# 3 dB frequency of first-order active high-pass filter

Using the KCL law in the red dot in the circuit shown above, I got this equation:

$$\frac{V_\text{in}}{R_1 + \frac{1}{j\omega C}} + \frac{V_\text{out}}{R_2} = 0$$

From this equation the transfer function of this circuit is:

$$T(s) = \frac{V_\text{out}(s)}{V_\text{in}(s)} = -\frac{R_2}{R_1} \times \frac{j\omega}{j\omega + \frac{1}{R_1C}}$$

How can I get 3 dB frequency from the transfer function?

• What have you tried so far? Commented Sep 29, 2022 at 15:16
• Do you know how to convert -3 dB to a voltage ratio? Or, there is a shortcut for first order filters if you remember the relationship between real and imaginary at the cutoff frequency. Commented Sep 29, 2022 at 15:33
• If you write $T(s)$ then you have to follow with $s$, not $j\omega$, otherwise you need to write $T(j\omega)$ (this is also a hint). Commented Sep 29, 2022 at 15:46
• What is difference between s and jw ?? I thought they are same. Commented Sep 29, 2022 at 16:17
• @moonjy1120 they are just different notations. s comes from the Laplace transform while $jw$ is the Fourier representation. While they are equivalent you are mixing domains Commented Sep 29, 2022 at 17:08

Calculate the magnitude of $$\T(s = j\omega)\$$ when it is not acting as a filter (which is as $$\\omega \to \infty\$$ for a high pass filter).

$$|T(j\omega)| = \left|-\frac{R_2}{R_1}\right| \times \frac{|j\omega|}{\left|j\omega + \frac{1}{R_1C}\right|} = \frac{R_2}{R_1} \times \frac{\omega}{\sqrt{\omega^2 + \left(\frac{1}{R_1C}\right)^2}}$$

$$\lim_{\omega \to \infty}|T(j\omega)| = \frac{R_2}{R_1}$$

since the $$\\omega\$$ term in the denominator dominates the $$\1/(R_1C)\$$ term and then the fraction with $$\\omega\$$ simplifies to 1. Alternatively, you can find this by inspection since the capacitor is a short circuit at high frequencies and the circuit becomes just an inverting op amp amplifier.

Next, set the transfer function magnitude equal to $$\-3\text{ dB} \approx 1/\sqrt{2}\$$ times the previously calculated magnitude.1 In this case the $$\-R_2/R_1\$$ terms cancel so you are left with

$$\frac{\omega}{\sqrt{\omega^2 + \left(\frac{1}{R_1C}\right)^2}} = \frac{1}{\sqrt{2}}$$

Solve this equation for $$\\omega\$$ to obtain the -3 dB frequency, which turns out to be $$\omega = \frac{1}{R_1C}$$

1The power gain of a circuit is given in dB as $$10 \log_{10}\left(\frac{P_\text{out}}{P_\text{in}}\right)$$ and the cutoff frequency is typically defined as the frequency at which the output is at half the power of the input: $$10 \log_{10}\left(\frac{1/2 \times P_\text{in}}{P_\text{in}}\right) = 10 \log_{10}(1/2) = -3.0103\text{ dB} \approx -3\text{ dB}$$

The transfer function is given in terms of voltage rather than power so we need to use the fact that $$\P = V^2/R\$$: $$10 \log_{10}\left(\frac{P_\text{out}}{P_\text{in}}\right) = 10 \log_{10}\left(\frac{V_\text{out}^2 / R}{V_\text{in}^2 / R}\right) = 10 \log_{10}\left(\frac{V_\text{out}^2}{V_\text{in}^2}\right) = 10 \log_{10}\left(\frac{V_\text{out}}{V_\text{in}}\right)^2 = 20 \log_{10}\left(\frac{V_\text{out}}{V_\text{in}}\right)$$

where the last equality follows from the fact that $$\\log_b(x^p) = p \log_b(x)\$$ and explains why we multiply the logarithm by 10 when working with power but 20 when working with voltage (or current).

Solving for $$\V_\text{out}/V_\text{in}\$$ at the cutoff frequency:

$$10 \log_{10}\left(\frac{V_\text{out}}{V_\text{in}}\right)^2 = 10 \log_{10}(1/2)$$

$$\left(\frac{V_\text{out}}{V_\text{in}}\right)^2 = 1/2$$

$$\frac{V_\text{out}}{V_\text{in}} = \frac{1}{\sqrt{2}}$$

How can I get 3 dB frequency from transfer function?

It's simpler than that; the half power point (commonly know as the -3 dB point) is when: -

$$R_1 = X_{C}$$.

And we know that $$\X_{C} = \dfrac{1}{\omega C}\$$ therefore: -

$$\omega = \dfrac{1}{R_1 C}\hspace{1cm}\text{or}\hspace{1cm} f = \dfrac{1}{2\pi R_1 C}$$

From the transfer function DC gain is 1 so

• Your answer for the -3 dB frequency is correct but the DC gain is not 1, it is 0. The high frequency gain is $-R_2/R_1$.
– Null
Commented Sep 29, 2022 at 18:18