Stanford Research Systems SR844, User Manual

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SR844 Basics 2-19
SR844 RF Lock-In Amplifier
Sources of Error
Spurious Responses
It is useful to consider the signal in the frequency domain. Fourier’s theorem states that
any signal can be represented as an infinite sum of sine waves, each with different
frequency, amplitude and phase. In the frequency domain, a signal is described in terms of
its individual frequency components. This is in contrast to the time domain description,
where the signal is described by its value at each point in time, just like one would see on
an oscilloscope. The SR844 circuitry is linear, which means that the signal at any point is
the sum of the signals due to each frequency component. For the purpose of analysis the
individual frequency components may be treated independently.
The SR844 multiplies the signal by a (chopped) square wave at the reference frequency.
All components of the input signal are multiplied by the reference frequency
simultaneously. Signal and noise at the reference frequency give rise to (chopped) DC. In
general other frequency components give rise to mixer outputs at other frequencies, and
are not detected. There are a few exceptions, which constitute the spurious response of
the instrument. It is good for a user to be aware of and understand these limitations.
Spurious responses are outputs due to signals at frequencies other than the reference
frequency. These outputs are indistinguishable from the output due to a signal at the
reference frequency.
Square Wave Response
The first class of spurious responses are harmonics of the reference frequency. Recall that
the mixer multiplies the input signal by the lock-in reference, which is really a square
wave. A square wave at the reference frequency may be written as
sin(
ω
R
t) + (1/3) sin (3
ω
R
t) + (1/5) sin (5
ω
R
t) + … (2-18)
The lock-in is detecting signals at all odd harmonics of the reference simultaneously. An
input signal at 3
ω
R
yields an output 1/3 as large as a signal at
ω
R
. This –10 dB response
at the third harmonic is a fundamental limitation of the technique employed in the SR844.
When the input signal is also a square wave (at the reference frequency), all of the odd
harmonics of the signal coincide with the harmonics of the reference and are detected. In
the frequency domain, this is simply multiplying eqn. 2-18 by itself and keeping those
resultant terms which are at DC. Thus, the contribution from all odd harmonics is
1 + (1/3)
2
+ (1/5)
2
+ (1/7)
2
+ ...
≈
1.2
The amplitude of the fundamental sine component of a square wave is 4/
π
x
the peak
amplitude of the square wave. The detected amplitude is 4/
π
x peak x 1.2 or 1.53 x peak.
The SR844 reads the signal in units of Vrms (0.707
x
1.53
x
peak) or 1.08
x
peak (Vrms).
IF Sidebands
The second class of spurious responses are chopping sidebands at
ω
R
±2N
ω
C
, where
ω
C
is
the chopping frequency (IF) and N is an integer. To understand these spurious responses,
we need to understand a little more about how chopping works. The chopping operation