# High pass filter - Cut-off frequency calculation doesn't match simulation

I have made the following simulation with the circuit you can see below: I'm asked to find the cut-off frequency for this filter. From the simulation, it looks like it $$\f_c = 500 \text{Hz}\$$ or somewhere around there.

However, if I use the formula for cut-off frequency I get something totally different.

$$\f_c=\frac{R}{2\pi L}=\frac{500 \Omega}{2\pi \cdot 10 \text{mH}}=7960 \text{Hz}\$$

Am I using the correct formula, or is my interpretation of the simulation wrong?

I hope someone can help me.

• $f_c$ is the frequency at which the gain is $-3db$; why do you think $f_c$ is 500Hz? – across Apr 2 '20 at 9:07
• Huh,I guess have misunderstood some things then. However, if $f_c$ occurs when gain is -3dB, then the simulation and calculation still doesn't match. – Carl Apr 2 '20 at 9:13
• maybe the frequency is not f rather w? – Mike Apr 2 '20 at 9:15
• Are you perhaps looking at the phase line (the dotted line) thinking it is the gain line? – Justin Apr 2 '20 at 17:41

From the simulation, it looks like it fc=500Hz or somewhere around there.

Your simulation looks more like 15 kHz (3 dB point) and the reason is because you haven't used enough resolution in your AC analysis and the graph is inaccurately interpolating between points that are too far apart: - Use more resolution in your AC analysis like in this simulation I did: - The 3 dB point is around 7900 Hz.

Also try using a logarithmic X axis.

• Very often it is more exact to use the phase shift of 45 deg. as an indication for 3dB cutoff. – LvW Apr 2 '20 at 11:06
• For any 1st-order lowpass or highpass the phase shift at the 3dB frequency is ALWAYS 45 deg. Look into your phase response diagram. In many cases. this point is easier to find than the 3dB drop of the magnitude. – LvW Apr 2 '20 at 11:16

Well, mathematically speaking we can write:

$$\mathcal{H}\left(\text{s}\right)=\frac{\text{sL}}{\text{sL}+\text{R}}\tag1$$

Using $$\\text{s}=\text{j}\omega\$$, we get:

$$\left|\underline{\mathcal{H}}\left(\text{j}\omega\right)\right|=\left|\frac{\text{j}\omega\text{L}}{\text{j}\omega\text{L}+\text{R}}\right|=\frac{\left|\text{j}\omega\text{L}\right|}{\left|\text{j}\omega\text{L}+\text{R}\right|}=\frac{\omega\text{L}}{\sqrt{\text{R}^2+\left(\omega\text{L}\right)^2}}\tag2$$

Now, we get:

• $$\lim_{\omega\to0}\left|\underline{\mathcal{H}}\left(\text{j}\omega\right)\right|=0\tag3$$
• $$\lim_{\omega\to\infty}\left|\underline{\mathcal{H}}\left(\text{j}\omega\right)\right|=1\tag4$$

Solving for the $$\-3\space\text{dB}\$$ point gives:

$$\frac{\omega\cdot10\cdot10^{-3}}{\sqrt{500^2+\left(\omega\cdot10\cdot10^{-3}\right)^2}}=\frac{1}{\sqrt{2}}\space\Longleftrightarrow\space$$ $$\omega=50000\space\text{rad/sec}\tag5$$

Which is the same as:

$$\frac{25000}{\pi}\space\text{Hz}\approx7957.75\space\text{Hz}\tag6$$

• Jan - just a formal correction: The last line must not be written as one single equation. Instead you should write down two separate equations. – LvW Apr 2 '20 at 11:05
• @LvW I do not see what you mean, you can edit my answer is you want. – Jan Apr 2 '20 at 11:08
• Jan...it is pure math: On the left side of the equations is w....and on the right side we see the frequency f. That means w=f ???. It should read: 50000rad/sec are equivalent to f=7957.75 Hz (not equal) – LvW Apr 2 '20 at 13:01
• @LvW I've studied mathematics and I see what you mean, but using rad/sec and Hz make a difference – Jan Apr 2 '20 at 13:21
• Yes - and therefore, I think, we must not equate the two ..(different units on both sides) – LvW Apr 2 '20 at 13:26