# Why does integrating my signal attenuate it?

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Janith Wijekoon on 18 Apr 2022
Commented: Janith Wijekoon on 18 Apr 2022
I fed a sine wave through an continuous time integral block as follows and noticed that the amplitude of the integrated output is much lower than the input signal, and it's also shifted up in the Y-axis. Why is this, and how should I account for it every time I include an integral in a model?  I'd also appreciate it if someone could tell me why the first few periods of the wave plotted by the scope here is much less smoother compared to the ones it plots later. Thanks!

Paul on 18 Apr 2022
Hi Janith,
If the input to the integrator block is u(t) = sin(w*t), the output of the integrator will be y(t) = -1/w*cos(w*t) + C. The constant, C, will be determined by the Initial Condition of the integrator block, which in this case is 0 based on the yellow curve. So
y(0) = 0 = -1/w*cos(w*0) + C = -1/w + C. So we have C = 1/w. Using that value of C the output y(t) is
y(t) = -1/w*cos(w*t) + 1/w = 1/w*(1 - cos(w*t))
The smoothness issue is probably related to the solver settings, assuming the model is using a variable step solver.
Janith Wijekoon on 18 Apr 2022
Thanks!

R2021a

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