SRS Labs manual What does a LOCK-IN MEASURE?, RMS or Peak?, What does the SR850 measure?

SR850 Basics

WHAT DOES A LOCK-IN MEASURE?

So what exactly does the SR850 meas- ure?Fourier's theorem basically states that any input signal can be represented as the sum of many, many sine waves of differing amplitudes, frequencies and phases. This is generally consid- ered as representing the signal in the "frequency domain". Normal oscilloscopes display the signal in the "time domain". Except in the case of clean sine waves, the time domain representation does not convey very much information about the vari- ous frequencies which make up the signal.

lowing the multiplier. This "bandwidth narrowing" is the primary advantage that a lock-in amplifier pro- vides. Only inputs at frequencies at the reference frequency result in an output.

RMS or Peak?

Lock-in amplifiers as a general rule display the input signal in Volts RMS. When the SR850 dis- plays a magnitude of 1V (rms), the component of the input signal at the reference frequency is a sine wave with an amplitude of 1 Vrms or 2.8 V pk-pk.

What does the SR850 measure?

The SR850 multiplies the signal by a pure sine wave at the reference frequency. All components of the input signal are multiplied by the reference simultaneously. Mathematically speaking, sine waves of differing frequencies are orthogonal, i.e. the average of the product of two sine waves is zero unless the frequencies are EXACTLY the same. In the SR850, the product of this multiplica- tion yields a DC output signal proportional to the component of the signal whose frequency is exact- ly locked to the reference frequency. The low pass filter which follows the multiplier provides the aver- aging which removes the products of the reference with components at all other frequencies.

The SR850, because it multiplies the signal with a pure sine wave, measures the single Fourier (sine) component of the signal at the reference frequen- cy. Let's take a look at an example. Suppose the input signal is a simple square wave at frequency f. The square wave is actually composed of many sine waves at multiples of f with carefully related amplitudes and phases. A 2V pk-pk square wave can be expressed as

S(t) = 1.273sin(ωt) + 0.4244sin(3ωt) +

0.2546sin(5ωt) + ...

where ω = 2πf. The SR850, locked to f will single out the first component. The measured signal will be 1.273sin(ωt), not the 2V pk-pk that you'd meas- ure on a scope.

In the general case, the input consists of signal plus noise. Noise is represented as varying signals at all frequencies. The ideal lock-in only responds to noise at the reference frequency. Noise at other frequencies is removed by the low pass filter fol-

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Thus, in the previous example with a2 V pk-pk square wave input, the SR850 would detect the first sine component, 1.273sin(ωt). The measured and displayed magnitude would be 0.90 V (rms) (1/√2 x 1.273).

Degrees or Radians?

In this discussion, frequencies have been referred to as f (Hz) and ω (2πf radians/sec). This is because people measure frequencies in cycles per second and math works best in radians. For purposes of measurement, frequencies as meas- ured in a lock-in amplifier are in Hz. The equations used to explain the actual calculations are some- times written using ω to simplify the expressions.

Phase is always reported in degrees. Once again, this is more by custom than by choice. Equations written as sin(ωt + θ) are written as if θ is in radians mostly for simplicity. Lock-in amplifiers always manipulate and measure phase in degrees.