# Circuit Theory/Frequency Response

If

The Output Voltage is Vo
The Input Voltage is Vi
ω = 2πf
The Impedance of Resistor : ZR = R
The Impedance of Capacitor : ZC = RC + 1 / jωC
The Impedance of Inductor : ZL = RL + jωL
The Respond Frequency is the Frequency when Capacitor or Inductor starts to React or conduct current : ωo
ω : 0 → ωo : Low Frequency Range
ω : ωo → 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

$V_o = V_i - I * Z_C$ = Vi - I * [RC + 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

$V_o = V_i - I * Z_L$ = Vi - I * [RL - 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