For a series RC low-pass filter with transfer function H(jω) = 1 / (1 + j ω R C), what is the magnitude |H(jω)|?

For a series RC low-pass filter with transfer function H(jω) = 1 / (1 + j ω R C), what is the magnitude |H(jω)|?

• A. 1
• B. 1 / sqrt(1 + (ω R C)^2)
• C. 1 / (1 + j ω R C)
• D. sqrt(1 + (ω R C)^2)

Answer: (C)

Understanding how to take the magnitude of a complex transfer function is key here. For H(jω) = 1 / (1 + j ω R C), the denominator is the complex number 1 + j(ωRC). Its magnitude is sqrt(1^2 + (ωRC)^2). Since the numerator is real and equal to 1, the magnitude of the whole expression is the numerator over the magnitude of the denominator: |H(jω)| = 1 / sqrt(1 + (ωRC)^2). This shows the RC low-pass behavior: at low frequencies the magnitude is near 1, and it rolls off as 1/√(1 + (ωRC)^2) as frequency increases. The form 1 / (1 + j ω R C) is the transfer function itself, not its magnitude, so it isn’t the correct magnitude.

Understanding how to take the magnitude of a complex transfer function is key here. For H(jω) = 1 / (1 + j ω R C), the denominator is the complex number 1 + j(ωRC). Its magnitude is sqrt(1^2 + (ωRC)^2). Since the numerator is real and equal to 1, the magnitude of the whole expression is the numerator over the magnitude of the denominator: |H(jω)| = 1 / sqrt(1 + (ωRC)^2). This shows the RC low-pass behavior: at low frequencies the magnitude is near 1, and it rolls off as 1/√(1 + (ωRC)^2) as frequency increases. The form 1 / (1 + j ω R C) is the transfer function itself, not its magnitude, so it isn’t the correct magnitude.