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I want to measure a pulse signal using an oscilloscope. The expected rising edge of the signal is in the 1 to 5 ns scale. I have a 20 MHz oscilliscope. Is it correct that the rising edge of the oscilloscope would be limited to 17.5 ns? In other words, do I need to use a faster oscilloscope? How does a 20MHz oscilloscope measure in the ns scale and what is the limit?

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The standard rule of thumb is the the rise time limit of an oscilloscope is about equal to (0.35)/(bandwidth). This means you can't resolve a 5 nanosecond rise event with a 20MHz scope.

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