Lockin Amplifier's amplitude R output meaning
Explanation of the R amplitude

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The Lockin Amplifier is a widely used instrument for small signal analysis in environments with significant noise. The Lockin Amplifier offers several output modes, including X/Y and R/θ. However, due to the complex processing chains within the Lockin Amplifier, there is no direct relationship between the input signal amplitudes and the R output.
In the specific case where the input signal consists of a single frequency tone, the output signal amplitude can be directly determined if the input signal is demodulated at its own frequency.
1. Singletone input signals:
Consider an input signal of the form: Adouble angle formulas, this signal can be expressed as: [1  cos(2f t)] A/2. The component 2f will be filtered by the lowpass filter, leaving A/2. Thus, the output R is half of the signal amplitude A, and onequarter of the peaktopeak amplitude value.
sin(f t), demodulated at the same frequency f. The output of the mixer is: A sin(f t) sin(f t). Using theExperimental results confirm this relationship: for a 1 MHz signal with a 500 mVpp amplitude signal, the demodulated results R is 125 mV.
The input signal frequency does not need to match the demodulation frequency exactly. As long as the frequency difference is smaller than the corner frequency of the lowpass filter, the mapping relationship remains valid.
For an input signal A⋅sin(f1⋅t), demodulated at the frequency f2, the resulting signal after mixing is: A/2⋅{cos[(f1 f2)⋅t]  cos[(f1 + f2)⋅t]}. After passing through the lowpass filter, the highfrequency component is attenuated, leaving: A/2⋅cos[(f1 f2)⋅t]. Given that the Moku Lockin Amplifier employs a dualphase demodulator (inphase and quadraturephase), the computed amplitude R is sqrt({A/2⋅cos[(f1 f2)⋅t]}^2 + {A/2⋅sin[(f1 f2)⋅t]}^2) equals to A/2.
A 10.000 1 MHz signal with a 1 Vpp amplitude is demodulated using a 10 MHz demodulation signal. The output exhibits the expected amplitude and undergoes periodic phase changes at a frequency of 100 Hz.
However, please note that this mapping relationship only applies to single frequency tone signals. For more complex signals, this relationship does not hold because the additional frequency components will be filtered out, resulting in signal loss and amplitude reduction.
For example, consider a test with an input signal containing two frequency tones: 10 MHz and 1 MHz. The baseline signal is a 1 MHz, 500 mVpp sine wave, with a 10 MHz, 500 mVpp signal superimposed on it. In this case, the R output amplitude remains 125 mV, even though the total input signal amplitude is now 1 Vpp. This occurs because the 10 MHz tone is filtered out by the 1 kHz bandwidth lowpass filter.
2. Amplitude modulated (AM) input signals:
For a message signal modulated on a fundamental carrier, the demodulated amplitude is expected to be A⋅d/2 of the original modulated signal, provided the message frequency is within the lowpass passband. Here, A is the amplitude of the carrier waveform, and d is the modulation depth. The modulated signal A⋅sin(f⋅t)⋅[1+d⋅sin(m⋅t)] can be expressed as (1−cos(2f⋅t))⋅A/2⋅[1+d⋅sin(m⋅t)] after demodulation, and the lowpassed signal is A/2⋅[1+d⋅sin(m⋅t)].
For example, consider an input signal with a 1 Vpp 10 MHz carrier wave, 100% modulated by a 1 kHz sine wave. The peaktopeak amplitude of the input signal is 2 Vpp because the maximum value of A⋅sin(f⋅t)⋅[1+sin(m⋅t)] is 1 and the minimum value is 1. The demodulated output peaktopeak amplitude is 500 mVpp, which is 1/2 of 1 Vpp since d is 100%.
In the case where the signal is 50% modulated, the peaktopeak amplitude of the input signal is 1.5 Vpp since the maximum value of A⋅sin(ft)⋅[1+0.5⋅sin(m⋅t)] is 0.75 and the minimum value is 0.75. Moreover, the demodulated output peaktopeak amplitude is 250 mVpp, which is d/2 of 1 Vpp.
In summary, the inputtooutput amplitude relationship follows a 1/4 mapping. However, this only applies when the signal consists of a single frequency tone. In other words, the R output represents a quarter of the peaktopeak amplitude of the input signal component that matches the demodulation frequency. If there is a message signal on the carrier wave, the amplitude of the demodulated message signal would be A⋅d/2, where A is the carrier amplitude, and d is the modulation depth.