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Calculators: LC Resonance

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Enter new numbers and see the remaining output value change. Floating point format ("1.1E-6") works; engineering units ("1.1u", etc.) do not.

Note that the units are simply ratios, so their actual units do not matter (as long as the same units are used for all steps). They're labeled in μF, μH, Ω and MHz for convenience. (Equally valid are F, H, Ω and Hz, without having to change any numbers.)

 

LC Resonance

An LC tank has a characteristic resonant frequency, at least if it is loaded lightly enough to resonate (more on that later). The resonant frequency occurs when the capacitive and inductive reactances cancel.

Resonant Network μH, μF, = F =
 
MHz
MHz, μH, = C =
 
μF
MHz, μF, = L =
 
μH

 

Resonant Impedance

An LC tank also has a characteristic impedance (this is different from the terminal impedance). The ratio of voltage to current circulating in the loop corresponds to the ratio of values.

Resonant Network μH, μF, = Z =
 
Ω
Ω, μH, = C =
 
μF
Ω, μF, = L =
 
μH

 

L or C by Resistor Divider

This isn't really resonant, but is handy enough to include. Set up one of the circuits shown below, using a sine wave generator with a well defined output impedance (usually 50Ω). Measure the source voltage by removing the capacitor/inductor from circuit (assuming a high-impedance probe, like a 10x scope probe). Then measure the voltage with the component connected. This only needs the amplitudes, not phase; as a result, the component type can't be determined. Both results are provided.

Impedance Divider R = Ω
F = MHz
V1 = V
V2 = V
Inductance L =
 
μH
or Capacitance C =
 
μF

 

Vector Impedance by Resistor Divider

The general case, given a phase measurement. Set up the circuit below, using a sine wave generator. Measure the source voltage with one probe (trigger source), and the load or divider tap voltage with another probe.

Impedance Divider R = Ω
V1 = V
V2 = V
θ = °
Z (rectangular) Z =
 
+ j
 
Ω
Z (polar) Z =
 
Ω ∠
 
°
Q Factor Q =
 
F = MHz
Series Equivalent Value
RC or LC Series Equivalent R =
 
Ω
L =
 
μH
C =
 
μF
Parallel Equivalent Value
RC or LC Series Equivalent R =
 
Ω
L =
 
μH
C =
 
μF

 

Frequency and Q Factor

Set up the circuit shown below, using a frequency generator (with reasonably constant voltage output over the testing range). L and C are the resonant tank under test, R1 represents the source resistance (including the generator's own output resistance, and any added if needed) and R2 represents the equivalent loss of the tank (don't connect a physical resistor here). Sweep the frequency until you find resonance. While probing the source and tank voltages with the oscilloscope (use 10× probes if possible), varying frequency above and below resonance slightly should look like this:

Sine Wave Sweep Animation

Measure the frequency under the middle condition, where the tank voltage amplitude is highest (the phase shift will also tend to line up, which may be easier to eyeball when Q is low). Measure the source and tank voltage. (If the source's resistance is a large part of R1, measure the open-circuit voltage with the tank disconnected.)

If the voltage drop is small (V2 / V1 > 0.5), you may get better measurements with a larger R1. Likewise if V2 / V1 ≈ 0, try a smaller R1.

RLC Divider R1 = Ω
Fo = MHz
Known Component:
Known Value μF
V1 = V
V2 = V
Unknown Value L =
 
μH
Resonant Impedance Zo =
 
Ω
Loss Resistance R2 =
 
Ω
Q Factor Q =
 

 

Parasitic Capacitance by Difference

This method is useful for finding the unknown capacitance across a coil, or the residual, parasitic or equivalent capacitance of any kind of resonator. First, measure the resonant frequency using the voltage divider circuit and peak-tracking method shown above. Then, connect a known capacitor in parallel, and repeat the measurement.

This method is exact for lumped constant components only. For equivalent parallel resonators, if the resonant mode is widely spaced from other modes, it should also work well (in which case, the resulting RLC values should correspond to the lumped equivalent circuit for that mode). If the modes are closely spaced (e.g., the higher modes of a stub transmission line or helical resonator), or repeated or overlapping (e.g., the nearly coincident poles of a filter), don't expect a useful result.

The voltage divider items are optional. They just repeat the above calculation, and are provided for convenience. V1 and V2 should be measured under the first condition (without Cx), though for small values of Cx, the voltage ratio should be similar at either frequency, so that it doesn't matter much.

LC Tank Natural Resonant Frequency Fo = MHz
LC Tank Plus Parallel Capacitor Loaded Resonant Frequency FL = MHz
Added Capacitance Cx = μF
Capacitance C =
 
μF
Inductance L =
 
μH
Resonant Impedance Zo =
 
Ω
Source Resistance R1 = Ω
Source Voltage V1 = V
Tank Voltage V2 = V
Loss Resistance R2 =
 
Ω
Q Factor Q =
 

 

L and Q by Matching Capacitor

This method potentially allows testing high-Q components at higher voltage or current levels, from a limited source. The above resistor divider method might only afford a few volts. A typical 15V function generator, delivers up to 0.5W. Suppose we wish to test a 100μH coil with Q of 500, which has an equivalent parallel resistance of 31.4kΩ. With perfect matching, that humble signal generator now develops 125Vrms across the coil! This can be helpful for testing power magnetics, where the loss value varies with amplitude.

Set up the circuit shown below, using a sine generator. Select C such that the resonant frequency is near the desired test frequency, and select Cs to give a good match (loaded V1 about half the unloaded Vs). Sweep frequency back and forth to find the notch frequency (where V1 reaches a minimum). Measure V1 and Fo at this condition. Disconnect Cs and measure V (open circuit voltage, OCV). Typically, the signal generator will be connected to an oscilloscope with a BNC tee connector, and the network connected to that in turn; OCV is then easily checked by unplugging the network.

Note that any reactance from the source acts to resonate in part with Cs, shifting the frequency of the minima. A signal generator is a good choice here as the output impedance should be flat and consistent. If you're getting inconsistent results, that may be something to check.

C Match Rs = Ω
Cs = µF
C = µF
Vs = V
V1 = V
Fo = MHz
Inductance L =
 
μH
Resonant Impedance Zo =
 
Ω
Loss Resistance R =
 
Ω
Q Factor Q =
 
Tank Voltage VT =
 
V

 

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