AMME1705 · Introduction to Electromechanical Systems
Time-Based Circuits: Capacitors, Inductors, RC/RL & Filters
This chapter covers the time-based circuits topic of AMME1705 Introduction to Electromechanical Systems at the University of Sydney — the reactive elements that store energy and shape a signal over time. You will work with the capacitor (C = Q/V, i = C dV/dt), the inductor (L = Φ/I, v = L dI/dt), the RC and RL time constants, and single-pole low-pass and high-pass filters. These formulae feed directly into the paper-based final exam and into the later motor-drive and power-supply chapters.
What this chapter covers
- 01Capacitance C = Q/V and the capacitor current law i = C dV/dt (why a capacitor blocks DC and passes high frequencies)
- 02Inductance L = Φ/I and the inductor voltage law v = L dI/dt (why an inductor shorts DC and generates back-EMF)
- 03Series/parallel combination rules for capacitors and inductors (and how they mirror or match resistors)
- 04The RC time constant τ = RC and the RL time constant τ = L/R, with correct SI units (seconds)
- 05Charging and discharging curves and the 63% (one τ) / settled-by-5τ rule of thumb
- 06Passive low-pass and high-pass filters, and how the output node decides which one you have
- 07The cutoff frequency f_c = 1/(2πRC) and the −3 dB (0.707) half-power point
- 08The first-order low-pass magnitude 1/√(1 + (ωRC)²) and reasoning by reactance X_C and X_L
- 09The RL turn-off transient: the reversed-polarity voltage spike and why inductive loads need a flyback diode
RC low-pass filter: time constant and cutoff frequency
- +1Identify the circuit: the output is taken across the capacitor, so this is a low-pass filter; use τ = RC and f_c = 1/(2πRC).
- +1Time constant: τ = RC = (10 × 10³ Ω)(10 × 10⁻⁹ F) = 1.0 × 10⁻⁴ s = 100 µs.
- +1Cutoff frequency: f_c = 1/(2πRC) = 1/(2π × 1.0 × 10⁻⁴) = 1.59 × 10³ Hz ≈ 1.59 kHz.
- +1At f = f_c the factor ωRC = 1, so |V_out/V_in| = 1/√(1 + 1²) = 1/√2 = 0.707 — the output amplitude is 0.707 of the input (the −3 dB half-power point).
Key terms
- Capacitance (C)
- The charge stored per volt across a capacitor, C = Q/V, measured in farads (F = C/V). Real values are usually µF, nF or pF.
- Capacitor current law (i = C dV/dt)
- The current into a capacitor is proportional to the rate of change of its voltage. In DC steady state dV/dt = 0, so the current is zero and the capacitor acts as an open circuit.
- Inductance (L)
- The magnetic flux linked per amp of current, L = Φ/I, measured in henries (H = Wb/A).
- Inductor voltage law (v = L dI/dt)
- The voltage across an inductor is proportional to the rate of change of its current (the back-EMF). In DC steady state dI/dt = 0, so the voltage is zero and the inductor acts as a short circuit.
- Time constant (τ)
- The characteristic settling time of a reactive circuit: τ = RC for an RC circuit and τ = L/R for an RL circuit, both in seconds. After one τ the response reaches about 63% of its final value.
- Cutoff frequency (f_c)
- The turnover frequency of a first-order filter, f_c = 1/(2πRC) for an RC filter (f_c = R/(2πL) for an RL filter) in Hz, where the output amplitude has dropped to 1/√2 ≈ 0.707 of the input — the −3 dB, or half-power, point.
- Low-pass filter
- An RC filter with the output taken across the capacitor: it passes low frequencies and attenuates high ones. Its amplitude response is 1/√(1 + (ωRC)²).
- High-pass filter
- An RC filter with the output across the resistor (or an RL filter with the output across the inductor): it passes high frequencies and blocks low ones. The cutoff is f_c = 1/(2πRC) for the RC version and f_c = R/(2πL) = 1/(2πτ) for the RL version (τ = L/R).
Time-Based Circuits: Capacitors, Inductors, RC/RL & Filters FAQ
Is this topic actually tested in the AMME1705 exam?
Yes. The AMME1705 final exam is paper-based and worth 33% of the unit, and time-based circuits are a standard part of it. Questions typically give you the component values in the stem and ask for a specified-precision number — a time constant, a cutoff frequency, a turn-off voltage, or a low- versus high-pass verdict. Treat the exam as a restricted-materials paper (calculator plus, per the exam cover sheet, one A4 double-sided handwritten note sheet), and confirm the exact open- or closed-book status and the date on your current Canvas page, as USyd wording has differed between the assessment table and the exam cover.
What is the single most common mistake on capacitors and inductors?
Mixing up the combination rules and the DC behaviour. Capacitors add in parallel and combine by reciprocals in series (the reverse of resistors), while inductors follow the same pattern as resistors. In settled DC a capacitor is an open circuit (zero current) and an inductor is a short circuit (zero volts). And remember τ = L/R is a division, so a bigger resistor makes an RL circuit faster, whereas τ = RC makes an RC circuit slower.
Can AI help me with capacitors, inductors and filters in AMME1705?
Yes — used the right way. Sia, the AskSia AI tutor, can explain step by step why a capacitor blocks DC, walk you through deriving a cutoff frequency, or check your reasoning on whether a circuit is low- or high-pass, so you understand the method rather than just copying a result. It will not hand you exam answers or guarantee a mark; the goal is to build the understanding and the unit discipline that the marked, specified-precision questions reward.
Studying with AI? Sia — free AI electrical engineering tutor works through AMME1705 step by step.
Exam move
Make this topic about two habits: reading the circuit and checking the units. First, always find the output node before you decide anything — the same resistor and capacitor give a low-pass (output across C) or a high-pass (output across R), so read the schematic rather than assuming from the parts. Second, keep the four defining relations at your fingertips (C = Q/V, i = C dV/dt, L = Φ/I, v = L dI/dt) and the two time constants (τ = RC, τ = L/R), and always confirm the units line up (Ω·F = s, H/Ω = s). Practise the specified-precision arithmetic: carry full accuracy through and round only the final line. Because the practice paper is about 80 marks across the 120-minute writing time, aim for roughly 1.5 minutes per mark, and confirm your paper's mark total on its cover. Finish by rehearsing the exam checklist — settled DC means capacitor open and inductor short; a timescale means one of the two time constants; a filter means read the output node then apply f_c = 1/(2πRC) (RC) or f_c = R/(2πL) (RL); and a switch-off means a reversed-polarity inductor spike of −IR.
