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. The function of frequency most usually being considered is the insertion loss of the network, but could in principle be any function of frequency related to the network. 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) or decibels per octave (dB/8ve). Decade being a x10 increase in frequency and octave being a x2 increase in frequency. 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.

First order roll-off

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

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

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.

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, Fundamentals of EEG Technology: Basic concepts and methods, pages 101-102, Lippincott Williams & Wilkins 1983 ISBN 089004385X.