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Integrator does averaging and gets that precise dc value in comparison to a low pass filter. Can we think of this as...... dc value => BW=0 => infinite gain as per Gain, BW relation. In case of Ideal Low pass, we have finite BW => finite Gain.
It is not related with bandwidth nor with averaging.
The relation gain=output/input is to be thaught when the output reaches steady state.
Maybe you are confused by the fact that an ideal integrator never reaches steady state (otherwise stated, it takes an infinite time). But in the limit (when t->infinity) the ouput goes to infinity for any (nonzero) d.c. input.
Reagrds
It is not related with bandwidth nor with averaging.
The relation gain=output/input is to be thaught when the output reaches steady state.
Maybe you are confused by the fact that an ideal integrator never reaches steady state (otherwise stated, it takes an infinite time). But in the limit (when t->infinity) the ouput goes to infinity for any (nonzero) d.c. input.
Reagrds
Please correct me if i m wrong. The Gain is usually a result of AC analysis(frequency response where we just apply a small signal) where we do not have any thing relating to transient or steady state. The explanation you have given deals with the time response of the circuit.
"AC analysis" means "small signal analysis". This includes d.c. as the case frequency=0. "DC small signal analysis" deals with small increments around, for example, the polarization point of the devices.
When you get gain as function of frequency in AC analysis, it is in fact a steady state analysis. For instance, its result does not say what happens if you suddendly apply an ac signal of some frequency, or a non-sinusoidal signal. For that, you usually perform a transient analysis. (A complete gain analysis would allow to calculate a transient response via transforms, but this is an indirect way.)
Regards
Although I do not understand the question (my answer: "infinite" means infinite and an ideal integrator does not exist), I completely agree with zorro. He is absolutely right.
LvW
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