All passive sensors discussed so far rely on DC excitation. DC has its charm in simplicity, low cost and fluid integration into existing digital technologies. However, when it comes to analog technologies, AC excitation remains a cornerstone of high-performance circuits employing inductive or capacitive elements.
For the most parts, the working principles of passive sensors are the same in either cases; their encoding might differ but the theory of operation should be the same. As such, AC sensors can use the same conditioning circuitry as their DC counterparts, plus some extra to deal with the additional information from frequency domain.
AC excitation characteristic
High tolerance to common resistive effects like corrosion and wear.
Immunity to electrical noise as information can be stored in frequency domain.
Immunity to thermoelectric effects.
Can use high-performance transformer, capacitive and inductive circuits.
Low power losses to self-heating as only the real part of the AC is responsible for heat dissipation.
Vulnerable to phase shifting and race conditions.
Problems with ground leakage / stray capacitance.
Complex interface with existing digital technologies, requiring rectification and rescaling to match the legacy DC signal.
Applying AC excitation to Wheatstone bridge circuits changes the quantity being measured from resistance to impedance. Impedance is a complex quantity containing resistance, inductance and capacitance as well. It wouldn’t hurt to revise the conversion of these DC values to AC complex values as follow:
In a nutshell, the operation of AC Wheatstone is similar to DC Wheatstone; an unknown quantity is weighed against known, adjustable quantities. The condition for a balanced bridge in AC domain is:
In other words, both the resistance and the phase factors must match on either arms for the bridge to be balanced. For measuring inductance and capacitance, a few bridge circuits can be used. The simplest and most straightforward bridge is the symmetrical bridge as seen below:
This symmetrical bridge does the trick illustrating the basic idea of impedance bridges but it is hardly appropriate in practice. In the real world, capacitors and inductors possess internal resistance and this must be properly addressed.
Wien bridge takes into account of the capacitor’s internal resistance, which is in series with the capacitive component, and counters it with a parallel resistor-capacitor pair as seen below:
The bridge is balanced when both conditions below are satisfied:
As Wien bridge’s frequency (or that of the AC source) can be calculated from
it is possible to use Wien bridge for measuring frequency of an unknown signal source.
Another thing about the real world is that, it is incredibly difficult to manufacture variable inductors. However, variable capacitors are readily available and fortunately there exists a bridge called Maxwell bridge that permits measurement of inductance using variable capacitors.
The full condition for balance is as follow:
Maxwell bridge’s balance condition is independent of source frequency. This permits some tolerance to mixed frequency AC voltage source. On top of that, Maxwell bridge uses only one inductor and avoids mutual inductance issue which plagues symmetrical bridge using two inductors. Not that two capacitors don’t cause any mutual capacitance; they do, but shielding against electric field is much easier than against magnetic field so capacitors are preferred.
Phase shifter is an important element of AC conditioning circuit. It allows (manual) phase matching of the input and output signal. Capacitive and inductive elements shift the phase by 90 and -90 degree respectively and they are the foundation of phase shifters.
Due to mutual inductance phenomenon as mentioned in Maxwell bridge’s discussion, RL phase shifters are possible though rarely employed in practice. For RC phase shifters, the phase angle is dependent on the value of R and C, and is calculated as follow:
Using phase shifters naturally introduces additional resistance, which in turn causes the signal to drift (off-set). This off-set should be compensated and hence phase shifters should be used before off-set compensation in the conditioning circuitry.
One important note is that, RC phase shifter is also a passive high pass filter and it will suppress low frequency signals; including the signal itself if it is low enough. This is one of the reasons why Phase-locked loop (PLL), albeit complicated in construction, is preferred in conditioning circuits over simple RC phase shifters for phase matching.
Using AC excitation, it is possible to encode information in both amplitude and frequency alteration. Because AC bridges are natural AM modulators, they produce an amplitude-modulated output signal of the carrier signal–the AC excitation–and the input signal–the impedance change due to physical changes. It is, therefore, imperative to demodulate the input signal from the output at a later stage for usage in a (possibly digital) control system.
One device that performs this demodulation (the term phase-sensitive detection is the same thing) is a lock-in amplifier. Essentially, a lock-in amplifier takes a modulated signal and a reference, carrier signal and attempt to extract the amplitude “shape” of the modulated signal at the reference’s frequency, reconstructing the sensor’s signal as a result. This is the description of its operation in time domain though it is a mere corollary of its frequency domain’s operation.
In frequency domain, the lock-in amplifier acts as a very narrow band-pass filter which locks in on a specific frequency (based on the reference) and suppresses all other frequencies. The output is a DC signal comprising of the signal’s amplitude at reference frequency and its phase shift from the reference.
This DC output can be used as a representation of the phase shift and as control signal in self-tuning circuits as seen in a previous memo. If reconstructing the sensor’s signal is of interest, this phase shift component can be rectified via a phase shifter beforehand so that the output is a linear representation of only the sensor’s signal (its amplitude, specifically). In other words:
Uout = 1/2 Vsig
when = 0
The phase shifter can be an RC phase shifter as described in the previous section or, more commonly, a PLL circuit, which actively keeps the reference signal in-phase with the input signal without further intervention. The reference signal can be taken directly from AC excitation source (at the engineer’s discretion) or from an internal oscillator.
More advanced “dual-phase demodulation” lock-in amplifiers use a fixed 90 degree phase shifter to generate two components X and Y for mathematical calculation of both the phase shift and amplitude quantities of the signal separately.
The in-phase component X from input signal and unchanged reference signal:
The quadrature component Y from input signal and phase shifted reference signal:
The magnitude of the signal at reference frequency:
And the phase shift of the signal from the reference signal: