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Electronics Basics: Analyze an RC Low-Pass Filter

Connect an ideal circuit model to transient and frequency behavior.

All worksheets
5 questions0m 0s

Prerequisites

Kirchhoff's laws, first-order differential equations, complex frequency and Laplace transforms.

Learn the skill

Use an ideal input voltage source, series R, and capacitor C to the reference node; output is across C with no additional load. With zero initial state, H(s) = V_out(s)/V_in(s) = 1/(1 + sRC). Let tau = RC. For sinusoidal steady state, magnitude is 1/sqrt(1 + (omega tau)^2). Use diagrams and calculations only. Never use wall outlets, short batteries, open powered equipment or handle charged capacitors.

Worked example

For R = 1000 ohms and C = 0.000001 F, tau = 0.001 s. Cutoff angular frequency is 1000 rad/s; cutoff frequency is about 159.15 Hz. At cutoff, gain magnitude is about 0.707 and phase is -45 degrees.

Input voltage drives a series resistor R. A capacitor C connects the output node to the common reference line. Output voltage is measured across the capacitor with no other load; H(s) is one divided by one plus sRC.
Input voltage drives a series resistor R. A capacitor C connects the output node to the common reference line. Output voltage is measured across the capacitor with no other load; H(s) is one divided by one plus sRC.
Question 1 What is tau for the example?
Question 2 What is cutoff frequency to two decimals?
Question 3 At omega tau = 1, what is the magnitude to three decimals?
Question 4 At omega tau = 1, what is the phase?
Question 5 What must change if a finite resistance is added across the output?

Further inquiry

Derive RC dV_out/dt + V_out = V_in and its zero-state transfer function. Find the pole, step response and normalized sensitivity of cutoff to R and C. Then place a finite R_load across C and derive the changed DC gain and time constant. State why ideal-source and no-load assumptions cannot silently survive that change.

Review criteria

  • State the source, component, initial-state and load assumptions explicitly.
  • Show units, intermediate derivations and limiting-case checks.
  • Separate a mathematical circuit transformation from any physical equipment procedure.