Engineering Guide
LC Resonance and Tank Design: Q, Bandwidth, Parasitics, and Tolerance
Move from the ideal LC frequency equation to a tank that still behaves correctly after loss, loading, tolerance, bias, temperature, layout, and measurement are included.
Reading time: 19 min · Intermediate · Updated October 9, 2026
What the Ideal Equation Tells You
An ideal inductor stores magnetic energy and an ideal capacitor stores electric energy. At the natural frequency, energy transfers between them and their reactance magnitudes are equal. The LC Resonance Calculator solves frequency, inductance, or capacitance with full-precision values.
Formula reference
Ideal LC Resonance
Normalize L to henries, C to farads, and frequency to hertz before applying the equations.
f0 = 1 / (2π√(LC))ω0 = 2πf0 = 1/√(LC)L = 1 / ((2πf0)²C)C = 1 / ((2πf0)²L)XL = 2πfLXC = 1/(2πfC)|XL| = |XC| at f0Variable definitions
- L is effective inductance at the operating frequency and bias.
- C is effective capacitance at frequency, bias, and temperature.
- Equality of ideal reactances does not define real impedance magnitude or bandwidth.
Series and Parallel Tanks
| Topology | Ideal resonance | Real design question |
|---|---|---|
| Series LC | Reactive terms cancel; impedance approaches zero | What limits current and sets minimum impedance? |
| Parallel LC | Branch susceptances cancel; impedance approaches infinity | What loss and loading set finite tank impedance? |
| Loaded resonator | Natural frequency remains a starting point | How do source, load, coupling, and loss alter Q and center frequency? |
Series resonance can produce high branch current and component voltage magnification. Parallel resonance can produce circulating branch currents even when source current is modest. Check RMS current, peak voltage, capacitor ripple, inductor saturation, and thermal limits.
Q, Damping, and Bandwidth
Formula reference
Useful Resonator Relationships
Do not apply one Q formula across series and parallel equivalent circuits without converting the loss model.
Q ≈ f0/BWSeries RLC: Q = ω0L/Rseries = 1/(ω0CRseries)BW ≈ f0/QVariable definitions
- These forms assume a compatible second-order model and bandwidth definition.
- Rseries includes effective winding, capacitor, source, and reflected loading loss.
- Parallel-tank Q uses a different resistance representation.
High Q narrows bandwidth and increases energy circulation, but also increases sensitivity to tolerance, coupling, and drift. Low Q broadens response and reduces magnification. Filter response synthesis belongs to the Practical Filter Design Guide; this Guide owns the component-level LC tank.
Tolerance and Frequency Spread
Because frequency varies with the inverse square root of both components, a small positive change in L or C lowers frequency by about half the fractional component change. For production limits, calculate exact corners rather than averaging tolerances.
Formula reference
Tolerance Analysis
fmin = 1/[2π√(LmaxCmax)]fmax = 1/[2π√(LminCmin)]small-change: Δf/f ≈ -0.5(ΔL/L + ΔC/C)Variable definitions
- Worst-case analysis assumes component extremes can coincide.
- RSS or statistical analysis requires justified distributions and independence.
- Bias, temperature, and aging shifts must be combined with initial tolerance.
Loss, Parasitics, and Self-Resonance
| Nonideality | Effect | Required check |
|---|---|---|
| Inductor DCR / AC resistance | Lowers Q and heats winding | Impedance and Q versus frequency and bias |
| Inductor parallel capacitance | Creates component SRF | Keep intended tank behavior within valid region |
| Capacitor ESR / ESL | Adds damping and high-frequency resonance | Use impedance curves, not nominal C alone |
| PCB capacitance / trace inductance | Shifts frequency and impedance | Model or measure final layout |
| Source and load | Damps and may pull resonance | Include coupling and termination |
| Magnetic bias | Reduces effective L | Use DC-bias curves at peak current |
Design Workflow
- 1. Define target frequency and required bandwidth or Q.
- 2. Choose a practical impedance level and initial L/C pair.
- 3. Solve the missing component with unrounded values.
- 4. Check inductor SRF, Q, DCR, bias, and saturation.
- 5. Check capacitor dielectric, bias, ESR, ESL, and voltage.
- 6. Calculate tolerance, temperature, and aging corners.
- 7. Include source, load, coupling, and PCB parasitics.
- 8. Measure the assembled tank under operating conditions.
Worked Examples
| Case | Calculation | Result |
|---|---|---|
| 10 µH and 100 pF | 1/(2π√LC) | Approximately 5.0329 MHz |
| 1 mH and 1 µF | 1/(2π√LC) | Approximately 5.0329 kHz |
| Target 1 MHz, C = 1 nF | 1/((2πf)²C) | Approximately 25.330 µH |
| Target 10 MHz, L = 1 µH | 1/((2πf)²L) | Approximately 253.30 pF |
| At resonance | 2πf0L and 1/(2πf0C) | Equal ideal reactance magnitudes |
| f0 = 1 MHz, Q = 50 | BW = f0/Q | 20 kHz |
| L +10%, C nominal | 1/√1.10 | Frequency about 4.65% lower |
| L and C each +5% | 1/1.05 | Frequency about 4.76% lower |
| Capacitor parasitic +5 pF on 100 pF | Use 105 pF effective | Frequency shifts lower |
| Inductor loses 20% L under bias | f proportional to 1/√L | Frequency rises about 11.8% |
Common Mistakes
- Using µH or pF directly in an SI formula.
- Treating series and parallel resonance as the same impedance response.
- Assuming f0 alone determines Q or bandwidth.
- Ignoring inductor self-resonant frequency.
- Using zero-bias inductance in a biased power circuit.
- Ignoring capacitor voltage coefficient and dielectric drift.
- Averaging L and C tolerances instead of calculating corners.
- Ignoring source, load, and coupling resistance.
- Rounding the solved component before round-trip verification.
- Assuming resonance guarantees oscillator startup.
Scope Boundaries
This workflow predicts and verifies a passive LC tank. It does not synthesize an arbitrary filter response, guarantee oscillator startup, model distributed transmission lines, or replace measured S-parameters. Use the actual component models and assembled-layout measurement when dimensions or frequencies invalidate lumped-element assumptions.
Support reference
FAQ
What is LC resonance?
LC resonance is the frequency where ideal inductive and capacitive reactance magnitudes are equal, allowing energy to exchange between magnetic and electric fields.
How do I calculate LC resonant frequency?
Use f0 = 1/(2π√LC), with inductance in henries and capacitance in farads.
What is the difference between series and parallel resonance?
An ideal series LC branch has minimum impedance at resonance. An ideal parallel LC tank has maximum impedance. Real losses and loading keep both values finite.
What happens to XL and XC at resonance?
Their magnitudes are equal: 2πf0L = 1/(2πf0C). Their reactive signs are opposite, so cancellation or circulating current depends on topology.
How are Q and bandwidth related?
For a lightly damped second-order resonator under a compatible bandwidth definition, Q is approximately f0 divided by the -3 dB bandwidth.
Does resonant frequency determine filter bandwidth?
No. Frequency is set ideally by L and C, while bandwidth also depends on resistance, source and load impedance, coupling, topology, and component loss.
How do component tolerances affect resonance?
For small independent changes, Δf/f is approximately -0.5(ΔL/L + ΔC/C). Worst-case corners should be calculated from actual L and C limits.
Why does measured resonance differ from the formula?
DCR, ESR, winding capacitance, capacitor ESL, PCB parasitics, fixture loading, coupling, and frequency-dependent component values shift the result.
Can I operate an inductor above its self-resonant frequency?
The physical part may become net capacitive above SRF. Use its impedance curve and a model valid for the intended frequency rather than the ideal L value alone.
Does LC resonance guarantee oscillator startup?
No. An oscillator also needs adequate loop gain or negative resistance, correct phase, amplitude control, and enough margin to overcome tank loss.
Can the same LC values be used in any topology?
They give the same ideal natural frequency, but impedance level, Q, loading, current, voltage magnification, and response depend on series or parallel connection and surrounding circuitry.
How should I verify an LC tank?
Measure impedance or frequency response with the intended layout, loading, bias, amplitude, temperature, and fixture, then compare against tolerance and parasitic models.
Summary
Use the ideal LC equation to establish a target, then design the real tank around topology, impedance level, Q, bandwidth, loss, loading, tolerance, bias, temperature, parasitics, self-resonance, and measurement.