Mixer
A mixer multiplies its two inputs. Fed a signal and a local oscillator at the same frequency, it demodulates: the product has a DC part set by their phase. It is the heart of a lock-in and of a Pound–Drever–Hall lock.
Ports
Two inputs, a and b, and one output, out.
The model
The output is the product of the inputs times a gain \(k\) (1/V):
\[v_\text{out} = k\,v_a\,v_b.\]Taken part by part: each input’s DC carries the other’s tones through, and each pair of tones makes a sum and a difference:
\[k\operatorname{Re}\!\left(a e^{i\omega_1 t}\right)\operatorname{Re}\!\left(b e^{i\omega_2 t}\right) = \frac{k}{2}\operatorname{Re}\!\left(ab\, e^{i(\omega_1 + \omega_2)t}\right) + \frac{k}{2}\operatorname{Re}\!\left(a b^*\, e^{i(\omega_1 - \omega_2)t}\right).\]Two tones at the same frequency give a DC level \(\tfrac{k}{2}\lvert a\rvert\lvert b\rvert\cos(\phi_a - \phi_b)\), which is how a mixer reads phase.
Noise
The noise on each input is moved by the other input’s DC and tones. A tone of amplitude \(c\) on one input moves the other’s noise density to either side of it:
\[S_\text{out}(f) = k^2\Big[v_{a,\text{DC}}^2 S_b(f) + v_{b,\text{DC}}^2 S_a(f) + \sum_{j}\frac{\lvert c_j\rvert^2}{4}\big(S(\lvert f - f_j\rvert) + S(f + f_j)\big)\Big],\]summed over each input’s tones against the other’s noise. So noise beside a reference frequency lands at DC after mixing, as in a lock-in. Noise times noise is left out, as are tones a million times weaker in power than the strongest.
Settings
| Field | Default | Meaning |
|---|---|---|
k | 1 | Gain (1/V) |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Readouts
The panel gives its output in the steady state: the DC level, the five strongest tones (each one’s amplitude, rms and the phase of its cosine), and the noise, as a density at 1 kHz and in rms from 1 Hz to 1 GHz. Any notes, such as an input left unwired, are listed under it.
Assumptions
- An ideal multiplier: no leakage of either input to the output, no conversion loss beyond its gain, no harmonics of the local oscillator.
- Noise multiplied by noise is left out.
Related
- Low-pass filter, to keep the DC part.
- Lock-in amplifier.
- Electronics: the mixer.