Phase-locked loops (PLLs) are closed-loop negative-feedback control systems that maintain the phases of two periodic signals in a well-defined phase relation. Consequently, PLLs are versatile tools for measuring and tracking a signal's frequency, for extracting a given frequency component of the original signal while eliminating noise and spurious components, or for synthesizing new signals based on the input signal.
In addition, PLLs can provide feedback to an external system to drive it at certain points of its transfer function, for example at resonance, or to synchronize two external oscillators by tracking their beat note as often done in optical PLLs. This versatility makes PLLs great tools for physics and engineering applications such as scanning probe microscopy, MEMS, NEMS and resonators, electronic engineering, optics and photonics.
The phase detector takes two periodic input signals and outputs their relative phase difference. Often one of the two inputs is generated internally by the PLL, as depicted in the figure to the left.
This element subtracts the phase difference from a user-defined setpoint to generate an error signal. From that, the feedback signal is calculated by adding three values obtained separately from the error signal and referred to as proportional (P), integral (I) and derivative (D).
Adjustable signal source
This component receives the feedback signal from the PID controller. Based on this signal, the frequency is adjusted to vary the phase difference on the phase detector so that it meets the setpoint of the PID. Typical examples for adjustable signal sources are numerically or voltage-controlled oscillators (NCOs or VCOs), but an entire laser system can also act as the signal source.
Depending on what serves as a frequency reference and where the feedback is provided, we distinguish between three PLL configurations. All examples are discussed by taking as a reference Zurich Instruments' PLLs, given that they can address all of these cases and be quickly adapted from one to the other as the need arises.
Track an external frequency reference
The aim of the configuration depicted above is to track a specific frequency, i.e., map it to the numerical oscillator of the instrument. As part of a lock-in amplifier, this configuration enables the demodulation at an external reference frequency. This functionality can be automated for most use cases but when the spectrum applied provides too many lines manual settings can limit the frequency range where the PLL operates to assure the correct frequency is tracked at all times.
Frequency tracking and external-reference mode are not the only use cases of this configuration. Locking the internal oscillator to external reference results in a jitter- and spurious-free signal at the tracked frequency. Therefore, this configuration is invaluable for applications that require spectral filtering. Another advantage of the all-digital implementation is the possibility to multiply or divide the tracked frequency to generate an internal reference at its harmonics and ratios (n/m). This allows for further measurement schemes based on the same external frequency reference.
Drive a device at a variable resonance frequency
PLLs can be used to drive a device at its resonance frequency or another point of its transfer function. It is very common, for example, to drive vibrating inertial devices such as AFM cantilevers, and micro-and nano-mechanical systems (MEMS/NEMS), including MEMS gyroscopes and accelerometers, at resonance to benefit from resonance enhancement and linearization of the measurement response. The PLL assures the device is always driven at the same working point even when its resonance frequency changes with time.
In this configuration, the PID controller feeds back to an internal oscillator that at the same time provides the signal to drive the device at the right working point. When it comes to optimizing the PLL parameters, the loop filter bandwidth deserves special attention. The higher the bandwidth is set the faster the PLL can react to changes in the device's response. That makes the operation robust to quick changes and disturbances but it also introduces more noise as compared to low feedback bandwidth operation. The best tradeoff depends on your application requirement and operation conditions.
To find a good set of parameters quickly, Zurich Instruments' PLLs come with the LabOne software, which includes a PID Advisor to help the user with the setup. It provides simulated step response and transfer function of the entire setup to assist in the design of a robust closed-loop operation. Further optimization of the PID parameters is possible with the Auto-Tune routine to minimize the residual PID error.
This straightforward workflow is possible thanks to the digital implementation of the entire loop and the PID advisor that allows configuring and maintaining the feedback loop under broader conditions. For instance, a Zurich Instruments' PLL can be easily configured to lock on a device's different resonances, each with its own transfer function. Additionally, a digital PLL's ability to multiply or divide the locked frequency is also useful in this configuration when a parametric drive is needed, for example, to characterize a nonlinearity of the device or with parametric feedback cooling.
Feedback on an external oscillator
In the configuration sketched above, the PLL maps its internal oscillator's frequency on an external system. This is often needed for larger systems where multiple devices are required to operate in sync, for example, multiple laser systems (such as optical PLLs) or atomic clocks. Even though the implementation of the individual elements in such systems can differ a lot, the entire system can often be understood as a voltage-controlled oscillator (VCO) where the variation of an input voltage leads to a change in output frequency.
The PLL captures this output frequency and synchronizes it with its internal oscillator by providing a feedback signal to the system. The spectral filtering of the demodulator plus the programmable output limiter of the analog voltage signal contributes to stable operation at the desired frequency component, even after relocking in case the lock was lost. Also, the system is not driven at any values outside where safe operation is guaranteed even if the integrator goes into saturation.
The two following features are particularly beneficial for this configuration:
D-gain low-pass filter
In many cases, introducing D-gain offers the potential to speed up the loop response. However, this does often lead to instability because it introduces increasing gains towards high frequencies. Introducing an adjustable low-pass filter in the D-branch helps to take advantage of the D-part without suffering the instability problem.
The phase-detection facilitated by the lock-in amplifier is usually limited by the arct2(y/x) function to ±π. This imposes a limitation to the "capture range" of the PLL and also to the stability of operation in the presence of external noise. In the digital domain, this limitation can be overcome relatively easily by applying a "phase unwrap" that detects whenever the phase-detection exceeds its limits and keeps track accordingly. Our instruments support a capture range of up to ±1024 π.
Go beyond a single PLL
In the case in addition to the frequency also the amplitude of a signal needs to be controlled, one more PID controller is required. A common example is automatic gain control (AGC), as depicted in the figure. The PID controller below the PLL steers the amplitude of the drive signal based on a user-defined set point. Similar configurations that employ cascaded or parallel running control loops include:
- Force-to-rebalance (FTR) control for MEMS gyroscopes.
- Dual-frequency resonance tracking (DFRT) for piezo force microscopy.
- FM-KPFM for Kelvin probe force microscopy.
- Interferometer stabilization for optical applications.
PLLs for Lock-in Amplifiers
All Zurich Instruments' PLLs are implemented by means of digital signal processing on an FPGA with multiple numerically controlled oscillators available as signal sources. The phase detector is realized as the dual-phase demodulator of a lock-in amplifier with a low-pass filter that rejects many of the unwanted spectral components.
Providing a clean signal to the PID controller increases the stability of the PLL. Zurich Instruments' PLLs can realize basic PLL configurations as well as more complex measurement and control schemes with a single instrument, because the PLLs are upgrade options for our lock-in amplifiers and can run in parallel with other built-in functionality such as feedback controllers, demodulators, and data capture and analysis tools. This white paper provides a more detailed discussion of lock-in amplifiers and phase detection.
LabOne instrument control software
All Zurich Instruments' lock-in amplifiers are equipped with the LabOne® toolset that allows users to fully characterize their system with a parametric sweeper, an oscilloscope, and many other data acquisition tools. For instance, it is possible to visualize the PID error as a histogram to spot deviations from a normal distribution, which may indicate that something in the setup does not work as expected. Further, the PLL's bandwidth can be measured under real experimental conditions using a frequency modulation method, as shown in the figure to the right.
- An all-digital PLL integrated in a lock-in amplifier provides a straightforward implementation of phase detection, closed-loop control, and signal generation within a single instrument, thus reducing the overall complexity of the experimental setup.
- The PID Advisor makes it possible to model the setup and calculate sensible starting parameters.
- The LabOne toolset consisting of Scope, Spectrum Analyzer, Sweeper and Plotter facilitates an integrated analysis and monitoring of the locking quality.
- The phase unwrap functionality over the range ±1024π expands the 'capture range' of the PLL from the typical ±π and ensures robust operation.