Stanford Research Systems SR844, User Manual

Page: 34 / 246

2-4 SR844 Basics
SR844 RF Lock-In Amplifier
which is proportional to the cosine of the phase difference between the input and the
reference. Hence the term phase sensitive detection.
In order to measure V
I
using Eqn (2-3), the phase difference between the signal and
reference,
θ
R
–
θ
I
, must be stable and known. The SR844 solves this problem by using two
mixers, with the reference inputs 90
°
out of phase. The reference input to the second
mixer is V
R
sin(
ω
R
t+
θ
R
–
π
/2) and the output of the second mixer is
V
M2
=
½ V
I
V
R
cos(
θ
R
–
θ
I
–
π
/2) + ½ V
I
V
R
sin(2
ω
R
t+
θ
R
+
θ
I
–
π
/2)
(2-4)
After filtering,
V
M2+FILT
=
½ V
I
V
R
cos(
θ
R
–
θ
I
–
π
/2)
(2-5)
=
½ V
I
V
R
sin(
θ
R
–
θ
I
)
(2-6)
The amplitude and phase of the input signal can be determined from the two mixer
outputs, Eqn (2-3) and (2-6). These computations are handled by the DSP chip in the
SR844.
Amplitude R =
(2/V
R
)
× √
[ (V
M1+FILT
)
2
+ (V
M2+FILT
)
2
]
(2-7)
Phase
θ
R
–
θ
I
= tan
-1
(V
M2+FILT
/V
M1+FILT
) (2-8)
In-Phase
Component X = R cos (
θ
R
–
θ
I
) (2-9)
Quadrature
Component Y = R sin (
θ
R
–
θ
I
) (2-10)
Units
RMS or Peak ?
Lock-in amplifiers as a general rule measure the input signal in Volts rms. When the
SR844 displays a magnitude of 1 V (rms), the sine component of the input signal at the
reference frequency has an amplitude of 1 Vrms or 2.8 Vpk-pk. This is important to
remember whenever the input signal is
not a sine wave. For example, if the signal input is
a square wave with a 1 Vpk (2 Vpk-pk) amplitude, the sine component at the fundamental
frequency has a peak amplitude of 4/
π ×
1 Vpk. The lock-in displays the rms amplitude
(Vpk/
√
2) or 0.9 Vrms.
Degrees or Radians ?
In this discussion, frequencies have been referred to as f [Hz] and
ω
[radian/sec].
ω
=
2
π
f
(2-19)
This is because it is customary to measure frequency in Hertz, while the math is most
convenient using
ω
. For purposes of measurement, the SR844 reports frequency in kHz
and MHz. The equations used to explain the calculations are often written using
ω
to
simplify the expressions.