www.lansdale.com

Page 7 of 11

Issue A

LANSDALE Semiconductor, Inc.

ML12179

Legacy Applications Information

Appendix: Derivation of Loop Filter Transfer Function

The purpose of the loop filter is to convert the current from the

phase detector to a tuning voltage for the VCO. The total transfer

function is derived in two steps. Step 1 is to find the voltage gen-

erated by the impedance of the loop filter. Step 2 is to find the

transfer function from the input of the loop filter to its output.

The “voltage” times the “transfer function” is the overall transfer

function of the loop filter. To use these equations in determining

the overall transfer function of a PLL multiply the filter's imped-

ance by the gain constant of the phase detector then multiply that

by the filter's transfer function (which is unity in the 2nd and 3rd

order cases below).

Figure 10. Transfer Function for the Crystal Noise in the Frequency Plane

Figure 11. Transfer Function for the VCO Noise in the Frequency Plane

T(j

ω=

•

)N

1+ j

2

ζ

o

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

1

(

((

(

(

(

(

(

(

(

(

(

+–

+

+–

K3K4

4

o4

B

2

o2

j2

ζ

()

o

(AK4

K5)

3

o3

T(j

ω=

)

K3K4

4

o4

B

–

2

o2

j

–+

(AK4

K5)

3

o3

1+

–

–

K3K4

4

o4

B

2

o2

j

++

2

ζ

o

(AK4

K5)

3

o3

Figure 12. Overall Transfer Function of the PLL

For the 2nd Order PLL:

Vp

Co

Ro

Vt

ZLF(s)

RoCos1

Cos

TLF(s)

Vt(s)

Vp(s)

1,

==

=

=

+

Vp(s)

Kp(s)ZLF(s)

For the 3rd Order PLL:

Vp

Co

Ro

Vt

ZLF(s)

RoCos1

+

CoRoCas2

(Co

Ca)s

++

TLF(s)

Vt(s)

Vp(s)

1,

=

=

=

=

Vp(s)

Kp(s)ZLF(s)

Ca

For the 4th Order PLL:

Vp

Co

Ro

Vt

ZLF(s) =

(RoCos1)

++

(RxCxs1)

CoRoCaRxCxs3

[

++

+

+

+

+

+

(Co

Ca)RxCx

CoRo(Cx

Ca) ] s2

(Co

Ca

Cx)s

TLF(s) =

=

Vt(s)

Vp(s)

1

(RxCxs1)

+

,Vp(s) = Kp(s)ZLF(s)

Ca

Rx

Cx