Circuit Theory/Frequency Response

If

The Output Voltage is V_o

The Input Voltage is V_i

ω = 2πf

The Impedance of Resistor : Z_R = R

The Impedance of Capacitor : Z_C = R_C + 1 / jωC

The Impedance of Inductor : Z_L = R_L + jωL

The Respond Frequency is the Frequency when Capacitor or Inductor starts to React or conduct current : ω_o

ω : 0 → ω_o : Low Frequency Range

ω : ω_o → Infinity  : High Frequency Range

Resistor

The Voltage or Current of the resistor does not change with frequency or time only the magnitude change

$V_{o}=V_{i}-I*Z_{R}=V_{i}-I*R$

Capacitor

The Voltage of the Capacitor lags the applied Voltage one angle of 90°

The Capacitor has a respond frequency equal to 1 / RC (The frequency when Capacitor starts to react i.e. starts to conduct ). And the time it takes to reach this frequency equals RC

The Voltage or Current of the Capacitor will change with frequency or time

$V_{o}=V_{i}-I*Z_{C}$ = V_i - I * [R_C + 1 / jωC]

ω = 0	ω = 0 → 1 / RC	ω = 1 / RC	ω = 1 / RC →	ω = infinity
$V_{o}=0$	O	$V_{o}=V_{i}$	Mo	$V_{o}=V_{i}$

The circuit of the Capacitor is more stable at high frequency . Frequencies from the response frequency 1 / RC to infinity

Inductor

The Voltage of the Inductor lead the applied Voltage one angle of 90°

The Inductor has a respond frequency equal to R / L . And the time it takes to reach this frequency equals L / R

The Voltage or Current of the Inductor will change with frequency or time

$V_{o}=V_{i}-I*Z_{L}$ = V_i - I * [R_L - jωL]

ω = 0	ω = 0 → 1 / RC	ω = 1 / RC	ω = 1 / RC →	ω = infinity
$V_{o}=V_{i}$	$V_{o}=V_{i}$	$V_{o}=V_{i}$	$V_{o}<V_{i}$	$V_{o}=0$

The circuit of the Inductor is more stable at low frequency (The Voltage of the Inductor Does change with Frequency). Frequencies from zero up to the response frequency R / L

With the right connection of the Resistor and Capacitor or Inductor the circuit can be used as a Low Pass Filter or High Pass Filter