The relationship BW = 0.35/(10-to-90% rise time) only applies to a first-order system (having a single-pole filter, which makes a 6-dB-per-octave roll-off with zero overshoot). The anti-aliasing filters and intrinsic roll-offs of input components in these scopes are likely greater than first-order, rendering this formula inaccurate. The best way to measure BW is to measure it as I described in an earlier post, by using an RF signal generator into 50 ohms. Granted, not everyone has access to these instruments, and I applaud the ingenious idea of discharging a cap. But the 11.44% overshoot demonstrates that the rule of thumb is not accurate for this case. See the response curves for the TI LMH6518 variable gain amplifier, alleged to be used in the DS2000A. The higher bandwidths choices have steeper and more complex roll-offs. Ironically, the rise time formula probably is most accurate when the BW is limited to 20MHz.
HP/Agilent/Keysight has this application note about the relation between BW and rise-time:
http://cp.literature.agilent.com/litweb/pdf/5988-8008EN.pdfUnderstanding Oscilloscope Frequency Response and Its Effect on Rise-Time Accuracy
Application Note 1420Introduction When you combine many circuit elements with similar frequency responses, you get a Gaussian response. Traditional analog oscilloscopes chain many analog amplifiers from the input to the cathode ray tube (CRT) display, and therefore exhibit a Gaussian response. The properties of a Gaussian-response oscilloscope are fairly well understood in the industry.
Less familiar, though, is the flat-response that is now more commonly exhibited by modern, high-bandwidth digital oscilloscopes. A digital oscilloscope has a shorter chain of analog amplifiers, and it can use digital signal processing techniques to optimize the response for accuracy. More importantly, a digital oscilloscope can be subject to sampling alias errors, which is not an issue with analog scopes. Compared to a Gaussian response, a flat response reduces sample alias errors, an important requirement in the design and operation of a digital oscilloscope.
This application note reviews the properties of both Gaussian- and flat-response oscilloscopes, then discusses rise-time accuracy for each response type. It shows that a flat-response oscilloscope gives more accurate rise-time measurements than a Gaussian-response oscilloscope of equal bandwidth, and how you can estimate the oscilloscope bandwidth you need.
This discussion refers to using a 1 GHz oscilloscope, but this analysis is scalable to other bandwidths with the same validity.
Properties of a Gaussian-Response OscilloscopeFigure 1 depicts a typical Gaussian frequency response for a 1 GHz oscilloscope. A Gaussian-response offers good pulse response without overshoot, regardless of how fast the input signal is. Figure 2 shows the pulse response of a 1 GHz Gaussian-response oscilloscope to a fast step input.
In a Gaussian-response oscilloscope, the oscilloscope's rise time is related to the oscilloscope's bandwidth using
the familiar formula:
Rise time = 0.35/bandwidth
Another common property of Gaussian systems is that the overall system bandwidth of the oscilloscope and its probe is the inverse root mean square (RMS) value of their individual bandwidths. The system bandwidth can be calculated using the familiar relationship:
System bandwidth = 1/(1/BW
probe2 + 1/BW
oscilloscope2)
0.5Often oscilloscope probes are designed to have sufficiently higher bandwidth than the oscilloscope bandwidth, so you do not need the above formula for derating the system bandwidth. Inversely, the measured rise time is commonly related to the system rise time and signal rise time using the formula:
Measured rise time = (RT
signal2 + RT
system2)
0.5Sometimes this relationship is used to estimate the actual signal rise time when the oscilloscope's system rise time is not sufficiently faster than the signal's rise time to make an accurate measurement.
Properties of a Flat-Response OscilloscopeFigure 1 compares a flat response to a Gaussian response. Note that the frequency response is much flatter below the –3 dB bandwidth, but then drops off very rapidly above the –3 dB bandwidth. This response shape is sometimes referred to as a maximally flat or brick wall response.
There are a couple of advantages to a flat-response. First, the frequency content of the signal below the –3 dB bandwidth is less attenuated, and thus measured more accurately. Secondly, the steeper roll helps reduce sampling alias errors in digital oscilloscopes (more on this later).
In the time domain, a flat response results in a pulse response with overshoot and ringing when the oscilloscope input is driven with a fast step input, as depicted in Figure 2. Such overshoot and ringing is often perceived as an undesirable effect in an oscilloscope. However, this ringing only occurs if the signal rise time is significantly faster than the oscilloscope can measure accurately, in which case you should use a higher-bandwidth oscilloscope.
Unlike Gaussian systems, the system bandwidth of a flat-response oscilloscope is not determined by the inverse RMS value of the sub-system parts.
The commonly used bandwidth and rise-time formulas for Gaussian-response oscilloscope systems do not apply to flat-response oscilloscope systems! Instead, you should rely on the oscilloscope vendor to specify the system bandwidth of an oscilloscope/probe combination.
In the case of a flat-response oscilloscope, the rise time is related to the bandwidth, as described in the formula:
Rise time = N/bandwidth
(where N = 0.4 to 0.5)
The larger N is, the steeper the frequency response is, or the more it approaches the "brick wall" configuration shown in Figure 1. The above relationship will sometimes be included in an oscilloscope's specifications, which can give you an indication of what type of response the oscilloscope has...
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