Instrumentation & Measurement Magazine 24-2 - 76

the mixer. Ideally, when the phase difference between the reference signal and the system output signal is equal to π/2, the
measured dc voltage at the output port of the synchronous demodulator should be zero. A dc offset is the deviation from the
null output expected when the phase difference is π/2. If the
dc offset is known, its effect can be negated, or at least minimized, by, for instance, applying an appropriate dc bias to the
port.

Mixer-induced Phase Shift
Another multiplier-related nonideality is mixer-induced
phase shifts. Even when the effects of dc offsets are minimized,
the output may still be zero at the output in cases where the
phase difference between the signals is something other than
π/2. This is because the mixer itself may change the relative
phase of the two input signals. For instance, in-phase input signals may produce a dc output voltage that indicates a relative
phase difference between them. Such issues are often related to
the symmetry of the circuit layout and the differences between
matched components. It is thus possible to measure and calibrate them out.

Square-wave Excitation
Another consideration while working with synchronous demodulators is the waveform of the reference signal. A square
wave may be used for ac excitation of the system, where it is
impractical to generate a sine wave to modulate the signal.
Generating a square-wave is much simpler than producing a sine wave, where one can simply use a microcontroller
pin to control an analog switch or a switching transistor. Although it is easier to use a square-wave-based synchronous
demodulator, the performance of the system in terms of noise
rejection is inferior to those using sine waves [9]. The reason
behind this drawback is that square waves contain many harmonics, and the noise in each harmonic is modulated back to
dc. On the other hand, using a sine wave makes the method
less prone to noise and can retrieve very small signals in the
presence of high noise or interference. However, the simplicity and low cost of the system makes it more convenient to
use the square-wave in many applications, even with this
drawback.

Intermodulation
So far, we have assumed the transfer function of the DUT to be
linear. However, if the transfer function has nonlinear properties, it can create higher harmonics. Various physical effects
feature a nonlinear dependence, such as Kerr effects in optics,
Joule heating in resistors, and electrostatic actuation in microsystems. A synchronous demodulator may be used to study
different harmonics and to explore these nonlinearities. The
nonlinearity described by a quadratic transfer function, for instance, can be quantified using a harmonic excitation signal
and lock-in detection by locking to the second harmonic of the
reference signal.
If the input to the DUT is a combination of several sinusoidal signals, nonlinear characteristics of the DUT can result in
76	

the generation of intermodulation signals [10]. This situation,
for instance, can arise if the excitation provided by the oscillator is distorted and contains higher harmonics in addition
to the fundamental sine wave (for example, a square wave).
The intermodulation frequencies that are close to dc and
pass through the low-pass filter can cause beating and other
nonidealities.

Noise
The synchronous demodulator will inevitably add its noise to
the noise generated by the DUT. Different types of noise, which
are stimulated by the different components of the synchronous
demodulators, determine the noise added by the synchronous
demodulator during the measurement.

Noise from the Linear Components
Most studies on synchronous demodulator interfaces have focused on reducing the noise in linear components along the
signal chain. In particular, the first amplifier stage before the
mixer often contributes the most amount of noise to the signal
and necessitates careful design. As the output of a synchronous demodulator is a dc signal, it is prone to excessive flicker
(1/f) noise. Therefore, the noise from the low-pass filter at the
output of the system also requires special attention.
In the past decades, many integrated synchronous demodulators for sensor applications have been developed with low
input noise voltage and high gain to measure input signals of
the order of 100 nV [11], [12]. These implementations meet the
requirements for portable conditioning systems, such as a low
power consumption below milliwatt, battery-compatible design, low cost, and accuracy above 95% [13].

Noise from the Nonlinear Components
With an increasing need for low-cost, high-performance interfaces, the noise contributions of other system components
such as reference oscillator and demodulator need to be studied more closely. The instability and noise of simple oscillators
can adversely affect the performance of a synchronous demodulator. For instance, phase noise is one of the parameters that
identify the quality of the oscillator. Narrowband phase noise
can be used to model the components near the oscillator carrier
on the output noise profile in measurement systems. In [14],
the effect of oscillator phase noise on synchronous demodulation measurement systems was studied.
We can use the sinusoidal representation of narrowband
Phase Modulated (PM) noise to model phase noise in the reference signal (carrier). Such representation will result in an
infinite number of sidebands around the carrier. To determine
the amplitude of these sidebands, we can use (3) where β is
the maximum phase deviation, fm is the offset frequency of the
sideband, fc is the carrier frequency, ϕn is the non-deterministic
term for the noise and Jk is the Bessel function of the first kind:

	



cos 2 fct   sin  2 fmt  n 





  J k    cos 2 fc t  k 2 f mt  n
k

IEEE Instrumentation & Measurement Magazine	

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

April 2021



Instrumentation & Measurement Magazine 24-2

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