# Real-Life RLC circuit - why so lopsided?

I am using a superconducting $RLC$-circuit for detecting weak electrical signals. The center-frequency is around 650 kHz, and the $Q$-value approximately 50k. I am trying to better understand my signal background. Without the signal source, the electrical schematic be simplified to this:

Note that there is no input to this circuit: We are looking at the Johnson-noise of the detection circuit in combination with the noise-floor of the amplifier's input stage.

Ideally, the power spectrum of the Johnson noise is given by $|Z_{RLC}|^2$, which can be shown to be a Lorentzian:

$$\text{noise in dB} = 10 \log_{10}\left( 10^{A/10} \frac{\left(\frac{\nu_0}{2 Q}\right)^2}{\left(\nu-\nu_0\right)^2 + \left(\frac{\nu_0}{2 Q}\right)^2} + 10^{b/10}\right) \quad .$$

In this form, the amplitude $A$ and the background $b$ are given in dB. The center of the resonance is given by $\nu_0$, and the width given by the $Q$-value with $Q = \nu_0/\text{FWHM}$.

In reality, the resonance is almost a Lorentzian, but is lopsided. This becomes especially obvious when using a Lorentzian as a fit-function:

I checked a bit further away from the resonance to make sure that the transfer function of my amplifier is reasonably flat. Therefore, I don't think it can explain the lopsidedness. I suspect the lopsidedness is caused by an additional resonance, for example some by parasitic $R$, $L$ or $C$ somewhere in the system, like the series/parallel resonances in a Quartz oscillator. But I cannot find a reasonable model for that. What kind of parasitic component could cause this lopsidedness?

Possibly relevant: The amplifier is not directly connected to the "hot end" of the resonator, but to a 1:10 tap close to the "cold end" (ground).

• I agree that your explanation (another resonance) is a good one to investigate - it was the thing that sprang to my mind on seeing the plot before I read your last paragraph. You've checked the amplifier presumably with a signal generator - but have you checked that the amplifier's input impedance is also flat? These are fairly low frequencies, so matching is unlikely to be too important. – WetSavannaAnimal Jun 14 '17 at 0:23
• That looks a lot like another resonance. – DanielSank Jun 14 '17 at 2:48
• This kind of lopside can come from a mismatch of impedance somewhere (typically the amplifier). empirically, adding a parameter of asymetrie in the numerator (v/2Q -> v/2Q + u in the numerator) should make the model more fitting – sailx Jun 14 '17 at 7:26
• This is a good question, I hope the site gets more like this. – Emilio Pisanty Jun 14 '17 at 7:31

A typical order of magnitude for the maximum sweep rate to get acceptable results is $f^2/Q^2$ Hz/sec where $f$ is the resonance frequency in Hz. A simple check is to compare two frequency sweeps with the frequency increasing and decreasing through the resonance.