Electrical Engineering Asked on December 13, 2020

In a typical resonant bandpass filter, resonance is set as none at 0 and full at 1, such that the filter rings infinitely at 1 and not at all at 0.

If there is a two-pole resonant bandpass filter, with an impulse excitation to a maximum amplitude of “1” and resonance setting between 0 and 1:

What is the mathematical equation that would allow you to predict the decay time of the resonant filter based on its resonance setting? For example, to an arbitrary level of “1/e”? ie. To 36.7879% its original amplitude?

Thanks.

**Update June 3, 2018 @ 5:17 PM**

Thanks to some good answers below, I now better understand the problem as follows:

- I need an equation to solve for Q or ζ of the filter based on an input of time (where that time represents seconds to reach 1/e amplitude relative to a starting 1 amplitude).
- The bandpass filter will only be needed in an “underdamped” condition, ie. Q>0.5.

Thanks again. This has been very informative and I think this should be a clearer and more answerable question.

Andy's answer is good, but it can be generalized. For the transfer function:

$$H(s)=frac{2zetaomega_n s}{s^2+2zetaomega_n s+omega_n^2}$$

the inverse Laplace for all of the three mentioned cases have the same exponential decaying term in them:

$$exp(-zetaomega_n t)$$

That will be the envelope of the whole response. The rest of the time response is either of the form $sin{t}+cos{t}$ (oscillating, for underdamped), or $sinh{t}+cosh{t}$ (for overdamped), or a fixed term as a function of time, involving $omega_n$ and $zeta$.

If you really want them in full, here they are:

- overdamped:

$$h(t)=exp(-omega_nzeta t)left[2omega_nzetacosh(sqrt{zeta^2-1}omega_nt)-frac{2omega_nzeta^2sinh(sqrt{zeta^2-1}omega_nt)}{sqrt{zeta^2-1}}right]$$

- critically damped:

$$h(t)=exp(-omega_nzeta t)left[2omega_nzeta(1-omega_nzeta t)right]$$

- underdamped:

$$h(t)=exp(-omega_nzeta t)left[2omega_nzetacos(sqrt{1-zeta^2}omega_nt)-frac{2omega_nzeta^2sin(sqrt{1-zeta^2}omega_nt)}{sqrt{1-zeta^2}}right]$$

If you need to find the time when the *envelope* (alone) reaches a certain value x, then it's as simple as:

$$exp(-omega_nzeta t)=x => -omega_nzeta t=log x => t=-frac{log x}{omega_nzeta}$$

Update:This is only valid for theunderdampedcase, as there is a clear case of an oscillating term -- cos()-sin() -- multiplied by an inverse exponential. For the other two cases, it's impossible to do it analitically, as the time, t, is in more than one term, and that term is not oscillating, but either linearly variable (critically damped), or exponential (overdamped), so simplification is not possible. Also see the last pargraph.

Don't forget that, in all three cases, there is an additional term that can be factored out: $2omega_nzeta$, which gives the amplitude of the initial decay.

Note the $omega_n$ in there, which tells you that the higher the frequency the higher the amplitude(!). This is related to an impulse that has the area of 1 (e.g. for a 1ms impulse, the amplitude is 1000). If you want to consider only a unity amplitude impulse, of zero duration (unity amplitude Dirac), then the extra term can be omitted (discarded).

*Caveat emptor*: the shape of the decay will *not* correspond to the final result in the case of $zeta$>1, due to the hyperbolic functions! If the sinh() term would have been just like the cosh() term, the whole expression would have been reduced to an exp(-t), but sinh() is multiplied by $frac{zeta}{sqrt{zeta^2-1}}$, which means that the whole cosh()-sinh() term will "explode" if it were all by itself. Being multiplied by the exp() in front limits the response, but the envelope is no longer the same (contrary to the underdamped case). Here is what I mean:

`G1, G2, G3`

together with `C1, C2`

form the bandpass filter having `V(o)`

as the output. `V(exp)`

is the exponential decay (together with the $omega_nzeta t$ term, see `Rpar=2*z*wn`

in `B1`

), `V(sinh_cosh)`

is the cosh()-sinh() term, `V(test)`

is the multiplication of the previous two, compared with `V(o)`

in the lower-most plot (they are the same). The upper two plots show the individual responses of the envelope and of the hyperbolic term -- this is what I meant when I said it "explodes", it goes to $-infty$. And here is why I said that the envelope will not match the response, only by itself, due to the hyperbolic term:

Neither the output (blue trace), or its absolute value (black) will match the envelope alone (red). Which means that the solution above will not work, and the whole expression of the impulse response has to be considered, which means that, sadly, you're out of luck, since the expression contains the time, t, in both the envelope and the hyperbolic term. Even if you were to equivalate this with its equivalents, cosh=[exp(-x)+exp(-x)]/2 (minus for sinh), you cannot apply a logarithm to each exp(), only as a whole, which means the time cannot be extracted analitically, so you're stuck to numerical methods.

Answered by a concerned citizen on December 13, 2020

A band pass filter would typically have a transfer function like this: -

$$H(s) = dfrac{s2zetaomega_n}{s^2 + s2zetaomega_n +omega_n^2}$$

Where $2zeta$ is 1/Q

And $omega_n$ is the natural resonant frequency of the filter.

For an LC bandpass filter, $omega_n$ = $dfrac{1}{sqrt{LC}}$

To convert this to the time domain (due to an impulse response) there will be three time-domain solutions: -

- $zeta$ < 1 (under-damped)
- $zeta$ = 1 (critically-damped)
- $zeta$ > 1 (over-damped)

There is, unfortunately not a one-size-fits-all solution. Life gets easier if you set $omega_n$ to 1 and then look at the inverse Laplace table for the three scenarios.

For a **critically** damped situation ($zeta = 1$): -

$$H(s) = dfrac{2s}{s^2 + 2s +1}$$

And, using inverse Laplace tables you find the solution as: -

$$h(t)=2e^{-t}-e^{-t}cdot :2t$$

For an **underdamped** situation with $zeta$ = 0.5 you get: -

$$H(s) = dfrac{s}{s^2 + s +1}$$

And this inverse transforms to: -

$$h(t)=e^{-frac{t}{2}}cos left(frac{sqrt{3}t}{2}right)-frac{1}{sqrt{3}}e^{-frac{t}{2}}sin left(frac{sqrt{3}t}{2}right)$$

For an **overdamped** situation where for example $zeta$ = 2 you get: -

$$H(s)=dfrac{4s}{s^2 + 4s+1}$$

And this inverse transforms to: -

$$h(t)=4e^{-2t}cosh left(sqrt{3}tright)-frac{8}{sqrt{3}}e^{-2t}sinh left(sqrt{3}tright)$$

You can use this really good interactive calculator instead of tables should you wish: -

What is the mathematical equation that would allow you to predict the decay time of the resonant filter based on its resonance setting?

The above formulas are the only ones I can recommend.

Answered by Andy aka on December 13, 2020

I think I've got it worked out using "bandwidth" in octaves as my primary control point, but would appreciate if anyone can check my work.

According to wikipedia, the Q of the filter can be calculated as follows:

https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d004126524d7bedf71098dc6d9c0ebe5935322

Again according to wiki, Q then represents the number of oscillations to reach e^-pi relative to original amplitude.

Thus: Q/fo = time in seconds for the filter to reach e^-pi amplitude.

I am interested in calculating time to reach 1/e amplitude. So I would then just need an equation to describe the general amplitude decay.

Would amplitude decay over time follow a curve of y=1/c^x?

If so, substituting in y = e^-pi would allow me to solve for c as: e^(pi/(Q/fo))

With c, I can then solve x at y=1/e (ie. time at my target amplitude):

y= 1/c^x

x= (ln(1/y))/ln(c)

x= (ln(1/(1/e))/ln(e^(pi/(Q/fo)))

x= (ln(e))/ln(e^((pi*fo)/Q)

x= 1/((pi*fo)/Q)

x= Q/(pi*fo)

This will then tell me the time to reach amplitude 1/e at the given bandwidth setting, which is what I want.

It really simplified to something very easy and elegant. Is this correct? Or did I screw something up in my reasoning or approach?

Thanks again

Answered by mike on December 13, 2020

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