 9 of 12 Butterworth Low-Pass

• The VCVS can be used to make an active filter version of the .

R R C 4 3

R − vout + v in R C K = 1 + R3/R4

− • The desired damping is d0 = 1.414, so K = 3 d0 = 1.586. − The ratio R3 / R4 = 2 d0 = 0.586.

ω 3 -1 • For a cutoff frequency at 1 KHz, B = 6.28 x 10 s . Resistors are best in range from 1 KΩ to 10 KΩ ω μ Ω So typical C = 1/R B; about 0.1 F at 1.5 K .

• High pass just inverts R and C in the circuit. • Higher order filters requires multiple stages, with K-values set for each stage.

LABORATORY ELECTRONICS II 10 of 12 Bessel and Chebyshev VCVS

• The VCVS can also be used to make other special purpose active filters.

ω • The needs a normalizing factor fn for the frequency: RC = 1/fn B.

The gain for a Bessel filter requires K = 1.268, and the frequency factor is fn= 1.272.

• The has an additional variable that controls the ripple band size.

For a ripple band of 0.5 dB: K = 1.842, fn= 1.231.

For a ripple band of 2.0 dB: K = 2.114, fn= 0.907.

• Users of specific values and higher order filters rely on tanles of K and fn values for circuit design.

LABORATORY ELECTRONICS II 11 of 12 Bandpass VCVS Filter

• The VCVS circuit can also be used to create a bandpass filter. • Use one-pole for each cutoff frequency.

R R R1 4 3

R5 − v vin + out

C2 R2 C1

• The cutoff frequencies are at 1/R1C1 and 1/R2C2.

LABORATORY ELECTRONICS II 12 of 12