What is the cutoff frequency for follwoing RC circuit?
The correct answer and explanation is :
The cutoff frequency ( f_c ) for an RC circuit is the frequency at which the output voltage drops to ( \frac{1}{\sqrt{2}} ) (approximately 0.707) of its maximum value. For a simple RC low-pass or high-pass filter, the cutoff frequency can be determined using the following formula:
[
f_c = \frac{1}{2 \pi RC}
]
Explanation:
- RC Circuit Basics:
An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel, depending on the type of filter being created (low-pass or high-pass). The resistor controls the rate of charge and discharge of the capacitor. - Impedance of Components:
- The resistor offers resistance ( R ) to current flow, which does not depend on the frequency of the input signal.
- The capacitor has an impedance that decreases with increasing frequency. The capacitive reactance ( X_C ) is given by:
[
X_C = \frac{1}{2 \pi f C}
]
where ( f ) is the frequency of the signal.
- Low-Pass and High-Pass Filters:
- In a low-pass filter, the output is taken across the resistor, and high-frequency signals are attenuated due to the impedance of the capacitor.
- In a high-pass filter, the output is taken across the capacitor, and low-frequency signals are attenuated because the capacitor acts as a short at low frequencies.
- Cutoff Frequency:
The cutoff frequency is the point where the reactance of the capacitor equals the resistance of the resistor, and the total impedance of the circuit causes the output signal to decrease by 3 dB (to 70.7% of the maximum value). This results in the following equation for the cutoff frequency:
[
f_c = \frac{1}{2 \pi RC}
]
where:
- ( R ) is the resistance in ohms,
- ( C ) is the capacitance in farads.
- Importance of Cutoff Frequency:
The cutoff frequency defines the boundary between the pass band (where signals are not attenuated) and the stop band (where signals are significantly attenuated). For frequencies above or below the cutoff frequency (depending on the filter type), the signal is attenuated by 20 dB per decade.
Example Calculation:
For an RC circuit with a resistance of 1 kΩ and a capacitance of 0.1 µF, the cutoff frequency is calculated as:
[
f_c = \frac{1}{2 \pi \times 1000 \times 0.1 \times 10^{-6}} \approx 1,591 \, \text{Hz}
]