# Fundamental Physics/Electricity/Electric circuits/RC Circuits

## RC Circuits

RC circuits are circuits that contain a resistor and a capacitor. These circuits are primarily used as frequency filters. There are two basic arrangements: high-pass and low-pass. A high-pass filter allows frequencies above the cut-off frequency to pass, while a low-pass filter allows frequencies beneath the cut-off frequency to pass. The arrangement of the resistor and the capacitor is what determines their behaviour.

Note that at a particular frequencly, called the cut-off frequency, the Capactive Reactance is equal to the Resistance value. (There is also an associated phase shift of 45 degrees.)

${\displaystyle R=X_{c}}$

Substituting

${\displaystyle X_{C}={\frac {1}{2\pi fC}}}$ we then have:
${\displaystyle R={\frac {1}{2\pi fC}}}$

The cut-off frequency, defined as the frequency at which the signal power is attenuated by 50% (or 3.01 dB), is a function of the resistive and capacitive values. We can rearrange the above formula to solve for ${\displaystyle f}$ as follows:

${\displaystyle f_{cut-off}={\frac {1}{2\pi RC}}}$

## RC series

A circuit of 2 component a resistor and a capacitor connected in series

${\displaystyle V_{C}+V_{R}=0}$
${\displaystyle C{\frac {d}{dt}}v(t)+{\frac {v}{R}}v(t)=0}$
${\displaystyle {\frac {d}{dt}}v(t)=-{\frac {1}{RC}}v(t)}$
${\displaystyle \int {\frac {dv(t)}{v(t)}}=-\int {\frac {1}{RC}}dt}$
${\displaystyle Lnv(t)=-{\frac {1}{RC}}t+c}$
${\displaystyle v(t)=e^{(-{\frac {1}{RC}}t+c)}}$
${\displaystyle v(t)=Ae^{-{\frac {1}{RC}}t}}$

## Filter circuit

### Low pass filter

When the capacitor is in parallel with the load while the resistor is in series with the capacitor and load, this creates a low pass filter.

Low pass filter has a transfer function

${\displaystyle H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}=RC}$

Frequency response of Low pass filter

${\displaystyle \omega =0.v_{o}=v_{i}}$
${\displaystyle \omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega =00.v_{o}=0}$

Cut off frequency, ${\displaystyle \omega _{o}}$ , frequency at which ${\displaystyle v_{o}={\frac {1}{2}}v_{i}}$

${\displaystyle \omega _{o}={\frac {1}{T}}}$

### High pass filter

When the resistor is in parallel with the load and the capacitor is in series with the resistor, a high pass filter is created.

High pass filter has a transfer function

${\displaystyle H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {j\omega T}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}=RC}$

Frequency response of High pass filter

${\displaystyle \omega =0.v_{o}=0}$
${\displaystyle \omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega =00.v_{o}=v_{i}}$

Cut off frequency, ${\displaystyle \omega _{o}}$ , frequency at which ${\displaystyle v_{o}={\frac {1}{2}}v_{i}}$

${\displaystyle \omega _{o}={\frac {1}{T}}}$

A single RC circuit creates a filter with a 20.0 dB/decade, or 6.02 dB/octave, slope.