In a RC low-pass filter, which arrangement of components is correct?

In a RC low-pass filter, which arrangement of components is correct?

• A. Output across resistor
• B. Resistor in series with capacitor to ground; output across capacitor
• C. Output across capacitor; input coupling arrangement
• D. Resistor in parallel with capacitor; output across both

Answer: (B)

This question hinges on how an RC network creates a low-pass response. If you place the resistor in series with the input and connect a capacitor from the node to ground, taking the output across that capacitor, you get the standard RC low-pass filter. The capacitor presents an impedance Zc = 1/(jωC). With the output taken at the node between the resistor and capacitor, the voltage divider gives Vout = Vin · Zc / (R + Zc). This simplifies to Vout/Vin = 1 / (1 + jωRC). At low frequencies, where ω is small, Vout ≈ Vin, so the signal passes. At high frequencies, where ω is large, Vout diminishes roughly as 1/(ωRC), attenuating high-frequency components. If the output were across the resistor instead, the transfer would be Vin · R / (R + Zc) = Vin · (jωRC) / (1 + jωRC), which is a high-pass response, not what we want for a low-pass. Having the resistor and capacitor in parallel or taking the output across both would not yield the same simple single-pole low-pass behavior, so the described arrangement uniquely provides the desired low-pass effect.

This question hinges on how an RC network creates a low-pass response. If you place the resistor in series with the input and connect a capacitor from the node to ground, taking the output across that capacitor, you get the standard RC low-pass filter.

The capacitor presents an impedance Zc = 1/(jωC). With the output taken at the node between the resistor and capacitor, the voltage divider gives Vout = Vin · Zc / (R + Zc). This simplifies to Vout/Vin = 1 / (1 + jωRC). At low frequencies, where ω is small, Vout ≈ Vin, so the signal passes. At high frequencies, where ω is large, Vout diminishes roughly as 1/(ωRC), attenuating high-frequency components.

If the output were across the resistor instead, the transfer would be Vin · R / (R + Zc) = Vin · (jωRC) / (1 + jωRC), which is a high-pass response, not what we want for a low-pass.

Having the resistor and capacitor in parallel or taking the output across both would not yield the same simple single-pole low-pass behavior, so the described arrangement uniquely provides the desired low-pass effect.