Practical Alternating Current Circuit Analysis: Designing an RLC Series Resonant Tank

When performing alternating current circuit analysis for high-frequency power applications—like induction heating, wireless power transfer, or ultrasonic transducer driving—the RLC series resonant tank is the foundational topology. To build a 100 kHz series resonant tank driven by a 12V AC source, you need a 2.5 µH inductor (e.g., Coilcraft XEL2520-252ME) and a 1.0 µF polypropylene film capacitor (e.g., WIMA FKP1U021006B00KSSD). At resonance, the reactive impedances cancel, leaving only the parasitic resistance to limit current, which creates massive voltage multiplication across the reactive components.

The Core Topology: Series vs. Parallel Resonance

Before picking parts, we must define the node topology and justify the series configuration over the parallel alternative.

Why Series Over Parallel?

In alternating current circuit analysis, a series RLC circuit drops its total impedance to a minimum at the resonant frequency ($Z = R$). This allows massive current to flow from a low-impedance voltage source (like a MOSFET H-bridge). Conversely, a parallel RLC circuit spikes its impedance to a maximum at resonance, requiring a high-impedance current source to drive it effectively. Because most DIY and industrial solid-state drivers are voltage-source inverters, the series topology is the default choice for efficient power transfer without complex impedance-matching transformers.

Design Walkthrough: Picking Real Component Values

Let’s design for a target resonant frequency ($f_r$) of 100 kHz. The fundamental resonance formula is:

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

If we select a standard off-the-shelf inductor of 2.5 µH, we can solve for C:

$$C = \frac{1}{(2\pi \cdot 100,000)^2 \cdot 2.5 \times 10^{-6}} \approx 1.013 \mu F$$

We will round to a standard 1.0 µF capacitor value, shifting our actual resonant frequency to roughly 100.6 kHz.

The Hidden Danger: Q-Factor and Voltage Multiplication

The most critical step in AC resonant analysis is calculating the Quality Factor (Q), which dictates voltage multiplication. At resonance, the voltage across the inductor and capacitor is $Q$ times the input voltage.

$$Q = \frac{1}{R_{total}}\sqrt{\frac{L}{C}}$$

Assuming our Coilcraft inductor has a DC resistance (DCR) of 12 mΩ, the capacitor has an equivalent series resistance (ESR) of 8 mΩ, and PCB traces add 10 mΩ, our $R_{total}$ is roughly 0.03 Ω.

$$Q = \frac{1}{0.03}\sqrt{\frac{2.5 \mu H}{1.0 \mu F}} = 33.3 \times 1.58 = 52.6$$

If you drive this circuit with a 12V RMS AC source, the voltage across the capacitor at Node C will be 12V × 52.6 = 631V RMS. This is why you must never use a standard 50V ceramic capacitor here. You must select a high-voltage film capacitor rated for at least 630VDC / 250VAC, such as the WIMA FKP1 series, to prevent catastrophic dielectric breakdown.

Behavior Matrix: Parameter Shifts in AC Analysis

Understanding how the circuit behaves when components drift or the driving frequency shifts is essential for robust design. Refer to this behavior matrix when debugging phase and amplitude issues on the bench.

Parameter Change	Effect on Impedance (Z)	Effect on Phase Angle	Effect on Power Transfer
Drive Frequency > $f_r$	Increases (Inductive dominance)	Current lags voltage (Positive)	Drops sharply; reactive power increases
Drive Frequency < $f_r$	Increases (Capacitive dominance)	Current leads voltage (Negative)	Drops sharply; risk of ZVS failure in H-bridge
Capacitance Decreases (e.g., thermal drift)	Shifts $f_r$ higher	Circuit appears inductive at original drive freq	Reduces unless drive freq is re-tuned
Inductance Increases (e.g., core saturation)	Shifts $f_r$ lower	Circuit appears capacitive at original drive freq	Can cause shoot-through in MOSFET drivers

Failure Modes: What Breaks at the Extremes

Standard textbook alternating current circuit analysis rarely covers what happens when a component fails hard. Here is the failure-mode contrast for the series RLC topology:

• Capacitor Shorts (Node C to Node D): The inductor is now connected directly across the AC source. Because an inductor's impedance at low frequencies (or DC during dead-time) is near zero, massive current will flow, instantly destroying the driving MOSFETs unless protected by a fast-blow fuse or hardware overcurrent desaturation (DESAT) detection.
• Capacitor Opens: The resonant path is broken. Current drops to zero. The inductor's stored magnetic energy has nowhere to go during switching transitions, resulting in massive voltage spikes (inductive kickback) that will avalanche the driver transistors.
• Inductor Shorts: The capacitor is connected directly across the AC source. At 100 kHz, a 1.0 µF capacitor has an impedance of roughly 1.59 Ω. This will draw massive current, likely tripping the source protection or melting the PCB traces.
• Inductor Opens: Circuit broken, zero current. Safest failure mode, but the system ceases to function.

Breadboard Testing Protocol

Never apply full power to a resonant tank without a low-voltage sweep. Follow this numbered sequence to validate your alternating current circuit analysis math on the bench.

1. Assemble the Low-Power Test Bed: Connect the 2.5 µH inductor and 1.0 µF capacitor in series on a breadboard. Do not use long jumper wires; keep leads under 1 cm to minimize stray inductance.
2. Inject a Small Signal: Use a function generator (e.g., Siglent SDG1032X) set to a 1V peak-to-peak sine wave. Connect the generator output to Node A and ground to Node D.
3. Probe Node C: Connect an oscilloscope probe (set to 10x attenuation) across the capacitor (Node C to Node D).
4. Sweep the Frequency: Start at 50 kHz and slowly sweep up to 150 kHz. Watch the oscilloscope amplitude.
5. Identify the Peak: Note the exact frequency where the voltage at Node C peaks. This is your actual $f_r$. It should be within 5% of your calculated 100.6 kHz. If it is significantly lower, your breadboard is introducing excessive stray capacitance.
6. Measure Phase: Connect a second probe to Node A. At the exact resonant peak, the waveforms at Node A and Node C should cross zero at the exact same time (0° phase shift between source voltage and tank current, though capacitor voltage lags current by 90°).

Component Selection Decision Tree

Use this decision matrix to finalize your bill of materials. It eliminates the "it depends" ambiguity and terminates in a concrete hardware selection.

Design Constraint	If True...	If False...	Final Concrete Pick
Is the load highly variable (e.g., melting metal)?	Add a swamping resistor in series to lower Q and broaden the bandwidth.	Rely on the natural ESR of the components for high Q.	No explicit swamping resistor needed for fixed wireless power.
Will the ambient temperature exceed 85°C?	Select a C0G/NP0 ceramic or high-temp polypropylene film.	Standard X7R ceramics or standard film are acceptable.	WIMA FKP1 Polypropylene (stable up to 100°C).
Is the drive frequency > 200 kHz?	Ferrite core inductors will suffer excessive eddy current losses.	Standard powdered iron or ferrite shielded cores are fine.	Coilcraft XEL series (composite core, low loss at 100kHz).
Does the calculated Q-factor voltage exceed 200V?	You must use a capacitor rated > 400VDC.	A standard 50V or 100V rated capacitor is sufficient.	WIMA FKP1 rated at 630VDC / 250VAC.

By following this framework, you move beyond abstract phasor diagrams and build a resonant tank that survives real-world parasitics, thermal drift, and high-Q voltage multiplication. For deeper mathematical proofs on impedance matching and complex power, refer to the foundational guides on Series Resonant Circuits at Electronics Tutorials and the comprehensive RLC Circuit analysis on Wikipedia. Always validate your math with a low-voltage frequency sweep before applying mains or high-current DC bus power.