Electronic – How to calculate phase shift of a low-pass filter’s output

filterlow passpassive-filterphase shiftproteus

I'm simulating a basic low-pass RC filter in Proteus, and I need to find the phase shift of the output at varying frequencies. I've plotted a frequency response graph which shows the phase against frequency, but it doesn't show specific values. I can get rough values by just looking at the graph, but I need a precise answer.
Can anyone show me a way to actually calculate the phase shift of the output? (Using Proteus, MATLAB, or just a calculation if possible)

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Edit: I've tried a couple of calculations that I've seen while researching this, but I believe that the phase should be 0 at 10Hz, and neither calculation gives me that.
I tried: arctan(fRC) and -arctan(f/20)

Best Answer

Well, we know that:

$$\mathcal{H}\left(\text{s}\right):=\frac{\text{V}_\text{out}\left(\text{s}\right)}{\text{V}_\text{in}\left(\text{s}\right)}=\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{R}}=\frac{1}{1+\text{sCR}}\tag1$$

Now, substitute \$\text{s}=\text{j}\omega\$ where \$\text{j}^2=-1\$ and \$\omega=2\pi\text{f}\$. And after that find \$\left|\underline{\mathcal{H}}\left(\text{j}\omega\right)\right|\$ and \$\arg\left(\underline{\mathcal{H}}\left(\text{j}\omega\right)\right)\$.

EDIT

So, we get:

  • $$\left|\underline{\mathcal{H}}\left(\text{j}\omega\right)\right|=\frac{1}{\left|1+\text{j}\omega\text{CR}\right|}=\frac{1}{\sqrt{1+\left(\omega\text{CR}\right)^2}}\tag2$$
  • $$\arg\left(\underline{\mathcal{H}}\left(\text{j}\omega\right)\right)=\arg\left(1\right)-\arg\left(1+\text{j}\omega\text{CR}\right)=-\arctan\left(\omega\text{CR}\right)\tag3$$
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