Roll-off

Last updated

Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband. It is most typically applied to the insertion loss of the network, but can, in principle, be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of logarithmic frequency; consequently, the units of roll-off are either decibels per decade (dB/decade), where a decade is a tenfold increase in frequency, or decibels per octave (dB/8ve), where an octave is a twofold increase in frequency.

Contents

The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the cut-off point of the frequency curve. Roll-off enables the cut-off performance of such a filter network to be reduced to a single number. Note that roll-off can occur with decreasing frequency as well as increasing frequency, depending on the bandform of the filter being considered: for instance a low-pass filter will roll-off with increasing frequency, but a high-pass filter or the lower stopband of a band-pass filter will roll-off with decreasing frequency. For brevity, this article describes only low-pass filters. This is to be taken in the spirit of prototype filters; the same principles may be applied to high-pass filters by interchanging phrases such as "above cut-off frequency" and "below cut-off frequency".

First-order roll-off

First-order RC filter low-pass filter circuit. First order RC circuit.svg
First-order RC filter low-pass filter circuit.
Roll-off of a first-order low-pass filter at 6 dB/octave (20 dB/decade) Roll-off graph 6dB.svg
Roll-off of a first-order low-pass filter at 6 dB/octave (20 dB/decade)

A simple first-order network such as a RC circuit will have a roll-off of 20 dB/decade. This is approximately equal (to within normal engineering required accuracy) to 6 dB/octave and is the more usual description given for this roll-off. This can be shown to be so by considering the voltage transfer function, A, of the RC network: [1]

Frequency scaling this to ωc = 1/RC = 1 and forming the power ratio gives,

In decibels this becomes,

or expressed as a loss,

At frequencies well above ω=1, this simplifies to,

Roll-off is given by,

For a decade this is;

and for an octave,

Higher order networks

Multiple order RC filter buffered between stages. High order buffered RC circuit.svg
Multiple order RC filter buffered between stages.
Roll-off graph of higher-order low-pass filters showing various rates of roll-off Roll-off graph multiple.svg
Roll-off graph of higher-order low-pass filters showing various rates of roll-off

A higher order network can be constructed by cascading first-order sections together. If a unity gain buffer amplifier is placed between each section (or some other active topology is used) there is no interaction between the stages. In that circumstance, for n identical first-order sections in cascade, the voltage transfer function of the complete network is given by; [1]

consequently, the total roll-off is given by,

A similar effect can be achieved in the digital domain by repeatedly applying the same filtering algorithm to the signal. [2]

LC low-pass ladder circuit. Each element (that is L or C) adds an order to the filter and a pole to the driving point impedance. LC ladder circuit.svg
LC low-pass ladder circuit. Each element (that is L or C) adds an order to the filter and a pole to the driving point impedance.

The calculation of transfer function becomes somewhat more complicated when the sections are not all identical, or when the popular ladder topology construction is used to realise the filter. In a ladder filter each section of the filter has an effect on its immediate neighbours and a lesser effect on more remote sections so the response is not a simple An even when all the sections are identical. For some filter classes, such as the Butterworth filter, the insertion loss is still monotonically increasing with frequency and quickly asymptotically converges to a roll-off of 6n dB/8ve, but in others, such as the Chebyshev or elliptic filter the roll-off near the cut-off frequency is much faster and elsewhere the response is anything but monotonic. Nevertheless, all filter classes eventually converge to a roll-off of 6n dB/8ve theoretically at some arbitrarily high frequency, but in many applications this will occur in a frequency band of no interest to the application and parasitic effects may well start to dominate long before this happens. [3]

Applications

Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM systems. [4] Roll-off is also significant on audio loudspeaker crossover filters: here the need is not so much for a high roll-off but that the roll-offs of the high frequency and low-frequency sections are symmetrical and complementary. An interesting need for high roll-off arises in EEG machines. Here the filters mostly make do with a basic 6 dB/8ve roll-off, however, some instruments provide a switchable 35 Hz filter at the high frequency end with a faster roll-off to help filter out noise generated by muscle activity. [5]

See also

Notes

  1. 1 2 J. Michael Jacob, Advanced AC circuits and electronics: principles & applications, pages 150-152, Cengage Learning 2003 ISBN   0-7668-2330-X.
  2. Todd, pp 107–108
  3. Giovanni Bianchi, Roberto Sorrentino, Electronic filter simulation & design, pages 129–130, McGraw-Hill Professional 2007 ISBN   0-07-149467-7.
  4. Lundheim, L, "On Shannon and "Shannon's Formula", Telektronikk, vol. 98, no. 1, 2002, pp. 24–25.
  5. Mayer et al, pp 104–105.

Related Research Articles

<span class="mw-page-title-main">Cutoff frequency</span> Frequency response boundary

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter.

<span class="mw-page-title-main">Bode plot</span> Graph of the frequency response of a control system

In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.

<i>Q</i> factor Parameter describing the longevity of energy in a resonator relative to its resonant frequency

In physics and engineering, the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher Q indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping, so that they ring or vibrate longer.

A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters".

<span class="mw-page-title-main">Short-time Fourier transform</span> Fourier-related transform suited to signals that change rather quickly in time

The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time, known as a spectrogram or waterfall plot, such as commonly used in software defined radio (SDR) based spectrum displays. Full bandwidth displays covering the whole range of an SDR commonly use fast Fourier transforms (FFTs) with 2^24 points on desktop computers.

Particle velocity is the velocity of a particle in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.

<span class="mw-page-title-main">Butterworth filter</span> Type of signal processing filter

The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

<span class="mw-page-title-main">LC circuit</span> Electrical "resonator" circuit, consisting of inductive and capacitive elements with no resistance

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

A resistor–inductor circuit, or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, either in series driven by a voltage source or in parallel driven by a current source. It is one of the simplest analogue infinite impulse response electronic filters.

<span class="mw-page-title-main">Isotropic radiator</span>

An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source. Isotropic radiators are used as reference radiators with which other sources are compared, for example in determining the gain of antennas. A coherent isotropic radiator of electromagnetic waves is theoretically impossible, but incoherent radiators can be built. An isotropic sound radiator is possible because sound is a longitudinal wave.

Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

<span class="mw-page-title-main">Fundamental thermodynamic relation</span>

In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G or H (enthalpy). The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.

<span class="mw-page-title-main">Zobel network</span>

Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

<span class="mw-page-title-main">Primary line constants</span> Parameters of transmission lines

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

<span class="mw-page-title-main">RLC circuit</span> Resistor Inductor Capacitor Circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

In physics, and especially scattering theory, the momentum-transfer cross section is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

Staggered tuning is a technique used in the design of multi-stage tuned amplifiers whereby each stage is tuned to a slightly different frequency. In comparison to synchronous tuning it produces a wider bandwidth at the expense of reduced gain. It also produces a sharper transition from the passband to the stopband. Both staggered tuning and synchronous tuning circuits are easier to tune and manufacture than many other filter types.

References