Transient Response Suppression

In my project i am using an ultrasonic transducer. The transducer has a high Q-factor so that it's zero input transient responses decay rate is quite low and i am suffering from that because the transient is interfering with the reflected signal.

Here is the circuit that i am using, C0 L1 R1 and C1 accounts for the transducer

It has the response

where green is Vin and red is Vout

In a linear time invariant circuit,

$\frac{d}{dx} \overline{x}(t) =\overline{\overline{A}} \overline{x}(t) + \overline{b} v_{s}(t)$ where $v_{s}$ is the source and $\overline{x}$ is the state vector then we have,

$\overline{x}(t) = e^{ \overline{\overline{A}}t} \overline{x}(0) + \int_0^te^{ \overline{\overline{A}}(t - \tau) } \overline{b} v_{s}( \tau )d \tau$

, i wonder if for any input $v_{s1}(t), v_{s1}(t)= 0~~ for~~t > t_{1}$ there exists a function $v_{s2}(t), ~~v_{s2}(t) \neq 0~~ for ~~t_{1}\leq t\leq t_{2}$ such that $\left\{ \begin{array}{ll} v_{s}(t) = v_{s1}(t) & t\leq t_{1} \\ v_{s}(t) = v_{s2}(t) & t_{1}\leq t\leq t_{2} \\ 0 & o.w. \\ \end{array} \right.$ $,\overline{x}(t_{2}) = 0$

That is to say, i want to apply a signal right after the original signal so that the latter will suppress the transient created by the first. I have found analytical solution for $v_{s2}(t)$ if $\overline{b}$ is an eigenvector of $\overline{\overline{A}}$ but i am seeking a general solution. Here is an example i found with trial and error:

Where the green signal is the input and the red signal is the response of a high Q bandpass filter the signal is a 250Hz pulse with 5 cycles. if we apply a 1 kHz for 2 cycles at 22ms we have,

Observe that the transient response is immediatelly suppressed

I would be glad if you propose a method to find a $v_{s2}(t)$ preferably a square one.

• It looks like you are applying a "counter voltage" that allows the energy in the reactive components to be immediately turned to heat in the lossy part of the device. This is not a new thing but the problem with your question is that it is dealing a lot with formulas and lacks EE content like the equivalent circuit of your device and your method of driving the device. In other words I have no idea what you have done electrically and therefore no way to suggest a better or more proven alternative method. Jul 23, 2018 at 13:48
• Yeah, what Andy said. I have no interest in wading thru the equations to see what they are trying to say. After skipping over them, I'm not sure what you are asking. Ask what you want in words. Use equations to back the words to give details if really needed. However, the concepts shouldn't need that level of detail. It's also hard to see what the graphs are, with them aliased to oblivion like that. Again, there are too many details here obscuring (if you ever said it at all) what you are actually trying to accomplish and what your question is. Jul 23, 2018 at 14:01
• I haven't waded through the equations either, but as @Andyaka mentioned the general problem has been solved by active damping. If you Google "active damping for ultrasonic transducers" you should be able to find some references. Jul 23, 2018 at 14:34