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 (PLL)- Phase-locked loop

# Phase-locked loop

In electronics, a phase-locked loop (PLL) is a closed-loop feedback control system that maintains a generated signal in a fixed phase relationship to a reference signal.

This technique is widely used in radio, telecommunications, computers and other electronic applications where it is desired to stabilize a generated signal or to detect signals in the presence of noise. Since an integrated circuit can hold a complete phase-locked loop building block, the technique is widely used in modern electronic devices, with signal frequencies from a fraction of a cycle per second up to many gigahertz.

## Contents

1 History
• 2 Basic operation
• 2.1 Mechanical analogy
• 2.2 Basic PLL
• 2.3 (Indirect) synthesizers
• 3 Applications
• 4 Elements
• 4.1 Phase detector
• 4.2 Oscillator types
• 5 Usage
• 5.1 Clock distribution
• 6 Jitter and noise
• 7 Analog phase-locked loop
• 8 Control system analysis
• 10 References

//

## History

Earliest research towards what became known as the phase-locked loop goes back to 1932, when British researchers developed an alternative to Edwin Armstrong's superheterodyne receiver, the Homodyne. In this so-called homodyne or synchrodyne system, a local oscillator was tuned to the desired input frequency and multiplied with the input signal. The resulting output signal included the original audio 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 H.de Bellescise, in the French journal Onde Electrique. [1]

When Signetics introduced a line of monolithic integrated circuits that were complete phase-locked loop systems on a chip, applications for the technique multiplied.

The "CD4046" CMOS Micropower Phase-Locked Loop became a popular integrated circuit including two different phase detectors built in.

## Basic operation

### Mechanical analogy

Tuning a string on a piano can be compared to the operation of a phase-locked loop. Using a tuning fork or pitchpipe to provide a reference frequency, the tension of the string is adjusted up or down until the beat frequency is inaudible. Strictly, this is an example of a frequency "locked" loop, since the phase of the reference and tuned string are immaterial to piano tuning. A phase-locked loop responds both to the frequency and phase of the input signals, automatically raising or lowering the frequency of a controlled oscillator until it is matched to the reference in both frequency and phase.

### Basic PLL

PLLs are generally built of a phase detector, low pass filter and voltage-controlled oscillator (VCO) placed in a negative feedback configuration. There may be a divider in the feedback path or in the reference path, or both, in order to make the PLL's output clock a rational multiple of the reference. By replacing the simple divide-by-N counter in the feedback path with a programmable pulse swallowing counter, it is possible to obtain fractional multiples of the reference frequency out of the PLL.

The oscillator generates a periodic output signal. Assume that initially the oscillator is at nearly the same frequency as the reference signal. Then, if the phase from the oscillator falls behind that of the reference, the phase detector causes the charge pump to change the control voltage, so that the oscillator speeds up. Likewise, if the phase creeps ahead of the reference, the phase detector causes the charge pump to change the control voltage to slow down the oscillator. The low-pass filter smooths out the abrupt control inputs from the charge pump. 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.

### (Indirect) synthesizers

Most PLLs also include a divider between the VCO and the feedback input to the phase detector to produce a Frequency synthesiser. 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 this divider divides by M, it allows the VCO to multiply the reference frequency by N / M. 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.

## Applications

Phase-locked loops are widely used for synchronization purposes; in space communications for coherent carrier tracking and threshold extension, bit synchronization, and symbol synchronization. Phase-locked loops can also be used to demodulate frequency-modulated 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 of both FM and AM signals
• Recovery of small signals that otherwise would be lost in noise (lock-in amplifier)
• 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, modems, and other tone decoders, for remote control 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. 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; 8B10B is very common.

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.

Clock generation: Most 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 in the United States) put limits on this 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 by FM receivers which have a bandwidth of tens of kilohertz.

## Elements

### Phase detector

An important part of a phase-locked loop is the phase detector. This compares the phase of the local oscillator to that of the reference signal.

There are several types of phase detectors. The simplest is an exclusive OR gate, which maintains a 90° phase difference, but cannot lock the signal unless it is already near frequency. A more complex phase detector uses a simple state machine to determine which of the two signals has a zero-crossing earlier or more often. This brings the PLL into lock even when it is off frequency. This type is known as a phase-frequency detector.

A four-quadrant multiplier, also known as a mixer can be used as a phase detector. By multiplying the oscillator and the reference signals, this generates an output consisting of a low-frequency signal whose amplitude is related to the phase difference, or phase error, between the oscillator and the reference, and a second unwanted signal at twice the oscillator frequency that is eliminated by a low-pass filter.

A PLL with a bang-bang charge pump phase detector supplies current pulses with fixed total charge, either positive or negative, to the capacitor acting as an integrator. A phase detector for a bang-bang charge pump must always have a dead band where the phases of the reference and feedback clocks are close enough that the detector fires either both or neither of the charge pumps, for no total effect. Bang-bang control systems are simple, but are associated with significant minimum peak-to-peak jitter, because once in lock the phase offset hunts between the two extrema values of the dead band.

A proportional phase detector directs the charge pump to supply charge amounts in proportion to the phase error detected. Although some proportional phase detectors have dead bands, some do not. Specifically, some designs arrange to produce both "up" and "down" control pulses when the phase offset is zero. These pulses are small, nominally the same duration, and cause the charge pump to produce equal-charge positive and negative current pulses. A proportional phase detector does not necessarily hunt while in lock, and so PLLs with this kind of control system typically have lower minimum peak-to-peak jitter that is determined by other limiting factors.

### Oscillator types

Inductive oscillators (LC oscillators) are built of an LC "tank" circuit, which oscillates by charging and discharging a capacitor through an inductor. These oscillators are typically used when a tunable precision frequency source is necessary, such as with radio transmitters and receivers. Most LC oscillators use off-chip inductors. On-chip inductors suffer large resistive losses, so that the Q of the resulting tank circuit is generally less than 10. As processes have made larger numbers of metal layers available, on-chip inductors have become more useful.

A voltage-controlled capacitor is one method of making an LC oscillator vary its frequency in response to a control voltage. Any reverse-biased semiconductor diode displays a measure of voltage-dependent capacitance and can be used to change the frequency of an oscillator by varying a control voltage applied to the diodes. Special-purpose variable capacitance varactor diodes are available with well-characterized wide-ranging values of capacitance. Such devices are very convenient in the manufacture of voltage-controlled oscillators (a voltage-controlled inductor would be in principle as useful, but such devices are unsatisfactory at the frequencies usually desired).

Crystal oscillators are piezoelectric quartz crystals that mechanically vibrate between two slightly different shapes. Crystals have very high Q, and can only be tuned within a very small range of frequencies. Crystal oscillators are typically used as the frequency reference for other PLLs, and can be found in nearly every consumer electronic device. Because the crystal is an off-chip component, it adds some cost and complexity to the system design, but the crystal itself is generally quite inexpensive.

Surface-acoustic-wave devices (SAWs) are a kind of crystal oscillator, but achieve much higher frequencies by establishing standing waves on the surface of the quartz crystal. These are more expensive than crystal oscillators, and are used in more specialized applications which require a direct and very accurate high frequency reference, for example, in cellular telephones.

For a PLL built into a microprocessor chip, ring oscillators can be used as voltage-controlled oscillators (VCOs). They are built of a ring of active delay stages. Generally the ring has an odd number of inverting stages, so that there is no single stable state for the internal ring voltages. Instead, a single transition propagates endlessly around the ring. The frequency is controlled by varying either the supply voltage or the capacitive loading on each stage. VCOs generally have the lowest Q of the used oscillators, and so suffer more jitter than the other types. The jitter can be made low enough for many applications (such as driving an ASIC), in which case VCOs enjoy the advantages of having no off-chip components (expensive) or on-chip inductors (low yields on generic CMOS processes). These oscillators also have larger tuning ranges than the other kinds, which improves yield and is sometimes a feature of the end product (for instance, the dot clock on a graphics card which drives a wide range of monitors).

## Usage

### 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. From a control theory perspective, the PLL is a special case of the Kalman filter.

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

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. The variance between these phases is called tracking jitter. Ideally, the static phase offset should be zero, and the tracking jitter should be as low as possible.

Phase noise is another type of jitter observed in PLLs, and is mostly caused by the amplifier elements used in the circuit. Some technologies are known to perform better than others in this regard. The best digital PLLs are constructed with emitter-coupled logic (ECL) 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) or CMOS.

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 supply and substrate noise rejection.

## Analog phase-locked loop

The equations governing a phase-locked loop with an analogue multiplier as the phase detector may be derived as follows. Let the input to the phase detector be xc(t) and the output of the voltage-controlled oscillator (VCO) is xr(t) with frequency ωr(t), then the output of the phase detector xm(t) is given by

$x_m(t) = x_c(t) \cdot x_r(t)$

the VCO frequency may be written as a function of the VCO input y(t) as

$\omega_r(t) = \omega_f + g_v y(t)\,$

where gv is the sensitivity of the VCO and is expressed in Hz/V.

Hence the VCO output takes the form

$x_r(t) = A_r \cos\left( \int_0^t \omega_r(\tau)\, d\tau \right) = A_r \cos(\omega_f t + \varphi(t) )$

where

$\varphi(t) = \int_0^t g_v y(\tau)\, d\tau$

The loop filter receives this signal as input and produces an output

xf(t) = Ffilter(xm(t))

where FFilter is the operator representing the loop filter transformation.

When the loop is closed, the output from the loop filter becomes the input to the VCO thus

y(t) = xf(t) = Ffilter(xm(t))

We can deduce how the PLL reacts to a sinusoidal input signal:

xc(t) = Acsin(ωct).

The output of the phase detector then is:

$x_m(t) = A_c \sin( \omega_c t ) A_r \cos(\omega_f t + \varphi(t)).$

This can be rewritten into sum and difference components using trigonometric identities:

$x_m(t) = {A_c A_f \over 2} \sin( \omega_c t - \omega_f t - \varphi(t) ) + {A_c A_f \over 2} \sin( \omega_c t + \omega_f t + \varphi(t) )$

As an approximation to the behaviour of the loop filter we may consider only the difference frequency being passed with no phase change, which enables us to derive a small-signal model of the phase-locked loop. If we can make $\omega_f \approx \omega_c$, then the $\sin(\cdot)$ can be approximated by its argument resulting in: $y(t)=x_f(t) \simeq - A_c A_f \varphi (t) / 2$. The phase-locked loop is said to be locked if this is the case.

Some parts of this article are derived from public domain parts of Federal Standard 1037C in support of MIL-STD-188.

## Control system analysis

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

$\frac{\theta_o}{\theta_i} = \frac{K_p K_v F(s)} {s + K_p K_v F(s)}$

Where

• θo is the output phase in radians
• θi is the input phase in radians
• Kp is the phase detector gain in Volts/radian
• Kv is the VCO gain in radians/Volt-second
• F(s) 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. The loop transfer function in this case is:

$F(s) = \frac{1}{1 + s R C}$

The loop response becomes:

$\frac{\theta_o}{\theta_i} = \frac{\frac{K_p K_v}{R C}}{s^2 + \frac{s}{R C} + \frac{K_p K_v}{R C}}$

This is the form of a classic harmonic oscillator. The denominator can be related to that of a second order system:

$s^2 + 2 s \zeta \omega_n + \omega_n^2$

Where

• ζ is the damping factor
• ωn is the natural frequency of the loop

For the one-pole RC filter,

$\omega_n = \sqrt{\frac{K_p K_v}{R C}}$
$\zeta = \frac{1}{2 \sqrt{K_p K_v R C}}$

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,

$R C = \frac{1}{2 K_p K_v}$
$\omega_c = K_p K_v \sqrt{2}$

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

$F(s) = \frac{1+s C R_2}{1+s C (R_1+R_2)}$

This filter has two time constants

τ1 = C(R1 + R2)
τ2 = CR2

Substituting above yields the following natural frequency and damping factor

$\omega_n = \sqrt{\frac{K_p K_v}{\tau_1}}$
$\zeta = \frac{1}{2 \omega_n \tau_1} + \frac{\omega_n \tau_2}{2}$

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

$\tau_1 = \frac{K_p K_v}{\omega_n^2}$
$\tau_2 = \frac{2 \zeta}{\omega_n} - \frac{1}{K_p K_v}$

## References

1. ^ Notes for a University of Guelph course describing the PLL and early history