What is the final step for relating rise time and 3 dB bandwidth?
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Figure 9: Rise time is the time separating two points on the rising edge of a curve. Points located 10% and 90% up the curve are commonly used, although other points are sometimes chosen.
First, find an expression for rise time.
The time-dependent equation for the rising edge of a pulse,
was used to find a relationship between rise time and the RC product. The relationship was found by referencing Figure 9 and noting that the amplitude at t1 is 0.1, the amplitude at t2 is 0.9, and the amplitude at infinite time is 1. These amplitude values and the defintion for rise time (t2 - t1 ) were used to find an expression,
which includes only rise time and the RC product.
Second, find an expression for the 3 dB bandwidth.
The frequency-dependent equation for the power scaling factors,
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Figure 10: The 3 dB bandwidth of the RC low-pass electrical filter, which is a first-order low-pass filter. An amplitude scaling factor of 0.707 corresponds to a power scaling factor of (0.707)2 = 0.5.
was used to find a relationship between the 3 dB bandwidth (f3dB ) and the RC product. As illustrated by Figure 10, at the upper-frequency boundary of the 3 dB bandwidth, the power scaling factor is half the maximum value. When the left-hand side of the equation is set to 0.5, the resulting expression,
relates the 3 dB bandwidth and the RC product.
Then, combine both expressions to eliminate the RC constant.
The results of the first and second steps, can be combined to find an expression that relates the rise time and 3 dB bandwidth to each other. This expression,
is a convenient way to estimate rise time when the 3 dB frequency is known, or vice versa.
Since this relationship was derived using a low-pass filter model, it is intended to be used only when the system behaves like a low-pass filter. Due to it being unlikely that the ideal RC low-pass filter circuit perfectly models the behavior of the system under consideration, this expression is useful for estimating parameters of real systems but should not be expected to provide exact results. When accurate values are needed, the parameter of interest should be directly measured. If both parameters are needed, but only one can be measured, the other can be calculated using rigorous Fourier transform-based techniques.
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