Electrical – Confusion with the DC response of an RC integrator


Below is the beginning section from a tutorial on filters:

An integrator (Figure 1a) is the simplest filter mathematically, and
it forms the building block for most modern integrated filters.
Consider what we know intuitively about an integrator. If you apply a
DC signal at the input (i.e., zero frequency), the output will
describe a linear ramp that grows in amplitude until limited by the
power supplies. Ignoring that limitation, the response of an
integrator at zero frequency is infinite, which means that it has a
pole at zero frequency. (A pole exists at any frequency for which the
transfer function's value becomes infinite.

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Can you explain what is meant by "the response of an integrator at zero frequency is infinite, which means that it has a pole at zero frequency."?
What does it mean the response to be infinite? And what does the term pole represent here?

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In this document there's a plot for s = jw+sigma. But if s has sigma component, doesn't that mean a damped oscillation? I can understand that for steady state the plot of freq. response is independent of time. But damped input is time dependent and not periodic but yet they can plot it as if it is time independent in that 3D plot for RC passive filter.

3D plot for s=jω+sigma there is still a point on the surface. So surface point at s=jω+sigma is not the same point with s=jω. I can understand s=jω is a sinusoidal input's frequency and has a fixed amplitude over time(time independent) but what is s=jω+sigma? If it is damped oscillation, how can it even be plotted on a surface which is time independent?

hen I look at s = jw axis(when sigma is zero) and for example when jw=628j which is f= 100Hz the vertical axis form that point on that 3D surface will be the response of the filter. This is the amplitude of the steady state response. So we have steady state input output like in Fourier. Now imagine s=jω+sigma = 628j+10. This is not a steady state input, and the vertical axis form that point on that surface will have a single value. But the response should be damped. So what is that value of the damped response on the surface? Is that the max amplitude or the mean value of the damped response?

Best Answer

The response at D.C. means the value of \$H(0)\$ where \$H(s)\$ is the frequency response. Imagine that you apply a fixed voltage to the integrator input. You will get a ramp as an output of the integrator ignoring the opamp supply rails. This ramp will have various harmonics, if you take its frequency response. Since it goes to infinity its D.C. component, which is the average value, is infinite too. Therefore there is a pole at \$s=0\$. This is a more qualitative way of explaining why \$H(s) = a / s\$.

To see why the ramp \$f(x) = a x\$ has an infinite response at D.C. consider the Laplace transform: $$H(0) = \int_0^\infty a x \ \mathrm{d}x = \left.\frac{a x^2}{2}\right|_0^\infty = a \infty$$