The inventor of the Phase-Locked Loops (PLL) circuit is H. de Bellescize of France. In order to simplify
the structure of the superheterodyne wireless receiver that was widely used at that time and eliminate the noise caused by the drift of the local oscillator frequency of the receiver,
Bellescize proposed the synchronous detection theory in 1932 and published the first public description of the phase-locked loop. However, It did not attract widespread attention at the time. It was not until the 1950s, with the practicality and popularization of televisions, that PLL circuits were widely used in the vertical and horizontal synchronization circuits of televisions.
The unique performance of the phase-locked loop circuit is that it can effectively track the phase of the input signal, thereby extracting an almost completely pure signal from the noise, and completing some functions that other circuits cannot complete. This is the purpose of the phase-locked loop circuit. The magic. In this chapter, we borrow basic knowledge such as the simplest feedback circuit to uncover the mystery of phase-locked loop tracking signal phase.
3.1 Composition of phase locked loop
3.1.1 Pay attention to the phase component of the signal
First of all, it needs to be explained that according to different application situations, there are many kinds of phase-locked loops in actual engineering design, such as
narrow-band tracking circuits for carrier extraction, demodulation loops for demodulators, and local oscillator improvement circuits. Frequency division or frequency multiplication phase-locked loop circuit for frequency stability
, etc. Although there are various forms of loops, they are all composed of three basic components:
Phase Detector (PD), Loop Filter (LF) and Voltage Controlled Oscillator (Voltage Controlled Oscillator, VCO), as shown in Figure 3-1. 
In order to facilitate the subsequent in-depth discussion of the loop model, it is first necessary to
clarify the conversion relationship between the signals of each node in Figure 3-1. Among them, 1( ) t is the instantaneous phase of the input signal, in radians (rad); 2( ) t is the instantaneous phase of the VCO output
frequency signal, in radians (rad); utd ( ) is 1 The difference between ( ) t and 2( ) t e ( ) t (let e ( ) t = 1( ) t - 2( ) t ) is a function of a voltage signal, the unit is volts (V ); utc( ) is the LF filtered voltage signal of utd ( ), in volts (V).
Please note that we describe the units of each signal in a somewhat verbose manner, not only to facilitate a more accurate description of the model, but also to more accurately calculate the loop parameters during engineering design. This point is particularly important. In actual operations, problems often occur due to non-uniform units of various parameters, resulting in loop design results that are far from expected values. Secondly, as can be seen from Figure 3-1, the loop input/output signal we are concerned about is not the amplitude of the voltage or current, but the instantaneous phase of the voltage signal (since the instantaneous frequency can be obtained by differentiating the instantaneous phase of the signal, So it can also be said that the loop focuses on the frequency of the input/output signal). Readers are also asked to keep this in mind from now on when reading the subsequent content of this book. Of course, the phase-locked loop itself is also a closed-loop control circuit. You can also establish a transfer function for the amplitude of the voltage or current of the input/output signal to analyze according to the method learned in the circuit analysis course. However, for a phase-locked loop, changes in signal amplitude are meaningless, and it is not helpful for loop analysis.
In fact, the amplitude of the VCO output signal has nothing to do with the input signal amplitude and is completely determined by the characteristics of the VCO itself. As we discuss the loop later in this chapter, we can see that if there is no loop filter in the loop, the phase-locked loop can work normally under certain conditions, but its performance cannot meet most engineering design requirements, so rarely use.
3.1.2 VCO is an integrating device
Readers who are exposed to phase-locked loops for the first time may find it a bit confusing to regard VCO as an integrating device. In fact, from the perspective of the function of the VCO, it is a voltage and frequency conversion device. It serves as a voltage-controlled oscillator in the loop. Its oscillation frequency changes linearly with the input control voltage, and its changing relationship is:
In the formula, v ( ) t is the instantaneous angular frequency of the voltage-controlled oscillator, in rad/s; o is the natural oscillation angular frequency of the voltage-controlled oscillator, that is, the oscillation angular frequency without input control voltage, in unit is rad/s; K0 is the frequency control sensitivity
or gain coefficient of the VCO, and the unit is rad/(s V) . In practical applications, the control characteristics of analog voltage-controlled oscillators have only a limited linear control range, and beyond this range the control sensitivity will decrease. When we discuss digital phase-locked loop circuits later in this book, we can see that for digital frequency controlled oscillators, their control characteristics are always linear.
The control characteristics of VCO are shown in Figure 3-2.
It can be seen from Figure 3-2 that when the VCO has no input voltage, its oscillation angular frequency is o; in the linear control area, the greater the control sensitivity, the greater the slope of the linear area. In order to further understand the control characteristics of VCO, we use SystemView to simulate the time domain waveform relationship between the VCO input signal and the output signal.
Example 3-1: SystemView simulates the control characteristics of VCO
. Since we only need to simulate the input and output waveforms of the VCO, the simulation system is very simple.
As shown in Figure 3-3, the entire system only uses 1 source icon block (Token0: Main Libraries→Sources→
Aperiodic→ Time Function) and 1 function library icon block (Token2: Main Libraries→ Functions→
FM /VCO), and 2 sink icon blocks for observing signal waveforms (Token3/Token4: Main Libraries→Sinks→Real Time (Plot)). The input signal is a voltage signal that increases linearly with time; the natural oscillation frequency (Freq) of the VCO is 5 Hz, and the control sensitivity (Mod Gain) is 2 Hz/V.
When introducing Figure 3-1, we said that the units of signals and parameters are important. Careful readers have discovered that
the parameters of the icon block VCO are inconsistent with the units of equation (3-1). This is the difference between angular frequency (usually represented by the symbol ) and frequency (usually
represented by the symbol f), which are related by 2f . Therefore, for the VCO icon block,
converted into angular frequency units, the natural oscillation angular frequency of VCO o 31.4159 rad/s, and the control sensitivity
K0 = 12.5664 rad/(s · V).
Figure 3-4 is a VCO waveform diagram simulated using SystemView. It can be seen from the figure that as the input signal voltage increases, the frequency of the VCO output signal also increases, and the phase of the VCO output signal changes continuously. The continuous change of phase of the VCO signal is also particularly important for the entire phase-locked loop circuit.
Whether it is the explanation of equation (3-1) or looking at the waveform of the VCO through simulation, it seems that there is still no
connection between the VCO and the integrator. Let's go back and look at the model in Figure 3-1, and remember the key issue discussed in Section 3.1.1:
the phase-locked loop focuses on the phase of the input and output signals.
Therefore, we need to find the relationship between the VCO input voltage utc( ) and the phase of the output signal. This is not difficult, because
by differentiating the instantaneous phase of the signal, the instantaneous frequency of the signal can be obtained; conversely, by integrating the frequency of the signal, the
instantaneous phase of the signal can be obtained, that is,
It can be seen from equation (3-2) that the voltage controlled oscillator has an integration factor, which is formed by the integral relationship between phase and angular frequency. In addition to an integral factor, equation (3-2) also has a linear multiplication factor ot, so this equation does not seem to be a pure integrator.
Let us analyze again. According to the content discussed in Section 3.1.1, the output of the phase detector utd ( ) is a
function of the difference e ( ) t between 1( ) t and 2( ) t. That is to say, utd ( ) is not related to the instantaneous phase of the VCO, but to
the instantaneous phase difference between the VCO and the input signal. Since it is a phase difference, you can set a reference phase (i.e. reference phase) arbitrarily. The phases of the input signal and the output signal are compared with this reference phase, and then the comparison results are compared. In this way, not only will it not have any impact on the model analysis of the entire loop, but it will also facilitate the discussion of the loop model.
How to get this reference phase? For receivers, we usually only know a rough range of the frequency of the input signal, and cannot determine its specific frequency (obtaining the frequency of the input signal is one of the main functions of the phase-locked loop);
But for local VCO, once the VCO device is selected, its natural frequency is determined. Therefore, we can use
the phase ot generated by the natural oscillation angular frequency of the VCO as the reference phase.
3.1.3 Sine phase detector or cosine phase detector
After the previous discussion, we know that the input of the phase detector is the phase difference between the two signals. Therefore, we need to first understand the generation process of this phase difference. Without loss of generality, the input signal can be expressed as
Having determined the expression of the phase difference, let's continue to discuss how the phase detector works.
The phase detector, as the name suggests, is a phase comparison device used to detect the
phase difference e ( ) t between the input signal phase 1( ) t and the feedback signal (VCO output signal) phase 2( ) t . The output error signal utd ( ) is a function of the phase difference θe ( ) t. In other words, as long as the change in the output signal voltage value of a certain device can directly reflect the change in the phase difference θe ( ) t, the
phase detection function can be realized. Of course, the transformation of this change should preferably have a linear relationship, otherwise it will be very complicated to analyze and use. In fact, the characteristics of the phase detector can be diverse, including sinusoidal characteristics, triangular characteristics, zigzag characteristics, etc. The most commonly used one is the sinusoidal characteristic. In the subsequent discussion in this chapter, the reader can see that the phase detector with the sinusoidal characteristic has very excellent performance. The phase locked loops discussed in this book all use phase detectors with sinusoidal characteristics.
is the maximum output voltage of the phase detector.
The above derivation process uses the triangular product and difference formula learned in middle school. Do you still remember it? This book will use various trigonometric formulas many times for derivation. Therefore, if the reader is not familiar with these formulas, be sure to spend some time reviewing them. Confucius said: "By reviewing the past and learning the new, you can become a teacher." In the field of communication technology, many seemingly advanced theories can ultimately be perfectly expressed by some of the most basic mathematical formulas. Once you understand the application of these immutable formulas in phase-locking technology, the originally boring formulas and theorems seem to suddenly become more lively.
Equation (3-10) is the sinusoidal phase identification characteristic. Is this the sine phase detection characteristic? This is obviously a cosine function. Almost all books and papers on phase-locked loops will mention that a phase detector with a structure similar to Figure 3-5 has a sinusoidal phase detection characteristic. Careful readers may have discovered that in some classic textbooks on the principles of phase locking technology, such as "Phase Locking Technology" compiled by Zheng Jiyu and others, when deriving the sinusoidal phase identification characteristics, the expression of the VCO output signal is the same as (3 - 4) Different, for
Compared with Equation (3-11) and Equation (3-4), the phase difference between the two is 90°. In this way, the phase of the input signal in the entire loop and the reference phase of the VCO output signal phase are actually 90° different. . Because of this, after the loop is locked, the signal that maintains phase synchronization with the input signal is not the signal output by the VCO to the phase detector, but a signal that is 90° out of phase with it.
Since the VCO output signal has two forms: sine and cosine, let's take a closer look. If the form of the input signal is rewritten into cosine form, according to the above derivation process, the phase identification characteristic expression may be transformed into the following two forms.
According to the previous discussion, there are two issues involved here. We will sort out these two issues and raise them clearly here. We will put them here first. Readers are asked to read the follow-up content of this chapter with these two issues in mind.
Question 3-1: Under what circumstances does the sine phase detector operate in the positive slope range? Under what circumstances does it work in the negative slope range?
Question 3-2: If the input signal is in the form of sine (or cosine), and the signal entering the phase detector from the VCO is in the
form of sine (or cosine), after the loop is locked, which signal of the VCO is the in-phase branch of the input signal?
3.1.4 Function of loop filter
Simply put, a loop filter is a low-pass filter that averages the ripple-containing signal output from the phase detector and converts it into a DC signal with little AC component. When we were in primary school, our Chinese teacher would definitely ask us to do "sentence contraction" exercises. After the above sentence was shortened, it became "The loop filter is a low-pass filter." If this is the case, the function of the loop filter is not essentially different from the low-pass filter in the sine phase detector.
In fact, the loop filter is indeed a low-pass filter, but it is not just an ordinary low-pass filter. In addition to completing the low-pass filtering function, it also has an important role, which is to determine the transmission characteristics of the entire PLL circuit, and then Determines almost all important characteristics such as the stability of the PLL circuit, capture bandwidth, capture speed, etc. Because of this, Flord.M.Gardner specifically discussed the name of the loop filter in the form of a "Commentary" in the book "Phase-Locked Loop Technology (3rd Edition)":
In hindsight, the choice of loop filter The name filter is somewhat of a misnomer, although the term was circulated in the first two editions of this book. We may note in particular that these examples are not low-pass filters, although some authors mistakenly call them so. A more appropriate name might be called a loop controller, which is the term used by our control system colleagues. The main function of these circuits is to establish the dynamics of the feedback loop and provide appropriate control signals to the VCO. Any filtering of unwanted signals is a secondary task, and as will be mentioned later, this filtering task is completed by another unit circuit. But the term loop filter has become too familiar to be corrected, so this term will be used in this book.
Therefore, the loop filter has two functions: one is a low-pass filter, and the passband must be much lower than the bandwidth of the phase detector, because the VCO can only output spurious components when inputting a DC signal with a small ripple. Small high-quality sinusoidal signal; the second is to control the loop characteristics, and this is the main function of the loop filter.
Going further, for electronic engineers, the so-called phase-locked loop circuit design is actually designing a loop filter. The late French "Tiger Prime Minister" Georges Clemenceau famously said: War is too important to leave it to generals. To paraphrase this sentence, the loop filter is too important to be
discussed briefly. We will discuss it specifically in subsequent chapters.
3.2 Understanding phase-locked loops from negative feedback circuits
Now let's discuss the topic of this chapter: Why can a phase-locked loop track the phase of the input signal? The answer is simple, because the phase locked loop is a phase control system and a phase negative feedback system. The "Electronic Technology" course we studied in college conducted a detailed analysis of feedback circuits. Although the object of circuit analysis is voltage or current, its analysis methods and analysis conclusions are directly helpful in understanding phase-locked loops.
Next, let’s briefly review the basic knowledge of negative feedback circuits and understand why negative feedback circuits can improve the stability of amplifier circuits. Then, through the working mechanism of negative feedback circuits, we can conceptually understand why phase-locked loops can achieve phase tracking. .
3.2.1 Concept of feedback circuit
Feedback, also known as feedback, is a basic concept in control theory. It refers to the process of returning the output of a system to the input and changing the input in some way, thereby affecting the function of the system. Feedback can be divided into negative feedback and positive feedback. The former makes the output play the opposite role to the input, reducing the error between the system output and the system target, and the system tends to be stable; the latter makes the output play a similar role to the input, making the system The deviation continues to increase, causing the system to oscillate, which can amplify the control effect. The study of negative feedback is a core issue in cybernetics.
The block diagram of the feedback amplification circuit is shown in Figure 3-7. The amplification circuit before feedback is recorded as the basic amplification circuit, and its amplification factor is A. It can be a single-stage or multi-stage amplification circuit; the feedback network The amplification factor is F, and most electronic circuits are composed of resistor-capacitor components; the basic amplification circuit and the feedback network combined are an amplification circuit that introduces feedback, which is called a feedback amplification circuit. It can be seen that the basic amplifier circuit and the feedback network form a closed loop, so the basic amplifier circuit is usually called an open-loop amplifier circuit, and the feedback amplifier circuit is called a closed-loop amplifier circuit.
The output after the output signal is sampled is sent back to the input terminal to be superimposed (also called comparison) with the input signal Xi. X di represents the net input signal that is actually added to the input terminal of the basic amplifier circuit after
superposition : XXX di if , equation The “” in means that the feedback signal can have the same or opposite polarity as the input signal. "+" means that the feedback signal has the same polarity as the original input signal, which strengthens the net input signal, increases the corresponding output, and increases the gain, so it is called positive feedback; "-" means that the feedback signal has the opposite polarity to the original input signal, so that The net input signal weakens, the corresponding output decreases, and the gain also decreases, so it is called negative feedback. The sign of feedback is also called the polarity of feedback.
3.2.2 Control function of negative feedback circuit
Negative feedback was originally conceived for use in high-fidelity amplifiers used in telephone lines, which was invented and patented by Harold Stephen Black in 1927. Later, Hendrik W. Bode of Bell Labs conducted further research on negative feedback and published "Network Analysis and Feedback Amplifier Design" in 1945. Since then, the theory of negative feedback has been Established.
It should be noted that the gain (A, F) of each stage circuit in the loop is a function of frequency, and has different response characteristics at different frequencies. Contents such as voltage feedback, current feedback, series feedback, and parallel feedback are also analyzed in the "Electronic Technology" course. For phase-locked loop technology, mastering the basic working principles of negative feedback circuits is enough to analyze the working processes of some simple phase-locked loops.