# Fundamental Physics/Electronics/RL Circuit

RL Circuit refers to a circuit having combination of resistance(s) and inductor(s). They are commonly used in chokes of luminescent tubes. In an A.C. circuit, inductors helps in reducing voltage, without the loss of energy. Due to the inductive reactance, the higher the AC frequency, the greater the impeadence of the inductor. Under DC conditions, an inductor acts as a static resistance.

Like RC circuit , with one resistor and one coil can be connected to form a low pass filter or a high pass filter. 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 Inductive Reactance is equal to the Resitance value. (There is also an associated phase shift of 45 degrees.)

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

Substituting ${\displaystyle X_{L}={2\pi fL}}$ we then have:

${\displaystyle R={2\pi fL}}$

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:

${\displaystyle f_{cut-off}={\frac {R}{2\pi L}}}$

## RL Series

A circuit of 2 component a resistor and an inductor connected in series

${\displaystyle V_{L}+V_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i(t)+Ri(t)=0}$
${\displaystyle {\frac {d}{dt}}i(t)=-{\frac {R}{L}}i(t)}$
${\displaystyle \int {\frac {di(t)}{i(t)}}=-\int {\frac {R}{L}}dt}$
${\displaystyle Lni(t)=-{\frac {R}{L}}t+c}$
${\displaystyle i(t)=e^{-{\frac {R}{L}}t+c}}$
${\displaystyle i(t)=Ae^{-{\frac {R}{L}}t}}$

## RL Filters

### High pass filter

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

High pass filter has a transfer function

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

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 ${\displaystyle v_{o}={\frac {1}{2}}v_{i}}$

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

### Low pass filter

When the resistor is in parallel with the load while the inductor is in series with the resistor and load, a low pass filter is created.

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

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

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 RL circuit creates a filter with a 20.0 dB/decade, or 6.02 dB/octave, slope.