Phase-locked loop analysis

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Phase-locked loop analysis [Copy link]

1. Linearization of the loop

　1. Linearization condition: 500)this.style.width=500;" align=absMiddle>≤ 500)this.style.width=500;" align=absMiddle>

　2. Linearized loop equation: 500)this.style.width=500;" align=absMiddle>

　3. Linearization loop phase mode:

　500)this.style.width=500;">

2. Loop Transfer Function

　Linear networks can be described by transfer functions

　A linear loop system can be described by a loop transfer function

　Using complex frequency domain transfer function to describe the system can simplify

The complex　 frequency domain loop transfer function can be used to describe the

　PP PLL.

1. Linear loop equation in complex frequency domain (P→S substitution)

500)this.style.width=500;">

2. Loop transfer function definition:

The loop transfer function is the ratio of the output to input phase Laplace transform

3. Loop transfer function type

　· Open loop transfer function function 500)this.style.width=500;" align=absMiddle>

　· Closed loop transfer function function500)this.style.width=500;" align=absMiddle>

　Error transfer function function 500)this.style.width=500;" align=absMiddle>

4. Conclusion

　· 500)this.style.width=500;" align=absMiddle> . 500)this.style.width=500;" align=absMiddle> are all related to 500)this.style.width=500;" align=absMiddle>

Different loop filters 　have different F ( s ), and the loop transfer function is also different.

　　If the denominators are different, they can be compared after standardization.

3. Loop transfer function denominator standardization

1. Example:

Standardized transfer function of 　　L, R, C series oscillating network and plot

　　Output frequency characteristics

　· First find the transfer function H(S): 500)this.style.width=500;" align=right>

　Normalize the denominator to:

　500)this.style.width=500;" align=absMiddle>

　· 500)this.style.width=500;" align=absMiddle>500)this.style.width=500;" align=absMiddle>

　· Normalized transfer function 500)this.style.width=500;" align=absMiddle>

　· Transfer function in frequency domain 500)this.style.width=500;" align=absMiddle>

　· Draw 500)this.style.width=500;" align=absMiddle>Amplitude-frequency characteristics

If 500)this.style.width=500;" align=absMiddle>, corresponding to ξ＝0.707 medium amplitude-frequency characteristic

It drops by 3dB, called 500)this.style.width=500;" align=absMiddle> is the cutoff frequency, this frequency

Use subscript c, recorded as 500) this.style.width=500;" align=absMiddle>

in conclusion:

　Denominator normalization facilitates loop comparison

　· 500)this.style.width=500;" align=absMiddle>, showing low-pass filtering characteristics

　· Cutoff frequency 500)this.style.width=500;" align=absMiddle>

　

2. Loop transfer function normalized for different loop denominators

Different loops 500)this.style.width=500;" align=absMiddle>, ξ are all different

(as shown in the table below)

500)this.style.width=500;">

500)this.style.width=500;">

in conclusion:

Different loops have different loop transfer functions, and ξ is also different .

The loop without loop filter is a first-order loop. Commonly used loop filters

　The loop formed by taking any one section as the loop filter is

　Second-Order Loop,Second-order loops are widely used.

· A loop is usually called an *order* type loop, where "order" is represented by a capital 1.

　Second and third representation, it depends on the highest power of the denominator of the loop transfer function

　The "type" is represented by 1, 2, or 3, depending on the ideal integral loop in the loop.

　The number of sections.

4. Loop Tracking Performance Analysis

Take the first-order loop frequency response characteristics as an example (for the modulation frequency Ω of the input phase )

1. Closed-loop transfer function frequency response characteristics

500)this.style.width=500;">

500)this.style.width=500;" align=absMiddle> 500)this.style.width=500;" align=absMiddle>

visible:

The frequency response characteristics of the first-order closed-loop transfer function show low-pass filtering characteristics .

· Cut-off frequency 500)this.style.width=500;" align=absMiddle>

2. Error transfer function frequency response characteristics

500)this.style.width=500;">

500)this.style.width=500;" align=absMiddle> 500)this.style.width=500;" align=absMiddle>

visible:

The first-order loop error transfer function exhibits high-pass filtering characteristics

　· in conclusion:

　　a) The frequency response characteristics of the first-order closed-loop transfer function are low-pass filtering ;

　　　The frequency response characteristics of the error transfer function are high-pass filtering

　　　b) The first-order ring cutoff angular frequency is 500)this.style.width=500;" align=absMiddle>, in order to improve

　　　　The loop's ability to suppress interference and noise should be made smaller .

　　　　Good , this is related to improving loop stability. 500)this.style.width=500;" align=absMiddle>The bigger the better

　　　　A contradiction occurs, so the first-order phase-locked loop has no practical value.

c) In practice, the second-order phase-locked loop can take into account the loop interference , noise capability and loop stability performance, so it is widely used .