# Fundamental Physics/Electronics/Frequency Response

## Resistor

For lossless resistor

${\displaystyle X_{R}={\frac {v_{c}}{i_{c}}}=0}$

For lossy resistor

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

## Capacitor

AC response of a capacitor connected with AC voltage source

### Lossless capacitor

Its reactance is calculated by

${\displaystyle X_{C}={\frac {v_{c}}{i_{c}}}={\frac {{\frac {1}{C}}\int idt}{c{\frac {dv}{dt}}}}}$

take Fourier transform

${\displaystyle X_{C}={\frac {1}{j\omega C}}}$
For lossless capacitor , frequency response can be determin by
${\displaystyle X_{C}(\omega =0)=00}$ . At frequency equals to zero , capacitor acts as open circuit switch
${\displaystyle X_{C}(\omega =00)=0}$ . At frequency equals to infinite , capacitor acts as open short switch
${\displaystyle X_{C}(\omega =\omega _{o})=1}$ . Capacitor operates at cut off frequency${\displaystyle \omega _{0}={\frac {1}{C}}}$


### Lossy capacitor

${\displaystyle Z_{C}=X_{C}+R_{C}}$
${\displaystyle X_{C}={\frac {v_{c}}{i_{c}}}+R_{C}={\frac {{\frac {1}{C}}\int idt}{c{\frac {dv}{dt}}}}+R_{C}}$

take Fourier transform

${\displaystyle Z_{C}={\frac {1}{j\omega C}}+R_{C}={\frac {j\omega T+1}{R_{C}}}}$
${\displaystyle T=CR_{C}}$

Frequency response can be determin by

${\displaystyle X_{C}(\omega =0)=00}$ . At frequency equals to zero , capacitor acts as open circuit switch
${\displaystyle X_{C}(\omega =00)=0}$ . At frequency equals to infinite , capacitor acts as open short switch
${\displaystyle X_{C}(\omega =\omega _{o})=1}$ . Capacitor operates at cut off frequency${\displaystyle \omega _{0}={\frac {1}{C}}}$

## Inductor

AC response of a capacitor connected with AC voltage source

For lossless inductor , its reactance is calculated by

${\displaystyle X_{L}={\frac {v_{c}}{i_{c}}}={\frac {L{\frac {di}{dt}}}{{\frac {1}{L}}\int vdt}}}$

take Fourier transform

${\displaystyle X_{L}=j\omega L}$
For lossless inductor , frequency response can be determin by
${\displaystyle X_{L}(\omega =0)=00}$ . At frequency equals to zero , inductor acts as short circuit
${\displaystyle X_{L}(\omega =00)=0}$ . At frequency equals to infinite , inductor acts as open circuit
${\displaystyle X_{L}(\omega =\omega _{o})=1}$ . Inductor operates at cut off frequency ${\displaystyle \omega _{0}={\frac {1}{L}}}$


For lossy capacitor

${\displaystyle Z_{L}=X_{L}+R_{L}}$
${\displaystyle X_{L}={\frac {v_{c}}{i_{c}}}={\frac {L{\frac {di}{dt}}}{{\frac {1}{L}}\int vdt}}+R_{L}}$

take Fourier transform

${\displaystyle Z_{L}=j\omega L+R_{L}}$