I am reading "Optoelectronic Sensors" by Didier Decoster and Joseph Harari.
At section 1.8.2 they start explaining the relationship between rise time and bandwidth, and they give a very weird formula that I don't understand:
That's the whole thing, they don't explain anything more. Then they move on to some very interesting things about noise, but they mention BP in there and I don't understand what it is.
Now, my understanding about bandwidth and rise time came from this website, specifically equation 18 that basically says:
$$
t_r = \frac{0.35}{B}
$$
where \$t_r\$ is rise time and \$B\$ is bandwidth.
With that in mind, here is what I don't understand about the equation 1.7 in "Optoelectronic Sensors":
-
What is BP and how is it different from BW?
-
Is \$\tau\$ (tau) the rise/fall time? Or is it something else?
-
What are \$\tau_m\$ (tau_m) and \$\tau_d\$ (tau_d)? They didn't mention them anywhere before.
Best Answer
The equations shown are all derived from the same formula which you already gave:
$$t_r = \frac{0.35}{B}$$
A single-pole system has the frequency response
$$H(s) = \frac{A}{1 + \tau\cdot s}$$
This sytem has a pole at
$$p_d = \frac{1}{\tau} \Rightarrow BW = \frac{1}{2\pi\cdot \tau}$$
The transient step response of this system can be calculated as
$$h_{out} = A\left(1 - e^{-\frac{t}{\tau}}\right)$$
From which you can calculate a timepoint for each output value:
$$t = -\tau\cdot\ln\left(1 - \frac{h_{out}}{A}\right)$$
If you'd rather use the rise/fall time instead of \$\tau\$, you can then easily calculate that
$$\tau_r = -\tau\cdot\ln(1-0.9) + \tau\cdot\ln(1-0.1) = \tau \ln(9) \approx 2.1972\cdot \tau$$
Hence
$$BW = \frac{1}{2\pi\cdot \tau} = \frac{\ln(9)}{2\pi\cdot \tau_r} = \frac{\ln(9)}{2\pi\cdot \tau_f}$$
The abbreviations are the ones used in French. "Bande Passante" (BP) means bandwidth, "Monter" (\$\tau_m\$) means to rise, "Descendre" (\$\tau_d\$) means to fall.