# Fundamental Physics/Electronics/RC Circuit

## 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.)

$R=X_{c}$ Substituting

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

$f_{cut-off}={\frac {1}{2\pi RC}}$ ## RC series A circuit of 2 component a resistor and a capacitor connected in series

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

### Low pass RC 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

$H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T}}$ $T={\frac {L}{R}}=RC$ Frequency response of Low pass filter

$\omega =0.v_{o}=v_{i}$ $\omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}$ $\omega =00.v_{o}=0$ Cut off frequency, $\omega _{o}$ , frequency at which $v_{o}={\frac {1}{2}}v_{i}$ $\omega _{o}={\frac {1}{T}}$ ### High pass CR 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

$H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {j\omega T}{1+j\omega T}}$ $T={\frac {L}{R}}=RC$ Frequency response of High pass filter

$\omega =0.v_{o}=0$ $\omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}$ $\omega =00.v_{o}=v_{i}$ Cut off frequency, $\omega _{o}$ , frequency at which $v_{o}={\frac {1}{2}}v_{i}$ $\omega _{o}={\frac {1}{T}}$ A single RC circuit creates a filter with a 20.0 dB/decade, or 6.02 dB/octave, slope.