About

This tool designs and analyzes series resonant circuits containing an inductor, capacitor, and resistor (RLC). It calculates resonant frequency, input impedance, quality factor (Q), and damping ratio -- or it can work backwards to find component values for a target resonant frequency. Once the component values are defined, you can view the steady state sinusoidal response for a given input voltage or current.

$R$ accounts for resistive losses, such as an inductor's winding resistance. When using this calculator, you can see the impact of $R$ on quality factor, damping, and power consumption. It can be effectively removed from the circuit by setting its value to zero, resulting in a lossless series LC circuit.

Instructions

To get started, fill in any two inputs in the Resonant Frequency section. For instance, enter inductor (L) and capacitor (C) values to calculate resonant frequency. Or if you want to design for a particular resonant frequency, enter it, along with either L or C, and the remaining component value will be calculated. The following outputs are provided: Quality Factor (Q), Damping Ratio (ζ), Damping Type, and input impedance (at resonance).

When the Resonant Frequency section is complete, the Input Impedance section will appear. Enter the source frequency ($f_s$) to calculate the circuit's input impedance ($Z_{in}$) at your specified frequency.

Next, the Steady State Response section will appear. Enter the source voltage ($V_s$) or current ($I_s$) to compute the LC circuit's steady state response to a sinusoidal source. The following will be calculated: input impedance (at specified $f_s$), resistive dissipation, and input current / voltage.

Definitions

1. Quality Factor (Q) describes damping of a resonant circuit, with a larger $Q$ corresponds to less damping. It can be express as:

$$Q = 2\pi \frac{\text{peak energy storage}}{ \text{energy dissipated per cycle}}$$

2. Damping Ratio ($\zeta$) is another way to describe damping in a resonant circuit. Its value corresponds to various damping properties as follows:

   • $\zeta = 0$ : undamped
   • $\zeta = 1$ : critically damped
   • $\zeta \lt 1$ : underdamped
   • $\zeta \gt 1$ : overdamped

Theory

In the Resonant Frequency Section, the component values ($L, C$) and resonant frequency ($f_r$) are related by:

$$f_r = \frac{1}{2\pi \sqrt{LC}}$$

The Quality Factor (Q) is calculated using:

$$Q= \frac{1}{R} \sqrt{\frac{L}{C}}$$

References

1. Langford-Smith, Ed., Radiotron designer's handbook, 4th ed. Sydney: Wireless Press, 1953.