RLC Circuits

In one line: Resistors dissipate energy, while capacitors and inductors store it. Together they set how fast circuits respond, which frequencies pass, and where a circuit resonates.

Core idea

A resistor turns electrical energy into heat. The two other basic passive elements store energy instead:

• A capacitor stores energy in an electric field. For two parallel plates, $C=\epsilon A/d$: larger plates, closer spacing and better dielectrics give more capacitance. Its current is $i=Cdv/dt$, so its voltage can’t change instantly. It stores $\frac{1}{2}C{v}^2$.
• An inductor stores energy in a magnetic field. Its voltage is $v=Ldi/dt$, so its current can’t change instantly. It stores $\frac{1}{2}L{i}^2$.

Because they resist sudden change, these elements give circuits memory and timing.

How it works

First-order circuits have one storage element.

• An RC or RL circuit responds to a step with an exponential approach to its final value.
• The time constant is $\tau =RC$ or $L/R$. After about five time constants, the response is within 1% of final.
• Example: 1 kΩ with 1 µF gives τ = 1 ms, so the circuit settles in about 5 ms.

Second-order circuits have both a capacitor and an inductor. Energy sloshes between them, and the response may be:

• overdamped: slow, with no overshoot
• critically damped: the fastest response without overshoot
• underdamped: ringing that dies away

Resonance. In AC analysis, inductors and capacitors have impedances that depend on frequency and act in opposite directions. They cancel at the resonant frequency:

$f_0=\frac{1}{2\pi \sqrt{LC}}$

10 µH with 100 pF resonates near 5 MHz. The quality factor $Q$ measures how sharp the resonance is.

Filters. RC, RL and RLC networks pass some frequencies and block others:

• low-pass: smoothing and anti-aliasing
• high-pass: blocking DC
• band-pass: tuning a radio
• band-stop: rejecting hum

In practice

These ideas appear everywhere:

• power-supply smoothing, and decoupling capacitors next to every chip
• timing circuits
• sensor filtering
• the tuned circuits of every radio
• parasitic capacitance and inductance in wiring, which limit how fast digital signals can switch

Second-order behavior, a system with two kinds of storage that can overshoot and ring, is also a useful mental model far outside electronics (see Thinking in Systems).

Questions to ask the experts

• What sets this circuit’s response time or bandwidth?
• Where do parasitic capacitance and inductance matter at our operating speed?
• How much margin is there to the nearest resonance?

Connections

• The discipline: Electrical and Computer Engineering. Solving them: Nodal Analysis.
• Active filters: Operational Amplifiers. When wavelength matters: Smith Chart. Stocks, delays and oscillation: Thinking in Systems.
• Capabilities exercised: Inquiry, Connecting Across Disciplines, Decision Making.

Sources

• Course lectures on capacitors, inductors and RLC circuits; no public source.