How to calculate phase shift of a low-pass filter's output?

Question

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)

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)

Answer 1

Well, we know that:

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

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

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EDIT

So, we get:

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

Answer 2

If this is a theoretical exercise you can get precise values from the simulation, or from mathematical theory. For example, take the values used to generate that plot, and curve-fit or interpolate them.

But if you intend to actually build the thing, you'll have to use real components, with real tolerances, and the actual phase shift will be consequent on them, however accurately you calculated the theoretical value.

You need to think about this in context of how precise an answer you need (which you haven't told us). Is it even possible with 1% resistors and 2% capacitors, for example?

You might run a Monte-Carlo simulation with random variations within the tolerance of each component and look at the spread of results.

Answer 3

You need a bit of maths that uses arctangent, resistance, capacitance, and frequency. Oh, and conversion from Hertz to radians per second.