About

This tool designs and analyzes parallel inductor capacitor (LC) circuits with an optional parallel resistive load ($R_p$) and inductor series resistance ($R_L$). 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.

$R_L$ can account for resistive losses in an inductor. When using this calculator, you can see the impact of $R_L$ on quality factor, damping, and power consumption. large values of $R_L$ will also impact resonant frequency for parallel resonant circuits.

$R_p$ is resistance in parallel with the capacitor. It can account for losses or loads across the input of the LC circuit, or even losses in a magnetic core.

Instructions

To get started, fill in any three inputs in the Components & 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. You can also adjust the resistance values $R_L$ and $R_p$. The following outputs are provided: Quality Factor (Q), Damping Ratio (ζ), Damping Type, and input impedance (at resonance).

When the Components & Resonant Frequency section is complete, the Input Impedance section will appear. Enter the source frequency ($f_s$) to calculate the LC 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 component currents.

Notes:

• $R_L$ affects resonant frequency of a parallel LC circuit. The impact is larger for larger values of $R_L$.
• $R_L$ can be omitted by leaving its field blank.
• $R_L$ can be set to zero if desired.

Definitions

1. Resonant Frequency can be defined in several ways. We define it as the non-zero frequency at which the input to the LC circuit is purely resistive.
2. Quality Factor (Q) describes damping of a resonant circuit, with a larger $Q$ corresponds to less damping. It can be express as:

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

3. 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 Components & Resonant Frequency Section, the component values ($L, C, R_L$) and resonant frequency ($f_r$) are related by the following equations. Note that we define resonant frequency here as the non-zero frequency at which the input to the LC circuit is purely resistive.

$$f_r = f_o\sqrt{1-\frac{C{R_L}^2}{L}}$$

where:

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

The Quality Factor (Q) is calculated using:

$$Q= \frac{1}{R_L \sqrt{(C / L)} + \sqrt{(L / C)} / R_p}$$

References

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