RLC Circuit Diagram
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An RLC circuit diagram shows a resistor (R), inductor (L), and capacitor (C) connected together, forming a circuit with a natural resonant frequency f0 = 1/(2π√LC). At resonance, the inductive and capacitive reactances cancel each other, and the circuit exhibits unique impedance, current, and voltage behaviours. RLC circuits are fundamental to radio tuning, bandpass and bandstop filters, oscillators, and impedance matching.
The RLC circuit combines the energy-dissipating element (R), the magnetic energy storage element (L), and the electric energy storage element (C). The key parameter is the resonant angular frequency ω0 = 1/√(LC), or in Hz: f0 = 1/(2π√(LC)).
Series RLC circuit: R, L, and C are all in series with the AC source. The total impedance is Z = R + j(ωL − 1/(ωC)). At resonance (ω = ω0), the imaginary part vanishes (XL = XC), leaving Z = R (purely resistive and minimum). The current is maximum at resonance: Imax = V/R.
Voltage magnification at resonance: The voltage across L or C individually can be much larger than the source voltage: VL = VC = I×XL = I×XC = (V/R) × (1/(ω0C)) = V × Q, where Q is the quality factor. Q = ω0L/R = 1/(ω0CR) = √L/C / R. For high-Q circuits, individual component voltages greatly exceed the supply — a phenomenon important for filter selectivity and a safety concern.
Parallel RLC circuit: R, L, and C are all in parallel across the source. At resonance, the impedance is maximum (theoretically infinite for ideal L and C) and equals R in practice. Current in the L and C branches circulates as reactive current, while the source only supplies the resistive (loss) current. This is called an anti-resonant or tank circuit and is the basis of LC oscillators and RF tank circuits.
Quality factor Q and bandwidth: Bandwidth BW = f0/Q = R/(2πL). A high Q (small R) gives a narrow bandwidth (sharp resonance); low Q gives a wide bandwidth. The half-power (−3 dB) frequencies are f1 = f0 − BW/2 and f2 = f0 + BW/2.
Damping: The circuit response type depends on the damping ratio ζ = R/(2√(L/C)) = R/(2) × √(C/L). Three cases: - Overdamped (ζ > 1): slow exponential approach to steady state, no oscillation. - Critically damped (ζ = 1): fastest approach without oscillation; R = 2√(L/C). - Underdamped (ζ < 1): oscillatory step response with ringing at the damped frequency ωd = ω0 × √(1 − ζ²).
Transient response to a step input: An underdamped series RLC circuit driven by a voltage step VS produces current oscillations that decay as: I(t) = (VS/L) × (e^(−αt)/ωd) × sin(ωd t), where α = R/(2L) is the damping coefficient and ωd = √(ω0² − α²).
Bandpass filter (series RLC): Output taken across R. Frequencies near f0 pass through (Z is minimum = R), while frequencies away from f0 are attenuated as Z increases. Bandwidth = R/(2πL).
Bandstop (notch) filter (series RLC in shunt): The series LC in series connection is placed in shunt to ground; at resonance, Z = R → near zero → strong attenuation of f0. At all other frequencies, the LC impedance is high, attenuating less.
Design example — 100 kHz bandpass filter with Q = 10: f0 = 100 kHz, Q = 10. Choose L = 100 μH. Then C = 1/(ω0²L) = 1/((2π×10^5)²×10^-4) ≈ 25.3 pF. R = ω0L/Q = (2π×10^5×10^-4)/10 = 6.28 Ω. BW = f0/Q = 10 kHz.
Impedance matching: Parallel RLC tanks are used to match impedances between RF stages. The impedance at resonance is Rp = Q² × R (series to parallel conversion), allowing a low-impedance source to drive a high-impedance load efficiently.
Build and analyse an RLC circuit in the free circuit diagram editor at circuitdiagrammaker.com — place R, L, and C in series or parallel, set values, and observe the impedance and phase response.
How to wire rlc circuit diagram
- Define the target resonant frequency Choose f0 from the application requirement (e.g. a tuned radio frequency or a notch frequency for interference rejection).
- Select L and C using the resonant frequency formula f0 = 1/(2π√LC). Fix one value (e.g. L = 100 μH) and solve for C = 1/((2πf0)² × L).
- Choose R to set the required Q or bandwidth Q = ω0L/R, so R = ω0L/Q. For a bandpass filter BW, R = 2πf0 × L / Q = 2π × BW × L.
- Assemble series RLC circuit Connect R, L, and C in series between the signal source and ground. Take output across R for bandpass, or across L+C for bandstop.
- Measure the resonant frequency Sweep the signal generator from 0.1×f0 to 10×f0 and find the frequency where the output across R is maximum (series RLC) or current is maximum.
- Measure Q and bandwidth Determine the two −3 dB frequencies (where output = 0.707 × peak). BW = f2 − f1, Q = f0/BW.
- Adjust component values if needed If f0 is off, trim C (variable capacitor) or calculate the error and select a more accurate fixed component. Use 1% tolerance components for precision.
Specifications
| Resonant frequency | f0 = 1 / (2π × √(L×C)) |
|---|---|
| Angular resonant frequency | ω0 = 1 / √(L×C) rad/s |
| Quality factor (series) | Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C) |
| Bandwidth (series RLC) | BW = f0 / Q = R / (2πL) |
| Series impedance at resonance | Z = R (minimum, purely resistive) |
| Parallel impedance at resonance | Z = R×Q² (maximum) |
| Damping ratio | ζ = R/(2) × √(C/L) |
| Voltage across L or C at resonance | V_L = V_C = Q × V_source |
| Critically damped condition | R = 2 × √(L/C) |
| Half-power frequencies | f1,f2 = f0 ± BW/2 |
Safety warnings
- Component voltages across L and C in a high-Q series resonant circuit can far exceed the supply voltage — always rate capacitors and inductors to at least Q × Vsupply to prevent insulation breakdown.
- High-voltage resonant circuits store energy that persists after power removal — discharge the capacitor through a bleed resistor and measure before handling circuit nodes.
Tools needed
- Inductor (value from f0 formula, low DCR)
- Capacitor (film or C0G ceramic, rated above Q×Vsupply)
- Resistor (value from Q or BW requirement)
- Signal generator (frequency sweep capability)
- Oscilloscope or spectrum analyser
- LCR meter (to verify L, C, and DCR values)
Common mistakes
- Selecting component voltage ratings based on supply voltage alone — in resonant circuits, individual component voltages can be Q times higher than the supply.
- Neglecting the inductor's DCR when calculating Q — DCR adds to R and degrades Q significantly for low-inductance, high-frequency designs.
- Using electrolytic capacitors in resonant circuits — their high internal losses (ESR) degrade Q severely; use film or C0G/NP0 ceramics instead.
- Confusing series and parallel resonance conditions — series resonance gives minimum Z; parallel resonance gives maximum Z. Using the wrong topology reverses the filter behaviour.
Troubleshooting
- Resonant frequency is lower than calculated
- Cause: Actual L or C values are higher than nominal due to tolerance, or stray capacitance (PCB traces, wiring) adds to C. Fix: Measure L and C with an LCR meter and recalculate f0. Reduce stray capacitance by shortening circuit traces or subtract stray C from the target C value.
- Q is much lower than designed
- Cause: Inductor DCR is higher than assumed, or the capacitor has high ESR. Fix: Measure DCR and add it to R in Q = ω0L/(R+DCR). Use a lower-loss inductor or a capacitor type with lower ESR (film or C0G ceramic).
- Oscillations at switch-on that damage components
- Cause: Circuit is underdamped with high Q; the initial transient produces large voltage spikes across L and C. Fix: Add a damping resistor or reduce Q to ζ ≥ 0.5 by increasing R to at least √(L/C)/√(2) to control the peak voltage on start-up.
Frequently asked questions
What is the resonant frequency of an RLC circuit diagram?
The resonant frequency is f0 = 1/(2π√(LC)), where L is in henrys and C is in farads. At this frequency, the inductive and capacitive reactances are equal (XL = XC), and they cancel in series circuits, leaving only resistance R.
What is the Q factor of an RLC circuit?
The quality factor Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C). It is a dimensionless ratio that indicates selectivity: high Q means a sharp, narrow resonance peak; low Q means a broad, flat response. Q also equals the ratio f0/BW.
What is the difference between series and parallel RLC resonance?
In series resonance, impedance is minimum (= R) and current is maximum at f0. In parallel resonance, impedance is maximum (= R×Q²) and current drawn from the source is minimum at f0. Series resonance is used for bandpass filters; parallel resonance is used for bandstop filters and LC oscillator tanks.
Can voltages across L and C exceed the supply voltage in an RLC circuit?
Yes. At series resonance, the voltage across L and across C individually equals Q × Vsource. For Q = 10 and Vsource = 10 V, the component voltages reach 100 V. This is called voltage magnification or resonant rise and must be considered when selecting component voltage ratings.
What is damping in an RLC circuit?
Damping is determined by the damping ratio ζ = R/(2√(L/C)). If ζ < 1, the circuit is underdamped and oscillates with decaying amplitude. If ζ = 1 (critically damped), it settles fastest without oscillation. If ζ > 1 (overdamped), it settles slowly and exponentially without oscillation.
How is an RLC circuit used as a bandpass filter?
In a series RLC, taking the output across R gives a bandpass response centred on f0, with −3 dB bandwidth BW = R/(2πL). Frequencies at f0 pass with minimum attenuation; frequencies far from f0 are blocked by either XL or XC dominance.
What is the bandwidth of an RLC circuit?
Bandwidth BW = f0/Q = R/(2πL). For example, with f0 = 1 MHz, Q = 50: BW = 1×10^6/50 = 20 kHz. The circuit passes signals from 990 kHz to 1010 kHz and rejects all others.