OHMLENS // EDA

Lesson 2.3 · Module 2: Reactive Components in Time & Frequency

RC Low-Pass & High-Pass Filters (Bode Plot)

A resistor and capacitor used as a filter that lets low notes through and blocks high ones (or the other way round). Audio tone controls work like this.

The live, interactive simulation for this lesson opens here when JavaScript is on.

Key ideas

  • A capacitor lets fast changes (high frequency) pass easily, but blocks slow ones.
  • Arranged one way it is a low-pass filter (removes hiss); the other way it is a high-pass filter (removes hum).
  • The cut-off frequency f = 1 ÷ (2π·R·C) is where the signal is reduced to about 70 %.

Try this in the simulation

  • Sweep the test-tone frequency slider up and watch the output wave get smaller.
  • Switch to the Bode plot to see the whole picture at once: loudness against frequency.
  • Flip the topology to high-pass and repeat.

How do RC low-pass and high-pass filters work?

A capacitor passes fast-changing signals easily and blocks slow ones. Put it with a resistor and you get a filter. In a low-pass filter the output is taken across the capacitor, so low frequencies pass and high ones are weakened. In a high-pass filter the two parts swap places. The cutoff frequency f = 1 ÷ (2π × R × C) is where the output has dropped to about 70 % of the input, and a Bode plot shows the whole picture of loudness against frequency.

What you’ll learn

  • The difference between a low-pass and a high-pass RC filter.
  • How to calculate the cutoff frequency from R and C.
  • How to read a Bode plot (gain and phase against frequency).
  • Why a small capacitor next to a chip’s power pin, a decoupling capacitor, keeps its supply steady.

The formula, letter by letter

fc=12πRC=1.06 kHz∣H∣=UoutUin=11+(f/fc)2=-2.8 dB\begin{aligned}\htmlData{sym=f_c}{f_{\mathrm{c}}} &= \frac{1}{2\htmlData{sym=pi}{\pi }\htmlClass{mv mv-r}{\htmlStyle{font-size:1.14em}{\htmlData{sym=R}{R}}} \htmlClass{mv mv-c}{\htmlStyle{font-size:1.13em}{\htmlData{sym=C}{C}}}} = \htmlData{funit=kHz,v=1061.032953945969}{\text{1.06 kHz}}\\ |\htmlData{sym=H:gain}{H}| &= \frac{\htmlClass{mv mv-v_out}{\htmlStyle{font-size:1.00em}{\htmlData{sym=U_out}{U_{\mathrm{out}}}}}}{\htmlClass{mv mv-v_in}{\htmlStyle{font-size:1.00em}{\htmlData{sym=U_in}{U_{\mathrm{in}}}}}} = \frac{1}{\sqrt{1+(\htmlData{sym=f}{f}/\htmlData{sym=f_c}{f_{\mathrm{c}}})^2}} = \htmlData{funit=dB,v=-2.760628044361229}{\text{-2.8 dB}}\end{aligned}

Shown with this lesson’s starting values. In the simulation the numbers change as you move the sliders.

fcf_{\mathrm{c}}
Cutoff frequency (Hz · hertz) : Where a filter starts to cut: the output is down to 71 % (−3 dB).
π\pi
π (pi) ≈ 3.1416 (no unit) : A fixed number. It shows up whenever something goes round in circles, such as a sine wave.
RR
Resistance (Ω · ohm) : How strongly something opposes current.
CC
Capacitance (F · farad) : How much charge a capacitor stores per volt.
HH
Filter gain |H| (no unit) : Output voltage divided by input voltage at this frequency. The bars mean "size of". No unit (shown here in dB).
UoutU_{\mathrm{out}}
Output voltage (V · volt)
UinU_{\mathrm{in}}
Input voltage (V · volt)
ff
Frequency (Hz · hertz) : How many times per second something repeats.

Common questions

What is a cutoff frequency?

It is the frequency where a filter lets through about 70 % of the signal (a drop of 3 dB). Below it, a low-pass filter mostly passes the signal. Above it, the signal fades away.

What is a Bode plot?

A graph of how much a circuit amplifies or weakens a signal at each frequency, usually with a second graph for the phase shift. Both axes are logarithmic, so you can see a very wide range at once.

What is a decoupling capacitor for?

A chip switching on and off causes sudden bursts of current. A small capacitor placed right beside the chip’s power pin supplies those bursts locally, so the supply voltage stays steady and the noise does not spread to the rest of the board.

Keep going

Words used in this lesson

Parts in this lesson

All lessons · How components work · Glossary