Flinty,
in addition to MrAl's very detailed explanation I like to emphasize the following:
* An ideal integrator cannot be realized (because of "infinity" as mentioned by MrAl). Thus, a real integrator is always a lowpass - however, with a very low cut-off or pole frequency.
* Nevertheless, each lowpass can be used as an integrator (with some errors resp. restrictions) for operating frequencies far above the low pass cut-off frequency (factor 10 or so) because for these frequencies the phase shift is approximately close enough to 90 deg (which is the ideal phase shift of an integrator).
Hi again,
I would also like to add this...
An integrator that is used as a low pass filter will end up integrating any tiny DC offset to infinity. This means it amplifies the DC offset until the output of the circuit saturates. Thus, when we find a circuit being used as an integrator it is either inside a control loop or else has a reset capability where something resets the integrator back to zero to get it ready for another cycle of integration. So in effect it only works over a limited time period. In this application it does function as an integrator though, not a low pass filter.
The usual circuit is an op amp with a capacitor from output to inverting input, and a resistor from the signal input to the inverting input, and with the non inverting input grounded.
One thing to note here is that the circuit with a DC input produces a nearly perfect ramp output. With a resistor across the capacitor we end up with a low pass filter, which then means we dont get a ramp anymore but a curved ramp.
An Integrator like this that are used in a feedback control loop does not have the DC problem because the loop corrects it. In fact, the steady state error of the system goes to a very low value. This is worth looking into also.
Hi,
Well, a true integrator integrates the input and so the output is the time integral of the input. The RC value is irrelevant as to the accuracy because it will always be accurate. The RC value does help set the gain, but that still doesnt bother the accuracy.
However, using a low pass filter for an integrator it is a different story. The low pass filter capacitor has a tendency to discharge over time with zero input while with a true integrator this doesnt happen. This means if we want to "integrate" a pulse with a low pass filter we need to set the RC time constant high enough so we dont get too much ripple on the output. A true integrator will ramp up with each pulse, and never ramp down, while an RC filter will ramp up and then ramp down after the pulse is complete. It ramps up and down with each pulse so that creates a sawtooth output which is usually too objectionable to be useful. With a larger RC time constant (which is sometimes estimated to be 10 RC time constants) the ripple is reduced and the filter becomes usable to some degree. Depending on the actual application however we may even want to go higher than that, such as to 100 RC time constants which provides a much smoother output at the cost of speed of response.
The kind of circuit you are talking about could be made with a single resistor and single capacitor. This is sometimes called an integrator too but it is not a true integrator. It approximates the action of the true integrator. For certain applications though it can still be of great value, such as in finding the time average of a signal. For RC large enough relative to the signal bandwidth the output approaches the actual time average of the signal input with good accuracy.
one more quistion:
if we have simple LPF(single resistance and single capacitor),then we have a sine wave as an I/P (Vm sinωt)it will work as a LPF not as an intgrator which intgrate I/P to be -cos but with filter pathing band which ??
or both of them ??
i think if we intgrate the sine wave to negative cosine it mean that we shifted sine wave by -90 degree??
if you got an explaination please help
We can use a normal passive RC circuit as an integrator and it works exactly like an integrator. We input a sine wave sin(wt) into the resistor and across the capacitor we get -cos(wt). That's definitely true. So we have ourselves an integrator after all.
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But there's a catch here. The above only works when w=1/RC. With ANY other w, it does not work perfectly as it does at that exact value of w.
A true integrator always produces a 90 degree phase shift (which way depends on if there is an additional inversion). An RC network produces a 90 degree phase shift for one frequency only.
Hello MrAl,
Sorry, but I cannot resist to comment your explanations.
According to my knowledge a simple RC element produces a phase shift of -45 deg at w=1/RC (and not -90 deg).
More than that, a phase shift of -90 deg is generated for infinite frequencies only. That was the reason I have stated in my former contribution that an RC combination NEVER can act as an ideal integrator.
However, in many applications it can be used as an approximation to an integrator (for frequencies much larger than 1/RC)
For active integrators the situation is slightly different:
Because all integrating devices (including opamp integrators) in reality have lowpass characteristics (with a very low corner frequency) the phase starts at 0 deg and reaches -90 deg (+90 deg for the classical inverting Miller integrator) at ONE single frequiency only. For larger frequencies the phase goes beyond this value due to the real nature of the opamp.
However, normally the phase slope around the -90 deg is very small so that in can be used in a rather broad frequency region for integrating purposes.
Additional comment regarding the allpass:
Quote: For a constant phase shift another network that is used is called the "All Pass".
Just the opposite is true. An allpass provides a (nearly) constant amplitude and a frequency-dependent phase shift. This is the reason an allpass can be used to correct unwanted phase excursions.
ok now after we all agree that lpf is an intgrator always so if we want to fliter some frequencies
only what is the filter will act also as an integrator who could we recover those integrated frequencies??
Hi MrAl,
I am afraid, I have to continue the discussion, because I really cannot agree to all of your statements (Quotations in italics):
I noted that two RC passive sections could act as a very near perfect integrator, but actually a single section could do it too as i'll show next.
No, that is not the case. The transfer function of an ideal integrating function (as given by you later) has an amplitude slope of -20 dB/dec and, correspondingly, a constant phase shift of -90 deg
With one single RC sections this can be approximated for frequencies far above the corner frequency 1/RC only (my suggestion: Frequencies at leat w(min)=10/RC).
I really don't know how a second RC section could improve the situation. I can imagine what your goal is: To bring the phase closer to -90 deg. and to go through this phase at one discrete frequency.
However, I doubt if this can improve the integrating properties of the whole circuit. Have you ever seen such an integrating device using TWO RC sections?
We can actually get an integration with a single RC section. This isnt hard to do because the capacitor itself is an integrating element:
dv/dt=i/C
Thus a single capacitor can be used as an integrator even without a resistor if we measure the output as the voltage across it and the input as the current through it:
V=(1/C)*Integral(i)dt+K
Yes, exactly this is done (better: the current source is approximated by the resistor) by using a resistor to charge the capacitor. That`s the working principle of the RC "integrator" as well as the Miller integrator using opamps. Here the Miller effect enlarges the resistor causing a very small corner frequency of the resulting lowpass with gain (due to the opamp action).
If we use two RC sections where the second R value is 100 times greater than the first R value and the second C value is 100 times less than the first C value then we will get a 90 degree phase shift at the frequency w=1/RC as indicated. It's pretty close too.
Yes, every second order circuit is able to produce -90 deg. at one single frequency. But this is not sufficient to be an integrator. What about the gain slope at this point? It will be larger than 20 dB/dec and, thus, cause new errors.
But also, sometimes the passive single section RC filter can act as an integrator and depending on the signal bandwidth it works just fine just like that. For an example where it works very well is in a signal averaging circuit. With a 10k resistor and 1uf capacitor we can easily find the average of a pulse signal of high enough frequency with only a small amount of ripple. The DC value is very close to the actual average DC of the signal.
Yes, of course - for a "high enough frequency", which is far above w=1/RC.
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Perhaps it would help to show some simulation results (magnitude and phase for one and two RC sections as well as an active integrator).
I'll try it.
Regards
For example, *YOU* say that we can not use a double RC section to integrate, while *I* say that we can.
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Second, you say that we can not use a single RC section to integrate, but you havent replied to the case where R is either zero or close to zero. We can not leave this out of the theory either because it helps to understand these concepts as well. It's not a second order system either. Take a RC section with very small R (possibly zero). Put a sine voltage across it of some frequency. Measure the current on the scope. Note the relationship between the current and voltage.
Hi again Winter,
Im sorry i just dont see what you are trying to get at. Sure it doesnt work over a wide range of frequencies, but not every circuit has to do that. For example, a line operated circuit works at 50 or 60Hz, or in some cases 400Hz. The frequency wont vary much either.
Yes there are some circuits that can do some things better than others, depending once again on the application.
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