LC Resonance & Filter Analyzer

Commercial-grade RLC simulator. Calculates resonant frequency ($f_r$), Q-factor, and Bandwidth with parasitic resistance modeling. Features Bode Plots (Frequency Domain) and Step Response (Time Domain) to visualize transient ringing.

Quick Load Presets:

Topology

Inductance ($L$)

Capacitance ($C$)

Series Res ($R_s$/DCR)

Parallel Load ($R_p$)

Source Voltage ($V_{in}$)

Target Freq ($f_{op}$)

Circuit Schematic (Lossy Model)

Simulation Results & Stress Sizing Verification

Circuit Parameters & Sizing

Parameter	Value	Unit
Resonant Frequency ($f_r$ / lossy)	0.00	kHz
Ideal Resonance ($f_0$ / lossless)	0.00	kHz
Damped Natural Freq ($f_d$)	0.00	kHz
Q-Factor (Loaded / Overall)	0.00	-
Bandwidth (-3dB)	0.00	kHz
Lower Cutoff Freq ($f_1$)	0.00	kHz
Upper Cutoff Freq ($f_2$)	0.00	kHz
Damping Ratio ($zeta$)	0.00	-
Characteristic Imp ($Z_0$)	0.00	Ω
Resonant Impedance ($Z_r$)	0.00	Ω
Impedance magnitude at $f_{op}$	0.00	Ω
Phase Angle at $f_{op}$	0.00	deg
Capacitor Voltage at $f_r$	0.00	V RMS
Inductor Voltage at $f_r$	0.00	V RMS
Reactive Power at $f_r$ ($Q_{var}$)	0.00	Var

Damping Regime: Critically Damped

Frequency Response (Magnitude)

Step Response (Time Domain Ringing)

MathJax Step-by-Step Calculation Breakdown

Resonance Mechanics

Energy "Sloshing" Dynamics

Resonance in an RLC circuit is the periodic exchange of energy between the Magnetic Field of the inductor and the Electric Field of the capacitor. At the resonant frequency ($f_0$), these energy transfers perfectly synchronize, allowing the circuit to oscillate with minimal external drive.

Textbook models describe this as a lossless exchange, but in industrial reality, energy is continually bled off by the parasitic resistance ($R$) of the copper windings, leading to the "damping" of the resonant peak.

Quality Analysis

The Q-Factor & Selectivity

The Quality Factor (Q) represents the efficiency of the resonator. A high Q ($Q > 10$) indicates very low energy loss and a extremely sharp frequency response, whereas a low Q ($Q < 1$) suggests rapid energy dissipation and a "mushy" selectivity.

In radio applications, High Selectivity is required to reject adjacent channels. Conversely, in digital signal snubbing, a Low Q is preferred to prevent high-frequency "ringing" that can corrupt data bits.

Impedance Topology

Series vs. Parallel Nuance

The circuit topology fundamentally changes the "job" of resonance. Series RLC acts as an Acceptor, where impedance drops to its minimum ($R_s$) at resonance. Parallel RLC acts as a Rejector (Tank), where impedance rises to its maximum ($Z_p$) at resonance.

This "opposing" behavior makes Series circuits ideal for Harmonic Filters and Parallel circuits ideal for Wave Traps and oscillator feedback loops.

Real-World Parasitics

The Limit of Ideal Models

Professional engineering requires modeling of Parasitics. Inductors possess DC resistance (DCR) and inter-winding capacitance ($C_p$). Capacitors possess Equivalent Series Resistance (ESR) and leakage ($R_p$). These "hidden" components create the Self-Resonant Frequency (SRF).

When an inductor reaches its SRF, it technically becomes a capacitor, rendering it useless for filtering at higher frequencies. This tool accounts for these parasitics to provide "Industrial-Grade" accuracy.

Approved International Standards & Applicability Rules

RLC resonators and harmonic filters are configured to filter system distortion and avoid power system resonance. Sizing is governed by the following international and Indian codes:

Standard Designation	Scope / Core Mandate	Applicability Rules & Design Limits
IEEE Std 18	IEEE Standard for Shunt Power Capacitors. Dictates capacitor overload thresholds.	Mandatory for filter design. Limits continuous capacitor voltage to $\le 110\%\text{ RMS}$, peak voltage to $\le 120\%\text{ peak}$, current to $\le 180\%\text{ RMS}$, and kVAR to $\le 135\%$.
IEC 61642	Industrial AC networks affected by harmonics. Focuses on filter application and detuned reactors.	Applies to selection and tuning of harmonic filters. Helps avoid unwanted resonance between substation transformer reactance and capacitor banks.
IEEE Std 519	Recommended Practice and Requirements for Harmonic Control in Electric Power Systems.	Sets voltage and current harmonic limits at the Point of Common Coupling (PCC) to prevent disturbance on utility grids.
IS 13925	Indian Standard for Shunt capacitors for AC power systems having a rated voltage up to and including 1000V.	Required for industrial installations in India. Outlines safety, insulation, temperature drift, and discharge device standards.

Filter Design Engineering Reference

What This Tool Does

It performs a multi-physics verification of series and parallel RLC filter networks. It models realistic loss elements ($R_s$ coil DCR, $R_p$ load/leakage) to compute true resonant peaks, damping factors, cut-off bandwidth, and off-resonance system impedance profiles.

Who Uses This Tool

Power Quality Engineers: Sizing detuned reactor filter banks for substations.
RF Antenna designers: tuning bandpass/notch filtering elements.
Switchgear designers: Sizing RC snubber elements to suppress breaker switching transients.
Instrumentation engineers: Verifying sensor signal filter tuning.

Industrial Usefulness

Prevents design failures where filter components explode during switching or operation. Sizing voltages across L and C against insulation ratings avoids sparkover hazards, and Q-factor calculation ensures the filter rejects harmonics properly.

Step-by-Step Mock Calculation Reference Walk-through

To assist computational models and design engineers in verifying calculations, this manual walk-through showcases the exact logic for a standard lossy RLC filter case.

Input Parameters for the Reference Case:

• Inductance ($L$): $10\,\mu\text{H}$ ($10^{-5}\,\text{H}$)

• Capacitance ($C$): $100\,\text{nF}$ ($10^{-7}\,\text{F}$)

• Series Resistance ($R_s$): $0.5\,\Omega$ (DCR)

• Parallel Load ($R_p$): $10\,\text{k}\Omega$ ($10^4\,\Omega$)

• Source Voltage ($V_{in}$): $10\,\text{V}$ (RMS)

• Target Frequency ($f_{op}$): $50\,\text{Hz}$

• Circuit Topology: Series RLC

STEP 1 Ideal Resonant Frequency ($f_0$) Frequency Equation

The ideal resonant frequency is the point where the inductive reactance and capacitive reactance cancel each other out.

                        $$ f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{10^{-5} \cdot 10^{-7}}} = \mathbf{159.15\,\text{kHz}} $$
                    

STEP 2 Characteristic Impedance ($Z_0$) Component Ratio

Characteristic impedance represents the surge impedance of the filter network, indicating energy density ratio between magnetic and electric branches.

                        $$ Z_0 = \sqrt{\frac{L}{C}} = \sqrt{\frac{10^{-5}}{10^{-7}}} = \mathbf{10\,\Omega} $$
                    

STEP 3 Coil Quality Factor ($Q_s$) Quality Value

Calculates the selectivity quality factor of the inductor winding coil. High coil resistance lowers this selectivity index.

                        $$ Q_s = \frac{Z_0}{R_s} = \frac{10}{0.5} = \mathbf{20} $$
                    

STEP 4 System Bandwidth ($BW$) Frequency Band width

Bandwidth represents the frequency spectrum width between the lower and upper half-power (-3dB) points.

                        $$ BW = \frac{f_0}{Q} = \frac{159.15\,\text{kHz}}{20} = \mathbf{7.96\,\text{kHz}} $$
                    

STEP 5 Damping Ratio ($\zeta$) Damping Coefficient

Damping ratio defines the decay speed of transients. A damping ratio $\zeta < 1.0$ is underdamped and will cause transient ringing.

                        $$ \zeta = \frac{1}{2Q} = \frac{1}{40} = \mathbf{0.025} \quad (\text{Underdamped}) $$
                    

STEP 6 Resonant Impedance ($Z_r$) Minimum Resistance

At series resonance, inductive and capacitive reactances cancel out, leaving the impedance equal to the winding resistance $R_s$.

                        $$ Z_r = R_s = \mathbf{0.5\,\Omega} $$
                    

STEP 7 Capacitor Voltage Stress at Resonance ($V_{C,res}$) Voltage Magnification

Under resonance, current is maximum. Voltage across the capacitor rises to $Q$ times the input source voltage.

                        $$ V_{C,res} = Q \cdot V_{in} = 20 \cdot 10\,\text{V} = \mathbf{200\,\text{V RMS}} $$
                    

Verdict: Overvoltage Risk! $200\,\text{V} > 1.1 \cdot 10\,\text{V} \Rightarrow \text{FAIL}$

STEP 8 Impedance Evaluation at Target $f_{op} = 50\,\text{Hz}$ Off-Resonance Sizing

Calculates the complex filter impedance magnitude at the operating line frequency (50 Hz).

                        $$ \omega = 2\pi \cdot 50 = 314.16\,\text{rad/s} $$
                        $$ X = \omega L - \frac{1}{\omega C} = 3.14\cdot 10^{-3} - 31830.98 = -31830.98\,\Omega $$
                        $$ |Z(j\omega)| = \sqrt{R_s^2 + X^2} = \mathbf{31,831\,\Omega} \quad (\text{Capacitive}) $$
                    

Industrial Resonance FAQ

1. Why is High Q vital for Radio Selectivity?

Selectivity is the ability of a receiver to separate the desired signal from noise. A High-Q resonant circuit acts like a narrow gate; it only allows a tiny sliver of the frequency spectrum through. Without high Q, radio stations would "bleed" into each other, creating a wall of unlistenable noise.

2. What is the danger of "Voltage Magnification"?

In Series resonance, the voltage across L and C can be Q times higher than the input. If your input is 230V and your Q is 10, the capacitor sees 2,300V! This frequently leads to insulation failure and explosions in high-power industrial filter banks.

3. How does Damping Ratio ($\zeta$) affect ringing?

The damping ratio ($\zeta$) determines the transient "ring-down" time. Underdamped ($\zeta < 1$) circuits oscillate before settling. Critical Damping ($\zeta = 1$) returns to zero in the shortest possible time without crossing the axis, which is the "Gold Standard" for sensor and instrument design.

4. What is "Self-Resonant Frequency" (SRF)?

Every inductor has internal parasitic capacitance. The SRF is the frequency where the inductor resonates with its own internal parasitics. Beyond this point, the inductor acts like a capacitor. For high-speed digital clocks, choosing components with an SRF higher than the clock speed is mandatory.

5. Why use Parallel Resonance for Wave Traps?

Parallel resonance creates a Maximum Impedance point. In power systems, "Wave Traps" are parallel LC filters tuned to block high-frequency communication signals from entering a substation while allowing the low-frequency (50/60Hz) power to pass through with zero resistance.

6. How does "Skin Effect" impact the Q-Factor?

At high frequencies, current only flows on the outer shell (skin) of a conductor. This increases the effective AC resistance ($R_{ac}$), which directly lowers the Q-factor. This is why high-performance RF inductors are often plated in silver or utilize Litz wire.

7. Do temperature shifts change Resonant Frequency?

Yes. Most capacitors and Ferrite cores have a Temperature Coefficient. As heat increases, the physical dimensions or permittivity change, shifting the resonance. In high-precision timing, NPO/C0G grade capacitors are used because they maintain stability across temperature swings.

8. When is Ferrite Shielding necessary?

Resonant loops act as efficient antennas. They broadcast EMI which can interfere with nearby electronics. Proper shielding or using "Closed-Core" toroidal inductors is necessary to contain the magnetic flux and fulfill EMC compliance (like FCC or CE) for commercial products.

9. How does capacitor ESR affect overall filter Q-factor?

Capacitor Equivalent Series Resistance (ESR) adds directly in series with the inductor coil resistance ($R_s$). The total active resistance becomes $R_{tot} = R_{coil_dcr} + R_{cap_esr}$. In high-frequency RF filters, ESR is the leading cause of filter detuning and internal heating.

10. How do component tolerances detune LC harmonic filters?

Manufacturing tolerances (typically $\pm 5\%$ or $\pm 10\%$) shift the actual inductance and capacitance values. This shifts the resonant frequency, causing detuning. Detuned filters block less harmonics, causing higher currents to flow to ground, which overloads the capacitors.