LCircuit: A Beginner’s Guide to Inductor–Capacitor CircuitsAn LC circuit (also called a resonant circuit, tank circuit, or tuned circuit) is one of the foundational building blocks in electronics. It consists of an inductor (L) and a capacitor © connected together. Despite their simplicity, LC circuits are central to radio-frequency design, filters, oscillators, impedance matching, and many other applications. This guide explains the essential theory, practical design considerations, common topologies, and simple examples to get you started.
What are inductors and capacitors?
- Inductor (L): a two-terminal passive component that stores energy in a magnetic field when current flows through it. Its impedance increases with frequency: Z_L = jωL. Inductors resist changes in current.
- Capacitor ©: a two-terminal passive component that stores energy in an electric field between conductors. Its impedance decreases with frequency: Z_C = 1/(jωC). Capacitors resist changes in voltage.
When combined, these components exchange energy back and forth between the inductor’s magnetic field and the capacitor’s electric field, producing resonance.
Resonance and resonant frequency
An ideal LC circuit resonates at the angular frequency ω0 where the reactances of L and C are equal in magnitude:
ω0 = 1 / sqrt(LC)
In hertz, the resonant frequency f0 is:
f0 = 1 / (2π sqrt(LC))
At resonance:
- The reactive impedances cancel (jωL + 1/(jωC) = 0 for a series LC), leaving only resistive elements.
- In a series resonant circuit the impedance is minimum (ideally zero), allowing maximum current.
- In a parallel resonant circuit the impedance is maximum (ideally infinite), allowing minimum current from a source.
Series vs. parallel LC circuits
Series LC:
- L and C in series across a source.
- At f0, series impedance → minimum; current is maximum.
- Used for narrow-bandpass filtering and frequency-selective current paths.
Parallel LC:
- L and C in parallel across a source.
- At f0, parallel impedance → maximum; circuit appears open.
- Used for narrow-band rejection, impedance peaking, and oscillators.
Comparison:
| Feature | Series LC | Parallel LC |
|---|---|---|
| Impedance at resonance | Minimum | Maximum |
| Common use | Band-pass, current selection | Band-stop, impedance peak, oscillators |
| Current/voltage behavior | Large circulating current, voltage shared | Large circulating current, voltage across branch |
Quality factor (Q) and bandwidth
Quality factor Q quantifies how “sharp” the resonance is and relates to the circuit’s energy loss versus stored energy.
For a series resonant circuit with series resistance R (representing losses): Q = ω0L / R = 1 / (ω0 C R)
Bandwidth (BW) is the frequency range where the power drops to half (–3 dB). For a resonant circuit: BW = f0 / Q
Higher Q → narrower bandwidth and larger voltage/current magnification at resonance; lower Q → broader response and more damping.
Energy exchange and damping
In an ideal LC with no resistance, energy oscillates indefinitely between L and C. Real circuits include resistance (winding resistance in inductors, dielectric loss in capacitors, and external resistances) that dissipate energy and damp the oscillation. Damping factor and Q determine how quickly oscillations decay when not driven.
Practical considerations when building LC circuits
- Parasitics:
- Inductors have series resistance (ESR) and parasitic capacitance; capacitors have ESR and parasitic inductance (ESL). These affect resonant frequency and Q.
- Component tolerances:
- L and C tolerances shift f0. Use precision components or allow tuning (trimmers) in sensitive designs.
- Frequency dependence:
- At high frequencies, stray capacitance and skin effect in inductors change behavior.
- Magnetic coupling:
- Nearby inductors can couple magnetically (mutual inductance), intentionally used in transformers or unintentionally causing interference.
- Layout:
- PCB layout matters: keep loop areas small to reduce radiated emissions and stray inductance; route ground planes properly.
Common applications
- Tuned amplifiers and RF front-ends: select desired frequency and reject others.
- Oscillators: LC tanks define the oscillation frequency (e.g., Colpitts, Hartley).
- Filters: used in band-pass, band-stop, and as building blocks in ladder filters.
- Impedance matching: reactive networks use L and C to transform impedances at a frequency.
- Pulse and timing circuits: shaping transient responses.
Simple design examples
-
Find components for f0 = 1 MHz using a 100 nF capacitor: f0 = 1 / (2π sqrt(LC)) → solve for L: L = 1 / ((2π f0)^2 C) For f0 = 1e6 Hz, C = 100e-9 F: L ≈ 1 / ((2π·1e6)^2 · 100e-9) ≈ 0.253 μH
-
Series resonant with R = 2 Ω, L = 10 μH, C = 100 pF: f0 = 1/(2π sqrt(10e-6·100e-12)) ≈ 5.03 MHz Q ≈ ω0 L / R (compute ω0 = 2πf0)
(Use precise calculators or SPICE for accuracy; these hand calculations are for estimation.)
Simulating LC circuits
Use SPICE-based tools (LTspice, ngspice, Qucs) or RF tools to:
- Sweep frequency for magnitude/phase response.
- Simulate transient ring-down to measure damping/Q.
- Include parasitic models for realistic results.
Troubleshooting tips
- If resonance is lower than expected: check for stray/parasitic capacitance or larger-than-expected capacitor/inductor value.
- If Q is poor: check component ESR, coil resistance, and PCB losses.
- Unexpected coupling or detuning: separate inductors, add shielding, or change layout.
Quick rules of thumb
- Doubling C halves the resonant frequency approximately by factor sqrt(2) relationship: f ∝ 1/√C.
- Small physical inductors at high frequency often have significant parasitic capacitance; consider distributed-element effects.
- For narrowband RF, aim for high-Q inductors and low-loss capacitors (NP0/C0G dielectrics).
Further learning resources
- Textbooks: “The Art of Electronics” (Horowitz & Hill) for practical context; RF/microwave texts for high-frequency behavior.
- Online: SPICE tutorials and RF filter design applets; datasheets for component parasitics.
In summary, LC circuits are a simple yet powerful concept: by exchanging energy between inductors and capacitors you get frequency-selective behavior used across electronics. Start with hand calculations for resonance and Q, simulate with SPICE including parasitics, and prototype with careful layout to achieve reliable results.
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