If you are asking what is resonance in a circuit, the direct answer is this: resonance occurs in an AC circuit containing both inductance (L) and capacitance (C) when their reactive effects perfectly cancel each other out. At this specific resonant frequency ($f_r$), the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$), leaving only the resistance (R) to oppose current flow. The governing formula is $f_r = \frac{1}{2\pi\sqrt{LC}}$.
On the bench, resonance is the mechanism behind radio tuners, induction heaters, and switching power supply snubbers. But designing a resonant circuit requires more than just plugging numbers into a formula. You must choose the correct topology, account for parasitic elements, and select components that won't detune under voltage stress. Below is a complete guide to designing, analyzing, and testing RLC resonance.
Series vs. Parallel RLC Topologies
The way you wire your resistor, inductor, and capacitor dictates whether your circuit magnifies voltage or current. Here is how the two primary topologies compare, including node mappings for your schematic.
Series RLC (Voltage Magnification)
In a series topology, components are daisy-chained. Current is identical through all elements, but voltage drops across each component vary wildly at resonance.
- Node Path: $V_{in}$ $\rightarrow$ Inductor (Node A) $\rightarrow$ Capacitor (Node B) $\rightarrow$ Resistor (Node C) $\rightarrow$ GND.
- Behavior at $f_r$: Impedance drops to its absolute minimum (equal only to R). Current peaks. The voltage across L and C can be many times higher than the source voltage (Q-factor magnification).
- Use Case: Bandpass filters, antenna matching networks, and series snubbers.
Parallel RLC (Current Magnification)
In a parallel topology (often called a tank circuit), all three components share the same two nodes.
- Node Path: $V_{in}$ connects to Node A. Inductor, Capacitor, and Resistor all bridge between Node A and GND.
- Behavior at $f_r$: Impedance peaks to its maximum (equal to R). Line current drops to a minimum, but a massive circulating current flows back and forth between L and C internally.
- Use Case: Oscillator tank circuits, bandstop (notch) filters, and RF amplifiers.
Element Behavior and Failure Mode Matrix
Understanding what happens when a component value drifts—or fails entirely—is critical for troubleshooting. The table below maps parameter changes and extreme failure modes across both topologies.
| Parameter / Fault | Effect on $f_r$ | Effect on Q-Factor | Series Topology Result | Parallel Topology Result |
|---|---|---|---|---|
| Increase L or C | Decreases | Changes slope | Passband shifts lower; peak current drops if R is constant. | Notch frequency shifts lower; circulating current increases. |
| Increase R | No change | Decreases (Series) / Increases (Parallel) | Bandwidth widens; peak current drops; filter becomes 'broad'. | Bandwidth widens; peak impedance drops; tank 'rings' less. |
| Inductor Opens | N/A (Circuit broken) | N/A | Current stops completely. Full $V_{in}$ appears across open L. | Circuit becomes a simple RC low-pass filter. No resonance. |
| Capacitor Shorts | N/A (Circuit broken) | N/A | Inductor and Resistor remain. Acts as RL high-pass. High DC current. | Source is shorted to GND. Breaker trips or source burns out. |
| Inductor Shorts | $f_r$ approaches infinity | N/A | Acts as RC low-pass. DC current limited only by R. | Source shorted through inductor wire. High current, thermal failure. |
Design Walkthrough: Building a 100 kHz Bandpass Filter
Let's design a series RLC bandpass filter targeting a resonant frequency of exactly 100 kHz. We will select real-world components and calculate the expected bandwidth.
1. Selecting the Inductor (L)
We need an inductor that won't self-resonate below our target. A standard 10 mH axial inductor (e.g., Bourns 78F103J-RC) has a self-resonant frequency (SRF) typically around 1.5 MHz, giving us plenty of margin at 100 kHz.
Value: $L = 10\text{ mH} (0.01\text{ H})$.
2. Calculating the Capacitor (C)
Rearranging the resonance formula to solve for C:
$C = \frac{1}{(2\pi f_r)^2 L} = \frac{1}{(2\pi \times 100,000)^2 \times 0.01} \approx 253.3\text{ pF}$.
Standard 5% capacitor values don't include 253 pF. We will parallel a 220 pF and a 33 pF capacitor to get exactly 253 pF.
Critical Spec: You must use C0G/NP0 dielectric MLCCs (e.g., Kemet C315C220J1G5). Do not use X7R or Y5V dielectrics. X7R capacitance drops significantly under AC voltage bias, which will dynamically detune your resonant frequency as the signal amplitude changes. For high-stability resonance, see the Electronics Tutorials guide on series resonance for more on dielectric selection.
3. Picking the Resistor (R) for Target Bandwidth
First, find the inductive reactance at 100 kHz:
$X_L = 2\pi f_r L = 2\pi \times 100,000 \times 0.01 = 6,283\text{ }\Omega$.
The Quality factor (Q) is $X_L / R$. If we want a moderately sharp filter with a Q of roughly 6, we need:
$R = \frac{X_L}{Q} = \frac{6283}{6} \approx 1,047\text{ }\Omega$.
We will use a standard 1 kΩ 1/4W metal film resistor (e.g., Vishay MFR-25).
Actual Q: $6283 / 1000 = 6.28$.
Bandwidth (BW): $f_r / Q = 100,000 / 6.28 = \mathbf{15.9\text{ kHz}}$. The filter will pass frequencies roughly between 92 kHz and 108 kHz.
Step-by-Step Breadboard Verification
Testing resonance requires sweeping an AC signal and measuring the amplitude and phase shift. Here is how to verify the design on the bench using a function generator (like a Siglent SDG1032X) and an oscilloscope (like a Rigol DS1054Z).
- Build the Circuit: Insert the 10 mH inductor, the parallel 220pF/33pF capacitor combo, and the 1 kΩ resistor in series on your breadboard. Connect the GND rail to the function generator's ground shield.
- Account for Parasitics: A standard solderless breadboard has roughly 2 pF to 5 pF of stray capacitance between adjacent rows. At 100 kHz, this shifts $f_r$ down by about 1-2%. Keep component leads short and avoid spreading the L and C across distant rows.
- Probe Setup: Connect Channel 1 of the scope to the function generator output ($V_{in}$). Connect Channel 2 across the 1 kΩ resistor ($V_{out}$). Use 10x probes to minimize probe capacitance loading the circuit.
- Frequency Sweep: Set the generator to a 2Vpp sine wave. Start at 50 kHz. Slowly increase the frequency while watching Channel 2. The amplitude on Ch2 will rise, peak, and then fall.
- Find the Peak: The peak voltage across the resistor should occur right around 98-100 kHz. Record the exact peak frequency.
- Verify Phase Shift: At resonance, $X_L$ and $X_C$ cancel, meaning the circuit is purely resistive. The voltage across the resistor (Ch2) should be exactly in phase with the source voltage (Ch1). Below $f_r$, Ch2 leads (capacitive); above $f_r$, Ch2 lags (inductive). Read more about phase relationships in All About Circuits' AC textbook.
Pushing the Extremes: When Resonance Breaks Down
Resonance is a double-edged sword. When you push the Q-factor to the extremes, physical component limits take over and the math breaks down.
The High-Q Voltage Trap (Series)
If you remove the 1 kΩ resistor and replace it with a 10 Ω resistor to achieve a Q of 628, the bandwidth narrows to just 159 Hz. However, at resonance, the voltage across the inductor and capacitor will be $Q \times V_{in}$. If your function generator outputs 2Vpp, the capacitor will experience $628 \times 2 = 1,256\text{V}$ peak-to-peak. Your 50V-rated ceramic capacitors will instantly arc over and fail, and the inductor's enamel wire insulation will break down. Always calculate internal component voltages before powering up a high-Q series tank.
The Low-Q Damping Wall (Parallel)
In a parallel tank circuit, a low Q (caused by high resistance or high inductor DCR) means the circuit loses its ability to 'ring'. If the parasitic resistance of your inductor's copper windings is too high relative to the reactance, the resonance peak flattens out entirely. The circuit stops acting like a filter and just acts like a messy, lossy inductor. For RF applications, you must use air-core or low-loss powdered iron inductors to keep the Q above 50.
