Rise Time

We have seen how the RC circuit responds to a step and how its behaviour can be characterised by the time constant (t). A related measurement is that of 'rise time'.

The definition of rise time is "that time taken for a linear network's output to rise from 10% to 90% of its final value when stimulated by a step input". This measurement is useful because it is easy to measure on an oscilloscope and can be applied to any linear network.

For our example of t = 1 second and for a 1 volt input step we have the following:

In case of the RC network, the 10% level is reached after 0.105 time constants and the 90% after 2.302 time constants; thus the rise time is (2.302 – 0.105) time constants which is 2.197 time constants or 2.197t. We'll call it 2.2t for simplicity.

In the frequency domain, the RC network offers no attenuation or loss at DC (0 Hz); attenuation rises with frequency and is 3 dB at a frequency of 1/(2pRC) or 1/(2pt).

This frequency is usually referred to as the 3 dB bandwidth or the half power bandwidth.

Now multiplying the 3 dB frequency by the rise time:

1/(2pt) x 2.2t = 2.2/2p = 0.35 (dimensionless)

OR:

Bandwidth x Rise Time = 0.35

The figure of 0.35 often is quoted when characterising the performance of oscilloscopes. A 'scope whose bandwidth is 100 MHz and whose step response is that of a simple RC network will have an internal rise time of 3.5 ns. Therefore, a pulse with rise time shorter than this cannot be measured on such a 'scope and it is fruitless to attempt to measure the rise time of a pulse if it is shorter than that of the 'scope's. Even signals whose rise time approaches that of the 'scope will be distorted so that their measured rise time is increased. The measured rise time is given by:

rt_measured =  {(rt_scope)² + (rt_pulse)²}^½

Rule: if you want to measure the rise time of a pulse; know what the rise time of your 'scope is otherwise you may end up merely measuring that of the 'scope and not that of the pulse.

Go to mathematical model

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Notes

1. Dennis Weller of Agilent Technologies says that with modern digital 'scopes the RT x BW = 0.35 rule is not accurate. It all boils down to: 'read the manual' - this should tell you how to get the best accuracy from any instrument. Remember that the probe used is also critical in determining the instrument's capabilities. (See: Relating wideband DSO rise time to bandwidth; Lose the 0.35!, EDN Europe December 2002.)