The following animation illustrates the behavior of a series RLC circuit in an AC circuit. Impedance, phase shift, and resonance behavior are visualized in real time.
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What Happens in This Circuit?
If terms like “impedance” or “reactance” sound unfamiliar, this section is for you. A series RLC circuit combines three basic components, each with a distinct role:
- Resistor (R) – like friction in a mechanical system. It converts electrical energy into heat and limits the current flow.
- Capacitor (C) – like a spring. It stores energy in an electric field and releases it again. The faster the voltage changes, the more current flows through it.
- Inductor (L) – like a heavy flywheel. It stores energy in a magnetic field and resists changes in current. The faster the current changes, the more it “pushes back.”
Together, the capacitor and inductor form an oscillating system – similar to a weight on a spring. Energy swings back and forth between the electric field (capacitor) and the magnetic field (inductor). The resistor dampens this oscillation, just as air resistance slows a pendulum.
At one specific frequency – the resonant frequency – the capacitor and inductor perfectly balance each other. At this point, the circuit behaves as if only the resistor were present, and the current reaches its maximum.
Animation Description
The animation shows a series RLC circuit with four synchronized displays: circuit diagram, formulas, phasor diagram, and time domain graph. The start button begins the animation. The phasors rotate and the sine waves move through the diagram.
The impedance of the circuit is calculated as:
\[ Z = \sqrt{R^2 + (X_L – X_C)^2} \]
With the reactances:
- \( X_L = 2\pi f L \) – inductive reactance
- \( X_C = \frac{1}{2\pi f C} \) – capacitive reactance
The phase shift φ between voltage and current is displayed as a double arrow in both the phasor diagram and the time domain graph.
Interactive Controls
The following parameters can be adjusted using the sliders:
- U (1–24 V): AC voltage amplitude
- f (1–200 Hz): AC voltage frequency
- R (1–1000 Ω): Resistance
- C (1–100 µF): Capacitance
- L (10–1000 mH): Inductance
The checkboxes L, C, and R allow individual components to be shown or hidden to examine different circuit configurations (e.g., pure RC or RL circuit).
Understanding the Phasor Diagram
The phasor diagram is a tool for analyzing AC circuits – but it takes some getting used to. Here is how to read it:
What is a phasor? A phasor is a rotating arrow (vector) that represents a sinusoidal quantity. Its length corresponds to the amplitude (peak value), and its angle indicates the phase position at any given moment.
What do the individual phasors show?
- The current phasor \( I \) serves as the reference – in a series circuit, the same current flows through all components.
- The voltage across R \( (U_R) \) points in the same direction as the current, because voltage and current are in phase at a resistor.
- The voltage across L \( (U_L) \) leads the current by 90° – the phasor points “ahead” of the current phasor.
- The voltage across C \( (U_C) \) lags behind the current by 90° – the phasor points in the opposite direction to \( U_L \).
Why do \( U_L \) and \( U_C \) point in opposite directions? Because inductor and capacitor react oppositely to current changes. Their effects partially cancel each other. The total voltage \( U \) results from the vector sum of all partial voltages.
Tip for the animation: Watch how the angle between the total voltage phasor and the current phasor (the phase angle φ) changes as you adjust the frequency. At resonance, both phasors point in the same direction – the phase angle becomes zero.
Physical Background
In a series RLC circuit, three effects combine: R converts energy into heat, L stores it in a magnetic field, C in an electric field. The inductor and capacitor periodically exchange energy. The resistor dampens this oscillation.
At the resonant frequency \( f_0 = \frac{1}{2\pi\sqrt{LC}} \), the reactances cancel out (\( X_L = X_C \)), the impedance reaches its minimum and equals the resistance. The current reaches its maximum and is in phase with the voltage.
Series vs. Parallel Resonant Circuit
This animation shows a series resonant circuit. In practice, parallel resonant circuits are common as well. The key differences:
| Property | Series Circuit | Parallel Circuit |
| Component arrangement | R, L, C in series | L and C in parallel (R often in series with L) |
| Impedance at resonance | Minimum (= R) | Maximum |
| Current at resonance | Maximum | Minimum |
| Typical application | Bandpass filter, frequency selection | Bandstop filter, signal blocking |
| Resonant frequency | \( f_0 = \frac{1}{2\pi\sqrt{LC}} \) | \( f_0 = \frac{1}{2\pi\sqrt{LC}} \) (ideal case) |
Both types share the same resonant frequency \( f_0 \), but they behave oppositely: where the series circuit lets maximum current through, the parallel circuit blocks it. A series circuit acts as a bandpass (passes the desired frequency), while a parallel circuit acts as a bandstop (blocks a specific frequency).
Common Misconceptions
The following questions come up when learning about RLC circuits. If you have wondered about any of them, you are not alone:
“How can the voltage across L or C be greater than the source voltage?”
This phenomenon is called resonance voltage rise. At resonance, energy oscillates between inductor and capacitor. The voltages across the components can be larger than the applied voltage – they point in opposite directions and cancel each other. The total voltage still equals the source voltage.
In the animation, you can observe this: set R to a low value and adjust f to the resonant frequency – the voltage phasors become longer than the total voltage phasor.
“Why is the impedance at resonance not zero?”
At resonance, only the reactive components cancel out (\( X_L = X_C \)). The resistance R remains. The impedance at resonance therefore equals R. Only in a lossless, idealized circuit (R = 0) would the impedance become zero – but such a circuit does not exist in practice.
“Why does the phase shift become zero at resonance?”
The phase shift φ depends on the ratio of reactive to resistive impedance: \( \tan(\varphi) = \frac{X_L – X_C}{R} \). At resonance \( X_L = X_C \), so the numerator becomes zero and with it the phase angle. The circuit then behaves resistively – current and voltage are in phase.
“Is a higher Q factor always better?”
Not necessarily. A high quality factor \( Q = \frac{1}{R}\sqrt{\frac{L}{C}} \) means sharp resonance and high selectivity – ideal for radio receivers that need to separate closely spaced stations. But for broadband applications (e.g., audio amplifiers), a lower Q factor is chosen to cover a wider frequency range. In the animation, you can see this effect: a small R value gives a sharp resonance peak, a large R value gives a flat, broad response.
Practical Applications
- Radio receivers: Tuned circuits select frequencies through resonance
- Filters: Highpass, lowpass, and bandpass filters in audio engineering
- Oscillators: LC oscillators for clock generation in digital circuits
- Power factor correction: Capacitors compensate for inductive loads in power grids
Real-World Examples with Concrete Values
FM radio tuning: An FM radio receives stations between 88 and 108 MHz. To tune to 100 MHz with an inductance of L = 0.1 µH, the required capacitance is:
\[ C = \frac{1}{(2\pi f_0)^2 \cdot L} = \frac{1}{(2\pi \cdot 100 \times 10^6)^2 \cdot 0.1 \times 10^{-6}} \approx 25 \text{ pF} \]
By varying the capacitance (e.g., with a variable capacitor), different stations can be selected.
Subwoofer crossover filter: A crossover filter for a subwoofer typically separates frequencies at around 200 Hz. With R = 8 Ω (speaker impedance) and C = 100 µF, an inductor of approximately L = 6.3 mH creates a resonant circuit at this cutoff frequency. Try entering these values in the animation to observe the resonance behavior.
Power factor correction in industry: An industrial motor with L = 50 mH at 50 Hz has an inductive reactance of \( X_L = 2\pi \cdot 50 \cdot 0.05 \approx 15.7 \text{ Ω} \). To compensate, a capacitor with \( C = \frac{1}{(2\pi \cdot 50)^2 \cdot 0.05} \approx 203 \text{ µF} \) is connected in the circuit.
Format and Use
Format: This animation runs in the browser (Chrome, Firefox, Edge, Safari). No download required.
Use case: Suited for computer labs, homework or self study. Students can adjust parameters independently.
For large screens: An HD version for download is available for projectors and large screens.
Related Animations
HD Animations
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Overview
| Title | RLC Circuits |
| Target Audience | Teachers and Lecturers |
| Features | Full screen mode Lossless scaling Large screens and projectors supported |
| License | MIT |
Contributors
C. Hein, S. Rikowski