Roll-off

This article is about roll-off in electrical network analysis. For the dumpster, see roll-off (dumpster).

Roll-off is a term commonly used in electrical network analysis, most especially in connection with filter circuits, to describe the steepness of a network's transmission function with frequency in the transition between a passband and a stopband. It is most typically applied to the insertion loss of the network, but could in principle be applied to any relevant function of frequency. 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 10-times increase in frequency, or decibels per octave (dB/8ve), where an octave is 2-times increase in frequency.

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. This depends 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.

Roll-off of a first order low-pass filter at 6dB/8ve (20dB/decade)

A simple first order network such as a RC circuit will have a roll-off of 20dB/decade. This is approximately equal (to within normal engineering required accuracy) to 6dB/8ve 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:

$A={\frac {V_{o}}{V_{i}}}={\frac {1}{1+i\omega RC}}$

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

$|A|^{2}={\frac {1}{1+\left({\omega \over \omega _{c}}\right)^{2}}}={\frac {1}{1+\omega ^{2}}}$

In decibels this becomes,

$10\log \left({\frac {1}{1+\omega ^{2}}}\right)$

or expressed as a loss,

$L=10\log \left({1+\omega ^{2}}\right)\ \mathrm {dB}$

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

$L\approx 10\log \left(\omega ^{2}\right)=20\log \omega \ \mathrm {dB}$

Roll-off is given by,

$\Delta L=20\log \left({\omega _{2} \over \omega _{1}}\right)\ \mathrm {dB/interval_{2,1}}$

For a decade this is;

$\Delta L=20\log 10=20\ \mathrm {dB/decade}$

and for an octave,

$\Delta L=20\log 2\approx 20\times 0.3=6\ \mathrm {dB/8ve}$

Higher order networks

Multiple order RC filter buffered between stages.

Roll-off graph higher orderr 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;

$A_{\mathrm {T} }=A^{n}$

consequently, the total roll-off is given by,

$\Delta L_{T}=n\Delta L=6n\ \mathrm {dB/8ve}$

Even if the sections are not identical, the roll-off will still tend to 6n dB/8ve at a frequency well above the highest ω_c when buffered sections are being considered. A similar effect can be achieved in the digital domain by repeatedly applying the same filtering algorithm to the signal.

Passive networks

LC low-pass ladder circuit

For entirely passive topologies the situation is a little more complicated. In the popular ladder topology construction of passive filters, each section of the filter has an effect on its immediate neighbours and a lesser effect on more remote sections. The nature of the transfer function is highly dependent on the design realisation method used. For an order n Butterworth filter the roll-off actually is asymptotic to 6n dB/8ve. This filter is designed to have a monotonically increasing loss with frequency tending towards the asymptotic value with increasing frequency. However, for other filters using exactly the same topology, but a different realisation method, the result can be radically different. Examples here are the Chebyshev filter and elliptic filter, both of which have faster roll-offs than the Butterworth and are anything but monotonic. These filters will all 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.

Applications

Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM systems. 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 6dB/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.

See also

• Bode plot

References

• J. William Helton, Orlando Merino, Classical control using H [infinity] methods: an introduction to design, pages 23-25, Society for Industrial and Applied Mathematics 1998 ISBN 0898714249.
• Todd C. Handy, Event-related potentials: a methods handbook, pages 89-92, 107-109, MIT Press 2004 ISBN 0262083337.
• Fay S. Tyner, John Russell Knott, W. Brem Mayer (ed.), Fundamentals of EEG Technology: Basic concepts and methods, pages 101-102, Lippincott Williams & Wilkins 1983 ISBN 089004385X.