Designing an RLC Parallel Circuit: Resonance, Component Values, and Testing

An RLC parallel circuit connects a resistor (R), inductor (L), and capacitor (C) across the same two nodes, sharing the same voltage but dividing the AC current. Unlike its series counterpart, which drops impedance to a minimum at resonance, the parallel topology spikes to a maximum impedance at its resonant frequency. This makes it the foundational building block for tank circuits, band-stop (notch) filters, and oscillator frequency-determining networks.

The RLC Parallel Circuit Topology and Node Behavior

In a standard parallel RLC topology, the circuit is defined by two primary nodes:

• Node A (Top Rail): The positive terminal of the AC voltage source. One lead of the resistor, inductor, and capacitor all tie into this node.
• Node B (Bottom Rail): The negative terminal or ground reference of the AC source. The remaining leads of all three components tie into this node.

Because all three components share the same voltage ($V_{AB}$), the total current drawn from the source is the phasor sum of the individual branch currents: $I_{total} = I_R + I_L + I_C$. At resonance, the inductive and capacitive branch currents are equal in magnitude but exactly 180 degrees out of phase. They cancel each other out within the LC loop, leaving only the resistive current to be drawn from the source.

Design Walkthrough: Picking Real Component Values

Abstract formulas are useless without real-world part numbers. Let us design a parallel RLC tank circuit targeting a resonant frequency ($f_r$) of 5 kHz, with a Quality factor (Q) of 10 to provide a reasonably sharp notch without excessive ringing.

1. Select the Capacitor (C)
Start with the capacitor, as standard values are more rigidly spaced than inductors. We will choose a 100 nF (0.1 µF) metallized polyester film capacitor, such as the WIMA MKS2 series. Film capacitors offer low equivalent series resistance (ESR), which is critical for maintaining a high Q.

2. Calculate the Inductor (L)
Using the resonant frequency formula $f_r = \frac{1}{2\pi\sqrt{LC}}$, we rearrange to solve for L:
$L = \frac{1}{(2\pi f_r)^2 C}$
$L = \frac{1}{(2\pi \times 5000)^2 \times 100 \times 10^{-9}} \approx 10.13 \text{ mH}$
We will select a standard 10 mH radial inductor, such as the Bourns 78FR10K-RC. Note that real inductors have DC resistance (DCR). The Bourns part has a DCR of roughly 5.6 Ω, which will slightly degrade our theoretical Q factor.

3. Calculate the Resistor (R)
In a parallel RLC circuit, the resistor dictates the bandwidth and the Q factor. The formula for parallel Q is $Q = R \sqrt{\frac{C}{L}}$.
Rearranging for R: $R = Q \sqrt{\frac{L}{C}}$
$R = 10 \times \sqrt{\frac{10 \times 10^{-3}}{100 \times 10^{-9}}} = 10 \times \sqrt{100,000} \approx 3,162 \text{ \Omega}$
We will use a standard 3.3 kΩ metal film resistor (e.g., Vishay MRS25 series). This slightly raises the Q to about 10.4, which is perfectly acceptable for bench testing.

Behavior Matrix: What Happens When Components Shift

Understanding how component drift affects the circuit is crucial for troubleshooting and tuning. The table below maps the behavioral shifts when a single variable is increased.

For deeper mathematical derivations on phasor interactions in these tanks, the Electronics Tutorials guide on parallel resonance provides excellent vector diagrams.

Failure Modes: Extremes, Shorts, and Opens

When components fail or are intentionally removed, the parallel RLC circuit behaves very differently than a series circuit. Here is what breaks at the extremes:

• Short the Capacitor: Node A is directly shorted to Node B through the capacitor branch. Total circuit impedance drops to near zero. The AC source sees a dead short, which will either trip a breaker, blow a fuse, or cause a function generator's output stage to fold back and shut down. Unlike a series circuit where a shorted capacitor simply shifts resonance, a parallel short destroys the circuit entirely.
• Open the Inductor: The L branch is removed. The circuit degrades into a simple parallel RC network. Resonance is destroyed. The circuit now acts as a basic low-pass filter with a -3dB cutoff determined by R and C.
• Short the Resistor: The parallel impedance is forced to near zero across all frequencies. The high-impedance peak at resonance is flattened completely. The Q factor drops to zero, and the circuit loses its frequency-selective properties.
• Open the Resistor: The damping is removed. Theoretically, Q approaches infinity and bandwidth narrows to a razor-thin spike. In reality, the peak impedance is limited solely by the inductor's parasitic DCR and the capacitor's ESR, resulting in a very high, but finite, voltage spike at resonance.

Step-by-Step Breadboard Testing Procedure

To verify your design on the bench, you need to measure the impedance peak. Since most multimeters cannot measure AC impedance directly at 5 kHz, we use a voltage divider technique with an oscilloscope.

Required Equipment:

• Function Generator (e.g., Siglent SDG1032X)
• Oscilloscope (e.g., Rigol DS1054Z)
• 100 Ω sense resistor (1% tolerance)
• Your 10mH inductor, 100nF capacitor, and 3.3kΩ resistor

Testing Steps:

1. Build the Divider: Connect the function generator output to the 100 Ω sense resistor. Connect the other end of the sense resistor to Node A of your parallel RLC tank. Connect Node B to the function generator ground.
2. Probe the Nodes: Connect Oscilloscope Channel 1 to the function generator output (source voltage, $V_{in}$). Connect Channel 2 to Node A (tank voltage, $V_{tank}$).
3. Set the Sweep: Configure the function generator for a 2V peak-to-peak sine wave. Set up a frequency sweep from 1 kHz to 10 kHz over 10 seconds.
4. Observe the Notch: Watch Channel 2. As the frequency approaches 5 kHz, the voltage across the tank ($V_{tank}$) will rise relative to $V_{in}$ because the tank's impedance is rising, taking a larger share of the voltage divider ratio.
5. Calculate Impedance: At the exact peak voltage on Channel 2, pause the sweep. Use the formula $Z_{tank} = R_{sense} \times (\frac{V_{tank}}{V_{in} - V_{tank}})$. With a 3.3kΩ resistor, your peak impedance should measure close to 3,300 Ω at roughly 5.03 kHz (slightly higher than 5 kHz due to the 10mH inductor being slightly under the calculated 10.13mH requirement).

For more on how parasitic elements affect high-frequency tank circuits, refer to the Georgia State University HyperPhysics module on parallel resonance.

RLC Parallel Circuit FAQ