# Fundamental Physics/Electronics/Frequency Response

## Resistor

For lossless resistor

$X_{R}={\frac {v_{c}}{i_{c}}}=0$ For lossy resistor

$Z_{L}=X_{R}+R=R$ ## Capacitor

AC response of a capacitor connected with AC voltage source

### Lossless capacitor

Its reactance is calculated by

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

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

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

$Z_{C}={\frac {1}{j\omega C}}+R_{C}={\frac {j\omega T+1}{R_{C}}}$ $T=CR_{C}$ Frequency response can be determin by

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

AC response of a capacitor connected with AC voltage source

For lossless inductor , its reactance is calculated by

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

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

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

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