Analog Design-filters Introduction and Motivation Filters are networks that process signals in a frequency-dependent manner. The basic concept of a filter can be explained by examining the frequency dependent nature of the impedance of capacitors and inductors. Consider a voltage divider where the shunt leg is a reactive impedance. As the frequency is changed, the value of the reactive impedance changes, and the voltage divider ratio changes. This mechanism yields the frequency dependent change in the input/output transfer function that is defined as the frequency response. Filters have many practical applications. A simple, single-pole, low-pass filter (the integrator) is often used to stabilize amplifiers by rolling off the gain at higher frequencies where excessive phase-shift may cause oscillations. A simple, singlepole, high-pass filter can be used to block dc offset in high gain amplifiers or single supply circuits. Filters can be used to separate signals, passing those of interest, and attenuating the unwanted frequencies. An ideal filter will have an amplitude response that is unity (or at a fixed gain) for the frequencies of interest (called the pass band) and zero everywhere else (called the stop band). The frequency at which the response changes from passband to stopband is the cut-off frequency. Prerequisites Basics like resistors and capacitors, op-amps. Learning Outcome Knowledge of Various Topologies of Filters. Transfer function of various filters. Ability to choose a filter for a particular application. Knowledge of design parameters and specifications for filters. Knowledge of characteristics of the filters discussed in chapter.
Ability to design a filter on given specifications and topologies. Suggested Time 12 hours Filter Parameters Hundreds of filters have been developed to meet the needs of various applications. Despite this variety, many filters can be described by a few common characteristics. The five parameters of a practical filter are defined below.
a. Corner Frequency FC : As a real filter rolls off gradually, the corner/cutoff frequency is specified as the frequency at which the response is times of that in the pass band. This frequency is also known as -3 db point. b. Stop-Band Frequency FS : The stop band frequency is the frequency at which the minimum attenuation in the stopband is reached. c. Pass-Band Ripple AMAX: For many types of filters, the response does not decrease monotonically as frequency moves from the center of the pass band out towards the stop band. The magnitude of the response may vary inside the pass band, and this variation is called the pass-band ripple. It is also specified in decibels. Pass-band ripple causes the frequency components of a signal to be amplified to different degrees. This has the effect of distorting the waveform of a signal passing through the filter. d. Minimum Stop-Band Attenuation AMIN : It defines the minimum signal attenuation within the stop-band. e. Attenuation Rate : The transition between the pass band and the stop band is a continuous function, and the rate at which this transition occurs is called the attenuation rate. The attenuation rate in expressed in decibels per decade. A high attenuation rate helps a filter distinguish between signals of similar frequency and is usually a desirable feature. The attenuation rate is also related to the order of a filter. A first-order filter will have an attenuation rate of 20 db/decade, while a fourth-order filter will have an attenuation rate approaching 80 db/decade. Many filters fall into one of the following response categories, based on the overall shape of their pass band.
a. Low-pass filters pass low-frequency signals while blocking high-frequency signals. The pass band ranges from DC (0 Hz) to a corner frequency FC. b. High-pass filters pass high-frequency signals while blocking DC and lowfrequency signals. The pass band ranges from a corner frequency (FC) to infinity. c. Band-pass filters pass only signals between two given frequencies, blocking lower and higher signals. The pass band lies between two frequencies, FL and FH. The pass band of these filters is often characterized as having a bandwidth that is symmetric around a center frequency. d. Band-stop filters block signals occurring between two given frequencies, FL and FH. The pass band is split into a low side (DC to FL) and a high side (FH to infinity). Band-stop filters are also called notch filters, especially when the stop band is narrow. The figure below shows how each of these filters operates on a swept-frequency input signal.
Filter Topology Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter topologies may be divided into passive and active types. Passive topologies contain passive devices in the circuit like capacitors, inductors and resistors. Active topologies include active components like amplifier. Passive Topologies Passive filters have been long in development and use. Most are built from simple two-port networks called "sections". A section must have at least one series component and one shunt component. Sections are invariably connected in a cascade or daisy-chain topology, consisting of either repeats of the same section or of completely different sections. Some passive filters, consisting of only one or two filter sections, are given special names including the L-section, T-section and Π-section, which are unbalanced filters, and the C-section, H-section and box-section, which are balanced. All are built upon a very simple "ladder" topology. A ladder network consists of cascaded asymmetrical L-sections (unbalanced) or C- sections (balanced). In low-pass form, the topology would consist of series inductors and shunt capacitors. Other band-forms would have an equally simple topology transformed from the low-pass topology.the figures below show the various ladder topologies.
The figure shown below uses L-Half Section topology to create low-pass and highpass filters.
Some other topologies used are the Bridger-T topologies and the lattice topology. Active Topologies An active filter is a type of analog electronic filter that uses active components such as an amplifier, while avoiding the need for inductors which are typically expensive, bulky and have poor tolerances compared with resistors and capacitors. Additionally, because inductors generate magnetic fields, they can cause unwanted interference with other nearby components. An amplifier prevents the load impedance of the following stage from affecting the characteristics of the filter. An active filter can have complex poles and zeros
without using a bulky or expensive inductor. The shape of the response, the Q (quality factor), and the tuned frequency can often be set with inexpensive variable resistors. Using active elements has some limitations. Basic filter design equations neglect the finite bandwidth of amplifiers. Available active devices have limited bandwidth, so they are often impractical at high frequencies. Amplifiers consume power and inject noise into a system. Sallen-Key Topology The Sallen-Key topology is an electronic filter topology used to implement secondorder active filters. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology. A VCVS filter uses a super-unity-gain voltage amplifier with practically infinite input impedance and zero output impedance to implement a 2- pole low-pass, high-pass, or bandpass response. The super-unity-gain amplifier allows for very high Q factor and passband gain without the use of inductors. A Sallen-Key filter is a variation on a VCVS filter that uses a unity-gain amplifier (a pure buffer amplifier with 0 db gain). Because of its high input impedance and easily selectable gain, an operational amplifier in a conventional non-inverting configuration is often used in VCVS implementations. Implementations of Sallen-Key filters often use an operational amplifier configured as a voltage follower; however, emitter or source followers are other common choices for the buffer amplifier. Higher-order filters can be obtained by cascading two or more stages. Generic Sallen-Key Topology The generic unity-gain Sallen Key filter topology implemented with a unity-gain operational amplifier is shown below.
Since the op-amp is connected in negative-feedback configuration, both the input terminals must have the same voltage. By choosing different passive components (resistors and capacitors) for Z1, Z2, Z3 and Z4 the filter can be made with low-pass, bandpass, and high-pass characteristics. Application: Low-Pass Filter An example of a unity-gain low-pass configuration is shown in the figure below.
The Q factor determines the height and width of the peak of the frequency response of the filter.
Application: High-Pass Filter A second-order unity-gain high-pass filter is shown in the figure below. Multiple Feedback Topology Multiple feedback topology is an electronic filter topology which is used to implement an electronic filter by adding two poles to the transfer function. A diagram of the circuit topology for a second order low pass filter is shown below.
The transfer function of the multiple feedback Topology circuit is: Biquad Filter A biquad filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name biquad is short for biquadratic. Given below is an example of Tow-Thomas Biquad. The basic configuration shown below can be used as either a low-pass or bandpass filter depending on where the output signal is taken from.