Frequency response

Prof. Dr. Thorsten Jungmann (last updated 2026-03-02)

Frequency response: shows how an electric circuit responds to different frequencies – the calling card of every filter and amplifier.

The frequency response (German: Frequenzgang) describes how an electrical system (e.g. a filter, amplifier or transfer element) changes signals as a function of frequency. It comprises two pieces of information: the amplitude response (German: Amplitudengang) (gain or attenuation as a function of frequency) and the phase response (German: Phasengang) (phase difference as a function of frequency).

1 Definition and quantity symbol

The frequency response is the complex transfer function $\underline{F}(j\omega )$, evaluated along the imaginary axis ($s=j\omega$):

$$\underline{F}(j\omega )=\frac{{\underline{U}}_{\mathrm{out}}(j\omega )}{{\underline{U}}_{\mathrm{in}}(j\omega )}$$

where:

• ${\underline{U}}_{\mathrm{out}}$ is the complex output voltage
• ${\underline{U}}_{\mathrm{in}}$ is the complex input voltage
• $\omega =2\pi f$ is the angular frequency

The frequency response has no unit of its own – it is a dimensionless complex ratio.

Decomposition into amplitude response and phase response:

• Amplitude response: $|F(\omega )|=|\underline{F}(j\omega )|$ (magnitude)
• Phase response: $\phi (\omega )=\arg (\underline{F}(j\omega ))$ (argument)

2 Representation in the Bode plot

The Bode plot shows the frequency response in two diagrams arranged one above the other:

• Amplitude response in decibels (dB) over a logarithmic frequency axis:

$$A(\omega )=20\cdot {\log }_{10}|F(\omega )|\mathrm{dB}$$

• Phase response in degrees (°) over a logarithmic frequency axis

Decibel values:

• $0\mathrm{dB}$: gain 1 (unchanged signal)
• $-3\mathrm{dB}$: gain $\frac{1}{\sqrt{2}}\approx 0.707$ (cut-off frequency)
• $-20\mathrm{dB}$: gain $0.1$ (attenuation to one tenth)
• $+20\mathrm{dB}$: gain $10$

3 Practical and worked examples

3.1 RC low-pass filter

A low-pass filter (low-pass filter) consists of $R=1\mathrm{k\Omega }$ in series with $C=100\mathrm{nF}$. The output voltage is taken across the capacitor.

Transfer function (complex voltage divider):

$$\underline{F}(j\omega )=\frac{{\underline{Z}}_C}{{\underline{Z}}_R+{\underline{Z}}_C}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}=\frac{1}{1+j\omega RC}$$

Cut-off frequency (at $-3\mathrm{dB}$):

$$f_c=\frac{1}{2\pi RC}=\frac{1}{2\pi \cdot 1000\mathrm{\Omega }\cdot 100\cdot 10^{-9}\mathrm{F}}=\underline{1.59\mathrm{kHz}}$$

Amplitude response:

$$|F(\omega )|=\frac{1}{\sqrt{1+{\left(\frac{f}{f_c}\right)}^2}}$$

Behaviour:

• At $f\ll f_c$: $|F|\approx 1$ ($0\mathrm{dB}$) – the signal passes unattenuated
• At $f=f_c$: $|F|=\frac{1}{\sqrt{2}}$ ($-3\mathrm{dB}$) – cut-off frequency
• At $f\gg f_c$: roll-off of $-20\mathrm{dB}$ per decade

3.2 RC high-pass filter

Same components as above, but the output voltage is taken across the resistor.

$$\underline{F}(j\omega )=\frac{j\omega RC}{1+j\omega RC}$$

The cut-off frequency is identical: $f_c=1.59\mathrm{kHz}$. Below it, low frequencies are attenuated; above it, high frequencies pass unattenuated.

4 Typical orders of magnitude

Filter type	Cut-off frequency	Typical application
Subwoofer crossover	$80$ bis $200\mathrm{Hz}$	Audio
Anti-aliasing filter	$f_c<\frac{f_{\mathrm{s}}}{2}$	A/D conversion
EMC mains filter	$50\mathrm{Hz}$ pass, $>100\mathrm{kHz}$ stop	Interference suppression
RF band-pass filter	$\mathrm{MHz}$ to $\mathrm{GHz}$	Radio engineering

5 Example phrasings

• ‘The amplitude response of the RC low-pass filter rolls off above the cut-off frequency at $-20dB/decade$.’
• ‘At the cut-off frequency, the attenuation is $-3\mathrm{dB}$.’
• ‘At $f=f_c$, the phase response shows a phase difference of $-45°$.’
• ‘In the Bode plot, the frequency axis is shown on a logarithmic scale.’

6 Related quantities

• Frequency and angular frequency: independent variable of the frequency response
• Impedance: $\underline{Z}(j\omega )$ is itself a frequency-dependent expression
• Resonance frequency: frequency of maximum gain in the resonant circuit
• Phase angle: frequency-dependent in the phase response
• Reactance: frequency-dependent, $X_L=\omega L$, $X_C=\frac{1}{\omega C}$

7 Related components

• Resistor: frequency-independent – reference element in filters
• Capacitor: frequency-dependent impedance – basic building block for low-pass and high-pass filters
• Inductor: frequency-dependent impedance – basic building block for high-pass and low-pass filters and resonant circuits