Oscilloscope Rise Time Calculation

An oscilloscope is a vital electronic test instrument that graphically displays varying signal voltages, usually as a two-dimensional plot of one or more signals as a function of time. Two important specifications for an oscilloscope are its bandwidth and its rise time.

• Bandwidth (BW): The bandwidth of an oscilloscope refers to the frequency range over which it can accurately measure a signal. It is typically defined as the frequency at which the input signal is attenuated by 3 dB (approximately 70.7% of its original amplitude). A higher bandwidth means the oscilloscope can capture faster signals more accurately.
• Rise Time (\(t_r\)): The rise time of an oscilloscope (or any electronic system) is the time it takes for a signal to change from a specified low value to a specified high value, typically from 10% to 90% of its final amplitude. It indicates how quickly the instrument can respond to a sudden change in input voltage.

Bandwidth-Rise Time Relationship

For a conventional oscilloscope, there is an inverse relationship between its bandwidth and its rise time. This relationship is often approximated by the formula:

$$t_r \approx \frac{0.35}{\text{BW}}$$

Where:

• \(t_r\) is the rise time (in seconds)
• BW is the bandwidth (in Hertz)
• The constant 0.35 is an empirical factor commonly used for systems with a single dominant pole response, which is a good approximation for many general-purpose oscilloscopes.

CRO Rise Time Calculation for 100 MHz Bandwidth

Given the bandwidth of the conventional oscilloscope is 100 MHz. We need to calculate its approximate rise time.

• Given Bandwidth (BW) = 100 MHz
• Convert Bandwidth to Hertz:
• $$ \text{BW} = 100 \times 10^6 \text{ Hz} $$
• Now, use the formula to find the rise time \(t_r\):
• $$t_r = \frac{0.35}{\text{BW}}$$
• Substitute the value of Bandwidth:
• $$t_r = \frac{0.35}{100 \times 10^6 \text{ Hz}}$$
• Perform the division:
• $$t_r = 0.0035 \times 10^{-6} \text{ s}$$
• To express this in nanoseconds (ns), recall that \(1 \text{ ns} = 10^{-9} \text{ s}\).
• $$t_r = 3.5 \times 10^{-3} \times 10^{-6} \text{ s}$$
• $$t_r = 3.5 \times 10^{-9} \text{ s}$$
• Therefore, the rise time is:
• $$t_r = 3.5 \text{ ns}$$

This calculation shows that for a 100 MHz oscilloscope, the approximate rise time is 3.5 ns. This means the CRO can respond to a fast-changing signal and reach 90% of its final value within 3.5 nanoseconds, which is characteristic of its ability to display high-frequency signals.

The final answer is 3.5 ns.