# RC Integrator

Discussion in 'General Electronics Chat' started by jag1972, Oct 18, 2015.

1. ### jag1972 Thread Starter Active Member

Feb 25, 2010
71
0
I am looking at the passive RC Integrator circuit, the Vout equation for this circuit is: $Vout= \frac {1}{RC} \int Vin \: dt$. This only works for certain conditions e.g. when $RC > T$. This is however misleading as most text books show a square wave which equals $5tau$. This is totally confusing IMO as the vout equation does not match the output voltage at all. Is it just me that thinks this or is it misleading or have I made a mistake in my assumption.

Jag.

2. ### StayatHomeElectronics AAC Fanatic!

Sep 25, 2008
1,021
71
The equation for Vout is valid when time, t, is less than the time constant, RC. That is t < RC = tau. This represents only the first portion of the response to the squarewave transitions. The circuits use as an integrator is limited in time. After time t it is no longer integrating.

Another equation for the capacitor charging is an exponential voltage rise (or decay depending on initial conditions) to the applied voltage. The amount of time it takes to get over 99% of the way there is 5tau. The first one fifth of this equation looks very much like a straight line.

Feb 25, 2010
71
0

4. ### jag1972 Thread Starter Active Member

Feb 25, 2010
71
0
I have a series RC Circuit where $R= 1 k\Omega\: and C = 10 nF$, the input to the circuit is a square wave with a period 1 ms.
The input square wave is: $\: x(t)=\: 5,\: 0\geq\:t\:> 0.5 ms\: 0,\: 0.5 ms\geq\:t\:> 1 ms$.

When I use the integrator formulae: $\frac{1}{RC} \int_{0}^{0.5 ms} \: x(t}\:dt$

$tau = 0.1 ms$, therefore after 5 tau, vC is 99.9% of 5 V. This can be determined from the following equation/model for vC:
$vC=5(1-\exp^{-5})$
However when I use the integrator formulae I do not get that value: $Vout = \frac{1}{RC} \int_{0}^{0.5 ms} 5\: dt$
The value using the integrator formulae is 25 V which is clearly wrong can some please tell me where I am going wrong. I am aware that the integrator only works as a true integrator when RC is large compared to T. But I would have thought that Vout should always be right.

5. ### StayatHomeElectronics AAC Fanatic!

Sep 25, 2008
1,021
71
Thanks for your reply, I thought that was the answer. BTW I made a mistake with the reistor value it was supposed to be $10 k\Omega$.