How does the crystal oscillator vibrate?

01 

Pierce crystal oscillator

At present, the pierce crystal oscillator is used most in our work , That is, the following structure .

CL1,CL2 Capacitance matching ,Rext Usually hundreds of ohms in series （ Sometimes not ）. The above structure may not look very familiar , Let's convert it , Become familiar with the following .

In the picture above, put RF Ignore it. , If you have used a passive crystal oscillator , You should know this RF The resistance value of is usually very large , Megaohm level , Its function is mainly to give the inverting amplifier a suitable bias . Like the triode circuit in our Analog Electronics , When it works, it needs appropriate DC bias , Here we ignore .

Equivalent circuit of crystal oscillator ：

I guess someone saw it and ran away , What is this lump , It's so complicated ？ It's not that hard , All the devices in it , In addition to the inverting amplifier , Are basic components , On the contrary, it is easier to analyze .

But before that ？ We'd better insert a knowledge point , That is the starting condition . Because only understand the starting conditions , We can know what it is and why , Theoretical analysis first , Then use simulation to verify , So much better .

02 

Starting condition

There are two starting conditions ： Phase and loop gain

In the figure above, the transfer function of the inverting amplifier is used A(s) Express , Crystal oscillator and its matching circuit are packed together , Its transfer function is regarded as F(s).

When the loop gain is greater than 1 when , It means that the input signal is sent to the input terminal after wandering around the loop , The signal amplitude is larger than before

The phase is 2nπ, Say that the input signal walks around the circuit , The phase is exactly the same as the original input signal , So the input signal is perfectly strengthened .

Combination of the two , This is the case ： After repeated amplification , Growing , In the end, it's what we see .

When the signal amplitude in the loop increases to a certain extent , Active device in oscillator （ Inverter in crystal oscillator circuit ） The existing nonlinearity will limit the continuous increase of amplitude , Make the output of the oscillator stable . Generally speaking, the amplitude of oscillation must not exceed the power supply voltage .

in other words , As long as those two conditions are met , The smallest signal , Infinite loop through the loop （ Input signal Vin Go around and become bigger Vin, Then go around again and become bigger Vin, Then go shopping again ...）, The final output amplitude will always rise .

Although we do not specifically provide the input signal of the corresponding frequency Vin, But there is always noise in the circuit , For example, white noise is full band , Although the signal is very small , But because of this constantly strengthening feature of the circuit , So there must be an output signal in the end . in other words , Vibration is inevitable .

All in all , These two conditions must be met for vibration initiation ：

About crystal vibration , There is a criterion called Barkhausen stability criterion , Students who want to go deep can check .

03 

Analysis of starting circuit

In the last section, we finished the vibration starting conditions , Then let's continue to analyze the previous circuit , Look at the phase first .

phase

The phase needs to meet 2nπ, obviously ,n It can't be equal to 0, Equal to zero, unexpectedly, there is no phase shift at all , There is a inverting amplifier in the circuit , The phase has shifted π, That is to say 180°. So the most likely thing is that the circuit of crystal oscillator also makes the signal phase shift 180 degree , The whole is enough 360°, That is to say 2π.

Rext And matching capacitance CL1 Form a low-pass filter , The resulting phase shift is less than 90° Of , It should be easy to know , The picture below is Rext=100Ω,CL1=10p Amplitude frequency curve of low-pass filter , You can see , The phase difference between input and output is ：0°~90°, Output lags input .

Rext And matching capacitance CL1 The resulting phase shift is less than 90°, Then crystal vibration and CL2 The resulting phase shift must be greater than 90°, Only in this way can we get enough 180°.

So when will Jingzhen and CL2 The resulting phase shift is greater than 90° What about ？ The answer is that it is possible only when the crystal oscillator equivalent circuit is inductive as a whole .

The equivalent circuit of crystal oscillator consists of three components , resistance , inductance , capacitance , The total equivalent impedance may have three cases ： Resistive , Capacitive , Perceptual .

From a mathematical point of view , After simplifying the formula of the total impedance of crystal oscillator, it can always be written in a complex form , Expressions containing real and imaginary numbers ：Z=R+jX. among R Is the resistance component ,X Is the reactance component .

When X=0：Z=R, The whole is resistive
When X>0：Z=R+jX, The whole is perceptual
When X<0：Z=R+jX, The whole is capacitive .

Here are three cases , Crystal oscillator and matching capacitor CL2 Phase of .

The impedance formula must be a big lump , No more , Let's look at the simulation directly , This is more intuitive （ resistance 100 Just symbolic value , You can take other , It doesn't affect the result , Capacitance value is also ）

You can see , Only when crystal oscillator is perceptual , The phase shift can be greater than 90°, Its phase shift range is ：0~180°.

in summary , When the crystal oscillator works, it is perceptual . that , What is the inductive frequency range of crystal oscillator ？

04 

The crystal oscillator is in the inductive frequency range

The impedance is in the inductive frequency range , Just list the total impedance formula of the crystal oscillator .

In theory, , Total impedance is a complex number , We can reduce it to real part and imaginary part , The imaginary part is greater than 0 when , That means the crystal oscillator is perceptual . But it's too hard , Let's take a look at the actual crystal oscillator parameters .

Most crystal oscillators are not given Lm,Rm,Cm Parametric , But I found that Epson's crystal oscillator was written , For example, the following one 25Mhz The crystal of .

You can see ,Rm The maximum is 80Ω, Dynamic capacitance Cm=1.94fF, Dynamic inductance Lm=20.91mH, Static capacitance C0=0.6pF.

We use these parameters directly , use LTspice Draw the impedance curve by simulation .

The principle of simulation is very simple , Give me a 1A The current source of , that Vz The voltage of is the impedance times the current , The current is 1A, that Vz The value of is the impedance value , therefore , On the left of the picture dB The amplitude value is the impedance value . The phase represents the phase angle of the voltage leading current , therefore , If the phase is greater than 0, Indicates that the overall impedance is inductive , If less than 0, Indicates that the overall impedance is capacitive .

As you can see from the diagram , The area between the two tips , The phase angle is greater than 0, Is the perceptual area , Crystal oscillation frequency 25Mhz Just in this area .

See from the graph , The frequencies of the two risers are 24.988623Mhz,25.028984Mhz, It's easy to guess , The frequency of two rising points is so special , It should be series resonance point and parallel resonance point .

We can verify , Calculate the series resonance and parallel resonance respectively .

Calculated value fs=24.988619Mhz, Reading value of simulation diagram fs=24.988623Mhz
Calculated value fa=25.0289844Mhz, Reading value of simulation diagram fa=25.028984Mhz
You can see , It can be said to be completely consistent .

therefore , The operating frequency range of the crystal is ：

Of course , The above just says that the actual working frequency of the crystal oscillator is between this , It's not that the frequency variation range is so large in actual work , There is a parameter called frequency offset to measure stability , Be careful not to misunderstand .

05 

Summary

That's all for this section , For the time being, it only explains the phase relationship , In fact, there are still some problems that have not been clarified .

Like this ： We all know that in practical application , Change the matching capacitance , You can fine tune the working frequency of the crystal oscillator , So the actual working frequency must have something to do with the matching capacitance , It should be relational . I have seen the specific relationship in some materials , But I only know the result , I haven't figured out how I got here , Let's put it for the time being .

The above is the content of this issue , If there are questions , Welcome to point out .

-END-

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