如何使用matlab对2阶PLL(锁相环)电路图形进行分析
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The open-loop transfer functions for the PLL are:
GH( s ) = Kφ⋅ R1⋅ C1⋅ s + 1 Kv 1 ωz ⋅ s + 1 Kv 1 ⋅ ⋅ = Kφ⋅ ⋅ ⋅ C1⋅ s s N C1⋅ s s N
The angle of open-loop gain is
(
)
N⋅ 1 + ωu_ ωz⋅ A(s ) = ωu_ ωz⋅
s
ωu
+ ωu_ ωz⋅ s + 1 ω ωu u
s
2
The simplified transfer function is acceptable for most situations, except when optimizing the bandwidth. It tends to underestimate the settling time for low phase margins.The inverse Laplace transform of the true transfer function is
Loop Filter
VCO KV s
_______________________________________
Table of Contents
I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. Introduction Inputs Initial Calculations Loop Filter Design Procedure 2nd Order PLL Design Function Optimal Settling Time for 2nd Order PLLs Outputs Noise, Transfer Function, and Settling Time Analysis Small Signal Transfer Functions Phase Noise Calculations Transient Step Response Plots Copyright and Trademark Notice
ωu_ ωz := tan( PM des ) ωu ωu_ ωz
and the zero location
ωz := ωz 2⋅ π = 0.364 kHz
The magnitude transfer function is given by:
MagGH ωu = 1 =
( )
2 ωu_ ωz + 1 Kv Kφ ⋅ ⋅ C1⋅ ωu ωu N
ferr( ωut , ωu_ ωz) := b← c← ωu_ ωz + 1 4⋅ ωu_ ωz + 1 − ωu_ ωz
2 2 2
1 −ωu_ ωz ωut − ωu_ ωz 2 −ωu⋅ exp ⋅ −2 + 5 ⋅ ωut ⋅ if ωu_ ωz = 2⋅ 2 + 5 2 ωu_ ωz 2 2 f ⋅ ωu_ ωz ⋅ sin c⋅ ωut − cos c⋅ ωut ⋅ exp −ωu_ ωz⋅ ωut if ωu_ ωz ≠ 2⋅ 2 + 5 step 2⋅ b c 2⋅ b 2⋅ b
C1 = 370.304 nF
R1 = 1.181 kΩ
Plugging these back into the transfer function we get: the following feedforward and feedback transfer functions.
ωu_ ωz⋅ G( s ) = s ωu
R RB Kφ Kv fO fN
Fig. 2: 2nd-order PLL with voltage-mode phase detector and active filter
1 N
Fig. 3: 2nd-order PLL with voltage-mode phase detector and passive filter
The closed-loop transfer function for a second order PLL is: s 1 N⋅ 1 + ωu_ ωz⋅ ω u 2 2 A(s ) = where b = ωu_ ωz + 1 2 s s b⋅ + ω u_ ω z ⋅ + 1 ω ωu u To simplified the transient response, we can assume ωu2 is much greater than ωz 2. In this case the closed loop transfer function simplifies to:
sec
ωu := 2⋅ π⋅ fu N := Kφ :=
Unity Gain Frequency Loop divider ratio Phase Detector Gain
3
_______________________________________
Loop Filter Design Procedure
2
_______________________________________
Introduction
Although the emphasis of PLL design is on 3rd-order PLLs, the settling time of second order PLLs is important, especially for boosted PLLs. In boost mode the transfer function of a third-order PLL sometimes reverts to a second order expression. As with 3rd-order PLL's there are different implementations, including voltage mode and those with operational-amplifier based phase detectors. This report will talk about the most popular, which uses a current-mode phase detector and a passive loop filter.
_______________________________________
Inputs
fr := 10⋅ MHz fo := 1⋅ GHz facc := 1kHz fstep := 25MHz I := 50µA fu := 1⋅ kHz Kv := 2⋅ π⋅ 10⋅ 10 ⋅ PM des := 70deg
_______________________________________
Initial Calculations
T := 1 fr fo fr I 2⋅ π T = 0.1 µS ωu = 6.283 × 10 N = 100 Kφ = 7.958 µA rad
3 rad
Reference Period
2⋅ π⋅ f ωu_ ωz + 1
2
G( f) = Kφ⋅
R1⋅ Kv
=
ωu_ ωz
⋅
N
H=
1 N
G( f) = N⋅
fu f
H=
1 N
Note that second order PLL only gives a first order roll off, because the stabilizing zero in the loop.
6
rad sec ⋅ volt
Reference oscillator frequency and period Output frequency Acceptable Frequency Error Maximum Frequency Step (output referred) Charge Pump Current Unity Gain Frequency VCO Gain Phase Margin (70 degrees recommended, 50 degrees if slewing is considered)
The inverse Laplace transform of the simplified transfer function is 4 ferr2( ωut , ωu_ ωz) := y ← −1 ωu_ ωz ω ut − 1 fstep ⋅ 2 − 1 ⋅ exp 2 ⋅ ωut if ωu_ ωz = 4 1 sin ⋅ y ⋅ ωut 1 2 − 1 fstep ⋅ −cos ⋅ ωut ⋅ exp if ωu_ ωz ≠ 4 2 ⋅ y ⋅ ωut + y 2 For comparison, we also plot an ideal settling with a bandwidth of ωu.
ωu ← 2⋅ π⋅ rad⋅ fu C1 ← ωu_ ωz + 1⋅ Kφ⋅ Kv ωu ⋅ N R1 ← ωu_ ωz ωu⋅ C1
2 2
x := pll2nd( fr , fo , I , fu , Kv , PM des ) C1 := x ⋅ farad
1 2
farad R1 ohm
C1
C1 = 370.304 nF R1 = 1.181 kohm
R1 := x ⋅ ohm
Main Loop Filter Capacitor Main Loop Resistor
5
2
_______________________________________
Optimal Settling-Time for 2nd Order PLLs
and set this equal to one to find C1, given ωu.
C1 := ωu_ ωz + 1⋅
2
Use C1 and ωz to find R1:
R1 = 1 ωz ⋅ C1 R1 := ωu_ ωz Kv Kφ ωu_ ωz + 1⋅ ⋅ ωu N
2
Kv Kφ ⋅ 2 N ωu
2nd Order PLL Design and Analysis
Phase Detector SREF Σ Kφ R1 C1 Loop Divider 1 N
Fig. 1: 2nd Order PLL with Current-Mode Phase Detector
useful functions and identities Units Constants
4
_______________________________________
2nd-Order PLL Design Function
The following function repeats the calculations above
pll2nd( fr , fo , I , fu , Kv , PM ) := ωu_ ωz ← tan( PM ) N← Kφ ← fo fr I 2⋅ π⋅ rad
AngleGH ωu = atan( ωu_ ωz) − 180deg
( )
and the phase margin is thus given by
PM ( ωu_ ωz) := atan( ωu_ ωz)
Use to solve for the zero/unity gain bandwidth spacing: ωu_ωz
2
+1 ⋅ N
2
H :=
1 N
s ω u For frequencies much greater than the unity gain bandwidth the magnitude response is
ωu_ ωz + 1
f f u 2 For ωu_ωz >>1, which is usually true:
GH( s ) = Kφ⋅ R1⋅ C1⋅ s + 1 Kv 1 ωz ⋅ s + 1 Kv 1 ⋅ ⋅ = Kφ⋅ ⋅ ⋅ C1⋅ s s N C1⋅ s s N
The angle of open-loop gain is
(
)
N⋅ 1 + ωu_ ωz⋅ A(s ) = ωu_ ωz⋅
s
ωu
+ ωu_ ωz⋅ s + 1 ω ωu u
s
2
The simplified transfer function is acceptable for most situations, except when optimizing the bandwidth. It tends to underestimate the settling time for low phase margins.The inverse Laplace transform of the true transfer function is
Loop Filter
VCO KV s
_______________________________________
Table of Contents
I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. Introduction Inputs Initial Calculations Loop Filter Design Procedure 2nd Order PLL Design Function Optimal Settling Time for 2nd Order PLLs Outputs Noise, Transfer Function, and Settling Time Analysis Small Signal Transfer Functions Phase Noise Calculations Transient Step Response Plots Copyright and Trademark Notice
ωu_ ωz := tan( PM des ) ωu ωu_ ωz
and the zero location
ωz := ωz 2⋅ π = 0.364 kHz
The magnitude transfer function is given by:
MagGH ωu = 1 =
( )
2 ωu_ ωz + 1 Kv Kφ ⋅ ⋅ C1⋅ ωu ωu N
ferr( ωut , ωu_ ωz) := b← c← ωu_ ωz + 1 4⋅ ωu_ ωz + 1 − ωu_ ωz
2 2 2
1 −ωu_ ωz ωut − ωu_ ωz 2 −ωu⋅ exp ⋅ −2 + 5 ⋅ ωut ⋅ if ωu_ ωz = 2⋅ 2 + 5 2 ωu_ ωz 2 2 f ⋅ ωu_ ωz ⋅ sin c⋅ ωut − cos c⋅ ωut ⋅ exp −ωu_ ωz⋅ ωut if ωu_ ωz ≠ 2⋅ 2 + 5 step 2⋅ b c 2⋅ b 2⋅ b
C1 = 370.304 nF
R1 = 1.181 kΩ
Plugging these back into the transfer function we get: the following feedforward and feedback transfer functions.
ωu_ ωz⋅ G( s ) = s ωu
R RB Kφ Kv fO fN
Fig. 2: 2nd-order PLL with voltage-mode phase detector and active filter
1 N
Fig. 3: 2nd-order PLL with voltage-mode phase detector and passive filter
The closed-loop transfer function for a second order PLL is: s 1 N⋅ 1 + ωu_ ωz⋅ ω u 2 2 A(s ) = where b = ωu_ ωz + 1 2 s s b⋅ + ω u_ ω z ⋅ + 1 ω ωu u To simplified the transient response, we can assume ωu2 is much greater than ωz 2. In this case the closed loop transfer function simplifies to:
sec
ωu := 2⋅ π⋅ fu N := Kφ :=
Unity Gain Frequency Loop divider ratio Phase Detector Gain
3
_______________________________________
Loop Filter Design Procedure
2
_______________________________________
Introduction
Although the emphasis of PLL design is on 3rd-order PLLs, the settling time of second order PLLs is important, especially for boosted PLLs. In boost mode the transfer function of a third-order PLL sometimes reverts to a second order expression. As with 3rd-order PLL's there are different implementations, including voltage mode and those with operational-amplifier based phase detectors. This report will talk about the most popular, which uses a current-mode phase detector and a passive loop filter.
_______________________________________
Inputs
fr := 10⋅ MHz fo := 1⋅ GHz facc := 1kHz fstep := 25MHz I := 50µA fu := 1⋅ kHz Kv := 2⋅ π⋅ 10⋅ 10 ⋅ PM des := 70deg
_______________________________________
Initial Calculations
T := 1 fr fo fr I 2⋅ π T = 0.1 µS ωu = 6.283 × 10 N = 100 Kφ = 7.958 µA rad
3 rad
Reference Period
2⋅ π⋅ f ωu_ ωz + 1
2
G( f) = Kφ⋅
R1⋅ Kv
=
ωu_ ωz
⋅
N
H=
1 N
G( f) = N⋅
fu f
H=
1 N
Note that second order PLL only gives a first order roll off, because the stabilizing zero in the loop.
6
rad sec ⋅ volt
Reference oscillator frequency and period Output frequency Acceptable Frequency Error Maximum Frequency Step (output referred) Charge Pump Current Unity Gain Frequency VCO Gain Phase Margin (70 degrees recommended, 50 degrees if slewing is considered)
The inverse Laplace transform of the simplified transfer function is 4 ferr2( ωut , ωu_ ωz) := y ← −1 ωu_ ωz ω ut − 1 fstep ⋅ 2 − 1 ⋅ exp 2 ⋅ ωut if ωu_ ωz = 4 1 sin ⋅ y ⋅ ωut 1 2 − 1 fstep ⋅ −cos ⋅ ωut ⋅ exp if ωu_ ωz ≠ 4 2 ⋅ y ⋅ ωut + y 2 For comparison, we also plot an ideal settling with a bandwidth of ωu.
ωu ← 2⋅ π⋅ rad⋅ fu C1 ← ωu_ ωz + 1⋅ Kφ⋅ Kv ωu ⋅ N R1 ← ωu_ ωz ωu⋅ C1
2 2
x := pll2nd( fr , fo , I , fu , Kv , PM des ) C1 := x ⋅ farad
1 2
farad R1 ohm
C1
C1 = 370.304 nF R1 = 1.181 kohm
R1 := x ⋅ ohm
Main Loop Filter Capacitor Main Loop Resistor
5
2
_______________________________________
Optimal Settling-Time for 2nd Order PLLs
and set this equal to one to find C1, given ωu.
C1 := ωu_ ωz + 1⋅
2
Use C1 and ωz to find R1:
R1 = 1 ωz ⋅ C1 R1 := ωu_ ωz Kv Kφ ωu_ ωz + 1⋅ ⋅ ωu N
2
Kv Kφ ⋅ 2 N ωu
2nd Order PLL Design and Analysis
Phase Detector SREF Σ Kφ R1 C1 Loop Divider 1 N
Fig. 1: 2nd Order PLL with Current-Mode Phase Detector
useful functions and identities Units Constants
4
_______________________________________
2nd-Order PLL Design Function
The following function repeats the calculations above
pll2nd( fr , fo , I , fu , Kv , PM ) := ωu_ ωz ← tan( PM ) N← Kφ ← fo fr I 2⋅ π⋅ rad
AngleGH ωu = atan( ωu_ ωz) − 180deg
( )
and the phase margin is thus given by
PM ( ωu_ ωz) := atan( ωu_ ωz)
Use to solve for the zero/unity gain bandwidth spacing: ωu_ωz
2
+1 ⋅ N
2
H :=
1 N
s ω u For frequencies much greater than the unity gain bandwidth the magnitude response is
ωu_ ωz + 1
f f u 2 For ωu_ωz >>1, which is usually true: