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.

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