The RC circuit shown in Fig. 30–39 is a low-pass filter because it passes low-frequency ac signals with less attenuation than high-frequency ac signals. (a) Show that the voltage gain is $A=V_{\text{out}}/V_{\text{in}}=1/(4{\pi }^2{f}^2{R}^2{C}^2+1{)}^{\frac{1}{2}}$ (b) Discuss the behavior of the gain A for f → 0 and f → ∞.

Verified step by step guidance

1. Step 1: Start by analyzing the RC circuit. The circuit consists of a resistor (R) and a capacitor (C) in series, with the output voltage (Vₒᵤₜ) taken across the capacitor. The input voltage (Vᵢₙ) is applied across the series combination of R and C.
2. Step 2: Write the impedance of the resistor and capacitor. The resistor has an impedance Z_R = R, and the capacitor has an impedance Z_C = 1 / (2πfC), where f is the frequency of the input signal.
3. Step 3: Calculate the total impedance of the series combination. The total impedance Z_total is given by Z_total = Z_R + Z_C = R + 1 / (2πfC).
4. Step 4: Use the voltage divider rule to find the output voltage Vₒᵤₜ. According to the voltage divider rule, Vₒᵤₜ = Vᵢₙ * (Z_C / Z_total). Substitute Z_C and Z_total into this equation.
5. Step 5: Simplify the expression for the voltage gain A = Vₒᵤₜ / Vᵢₙ. After simplification, you will find that A = 1 / √(1 + (2πfRC)²). Discuss the behavior of A as f → 0 (low frequency) and f → ∞ (high frequency). For f → 0, A approaches 1, meaning the circuit passes low-frequency signals with minimal attenuation. For f → ∞, A approaches 0, meaning high-frequency signals are heavily attenuated.

Key Concepts

RC Circuit

An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel. It is fundamental in filtering applications, where it can either pass or block certain frequencies of an input signal. In a low-pass filter configuration, the circuit allows low-frequency signals to pass while attenuating high-frequency signals, making it essential for applications in audio processing and signal conditioning.

Voltage Gain

Voltage gain (A) is a measure of how much an amplifier increases the voltage of a signal. It is defined as the ratio of the output voltage (Vₒᵤₜ) to the input voltage (Vᵢₙ). In the context of the RC low-pass filter, the voltage gain varies with frequency, and understanding its mathematical expression helps analyze how the circuit responds to different input frequencies.

Frequency Response

Frequency response describes how the output of a system varies with input frequency. For the RC low-pass filter, as frequency (f) approaches zero, the gain approaches its maximum value, indicating that low frequencies are passed with minimal attenuation. Conversely, as frequency approaches infinity, the gain approaches zero, demonstrating that high frequencies are significantly attenuated, which is crucial for understanding the filter's behavior in practical applications.