Summary of Low Pass Filters and High Pass Filters - RC and RL Circuits

Summary of "Low Pass Filters and High Pass Filters - RC and RL Circuits"

This video provides an in-depth explanation of low pass and high pass filters using RC (Resistor-Capacitor) and RL (Resistor-Inductor) circuits. It covers the fundamental principles, circuit configurations, frequency responses, and key formulas for cutoff frequencies and signal attenuation.

Main Ideas and Concepts

1. Low Pass Filters (LPF)

Function: Pass low frequency signals with minimal attenuation; impede or reduce amplitude of high frequency signals.
Key Behavior:
- Low frequency signals maintain amplitude.
- High frequency signals experience significant amplitude reduction.

a. RC Low Pass Filter

Circuit: Resistor and Capacitor arranged so output voltage is across the Capacitor.
Capacitive reactance \( X_C = \frac{1}{2 \pi f C} \) decreases with frequency.
At low frequencies, high capacitive reactance blocks signals less, allowing output.
At high frequencies, low capacitive reactance shunts signals to ground, reducing output.
Output voltage formula related to capacitive reactance and impedance.
Cutoff frequency: \( f_c = \frac{1}{2 \pi R C} \)
At cutoff frequency, output voltage is 70.7% of input (3 dB drop).
Decibel drop formula: \( 20 \log_{10} \left(\frac{V_{out}}{V_{in}}\right) \).

b. RL Low Pass Filter

Circuit: Resistor and Inductor with output across the Resistor.
Inductive reactance \( X_L = 2 \pi f L \) increases with frequency.
At low frequencies, low inductive reactance allows signals to pass.
At high frequencies, high inductive reactance blocks signals, reducing output.
Output voltage inversely related to inductive reactance.
Cutoff frequency: \( f_c = \frac{R}{2 \pi L} \)
Resistance directly proportional to cutoff frequency; inductance inversely proportional.

2. High Pass Filters (HPF)

Function: Pass high frequency signals; block or attenuate low frequency signals.
Output voltage increases with frequency.

a. RC High Pass Filter

Circuit: Positions of Resistor and Capacitor swapped compared to RC LPF; output across Resistor.
Capacitive reactance decreases with frequency, increasing output voltage.
At very high frequencies, output voltage approaches input voltage (no amplification).
Cutoff frequency: Same formula as RC LPF \( f_c = \frac{1}{2 \pi R C} \).
Frequency response graph shows output voltage increasing with frequency.

b. RL High Pass Filter

Circuit: Similar to RL LPF but output taken differently.
Inductor blocks high frequency signals (high \( X_L \)) and passes low frequency signals.
High frequency signals are forced to output.
Cutoff frequency: Same as RL LPF \( f_c = \frac{R}{2 \pi L} \).
Frequency response similar to RC HPF.

3. Reactance and Frequency Relationship

Capacitive Reactance \( X_C \): Inversely proportional to frequency.
- High frequency → low \( X_C \) → Capacitor passes signals easily.
- Low frequency → high \( X_C \) → Capacitor blocks signals.
Inductive Reactance \( X_L \): Directly proportional to frequency.
- High frequency → high \( X_L \) → Inductor blocks signals.
- Low frequency → low \( X_L \) → Inductor passes signals easily.

Methodologies / Key Formulas

Voltage Divider Output:
\( V_{out} = \frac{R_2}{R_1 + R_2} V_{in} \)
Capacitive Reactance:
\( X_C = \frac{1}{2 \pi f C} \)
Inductive Reactance:
\( X_L = 2 \pi f L \)
Impedance (general for RL or RC circuits):
\( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
Cutoff Frequency:
- RC Circuits: \( f_c = \frac{1}{2 \pi R C} \)
- RL Circuits: \( f_c = \frac{R}{2 \pi L} \)
Decibel Calculation:
\( \text{dB} = 20 \log_{10} \left(\frac{V_{out}}{V_{in}}\right) \)

Notable Quotes

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