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Phase-locked loop

Phase-locked loop

Overview
A phase-locked loop or phase lock loop (PLL) is a control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

 that generates an output signal
Signal (electrical engineering)
In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

 whose phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 is related to the phase of an input "reference" signal. It is an electronic circuit
Electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow...

 consisting of a variable frequency oscillator
Electronic oscillator
An electronic oscillator is an electronic circuit that produces a repetitive electronic signal, often a sine wave or a square wave. They are widely used in innumerable electronic devices...

 and a phase detector
Phase detector
A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a voltage signal which represents the difference in phase between two signal inputs...

. This circuit compares the phase of the input signal with the phase of the signal derived from its output oscillator and adjusts the frequency of its oscillator to keep the phases matched.
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Encyclopedia
A phase-locked loop or phase lock loop (PLL) is a control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

 that generates an output signal
Signal (electrical engineering)
In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

 whose phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 is related to the phase of an input "reference" signal. It is an electronic circuit
Electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow...

 consisting of a variable frequency oscillator
Electronic oscillator
An electronic oscillator is an electronic circuit that produces a repetitive electronic signal, often a sine wave or a square wave. They are widely used in innumerable electronic devices...

 and a phase detector
Phase detector
A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a voltage signal which represents the difference in phase between two signal inputs...

. This circuit compares the phase of the input signal with the phase of the signal derived from its output oscillator and adjusts the frequency of its oscillator to keep the phases matched. The signal from the phase detector is used to control the oscillator in a feedback loop.

Frequency is the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of phase. Keeping the input and output phase in lock step implies keeping the input and output frequencies in lock step. Consequently, a phase-locked loop can track an input frequency, or it can generate a frequency that is a multiple of the input frequency. The former property is used for demodulation
Demodulation
Demodulation is the act of extracting the original information-bearing signal from a modulated carrier wave.A demodulator is an electronic circuit that is used to recover the information content from the modulated carrier wave.These terms are traditionally used in connection with radio receivers,...

, and the latter property is used for indirect frequency synthesis.

Phase-locked loops are widely employed in radio
Radio
Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...

, telecommunications, computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

s and other electronic applications. They can be used to recover a signal from a noisy communication channel, generate stable frequencies at a multiple of an input frequency (frequency synthesis), or distribute clock timing pulses in digital logic designs such as microprocessor
Microprocessor
A microprocessor incorporates the functions of a computer's central processing unit on a single integrated circuit, or at most a few integrated circuits. It is a multipurpose, programmable device that accepts digital data as input, processes it according to instructions stored in its memory, and...

s. Since a single integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...

 can provide a complete phase-locked-loop building block, the technique is widely used in modern electronic devices, with output frequencies from a fraction of a hertz up to many gigahertz.

Automobile race analogy


For a practical idea of what is going on, consider an auto race. There are many cars, and each of them wants to go around the track as fast as possible. Each lap corresponds to a complete cycle, and each car will complete dozens of laps per hour. The number of laps per hour (a speed) corresponds to an angular velocity (i.e. a frequency), but the number of laps (a distance) corresponds to a phase (and the conversion factor is the distance around the track loop).

During most of the race, each car is on its own and is trying to beat every other car on the course, and the phase of each car varies freely.

However, if there is an accident, a pace car
Safety car
In motorsport, a safety car or pace car is a car which limits the speed of competing cars on a racetrack in the case of a caution period such as an obstruction on the track. During a caution period the safety car enters the track ahead of the leader...

 comes out to set a safe speed. None of the race cars are permitted to pass the pace car (or the race cars in front of them), but each of the race cars wants to stay as close to the pace car as it can. While it is on the track, the pace car is a reference, and the race cars become phase-locked loops. Each driver will measure the phase difference (a distance in laps) between him and the pace car. If the driver is far away, he will increase his engine speed to close the gap. If he's too close to the pace car, he will slow down. The result is all the race cars lock on to the phase of the pace car. The cars travel around the track in a tight group that is a small fraction of a lap.

Clock analogy


Phase can be proportional to time, so a phase difference can be a time difference. Clocks are, with varying degrees of accuracy, phase-locked (time-locked) to a master clock.

Left on its own, each clock will mark time at slightly different rates. A wall clock, for example, might be fast by a few seconds per hour compared to the reference clock at NIST
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

. Over time, that time difference would become substantial.

To keep his clock in synch, each week the owner compares the time on his wall clock to a more accurate clock (a phase comparison), and he resets his clock. Left alone, the wall clock will continue to diverge from the reference clock at the same few seconds per hour rate.

Some clocks have a timing adjustment (a fast-slow control). When the owner compared his wall clock's time to the reference time, he noticed that his clock was too fast. Consequently, he could turn the timing adjust a small amount to make the clock run a little slower. If things work out right, his clock will be more accurate. Over a series of weekly adjustments, the wall clock's notion of a second would agree with the reference time (within the wall clock's stability).

An early mechanical version of a phase-locked loop was used in 1921 in the Shortt-Synchronome clock
Shortt-synchronome clock
The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shortt in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK...

.

History


Automatic synchronization of electronic oscillators was described in 1923. Earliest research towards what became known as the phase-locked loop goes back to 1932, when British researchers developed an alternative to Edwin Armstrong
Edwin Armstrong
Edwin Howard Armstrong was an American electrical engineer and inventor. Armstrong was the inventor of modern frequency modulation radio....

's superheterodyne receiver
Superheterodyne receiver
In electronics, a superheterodyne receiver uses frequency mixing or heterodyning to convert a received signal to a fixed intermediate frequency, which can be more conveniently processed than the original radio carrier frequency...

, the Homodyne
Homodyne detection
Homodyne detection is a method of detecting frequency-modulated radiation by non-linear mixing with radiation of a reference frequency, the same principle as for heterodyne detection....

 or direct-conversion receiver
Direct-conversion receiver
A direct-conversion receiver , also known as homodyne, synchrodyne, or zero-IF receiver, is a radio receiver design that demodulates the incoming radio signal using synchronous detection driven by a local oscillator whose frequency is identical to, or very close to the carrier frequency of the...

. In the homodyne or synchrodyne system, a local oscillator
Local oscillator
A local oscillator is an electronic device used to generate a signal normally for the purpose of converting a signal of interest to a different frequency using a mixer. This process of frequency conversion, also referred to as heterodyning, produces the sum and difference frequencies of the...

 was tuned to the desired input frequency and multiplied with the input signal. The resulting output signal included the original modulation information. The intent was to develop an alternative receiver circuit that required fewer tuned circuits than the superheterodyne receiver. Since the local oscillator would rapidly drift in frequency, an automatic correction signal was applied to the oscillator, maintaining it in the same phase and frequency as the desired signal. The technique was described in 1932, in a paper by Henri de Bellescize, in the French journal L'Onde Électrique.

In analog television receivers since at least the late 1930s, phase-locked-loop horizontal and vertical sweep circuits are locked to synchronization pulses in the broadcast signal.

When Signetics
Signetics
Signetics, once a major player in semiconductor manufacturing, made a variety of devices which included integrated circuits, bipolar and MOS, the Dolby circuit, logic, memory and analog circuits. They developed microprocessors like the 2650, the bipolar 8X300 and had licensed Motorola 68000...

 introduced a line of monolithic integrated circuits that were complete phase-locked loop systems on a chip in 1969, applications for the technique multiplied. A few years later RCA introduced the "CD4046" CMOS
CMOS
Complementary metal–oxide–semiconductor is a technology for constructing integrated circuits. CMOS technology is used in microprocessors, microcontrollers, static RAM, and other digital logic circuits...

 Micropower Phase-Locked Loop, which became a popular integrated circuit.

Structure and function


Phase-locked loop mechanisms may be implemented as either analog or digital circuits. Both implementations use the same basic structure.

Both analog and digital PLL circuits include four basic elements:
  • Phase detector
    Phase detector
    A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a voltage signal which represents the difference in phase between two signal inputs...

    ,
  • Low-pass filter
    Low-pass filter
    A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

    ,
  • Variable-frequency oscillator
    Variable-frequency oscillator
    A variable frequency oscillator in electronics is an oscillator whose frequency can be tuned over some range. It is a necessary component in any tunable radio receiver or transmitter that works by the superheterodyne principle, and controls the frequency to which the apparatus is...

    , and
  • feedback
    Feedback
    Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

     path (which may include a frequency divider).

Variations


There are several variations of PLLs. Some terms that are used are analog phase-locked loop (APLL) also referred to as a linear phase-locked loop (LPLL), digital phase-locked loop (DPLL), all digital phase-locked loop (ADPLL), and software phase-locked loop (SPLL).

Analog or Linear PLL (LPLL): Phase detector is an analog multiplier. Loop filter is active or passive. Uses a Voltage-controlled oscillator
Voltage-controlled oscillator
A voltage-controlled oscillator or VCO is an electronic oscillator designed to be controlled in oscillation frequency by a voltage input. The frequency of oscillation is varied by the applied DC voltage, while modulating signals may also be fed into the VCO to cause frequency modulation or phase...

 (VCO).
Digital PLL (DPLL): An analog PLL with a digital phase detector (such as XOR, edge-trigger JK, phase frequency detector). May have digital divider in the loop.
All digital PLL (ADPLL): Phase detector, filter and oscillator are digital. Uses a numerically controlled oscillator (NCO).
Software PLL (SPLL): Functional blocks are implemented by software rather than specialized hardware.

Performance parameters

  • Type and order
  • Lock range: The frequency range the PLL is able to stay locked. Mainly defined by the VCO range.
  • Capture range: The frequency range the PLL is able to lock-in, starting from unlocked condition. This range is usually smaller than the lock range and will depend e.g. on phase detector.
  • Loop bandwidth: Defining the speed of the control loop.
  • Transient response: Like overshoot and settling time to a certain accuracy (like 50ppm).
  • Steady-state errors: Like remaining phase or timing error
  • Output spectrum purity: Like sidebands generated from a certain VCO tuning voltage ripple.
  • Phase-noise: Defined by noise energy in a certain frequency band (like 10kHz offset from carrier). Highly dependent on VCO phase-noise, PLL bandwidth, etc.
  • General parameters: Such as power consumption, supply voltage range, output amplitude, etc.

Applications


Phase-locked loops are widely used for synchronization
Synchronization
Synchronization is timekeeping which requires the coordination of events to operate a system in unison. The familiar conductor of an orchestra serves to keep the orchestra in time....

 purposes; in space communications
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...

 for coherent demodulation and threshold
Threshold
-Film and television:* Threshold * Threshold , an adaptation of the 1958 science fiction film It! The Terror from Beyond Space* Threshold , an American science fiction drama series...

extension, bit synchronization, and symbol synchronization. Phase-locked loops can also be used to demodulate frequency-modulated
Frequency modulation
In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...

 signals. In radio transmitters, a PLL is used to synthesize new frequencies which are a multiple of a reference frequency, with the same stability as the reference frequency.

Other applications include:
  • Demodulation
    Demodulation
    Demodulation is the act of extracting the original information-bearing signal from a modulated carrier wave.A demodulator is an electronic circuit that is used to recover the information content from the modulated carrier wave.These terms are traditionally used in connection with radio receivers,...

     of both FM
    Frequency modulation
    In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...

     and AM
    Amplitude modulation
    Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...

     signals
  • Recovery of small signals that otherwise would be lost in noise (lock-in amplifier
    Lock-in amplifier
    A lock-in amplifier is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment . It is essentially a homodyne with an extremely low pass filter...

    )
  • Recovery of clock timing information from a data stream such as from a disk drive
  • Clock multipliers in microprocessors that allow internal processor elements to run faster than external connections, while maintaining precise timing relationships
  • DTMF decoders, modem
    Modem
    A modem is a device that modulates an analog carrier signal to encode digital information, and also demodulates such a carrier signal to decode the transmitted information. The goal is to produce a signal that can be transmitted easily and decoded to reproduce the original digital data...

    s, and other tone decoders, for remote control
    Remote control
    A remote control is a component of an electronics device, most commonly a television set, used for operating the television device wirelessly from a short line-of-sight distance.The remote control is usually contracted to remote...

     and telecommunications

Clock recovery


Some data streams, especially high-speed serial data streams (such as the raw stream of data from the magnetic head of a disk drive), are sent without an accompanying clock. The receiver generates a clock from an approximate frequency reference, and then phase-aligns to the transitions in the data stream with a PLL. This process is referred to as clock recovery
Clock recovery
Some digital data streams, especially high-speed serial data streams are sent without an accompanying clock signal. The receiver generates a clock from an approximate frequency reference, and then phase-aligns to the transitions in the data stream with a phase-locked loop...

. In order for this scheme to work, the data stream must have a transition frequently enough to correct any drift in the PLL's oscillator. Typically, some sort of redundant encoding is used, such as 8b/10b encoding
8B/10B encoding
In telecommunications, 8b/10b is a line code that maps 8-bit symbols to 10-bit symbols to achieve DC-balance and bounded disparity, and yet provide enough state changes to allow reasonable clock recovery. This means that the difference between the count of 1s and 0s in a string of at least 20 bits...

.

Deskewing


If a clock is sent in parallel with data, that clock can be used to sample the data. Because the clock must be received and amplified before it can drive the flip-flops which sample the data, there will be a finite, and process-, temperature-, and voltage-dependent delay between the detected clock edge and the received data window. This delay limits the frequency at which data can be sent. One way of eliminating this delay is to include a deskew PLL on the receive side, so that the clock at each data flip-flop is phase-matched to the received clock. In that type of application, a special form of a PLL called a delay-locked loop
Delay-locked loop
In electronics, a delay-locked loop is a digital circuit similar to a phase-locked loop , with the main difference being the absence of an internal voltage-controlled oscillator...

 (DLL) is frequently used.

Clock generation


Many electronic systems include processors of various sorts that operate at hundreds of megahertz. Typically, the clocks supplied to these processors come from clock generator PLLs, which multiply a lower-frequency reference clock (usually 50 or 100 MHz) up to the operating frequency of the processor. The multiplication factor can be quite large in cases where the operating frequency is multiple gigahertz and the reference crystal is just tens or hundreds of megahertz.

Spread spectrum


All electronic systems emit some unwanted radio frequency energy. Various regulatory agencies (such as the FCC
Federal Communications Commission
The Federal Communications Commission is an independent agency of the United States government, created, Congressional statute , and with the majority of its commissioners appointed by the current President. The FCC works towards six goals in the areas of broadband, competition, the spectrum, the...

 in the United States) put limits on the emitted energy and any interference caused by it. The emitted noise generally appears at sharp spectral peaks (usually at the operating frequency of the device, and a few harmonics). A system designer can use a spread-spectrum PLL to reduce interference with high-Q receivers by spreading the energy over a larger portion of the spectrum. For example, by changing the operating frequency up and down by a small amount (about 1%), a device running at hundreds of megahertz can spread its interference evenly over a few megahertz of spectrum, which drastically reduces the amount of noise seen on broadcast FM radio
FM broadcasting
FM broadcasting is a broadcasting technology pioneered by Edwin Howard Armstrong which uses frequency modulation to provide high-fidelity sound over broadcast radio. The term "FM band" describes the "frequency band in which FM is used for broadcasting"...

 channels, which have a bandwidth of several tens of kilohertz.

Clock distribution




Typically, the reference clock enters the chip and drives a phase locked loop (PLL), which then drives the system's clock distribution. The clock distribution is usually balanced so that the clock arrives at every endpoint simultaneously. One of those endpoints is the PLL's feedback input. The function of the PLL is to compare the distributed clock to the incoming reference clock, and vary the phase and frequency of its output until the reference and feedback clocks are phase and frequency matched.

PLLs are ubiquitous—they tune clocks in systems several feet across, as well as clocks in small portions of individual chips. Sometimes the reference clock may not actually be a pure clock at all, but rather a data stream with enough transitions that the PLL is able to recover a regular clock from that stream. Sometimes the reference clock is the same frequency as the clock driven through the clock distribution, other times the distributed clock may be some rational multiple of the reference.

Jitter and noise reduction


One desirable property of all PLLs is that the reference and feedback clock edges be brought into very close alignment. The average difference in time between the phases of the two signals when the PLL has achieved lock is called the static phase offset (also called the steady-state phase error). The variance between these phases is called tracking jitter
Jitter
Jitter is the undesired deviation from true periodicity of an assumed periodic signal in electronics and telecommunications, often in relation to a reference clock source. Jitter may be observed in characteristics such as the frequency of successive pulses, the signal amplitude, or phase of...

. Ideally, the static phase offset should be zero, and the tracking jitter should be as low as possible.

Phase noise
Phase noise
Phase noise is the frequency domain representation of rapid, short-term, random fluctuations in the phase of a waveform, caused by time domain instabilities...

 is another type of jitter observed in PLLs, and is caused by the oscillator itself and by elements used in the oscillator's frequency control circuit. Some technologies are known to perform better than others in this regard. The best digital PLLs are constructed with emitter-coupled logic (ECL
Emitter coupled logic
In electronics, emitter-coupled logic , is a logic family that achieves high speed by using an overdriven BJT differential amplifier with single-ended input, whose emitter current is limited to avoid the slow saturation region of transistor operation....

) elements, at the expense of high power consumption. To keep phase noise low in PLL circuits, it is best to avoid saturating logic families such as transistor-transistor logic (TTL
Transistor-transistor logic
Transistor–transistor logic is a class of digital circuits built from bipolar junction transistors and resistors. It is called transistor–transistor logic because both the logic gating function and the amplifying function are performed by transistors .TTL is notable for being a widespread...

) or CMOS
CMOS
Complementary metal–oxide–semiconductor is a technology for constructing integrated circuits. CMOS technology is used in microprocessors, microcontrollers, static RAM, and other digital logic circuits...

.

Another desirable property of all PLLs is that the phase and frequency of the generated clock be unaffected by rapid changes in the voltages of the power and ground supply lines, as well as the voltage of the substrate on which the PLL circuits are fabricated. This is called substrate and supply noise rejection
Power supply rejection ratio
In electronics, power supply rejection ratio or PSRR is a term widely used in the electronic amplifier or voltage regulator datasheets; used to describe the amount of noise from a power supply that a particular device can reject.-Definition:The PSRR is defined as the ratio of the change in supply...

. The higher the noise rejection, the better.

To further improve the phase noise of the output, an injection locked oscillator can be employed following the VCO in the PLL.

Frequency Synthesis


In digital wireless communication systems (GSM, CDMA etc.), PLLs are used to provide the local oscillator for up-conversion during transmission and down-conversion
Digital down converter
In digital signal processing, a digital down-converter converts a digitized real signal centered at an intermediate frequency to a basebanded complex signal centered at zero frequency...

 during reception. In most cellular handsets this function has been largely integrated into a single integrated circuit to reduce the cost and size of the handset. However, due to the high performance required of base station terminals, the transmission and reception circuits are built with discrete components to achieve the levels of performance required. GSM local oscillator modules are typically built with a frequency synthesizer
Frequency synthesizer
A frequency synthesizer is an electronic system for generating any of a range of frequencies from a single fixed timebase or oscillator. They are found in many modern devices, including radio receivers, mobile telephones, radiotelephones, walkie-talkies, CB radios, satellite receivers, GPS systems,...

 integrated circuit and discrete resonator VCOs.

Frequency synthesizer manufacturers include Analog Devices, National Semiconductor and Texas Instruments. VCO manufacturers include Sirenza, Z-Communications, Inc. (Z-COMM).

Phase-locked loop block diagram


A phase detector compares two input signals and produces an error signal which is proportional to their phase difference. The error signal is then low-pass filtered and used to drive a VCO which creates an output phase. The output is fed through an optional divider back to the input of the system, producing a negative feedback loop. If the output phase drifts, the error signal will increase, driving the VCO phase in the opposite direction so as to reduce the error. Thus the output phase is locked to the phase at the other input. This input is called the reference.

Analog phase locked loops are generally built with an analog phase detector, low pass filter and VCO placed in a negative feedback
Negative feedback
Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...

 configuration. A digital phase locked loop uses a digital phase detector; it may also have a divider in the feedback path or in the reference path, or both, in order to make the PLL's output signal frequency a rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 multiple of the reference frequency. A non-integer multiple of the reference frequency can also be created by replacing the simple divide-by-N counter in the feedback path with a programmable pulse swallowing counter
Pulse swallowing counter
A pulse-swallowing counter is a component in an all-digital feedback system. The overall pulse-swallowing system is used as part of a fractional-N frequency divider. The overall pulse-swallowing system cancels beatnotes created when switching between N, N+1, or N−1 in a fractional-N...

. This technique is usually referred to as a fractional-N synthesizer or fractional-N PLL.

The oscillator generates a periodic output signal. Assume that initially the oscillator is at nearly the same frequency as the reference signal. If the phase from the oscillator falls behind that of the reference, the phase detector changes the control voltage of the oscillator so that it speeds up. Likewise, if the phase creeps ahead of the reference, the phase detector changes the control voltage to slow down the oscillator. Since initially the oscillator may be far from the reference frequency, practical phase detectors may also respond to frequency differences, so as to increase the lock-in range of allowable inputs.

Depending on the application, either the output of the controlled oscillator, or the control signal to the oscillator, provides the useful output of the PLL system.

Phase detector



The two inputs of the phase detector are the reference input and the feedback from the VCO. The PD output controls the VCO such that the phase difference between the two inputs is held constant, making it a negative feedback system. There are several types of phase detectors in the two main categories of analog and digital.

Different types of phase detectors have different performance characteristics.

For instance, the frequency mixer
Frequency mixer
In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...

 produces harmonics that adds complexity in applications where spectral purity of the VCO signal is important. The resulting unwanted (spurious) sidebands, also called "reference spurs" can dominate the filter requirements and reduce the capture range and lock time well below the requirements. In these applications the more complex digital phase detectors are used which do not have as severe a reference spur component on their output. Also, when in lock, the steady-state phase difference at the inputs using this type of phase detector is near 90 degrees. The actual difference is determined by the DC loop gain.

A bang-bang charge pump phase detector must always have a dead band where the phases of inputs are close enough that the detector detects no phase error. For this reason, bang-bang phase detectors are associated with significant minimum peak-to-peak jitter
Jitter
Jitter is the undesired deviation from true periodicity of an assumed periodic signal in electronics and telecommunications, often in relation to a reference clock source. Jitter may be observed in characteristics such as the frequency of successive pulses, the signal amplitude, or phase of...

, because of drift within the dead band. However these types, having outputs consisting of very narrow pulses at lock, are very useful for applications requiring very low VCO spurious outputs. The narrow pulses contain very little energy and are easy to filter out of the VCO control voltage. This results in low VCO control line ripple and therefore low FM sidebands on the VCO.

In PLL applications it is frequently required to know when the loop is out of lock. The more complex digital phase-frequency detectors usually have an output that allows a reliable indication of an out of lock condition.

Filter


The block commonly called the PLL loop filter (usually a low pass filter) generally has two distinct functions.

The primary function is to determine loop dynamics, also called stability
Nyquist stability criterion
When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...

. This is how the loop responds to disturbances, such as changes in the reference frequency, changes of the feedback divider, or at startup. Common considerations are the range over which the loop can achieve lock (pull-in range, lock range or capture range), how fast the loop achieves lock (lock time, lock-up time or settling time
Settling time
The settling time of an amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained within a specified error band, usually symmetrical about the final value.Settling time includes a...

) and damping
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

 behavior. Depending on the application, this may require one or more of the following: a simple proportion (gain or attenuation), an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 (low pass filter) and/or derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 (high pass filter). Loop parameters commonly examined for this are the loop's gain margin and phase margin
Phase margin
In electronic amplifiers, phase margin is the difference between the phase, measured in degrees, of an amplifier's output signal and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted,...

. Common concepts in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

 including the PID controller
PID controller
A proportional–integral–derivative controller is a generic control loop feedback mechanism widely used in industrial control systems – a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a...

 are used to design this function.

The second common consideration is limiting the amount of reference frequency energy (ripple) appearing at the phase detector output that is then applied to the VCO control input. This frequency modulates the VCO and produces FM sidebands commonly called "reference spurious". The low pass characteristic of this block can be used to attenuate this energy, but at times a band reject "notch" may also be useful.

The design of this block can be dominated by either of these considerations, or can be a complex process juggling the interactions of the two. Typical trade-offs are: increasing the bandwidth usually degrades the stability or too much damping for better stability will reduce the speed and increase settling time. Often also the phase-noise is affected.

Oscillator



All phase-locked loops employ an oscillator element with variable frequency capability. This can be an analog VCO either driven by analog circuitry in the case of an APLL or driven digitally through the use of a digital-to-analog converter
Digital-to-analog converter
In electronics, a digital-to-analog converter is a device that converts a digital code to an analog signal . An analog-to-digital converter performs the reverse operation...

 as is the case for some DPLL designs. Pure digital oscillators such as a numerically controlled oscillator are used in ADPLLs.

Feedback path and optional divider


PLLs may include a divider between the oscillator and the feedback input to the phase detector to produce a frequency synthesizer
Frequency synthesizer
A frequency synthesizer is an electronic system for generating any of a range of frequencies from a single fixed timebase or oscillator. They are found in many modern devices, including radio receivers, mobile telephones, radiotelephones, walkie-talkies, CB radios, satellite receivers, GPS systems,...

. A programmable divider is particularly useful in radio transmitter applications, since a large number of transmit frequencies can be produced from a single stable, accurate, but expensive, quartz crystal–controlled reference oscillator.

Some PLLs also include a divider between the reference clock and the reference input to the phase detector. If the divider in the feedback path divides by and the reference input divider divides by , it allows the PLL to multiply the reference frequency by . It might seem simpler to just feed the PLL a lower frequency, but in some cases the reference frequency may be constrained by other issues, and then the reference divider is useful.

Frequency multiplication in a sense can also be attained by locking the PLL to the 'N'th harmonic of the signal.

It should also be noted that the feedback is not limited to a frequency divider. This element can be other elements such as a frequency multiplier, or a mixer. The multiplier will make the VCO output a sub-multiple (rather than a multiple) of the reference frequency. A mixer can translate the VCO frequency by a fixed offset. It may also be a combination of these. An example being a divider following a mixer; this allows the divider to operate at a much lower frequency than the VCO without a loss in loop gain.

Time domain model


The equations governing a phase-locked loop with an analog multiplier
as the phase detector and linear filter may be derived as follows.
Let the input to the phase
detector be and the output of the VCO is
with phases and
. Functions and
describe waveforms of signals. Then the
output of the phase detector is given by


the VCO frequency is usually taken as a function of the VCO input
as
where is the sensitivity of the VCO and is
expressed in Hz / V;
is a free-running frequency of VCO.

The Loop Filter can be described by system of linear differential equations

where is an input of the filter,
is an output of the filter, is
-by- matrix,
. represents an initial state of
the filter. The star symbol is a conjugate transpose
Conjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...

.

Hence the following system describes PLL
where is an initial phase shift.

Phase domain model


Consider the input of pll and VCO output
are high frequency signals.
Then for any
piecewise differentiable -periodic functions
and there is a function
such that the output of Filter
in phase domain is assymptotically equal ( the difference is small with respect to the frequencies) to the output of the Filter in time domain model.
Here function is a Phase Detector characteristic.

Denote by the phase difference
Then the following dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

 describes PLL behavior
Here ;
is a frequency of reference oscillator( we assume that is constant).

Example


Consider sinusoidal signals
and simple one-pole RC circuit
RC circuit
A resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...

 as a filter.
Time domain model takes form
PD characteristics for this signals is equal to
Hence the phase domain model takes form
This system of equations is equivalent to the equation of mathematical pendulum

Linearized phase domain model


Phase locked loops can also be analyzed as control systems by applying the Laplace transform. The loop response can be written as:


Where
  • is the output phase in radian
    Radian
    Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

    s
  • is the input phase in radians
  • is the phase detector gain in volt
    Volt
    The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

    s per radian
  • is the VCO gain in radians per volt-second
    Second
    The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

  • is the loop filter transfer function (dimensionless)


The loop characteristics can be controlled by inserting different types of loop filters. The simplest filter is a one-pole RC circuit
RC circuit
A resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...

. The loop transfer function in this case is:


The loop response becomes:


This is the form of a classic harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

. The denominator can be related to that of a second order system:


Where
  • is the damping factor
  • is the natural frequency of the loop


For the one-pole RC filter,


The loop natural frequency is a measure of the response time of the loop, and the damping factor is a measure of the overshoot and ringing. Ideally, the natural frequency should be high and the damping factor should be near 0.707 (critical damping). With a single pole filter, it is not possible to control the loop frequency and damping factor independently. For the case of critical damping,


A slightly more effective filter, the lag-lead filter includes one pole and one zero. This can be realized with two resistors and one capacitor. The transfer function for this filter is


This filter has two time constants


Substituting above yields the following natural frequency and damping factor


The loop filter components can be calculated independently for a given natural frequency and damping factor


Real world loop filter design can be much more complex e.g. using higher order filters to reduce various types or source of phase noise. (See the D Banerjee ref below)

See also

  • Control theory
    Control theory
    Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

  • Direct-digital synthesis
  • Costas loop
    Costas loop
    A Costas loop is a phase-locked loop used for carrier phase recovery from suppressed-carrier modulation signals, such as from double-sideband suppressed carrier signals. It was invented by John P. Costas at General Electric in the 1950s...

  • Kalman filter
    Kalman filter
    In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...

  • Direct conversion receiver
  • Circle map
    Circle map
    In mathematics, a circle map is a member of a family of dynamical systems on the circle first defined by Andrey Kolmogorov. Kolmogorov proposed this family as a simplified model for driven mechanical rotors . The circle map equations also describe a simplified model of the phase-locked loop in...

     - a simple mathematical model of the phase-locked loop showing both mode-locking and chaotic behavior.
  • Carrier recovery
    Carrier recovery
    A carrier recovery system is a circuit used to estimate and compensate for frequency and phase differences between a received signal's carrier wave and the receiver's local oscillator for the purpose of coherent demodulation....

  • Delay-locked loop
    Delay-locked loop
    In electronics, a delay-locked loop is a digital circuit similar to a phase-locked loop , with the main difference being the absence of an internal voltage-controlled oscillator...

     (DLL)
  • PLL multibit
    PLL multibit
    A PLL multibit or Multibit PLL is a phase-locked loop which uses more of the information available in the PLL, to achieve improved performance. Unibit PLLs use only one bit of each counter's output bus, the most significant bit to measure the phase, whereas Multibit PLLs use more bits [1]...

  • Shortt-Synchronome clock
    Shortt-synchronome clock
    The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shortt in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK...

    - slave pendulum phase-locked to master (ca 1921).

Further reading


. (PDF version).. (provides useful Matlab scripts for simulation). (provides useful Matlab scripts for simulation). (FM Demodulation). An article on designing a standard PLL IC for Bluetooth applications.