Capacitor Inductor Equations: LC Resonance Derivation and Bench Examples

When you place a capacitor and an inductor together on a bench, you are no longer just storing energy; you are building a harmonic oscillator. The capacitor inductor equations govern everything from the output filter of a 5V buck converter to the tuning tank of a 13.56 MHz RFID reader. Yet, most hobbyists and junior engineers memorize the formulas, plug in the wrong micro-prefixes, and end up with melted MOSFETs or deafening audio crossover distortion.

This guide strips away the abstract textbook prose. We will derive the core LC equations, track the units through real solved problems, and dissect a bench failure where ignoring these exact equations destroyed a power supply.

The Core Capacitor Inductor Equations: Resonance and Impedance

When an inductor and capacitor are connected in series or parallel, they exchange energy. The inductor resists changes in current ($v = L \frac{di}{dt}$), while the capacitor resists changes in voltage ($i = C \frac{dv}{dt}$). At a specific frequency, their reactances perfectly cancel out (series) or peak (parallel). This is the resonant frequency ($f_r$). Equally important is the characteristic impedance ($Z_0$), which dictates the surge currents and voltage ringing during transient events.

The Resonant Frequency Equation:
$f_r = \frac{1}{2\pi\sqrt{LC}}$

The Characteristic Impedance Equation:
$Z_0 = \sqrt{\frac{L}{C}}$

Rearranged Forms for Component Selection

On the bench, you rarely solve for frequency. You usually have a target frequency and need to find the missing component. Here are the algebraic rearrangements you will actually use:

• Solving for L: $L = \frac{1}{4\pi^2 f_r^2 C}$
• Solving for C: $C = \frac{1}{4\pi^2 f_r^2 L}$
• Solving for L (from $Z_0$): $L = Z_0^2 \times C$
• Solving for C (from $Z_0$): $C = \frac{L}{Z_0^2}$

When These Formulas Apply (And When They Fail)

These equations assume ideal, lossless components. In reality, every inductor has series resistance (DCR) and parasitic parallel capacitance. Every capacitor has equivalent series resistance (ESR) and equivalent series inductance (ESL).

The formulas apply perfectly for calculating the theoretical center point of resonance. They fail when predicting the Q-factor (quality factor) or the actual peak amplitude of a ringing waveform. If your capacitor has high ESR (like a standard aluminum electrolytic), the resonance will be heavily damped, and the actual peak will be lower and broader than the ideal $f_r$ suggests. For high-Q circuits (like RF tanks), you must use low-ESR components (NP0/C0G ceramics or air-core inductors) for the math to match the oscilloscope trace.

Realistic Answer Magnitudes

If your calculator spits out a number outside these ranges, you have a unit error:

• Audio Crossovers: 20 Hz to 20 kHz. (Expect L in milliHenries, C in tens of microFarads).
• SMPS Output Filters: 1 kHz to 50 kHz. (Expect L in microHenries, C in hundreds of microFarads).
• RF / Antenna Matching: 1 MHz to 1 GHz. (Expect L in nanoHenries, C in picoFarads).

The "Micro-Micro" Unit Trap

The most common mistake that breaks these equations is the "micro-micro trap." If you multiply $10\mu H$ by $10\mu F$, human intuition says the answer is $100\mu$. It is not.

$10\mu H = 10 \times 10^{-6}$ and $10\mu F = 10 \times 10^{-6}$.
$(10 \times 10^{-6}) \times (10 \times 10^{-6}) = 100 \times 10^{-12}$ (which is 100 pico, not 100 micro). Forgetting to convert to base units (Henries and Farads) before multiplying will throw your square root off by a factor of 1,000, shifting your resonant frequency by a factor of 31.6.

Solved Problems: Tracking Units from Micro to Mega

Let's run two real-world scenarios, explicitly tracking the scientific notation to avoid the micro-micro trap.

Problem 1: Switch-Mode Power Supply (SMPS) Output Filter

Scenario: You are designing the LC output filter for a 12V-to-5V buck converter. You select a 47 $\mu$H shielded drum-core inductor and a 220 $\mu$F low-ESR polymer capacitor. What is the resonant frequency of this filter?

1. Convert to base units:
   $L = 47 \times 10^{-6}$ H
   $C = 220 \times 10^{-6}$ F
2. Multiply L and C:
   $LC = (47 \times 10^{-6}) \times (220 \times 10^{-6}) = 10,340 \times 10^{-12} = 1.034 \times 10^{-8}$
3. Take the square root:
   $\sqrt{1.034 \times 10^{-8}} = 1.0168 \times 10^{-4}$
4. Multiply by $2\pi$:
   $2 \times 3.14159 \times 1.0168 \times 10^{-4} = 6.389 \times 10^{-4}$
5. Invert for $f_r$:
   $f_r = \frac{1}{6.389 \times 10^{-4}} = 1565$ Hz

Result: The filter resonates at 1.56 kHz. This is a realistic magnitude for an SMPS filter, safely below a typical 100 kHz switching frequency.

Problem 2: RFID Tank Circuit Component Sizing

Scenario: You are building a 13.56 MHz NFC/RFID reader. Your PCB layout and parasitic capacitance dictate a total tank capacitance of 100 pF. What inductance do you need to hit exactly 13.56 MHz?

1. Convert to base units:
   $f_r = 13.56 \times 10^6$ Hz
   $C = 100 \times 10^{-12}$ F
2. Use the rearranged formula:
   $L = \frac{1}{4\pi^2 f_r^2 C}$
3. Square the frequency:
   $(13.56 \times 10^6)^2 = 1.8387 \times 10^{14}$
4. Calculate the denominator:
   $4 \times (3.14159)^2 \times 1.8387 \times 10^{14} \times 100 \times 10^{-12}$
   $= 39.478 \times 1.8387 \times 10^{14} \times 10^{-10}$
   $= 725,884$
5. Invert for L:
   $L = \frac{1}{725,884} = 1.377 \times 10^{-6}$ H

Result: You need a 1.38 $\mu$H inductor. (In practice, you would use a 1.2 $\mu$H fixed inductor and trim the capacitance with a variable trimmer cap to account for PCB parasitics).

Bench War Story: The LC Snubber That Melted a MOSFET

Equations on paper don't melt silicon; ignoring the physical realities behind the equations does. Here is a teardown of a failure that happened on my own bench.

The Setup: I was prototyping a high-current 24V to 12V synchronous buck converter switching at 200 kHz. To tame the output ripple, I needed an LC filter with a cutoff around 15 kHz. I calculated the values and selected a 4.7 $\mu$H molded inductor and a bank of three 22 $\mu$F X5R MLCC (Multi-Layer Ceramic Capacitors) in parallel.

The Numbers:
Total $C = 66 \mu F$.
Using $f_r = \frac{1}{2\pi\sqrt{4.7\mu H \times 66\mu F}}$, the math yielded a beautiful 9.03 kHz resonant frequency. The characteristic impedance $Z_0 = \sqrt{4.7\mu / 66\mu}$ was a mere 0.26 $\Omega$.

The Outcome: I applied 24V and enabled the controller. The high-side MOSFET (a robust 60V, 40A part) instantly shorted, popping the package and tripping my bench supply's over-current protection.

What Went Wrong: I had treated the capacitor inductor equations as the whole story. I ignored two critical real-world factors:

1. DC Bias Derating: X5R ceramic capacitors suffer massive capacitance loss under DC voltage. According to Murata's DC bias characteristics, a 22 $\mu$F X5R cap rated for 25V will lose up to 60% of its capacitance when 24V is applied. My 66 $\mu$F bank was actually acting like ~26 $\mu$F, shifting my resonance up to 14 kHz and altering the phase margin of the control loop.
2. The $Z_0$ Inrush Spike: Because the MLCCs have near-zero ESR, the characteristic impedance of the LC network was extremely low (0.26 $\Omega$). When the 24V rail stepped in, the inductor and capacitor formed a high-Q tank. The voltage across the capacitor rang to nearly $2 \times V_{in}$ (48V), but the real killer was the inrush current. The inductor's saturation current ($I_{sat}$) was only 8A. The instantaneous LC surge exceeded 30A, saturating the inductor core. Once saturated, the inductor's impedance dropped to just the DCR (a few milliohms), effectively placing a dead short across the MOSFET.

The Fix: I swapped the MLCCs for low-ESR polymer capacitors that don't suffer from DC bias derating, added a 100m$\Omega$ damping resistor in series with a bulk electrolytic capacitor to kill the high-Q ringing, and upgraded to a Wurth WE-HCI series inductor with a 15A saturation rating.

Practical Component Selection for LC Networks

To ensure your math translates to a working circuit, match your component chemistry to the application dictated by your equations.

• For High dV/dt Snubbers & RF: Use Polypropylene film capacitors (e.g., Cornell Dubilier 942C series). They handle massive current spikes without dielectric absorption issues. Pair them with air-core or powdered-iron inductors to avoid core saturation.
• For SMPS Output Filters: Use shielded ferrite-core inductors (e.g., Coilcraft XEL or XGL series) to minimize EMI. Always verify the $I_{sat}$ (saturation current) is higher than your peak switch current, not just your average DC load. For capacitance, use Polymer Aluminum or specialized high-stability ceramics (C0G/NP0 for low power, X7R with heavy derating margins for high power).
• For Audio Crossovers: Use non-inductive bipolar electrolytic capacitors and air-core or silicon-steel laminated inductors. The frequencies are low enough that parasitic ESL is negligible, but core linearity matters to prevent harmonic distortion in the audio band.

The capacitor inductor equations are the map, but component datasheets are the terrain. Always calculate your ideal $f_r$ and $Z_0$ first, then validate your chosen physical parts against their real-world parasitics and saturation limits before applying power.