第一個二階CP-PLL的數學模型是由佛洛依德·加德納 [2] [3] [4] [5]
CP-PLL的數學模型可以針對一些參數進行解析的預估，例如hold-in範圍（在VCO沒有過載的情形下，可能進行鎖相的輸入信號頻率範圍），及捕獲範圍（pull-in range，在CP-PLL任意初始狀態下，CP-PLL最終可以鎖相的輸入信號頻率範圍）[6]
二階CP-PLL的連續時間線性模型以及加德納的猜想
编辑
加德納的分析是以以下的近似為基礎[2]
t
p
=
|
θ
e
|
/
ω
r
e
f
,
θ
e
=
θ
r
e
f
−
θ
v
c
o
.
{\displaystyle t_{p}=|\theta _{e}|/\omega _{\rm {ref}},\ \theta _{e}=\theta _{\rm {ref}}-\theta _{\rm {vco}}.}
CP-PLL的PDF平均輸出為
i
d
=
I
p
θ
e
/
2
π
{\displaystyle i_{d}=I_{p}\theta _{e}/2\pi }
對應的傳遞函數為
I
d
(
s
)
=
I
p
θ
e
(
s
)
/
2
π
{\displaystyle I_{d}(s)=I_{p}\theta _{e}(s)/2\pi }
若用濾波器傳遞函數
F
(
s
)
=
R
+
1
C
s
{\displaystyle F(s)=R+{\frac {1}{Cs}}}
θ
v
c
o
(
s
)
=
K
v
c
o
I
d
(
s
)
F
(
s
)
/
s
{\displaystyle \theta _{\rm {vco}}(s)=K_{\rm {vco}}I_{d}(s)F(s)/s}
θ
e
(
s
)
θ
r
e
f
(
s
)
=
2
π
s
2
π
s
+
K
v
c
o
I
p
(
R
+
1
C
s
)
.
{\displaystyle {\frac {\theta _{e}(s)}{\theta _{\rm {ref}}(s)}}={\frac {2\pi s}{2\pi s+K_{\rm {vco}}I_{p}\left(R+{\frac {1}{Cs}}\right)}}.}
佛洛依德·加德納 [2] :1856 （加德納對CP-PLL的猜想）。
依照加德納的結果，也類似Egan在type 2 APLL捕獲範圍的猜想，Amr M. Fahim在其書中猜想[7] :6 ：為了要達到無限大的捕獲範圍，CP-PLL的迴路濾波器需要使用主動濾波器（Fahim-Egan在type II CP-PLL捕獲範圍的猜想）。
二階CP-PLL的連續時間非線性模型
编辑
為了簡化推導，但不失去通用性，假設VCO和參考信號在其相位為整數時為其下降緣。
令參考信號第一個下降緣的時間為
t
=
0
{\displaystyle t=0}
i
(
0
)
{\displaystyle i(0)}
i
(
0
−
)
{\displaystyle i(0-)}
θ
v
c
o
(
0
)
{\displaystyle \theta _{vco}(0)}
θ
r
e
f
(
0
)
{\displaystyle \theta _{ref}(0)}
若利用電阻和電容製作純PI（比例積分）的濾波器，其輸入電流
i
(
t
)
{\displaystyle i(t)}
v
F
(
t
)
{\displaystyle v_{F}(t)}
v
F
(
t
)
=
v
c
(
0
)
+
R
i
(
t
)
+
1
C
∫
0
t
i
(
τ
)
d
τ
{\displaystyle {\begin{aligned}v_{F}(t)=v_{c}(0)+Ri(t)+{\frac {1}{C}}\int \limits _{0}^{t}i(\tau )d\tau \end{aligned}}}
其中
R
>
0
{\displaystyle R>0}
C
>
0
{\displaystyle C>0}
v
c
(
t
)
{\displaystyle v_{c}(t)}
v
F
(
t
)
{\displaystyle v_{F}(t)}
θ
˙
v
c
o
(
t
)
=
ω
v
c
o
(
t
)
=
ω
v
c
o
free
+
K
v
c
o
v
F
(
t
)
,
{\displaystyle {\begin{aligned}{\dot {\theta }}_{vco}(t)=\omega _{vco}(t)=\omega _{vco}^{\text{free}}+K_{vco}v_{F}(t),\end{aligned}}}
其中
ω
v
c
o
free
{\displaystyle \omega _{vco}^{\text{free}}}
v
F
(
t
)
≡
0
{\displaystyle v_{F}(t)\equiv 0}
K
v
c
o
{\displaystyle K_{vco}}
θ
v
c
o
(
t
)
{\displaystyle \theta _{vco}(t)}
v
˙
c
(
t
)
=
1
C
i
(
t
)
,
θ
˙
v
c
o
(
t
)
=
ω
v
c
o
free
+
K
v
c
o
(
R
i
(
t
)
+
v
c
(
t
)
)
{\displaystyle {\begin{aligned}{\dot {v}}_{c}(t)={\tfrac {1}{C}}i(t),\quad {\dot {\theta }}_{vco}(t)=\omega _{vco}^{\text{free}}+K_{vco}(Ri(t)+v_{c}(t))\end{aligned}}}
其中有以下的不連續分段常數非線性
i
(
t
)
=
i
(
i
(
t
−
)
,
θ
r
e
f
(
t
)
,
θ
v
c
o
(
t
)
)
{\displaystyle i(t)=i{\big (}i(t-),\theta _{ref}(t),\theta _{vco}(t){\big )}}
初始條件為
(
v
c
(
0
)
,
θ
v
c
o
(
0
)
)
{\displaystyle {\big (}v_{c}(0),\theta _{vco}(0){\big )}}
二階CP-PLL的離散時間非線性模型
编辑
在時間區間內的PFD動態 假設參考信號頻率為常數:
θ
r
e
f
(
t
)
=
ω
r
e
f
t
=
t
T
r
e
f
,
{\displaystyle \theta _{ref}(t)=\omega _{ref}t={\frac {t}{T_{ref}}},}
T
r
e
f
{\displaystyle T_{ref}}
ω
r
e
f
{\displaystyle \omega _{ref}}
θ
r
e
f
(
t
)
{\displaystyle \theta _{ref}(t)}
令
t
0
=
0
{\displaystyle t_{0}=0}
t
0
m
i
d
d
l
e
{\displaystyle t_{0}^{\rm {middle}}}
i
(
0
)
=
0
{\displaystyle i(0)=0}
t
0
m
i
d
d
l
e
=
0
{\displaystyle t_{0}^{\rm {middle}}=0}
t
1
{\displaystyle t_{1}}
{
t
k
}
{\displaystyle \{t_{k}\}}
{
t
k
m
i
d
d
l
e
}
{\displaystyle \{t_{k}^{\rm {middle}}\}}
k
=
0
,
1
,
2...
{\displaystyle k=0,1,2...}
令
t
k
<
t
k
m
i
d
d
l
e
{\displaystyle t_{k}<t_{k}^{\rm {middle}}}
t
∈
[
t
k
,
t
k
m
i
d
d
l
e
)
{\displaystyle t\in [t_{k},t_{k}^{\rm {middle}})}
sign
(
i
(
t
)
)
{\displaystyle {\text{sign}}(i(t))}
±
1
{\displaystyle \pm 1}
τ
k
{\displaystyle \tau _{k}}
τ
k
=
(
t
k
m
i
d
d
l
e
−
t
k
)
sign
(
i
(
t
)
)
{\displaystyle \tau _{k}=(t_{k}^{\rm {middle}}-t_{k}){\text{sign}}(i(t))}
t
∈
[
t
k
,
t
k
m
i
d
d
l
e
)
{\displaystyle t\in [t_{k},t_{k}^{\rm {middle}})}
τ
k
=
0
{\displaystyle \tau _{k}=0}
t
k
=
t
k
m
i
d
d
l
e
{\displaystyle t_{k}=t_{k}^{\rm {middle}}}
若VCO的下降緣在參考信號的下降緣之前，則
τ
k
<
0
{\displaystyle \tau _{k}<0}
τ
k
>
0
{\displaystyle \tau _{k}>0}
τ
k
{\displaystyle \tau _{k}}
(
t
k
m
i
d
d
l
e
,
t
k
+
1
)
{\displaystyle (t_{k}^{\rm {middle}},t_{k+1})}
i
(
t
)
≡
0
{\displaystyle i(t)\equiv 0}
v
F
(
t
)
≡
v
k
{\displaystyle v_{F}(t)\equiv v_{k}}
t
∈
[
t
k
m
i
d
d
l
e
,
t
k
+
1
)
{\displaystyle t\in [t_{k}^{\rm {middle}},t_{k+1})}
將
(
τ
k
,
v
k
)
{\displaystyle (\tau _{k},v_{k})}
[8]
p
k
=
τ
k
T
r
e
f
,
u
k
=
T
r
e
f
(
ω
v
c
o
free
+
K
v
c
o
v
k
)
−
1
,
{\displaystyle p_{k}={\frac {\tau _{k}}{T_{\rm {ref}}}},u_{k}=T_{\rm {ref}}(\omega _{\rm {vco}}^{\text{free}}+K_{\rm {vco}}v_{k})-1,}
α
=
K
v
c
o
I
p
T
r
e
f
R
,
β
=
K
v
c
o
I
p
T
r
e
f
2
2
C
.
{\displaystyle \alpha =K_{\rm {vco}}I_{p}T_{\rm {ref}}R,\beta ={\frac {K_{\rm {vco}}I_{p}T_{\rm {ref}}^{2}}{2C}}.}
此處
p
k
{\displaystyle p_{k}}
u
k
+
1
{\displaystyle u_{k}+1}
ω
v
c
o
free
+
K
v
c
o
v
k
{\displaystyle \omega _{\rm {vco}}^{\text{free}}+K_{\rm {vco}}v_{k}}
1
T
r
e
f
{\displaystyle {\frac {1}{T_{\rm {ref}}}}}
最後，不考慮VCO過載的二階CP-PLL離散時間模型如下[4] [6]
u
k
+
1
=
u
k
+
2
β
p
k
+
1
,
p
k
+
1
=
{
−
(
u
k
+
α
+
1
)
+
(
u
k
+
α
+
1
)
2
−
4
β
c
k
2
β
,
for
p
k
≥
0
,
c
k
≤
0
,
1
u
k
+
1
−
1
+
(
p
k
mod
1
)
,
for
p
k
≥
0
,
c
k
>
0
,
l
k
−
1
,
for
p
k
<
0
,
l
k
≤
1
,
−
(
u
k
+
α
+
1
)
+
(
u
k
+
α
+
1
)
2
−
4
β
d
k
2
β
,
for
p
k
<
0
,
l
k
>
1
,
{\displaystyle {\begin{aligned}&u_{k+1}=u_{k}+2\beta p_{k+1},\\&p_{k+1}={\begin{cases}{\frac {-(u_{k}+\alpha +1)+{\sqrt {(u_{k}+\alpha +1)^{2}-4\beta c_{k}}}}{2\beta }},\quad {\text{ for }}p_{k}\geq 0,\quad c_{k}\leq 0,\\{\frac {1}{u_{k}+1}}-1+(p_{k}{\text{ mod }}1),\quad {\text{ for }}p_{k}\geq 0,\quad c_{k}>0,\\l_{k}-1,\quad {\text{ for }}p_{k}<0,\quad l_{k}\leq 1,\\{\frac {-(u_{k}+\alpha +1)+{\sqrt {(u_{k}+\alpha +1)^{2}-4\beta d_{k}}}}{2\beta }},\quad {\text{ for }}p_{k}<0,\quad l_{k}>1,\end{cases}}\end{aligned}}}
其中
c
k
=
(
1
−
(
p
k
mod
1
)
)
(
u
k
+
1
)
−
1
,
S
l
k
=
−
(
u
k
−
α
+
1
)
p
k
+
β
p
k
2
,
l
k
=
1
−
(
S
l
k
mod
1
)
u
k
+
1
,
d
k
=
(
S
l
k
mod
1
)
+
u
k
.
{\displaystyle {\begin{aligned}c_{k}=(1-(p_{k}{\text{ mod }}1))(u_{k}+1)-1,S_{l_{k}}=-(u_{k}-\alpha +1)p_{k}+\beta p_{k}^{2},l_{k}={\frac {1-(S_{l_{k}}{\text{ mod }}1)}{u_{k}+1}},d_{k}=(S_{l_{k}}{\text{ mod }}1)+u_{k}.\end{aligned}}}
此離散時間模型只在
(
u
k
=
0
,
p
k
=
0
)
{\displaystyle (u_{k}=0,p_{k}=0)}
[6]
若VCO過載，也就是
θ
˙
v
c
o
(
t
)
{\displaystyle {\dot {\theta }}_{\rm {vco}}(t)}
(
p
k
>
0
,
u
k
<
2
β
p
k
−
1
)
{\displaystyle (p_{k}>0,u_{k}<2\beta p_{k}-1)}
(
p
k
<
0
,
u
k
<
α
−
1
)
{\displaystyle (p_{k}<0,u_{k}<\alpha -1)}
[5]
高階CP-PLL的非線性模型
编辑
高階CP-PLL非線性模型推導和超越方程有關，無法求得解析解，需要用近似的方式計算[9]
.
