Active Filter

A. Marzuki

1 Introduction

2 First- Order Filters

3 Second-Order Filters

4 Other type of Filters

5 Real Filters

6 Conclusion

Table of Contents

•Consist of only amplifiers, resistors, and capacitors.

•Use resistors and capacitors in their feedback loops, to synthesize the desired filter characteristics.

•have high input impedance, low output impedance, and virtually any arbitrary gain.

•They are also usually easier to design than passive filters.

•They lack inductors.

Active filters Introduction

.•Performance at high frequencies is limited by the gain-bandwidth product of the amplifying.

•The op amp-based active filter can achieve very good accuracy, provided that low-tolerance resistors and capacitors are used.

•Active filters will generate noise due to the amplifying circuitry, but this can be minimized by the use of low-noise amplifiers and careful circuit design.

A standard form of the transfer function of a first-order low-pass filter is

Where TLP (j0) is the value of TLP (s) at DC and ωo pole frequency, It is common practice to normalize both the magnitude and frequency. Normalizing (1) yields

(1)

First-Order Transfer Functions

(2)

(3)

(4)

(5)

First order has 6dB roll off and increase rate

First-Order Filter Realizations

A first-order filter has only one pole and zero. For stability, the pole must be on the negative real axis but the zero may be on the negative or positive real axis. A singleamplifier low-gain realization of a first order low-pass filter is shown in below

The low-frequency (ω << ωo) gain magnitude and polarity are set by K. First-order high-pass filters can also be realized in a single-amplifier low-gain circuit.The transfer function for this filter is given in (4), and in this case THP(j∞) = K. Note that the first-order filters exhibit no peaking and the frequency-dependent part of the magnitude response approaches ±20 dB/decade.

The transfer function for this filter is

(6)

More….

Transfer function for low pass filter shown below is

DC Gain is And o is 1/CR2

More….

Alternative to low pass Filter with transfer function

Where the gain

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high pass Filter with transfer function

High Frequency Gain is o is 1/CR1

More….

Alternative to high pass Filter with transfer function

Where the gain, K =

o

Second-Order Transfer Functions

The standard form of a second-order low-pass filter is given as

The poles of the above transfer function (8) of are

Normalization of in both amplitude and frequency gives

(7)

(8)

(9)

(10)

(11)

(9)

(12)

(13)

Frequency Response (Magnitude and Phase)

(14)

(15)

If Q is greater than 0.707, then the normalized magnitude response has a peak value of

Butterworth1. The first, and probably best-known filter approximation is the Butterworth or maximally-flat

response.

2. nearly flat passband with no ripple. The rolloff is smooth and monotonic, with a low-pass or high-pass rolloff rate of 20 dB/decade (6 dB/octave) for every pole.

3. Thus, a 5th-order Butterworth low-pass filter would have an attenuation rate of 100 dB for every factor of ten increase in frequency beyond the cutoff frequency.

(16)

General Equation for amplitude response

Amplitude Response Curves of Low Pass FilterFor various order.

More Type Of FiltersChebyshev

Another approximation to the ideal filter is the Chebyshev or equal ripple response. As the latter name implies, this sort of filter will have ripple in the passband amplitude response. The amount of passband ripple is one of the parameters used in specifying a Chebyshev filter. The Chebyschev characteristic has a steeper rolloff near the cutoff frequency when compared to the Butterworth, but at the expense of monotonicity in the passband and poorer transient response.

Bessel exhibits approximately linear phase shift with frequency, so its action within the passband simulates a

delay line with a low-pass characteristic. The higher the filter order, the more linear the Bessel's phase response.

EllipticThe cutoff slope of an elliptic filter is steeper than that of a Butterworth, Chebyshev, or Bessel, but the

amplitude response has ripple in both the passband and the stopband, and the phase response is very non-linear. However, if the primary concern is to pass frequencies falling within a certain frequency band and reject frequencies outside that band, regardless of phase shifts or ringing, the elliptic response will perform that function with the lowest-order filter.The elliptic function gives a sharp cutoff by adding notches in the stopband. These cause the transfer function to drop to zero at one or more frequencies in the stopband. Ripple is also introduced in the passband. An elliptic filter function can be specified by three parameters (again excluding gain and cutoff frequency): passband ripple, stopband attenuation, and filter order n. Because of the greater complexity of the elliptic filter, determination of coefficients is normally done with the aid of a computer.

Second-Order Filters

These filters are easy to design and have been extensively used in many applications. The first realization is a low-pass Sallen and Key filter.

Low Pass Filter.

High Pass Filter

High Order FiltersThe filter order is the number of poles. An N-pole active filters have rolloff rate of Nx6dB/octave and increase rate of Nx6dB/octave. For Butterworth Filter, the solution is solved by taking first and second derivatives of the Transfer function.

/1.082

/0.9241

/2.613

/0.3825

4-pole Butterworth High Pass Filter

3dB =1/RC

More….

Low pass Filter

4-pole Butterworth Low Pass Filter

0.9241

1.082

0.3825

2.613

3dB =1/RC

Other type of FiltersSwitched Capacitor Filter,1. need no external resistors, capacitors or inductors,2. No internal resistor3. cutoff frequencies are set to a typical accuracy of 0.2% by an external clock frequency.4. filters whose cutoff frequencies are variable over a wide range simply by changing the clock

frequency.

Switched Current Filter,1. need no resistors, capacitors or inductors,2. filters whose cutoff frequencies are variable over a wide range simply by changing the clock

frequency.

Design Steps

1. Hand analysis for simple filter designs.2. Employ synthesis program for complex designs.3. Use analysis EDA tools such as SPICE and etc.4. Prototyping and Characterization.

Real Active Filters

Mores

Conclusion

1. Cover First Order and its basic circuit.2. Talked on 2nd Order and its basic circuit.3. Brief description on Switched Capacitor Filter.4. Active circuits are real components and there are many applications that use

them.

ReferencesSedra and Smith, Microelectronic Cicruits, 4th Edition, Oxford, 1998.

D. A. Neaman, Electronic Circuit Analysis and Design, 2th Edition, McGRAW-HILL, 2001.

The Circuits and Filters Handbook, Second Edition

Continuous-Time Active Filter Design

Kerry Lacanette, “A Basic Introduction to Filters–Active, Passive, and Switched-Capacitor,” National Semiconductor, Application Note 779, April 1991.